Properties

Label 8712.2.a
Level $8712$
Weight $2$
Character orbit 8712.a
Rep. character $\chi_{8712}(1,\cdot)$
Character field $\Q$
Dimension $136$
Newform subspaces $63$
Sturm bound $3168$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 8712 = 2^{3} \cdot 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8712.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 63 \)
Sturm bound: \(3168\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(5\), \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8712))\).

Total New Old
Modular forms 1680 136 1544
Cusp forms 1489 136 1353
Eisenstein series 191 0 191

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

\(2\)\(3\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(-\)\(-\)\(15\)
\(+\)\(-\)\(+\)\(-\)\(22\)
\(+\)\(-\)\(-\)\(+\)\(19\)
\(-\)\(+\)\(+\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(15\)
\(-\)\(-\)\(+\)\(+\)\(20\)
\(-\)\(-\)\(-\)\(-\)\(21\)
Plus space\(+\)\(66\)
Minus space\(-\)\(70\)

Trace form

\( 136q + 2q^{5} - 4q^{7} + O(q^{10}) \) \( 136q + 2q^{5} - 4q^{7} + 4q^{17} + 138q^{25} - 8q^{29} + 4q^{35} - 10q^{37} - 8q^{41} + 4q^{43} + 12q^{47} + 124q^{49} - 6q^{53} - 26q^{59} - 16q^{61} - 24q^{65} + 10q^{67} + 12q^{71} - 20q^{79} - 12q^{83} + 4q^{85} + 4q^{89} - 8q^{91} + 16q^{95} - 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8712))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 11
8712.2.a.a \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-4\) \(2\) \(+\) \(-\) \(-\) \(q-4q^{5}+2q^{7}-6q^{17}-4q^{19}+6q^{23}+\cdots\)
8712.2.a.b \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-3\) \(-4\) \(+\) \(-\) \(-\) \(q-3q^{5}-4q^{7}-3q^{13}+3q^{17}-4q^{19}+\cdots\)
8712.2.a.c \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-3\) \(4\) \(-\) \(-\) \(-\) \(q-3q^{5}+4q^{7}+3q^{13}-3q^{17}+4q^{19}+\cdots\)
8712.2.a.d \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-2\) \(-2\) \(-\) \(+\) \(+\) \(q-2q^{5}-2q^{7}-2q^{13}-2q^{17}-4q^{19}+\cdots\)
8712.2.a.e \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-2\) \(-2\) \(+\) \(-\) \(+\) \(q-2q^{5}-2q^{7}-6q^{17}+6q^{19}+4q^{23}+\cdots\)
8712.2.a.f \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-2\) \(0\) \(+\) \(-\) \(-\) \(q-2q^{5}-2q^{13}+6q^{17}-4q^{23}-q^{25}+\cdots\)
8712.2.a.g \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-2\) \(2\) \(-\) \(-\) \(+\) \(q-2q^{5}+2q^{7}+6q^{17}-6q^{19}+4q^{23}+\cdots\)
8712.2.a.h \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-2\) \(2\) \(+\) \(+\) \(+\) \(q-2q^{5}+2q^{7}+2q^{13}+2q^{17}+4q^{19}+\cdots\)
8712.2.a.i \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-1\) \(-4\) \(-\) \(-\) \(+\) \(q-q^{5}-4q^{7}-4q^{13}+4q^{17}+4q^{19}+\cdots\)
8712.2.a.j \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(-1\) \(4\) \(+\) \(-\) \(+\) \(q-q^{5}+4q^{7}+4q^{13}-4q^{17}-4q^{19}+\cdots\)
8712.2.a.k \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(0\) \(-2\) \(-\) \(-\) \(-\) \(q-2q^{7}-2q^{17}-8q^{19}+2q^{23}-5q^{25}+\cdots\)
8712.2.a.l \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(0\) \(-1\) \(+\) \(-\) \(-\) \(q-q^{7}+6q^{13}-4q^{17}-q^{19}+2q^{23}+\cdots\)
8712.2.a.m \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}-6q^{13}+4q^{17}+q^{19}+2q^{23}+\cdots\)
8712.2.a.n \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(-\) \(+\) \(-\) \(q+2q^{7}+6q^{13}-6q^{17}+2q^{19}-8q^{23}+\cdots\)
