Properties

Label 9216.2.a
Level $9216$
Weight $2$
Character orbit 9216.a
Rep. character $\chi_{9216}(1,\cdot)$
Character field $\Q$
Dimension $156$
Newform subspaces $46$
Sturm bound $3072$
Trace bound $67$

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Defining parameters

Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 46 \)
Sturm bound: \(3072\)
Trace bound: \(67\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(13\), \(17\), \(19\), \(67\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(9216))\).

Total New Old
Modular forms 1632 164 1468
Cusp forms 1441 156 1285
Eisenstein series 191 8 183

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim.
\(+\)\(+\)\(+\)\(28\)
\(+\)\(-\)\(-\)\(48\)
\(-\)\(+\)\(-\)\(36\)
\(-\)\(-\)\(+\)\(44\)
Plus space\(+\)\(72\)
Minus space\(-\)\(84\)

Trace form

\( 156q + O(q^{10}) \) \( 156q - 8q^{17} + 140q^{25} + 140q^{49} + 8q^{65} - 8q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(9216))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
9216.2.a.a \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-8\) \(-\) \(+\) \(q+\beta q^{5}-4q^{7}-4\beta q^{11}+3\beta q^{13}+\cdots\)
9216.2.a.b \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-8\) \(-\) \(-\) \(q+\beta q^{5}-4q^{7}+2\beta q^{11}-3\beta q^{13}+\cdots\)
9216.2.a.c \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-8\) \(-\) \(+\) \(q+\beta q^{5}-4q^{7}-4\beta q^{11}-3\beta q^{13}+\cdots\)
9216.2.a.d \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(-\) \(q+\beta q^{5}-2q^{7}+\beta q^{11}-\beta q^{13}-2q^{17}+\cdots\)
9216.2.a.e \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+2\beta q^{5}+3\beta q^{7}-4q^{11}-3\beta q^{13}+\cdots\)
9216.2.a.f \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{5}-4q^{11}+\beta q^{13}+4q^{19}+4\beta q^{23}+\cdots\)
9216.2.a.g \(2\) \(73.590\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+3\beta q^{5}-\beta q^{13}-8q^{17}+13q^{25}+\cdots\)
9216.2.a.h \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{5}-2\beta q^{7}+3\beta q^{13}-4q^{17}+\cdots\)
9216.2.a.i \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{5}+2\beta q^{7}+3\beta q^{13}-4q^{17}+\cdots\)
9216.2.a.j \(2\) \(73.590\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{5}+\beta q^{13}-2q^{17}-3q^{25}-3\beta q^{29}+\cdots\)
9216.2.a.k \(2\) \(73.590\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+3\beta q^{5}-5\beta q^{13}-2q^{17}+13q^{25}+\cdots\)
9216.2.a.l \(2\) \(73.590\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{5}-\beta q^{13}+2q^{17}-3q^{25}-3\beta q^{29}+\cdots\)
9216.2.a.m \(2\) \(73.590\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+3\beta q^{5}+5\beta q^{13}+2q^{17}+13q^{25}+\cdots\)
9216.2.a.n \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{7}-\beta q^{13}+6q^{17}-6q^{19}+6\beta q^{23}+\cdots\)
9216.2.a.o \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{7}+\beta q^{13}+6q^{17}+6q^{19}+6\beta q^{23}+\cdots\)
9216.2.a.p \(2\) \(73.590\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{5}+5\beta q^{13}+8q^{17}-3q^{25}+\cdots\)
9216.2.a.q \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+2\beta q^{5}-3\beta q^{7}+4q^{11}-3\beta q^{13}+\cdots\)
9216.2.a.r \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{5}+4q^{11}+\beta q^{13}-4q^{19}-4\beta q^{23}+\cdots\)
9216.2.a.s \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(4\) \(-\) \(-\) \(q+\beta q^{5}+2q^{7}-\beta q^{11}-\beta q^{13}-2q^{17}+\cdots\)
9216.2.a.t \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(8\) \(-\) \(+\) \(q+\beta q^{5}+4q^{7}+4\beta q^{11}+3\beta q^{13}+\cdots\)
9216.2.a.u \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(8\) \(-\) \(-\) \(q+\beta q^{5}+4q^{7}-2\beta q^{11}-3\beta q^{13}+\cdots\)
9216.2.a.v \(2\) \(73.590\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(8\) \(-\) \(+\) \(q+\beta q^{5}+4q^{7}+4\beta q^{11}-3\beta q^{13}+\cdots\)
