Response-Adapted Therapy With Copanlisib and Rituximab in Untreated Follicular Lymphoma

* INCLUSION CRITERIA: * Patients must have a confirmed histologic diagnosis of Follicular Lymphoma (FL), grade 1-2 or 3a, according to the criteria established by the most recent version of the World Health Organization (WHO) classification system. Pathologic diagnosis must be confirmed by Laboratory of Pathology, National Cancer Institute (NCI) * Stage II-IV disease. NOTE: Patients with stage I FL who have been treated with radiation therapy and have subsequently relapsed are eligible. * No prior systemic treatment for FL with chemotherapy, targeted small molecule therapy, or monoclonal antibody therapy prior to the first dose of copanlisib treatment. Patients may have received prior radiation therapy only; radiation therapy must have been completed \>12 weeks prior to the first dose of copanlisib. NOTE: Prior shortterm (less than or equal to 7 days) use of corticosteroids for acute medical complications related to sites of FL involvement is permitted. * Patients must meet standard criteria for initiation of systemic therapy as evidenced by presence of one of the following: * Development of symptomatic enlarged lymph nodes or spleen * Development of B symptoms (fever, night sweats, weight loss) or severe pruritus * Development of significant serous pleural or pericardial effusions (small effusions seen only on computed tomography (CT) scans are not indications for systemic therapy) * Development of bone marrow failure as a result of involvement by FL and not attributable to other causes; this would be manifest as a hemoglobin (Hgb) \< 9 g/dl, absolute neutrophil count \< 1 x 10\^9/L, or platelet count \< 75 x 10\^9/L * Critical organ involvement, organ compression (e.g., ureteric obstruction or epidural compression), or significant risk of future organ compressions * Increase in the size of lymph nodes on CT scans indicating progression of disease from previous CT scans * Adequate tissue from diagnostic biopsy; formalin fixed tissue block or 20 slides of tumor sample (archival or fresh) must be available for performance of correlative studies * Be greater than or equal to 8 years of age on day of signing informed consent. NOTE: Because no dosing or adverse event data are currently available on the use of (copanlisib) in patients \<18 years of age, children are excluded from this study * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Adequate organ function as evidenced by the following laboratory parameters: * Absolute neutrophil count (ANC): \>= 1,500 /mm\^3 (unless due to involvement by lymphoma or benign ethnic neutropenia) * Platelets: \>=75,000 / mcL (unless due to involvement by lymphoma; transfusions not permitted) * Hemoglobin: \>= 8 g/dL (transfusions permitted) * Renal function: Glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2 as estimated by the Modification of Diet in Renal Disease (MDRD) abbreviated formula. If not on target, a 24-hour urine creatinine clearance can be used to directly measure. * Serum total bilirubin: less than or equal to 1.5 X upper limit of normal (ULN) OR (\< 3 x ULN for patients with Gilbert syndrome, patients with cholestasis due to compressive adenopathies of the hepatic hilum or documented liver involvement or with biliary obstruction due to lymphoma) * Aspartate aminotransferase (AST) serum glutamic-oxaloacetic transaminase (SGOT) and alanine aminotransferase (ALT) serum glutamic-pyruvic transaminase (SGPT): less than or equal to 2.5 x ULN (less than or equal to 5 x ULN for patients with liver involvement by lymphoma) * Lipase: less than or equal to 1.5 x ULN * Women of childbearing potential (WOCBP) and men must agree to use effective contraception when sexually active. This applies for the time period between signing of the informed consent form and for at least 1 month after the last dose of copanlisib and 12 months after the last dose of rituximab, whichever is later, for WOCBP and for men after the last administration of study treatment. NOTE: A woman is considered of childbearing potential, (i.e., fertile), following menarche and until becoming post-menopausal unless permanently sterile. Permanent sterilization methods include but are not limited to hysterectomy, bilateral salpingectomy and bilateral oophorectomy. A postmenopausal state is defined as no menses for continuous 12 months without an alternative medical cause. A high follicle stimulating hormone (FSH) level in the postmenopausal range may be used to confirm a post-menopausal state in women not using hormonal contraception or hormonal replacement therapy. The investigator or a designated associate is requested to advise the patient how to achieve highly effective birth control (failure rate of less than 1%), e.g., intrauterine device (IUD), intrauterine hormone-releasing system (IUS), bilateral tubal occlusion, vasectomized partner and sexual abstinence. The use of condoms by male patients is required unless the female partner is permanently sterile. * Ability of patient to understand and the willingness to sign a written informed consent document EXCLUSION CRITERIA: * Known lymphomatous involvement of the central nervous system * History of any known primary or acquired immunodeficiency syndrome (e.g., human immunodeficiency virus (HIV)) * Cytomegalovirus (CMV) polymerase chain reaction (PCR) positive at baseline * Hepatitis B surface antigen (HbsAg) or core antibody (HbcAb) positive with a positive Hep B DNA Quantitative, HBV Viral Load result. NOTE: Subjects with positive hepatitis B serology (hepatitis B surface antigen (HbsAg) or Hepatitis B core antibody (HBcAb) may be enrolled onto the study but they must have a negative Hep B deoxyribonucleic acid (DNA) Quantitative, hepatitis B virus (HBV) Viral Load result before enrollment. * Uncontrolled intercurrent illness including, but not limited to the following that may limit interpretation of results or that could increase risk to the patient at the discretion of the investigator: * Active autoimmune disease that has required systemic treatment in the past 12 months (i.e., with use of disease modifying agents, corticosteroids or immunosuppressive drugs). NOTE: Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * History of (non-infectious) pneumonitis that required steroids, evidence of interstitial lung disease or active, non-infectious pneumonitis. * Active hepatitis C infection. NOTE: Subjects who are hepatitis C antibody positive will need to have a negative HCV PCR result before enrollment. Those with a positive PCR for hepatitis C are excluded. * Congestive heart failure \> New York Heart Association (NYHA) class 2 * Unstable angina * Myocardial infarction in the past 6 months * Uncontrolled hypertension despite optimal medical management * Arterial thromboembolic events such as cerebrovascular accident (including transient ischemic attacks), in prior 3 months * Uncontrolled Type I or II diabetes despite optimal medical management * Any second malignancy that requires active systemic therapy * Known mental or physical illness that would interfere with cooperation with the requirements of the trial or confound the results or interpretation of the results of the trial and, in the opinion of the treating investigator, would make the patient inappropriate for entry into the study. * Severe hepatic impairment (Child-Pugh C) * Requirement to continue on any of the medications that are excluded * Organ compromise that, in the opinion of the principal investigator (PI), necessitates immediate cytoreductive therapy * Pregnant or breast-feeding patients. Women of childbearing potential must have a serum pregnancy test performed a maximum of 7 days before start of treatment, and a negative result must be documented before start of treatment * Major surgical procedure or significant traumatic injury (as judged by the investigator) within 28 days before start of treatment, or have not recovered from major side effects, open biopsy within 7 days before start of treatment