9216.2.a.w \(4\) \(73.590\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(-8\) \(0\) \(-\) \(-\) \(q+(-2-\beta _{2})q^{5}+(-\beta _{1}+\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.x \(4\) \(73.590\) 4.4.4352.1 None \(0\) \(0\) \(-4\) \(-4\) \(-\) \(-\) \(q+(-1-\beta _{3})q^{5}+(-1-\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.y \(4\) \(73.590\) 4.4.4352.1 None \(0\) \(0\) \(-4\) \(4\) \(+\) \(-\) \(q+(-1-\beta _{3})q^{5}+(1+\beta _{2})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9216.2.a.z \(4\) \(73.590\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(0\) \(-8\) \(-\) \(-\) \(q+(\beta _{2}-\beta _{3})q^{5}+(-2-\beta _{3})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.ba \(4\) \(73.590\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(0\) \(-8\) \(+\) \(-\) \(q+(-\beta _{2}-\beta _{3})q^{5}+(-2+\beta _{3})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.bb \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta _{3}q^{5}+(-\beta _{2}-\beta _{3})q^{7}+(-3-\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.bc \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+(\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{7}+(-2+\beta _{3})q^{11}+\cdots\)
9216.2.a.bd \(4\) \(73.590\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+(-2+\beta _{3})q^{11}+\cdots\)
9216.2.a.be \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta _{2}q^{5}-\beta _{1}q^{7}+\beta _{3}q^{11}-\beta _{1}q^{13}+\cdots\)
9216.2.a.bf \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+(-2\beta _{1}-\beta _{2})q^{7}+(\beta _{1}+2\beta _{2})q^{13}+\cdots\)
9216.2.a.bg \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+(2\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+2\beta _{2})q^{13}+\beta _{3}q^{19}+\cdots\)
9216.2.a.bh \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+\beta _{2}q^{5}-\beta _{1}q^{7}+\beta _{3}q^{11}+\beta _{1}q^{13}+\cdots\)
9216.2.a.bi \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+(\beta _{1}+\beta _{2})q^{5}-\beta _{2}q^{7}+(2-\beta _{3})q^{11}+\cdots\)
9216.2.a.bj \(4\) \(73.590\) \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{2}q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(2+\beta _{3})q^{11}+\cdots\)
9216.2.a.bk \(4\) \(73.590\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(-\) \(q+\beta _{3}q^{5}+(\beta _{2}+\beta _{3})q^{7}+(3+\beta _{1})q^{11}+\cdots\)
9216.2.a.bl \(4\) \(73.590\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(0\) \(8\) \(-\) \(-\) \(q+(\beta _{2}-\beta _{3})q^{5}+(2+\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.bm \(4\) \(73.590\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(0\) \(8\) \(+\) \(-\) \(q+(-\beta _{2}-\beta _{3})q^{5}+(2-\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
9216.2.a.bn \(4\) \(73.590\) 4.4.4352.1 None \(0\) \(0\) \(4\) \(-4\) \(-\) \(-\) \(q+(1+\beta _{3})q^{5}+(-1-\beta _{2})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
9216.2.a.bo \(4\) \(73.590\) 4.4.4352.1 None \(0\) \(0\) \(4\) \(4\) \(+\) \(-\) \(q+(1+\beta _{3})q^{5}+(1+\beta _{2})q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
9216.2.a.bp \(4\) \(73.590\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(8\) \(0\) \(+\) \(-\) \(q+(2-\beta _{2})q^{5}+(\beta _{1}+\beta _{3})q^{7}+(2\beta _{1}-\beta _{3})q^{11}+\cdots\)
9216.2.a.bq \(8\) \(73.590\) 8.8.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-8\) \(+\) \(+\) \(q+\beta _{2}q^{5}+(-1-\beta _{5})q^{7}+(-\beta _{2}+\beta _{7})q^{11}+\cdots\)
9216.2.a.br \(8\) \(73.590\) 8.8.3288334336.1 None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q-\beta _{4}q^{5}-\beta _{7}q^{7}+\beta _{2}q^{11}+(-2+\beta _{5}+\cdots)q^{13}+\cdots\)
9216.2.a.bs \(8\) \(73.590\) 8.8.3288334336.1 None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q-\beta _{4}q^{5}+\beta _{7}q^{7}-\beta _{2}q^{11}+(2-\beta _{5}+\cdots)q^{13}+\cdots\)
9216.2.a.bt \(8\) \(73.590\) 8.8.\(\cdots\).1 None \(0\) \(0\) \(0\) \(8\) \(-\) \(+\) \(q-\beta _{5}q^{5}+(1-\beta _{6})q^{7}+(-\beta _{2}-\beta _{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(9216))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(9216)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 14}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(288))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(576))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1024))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2304))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3072))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4608))\)\(^{\oplus 2}\)