8712.2.a.o \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(0\) \(2\) \(+\) \(+\) \(-\) \(q+2q^{7}+6q^{13}+6q^{17}+2q^{19}+8q^{23}+\cdots\)
8712.2.a.p \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(-\) \(-\) \(-\) \(q+q^{5}-q^{13}-3q^{17}-4q^{23}-4q^{25}+\cdots\)
8712.2.a.q \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(1\) \(0\) \(+\) \(-\) \(-\) \(q+q^{5}+q^{13}+3q^{17}-4q^{23}-4q^{25}+\cdots\)
8712.2.a.r \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(2\) \(-4\) \(+\) \(-\) \(-\) \(q+2q^{5}-4q^{7}-6q^{13}+6q^{17}+8q^{19}+\cdots\)
8712.2.a.s \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(2\) \(-2\) \(+\) \(+\) \(+\) \(q+2q^{5}-2q^{7}-2q^{13}+2q^{17}-4q^{19}+\cdots\)
8712.2.a.t \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(2\) \(-2\) \(+\) \(-\) \(+\) \(q+2q^{5}-2q^{7}+4q^{13}+2q^{17}+2q^{19}+\cdots\)
8712.2.a.u \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(2\) \(0\) \(-\) \(-\) \(-\) \(q+2q^{5}+2q^{13}+2q^{17}+4q^{19}+8q^{23}+\cdots\)
8712.2.a.v \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(2\) \(2\) \(-\) \(-\) \(+\) \(q+2q^{5}+2q^{7}-4q^{13}-2q^{17}-2q^{19}+\cdots\)
8712.2.a.w \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(2\) \(2\) \(-\) \(+\) \(+\) \(q+2q^{5}+2q^{7}+2q^{13}-2q^{17}+4q^{19}+\cdots\)
8712.2.a.x \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(3\) \(2\) \(+\) \(-\) \(-\) \(q+3q^{5}+2q^{7}-6q^{17}-4q^{19}-q^{23}+\cdots\)
8712.2.a.y \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(4\) \(-4\) \(-\) \(-\) \(+\) \(q+4q^{5}-4q^{7}+6q^{13}-6q^{17}-6q^{19}+\cdots\)
8712.2.a.z \(1\) \(69.566\) \(\Q\) None \(0\) \(0\) \(4\) \(4\) \(+\) \(-\) \(+\) \(q+4q^{5}+4q^{7}-6q^{13}+6q^{17}+6q^{19}+\cdots\)
8712.2.a.ba \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-3\) \(-4\) \(-\) \(-\) \(+\) \(q+(-1-\beta )q^{5}+(-3+2\beta )q^{7}+(1-2\beta )q^{13}+\cdots\)
8712.2.a.bb \(2\) \(69.566\) \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-3\) \(2\) \(-\) \(-\) \(-\) \(q+(-1-\beta )q^{5}+(2-2\beta )q^{7}+2\beta q^{13}+\cdots\)
8712.2.a.bc \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-3\) \(4\) \(+\) \(-\) \(-\) \(q+(-1-\beta )q^{5}+(3-2\beta )q^{7}+(-1+2\beta )q^{13}+\cdots\)
8712.2.a.bd \(2\) \(69.566\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(-4\) \(+\) \(-\) \(+\) \(q-q^{5}-2q^{7}+(-2-\beta )q^{13}+(2-\beta )q^{17}+\cdots\)
8712.2.a.be \(2\) \(69.566\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(4\) \(-\) \(-\) \(+\) \(q-q^{5}+2q^{7}+(2+\beta )q^{13}+(-2+\beta )q^{17}+\cdots\)
8712.2.a.bf \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-1\) \(-3\) \(+\) \(-\) \(-\) \(q+(1-3\beta )q^{5}-3\beta q^{7}+(-2+2\beta )q^{13}+\cdots\)
8712.2.a.bg \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-1\) \(3\) \(-\) \(-\) \(+\) \(q+(1-3\beta )q^{5}+3\beta q^{7}+(2-2\beta )q^{13}+\cdots\)
8712.2.a.bh \(2\) \(69.566\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(+\) \(-\) \(q+\beta q^{5}+(-2+\beta )q^{7}+(-2-\beta )q^{13}+\cdots\)
8712.2.a.bi \(2\) \(69.566\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(-\) \(q+\beta q^{5}+(-2-\beta )q^{7}+(-2+\beta )q^{13}+\cdots\)
8712.2.a.bj \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(-2\) \(+\) \(-\) \(-\) \(q-\beta q^{5}+(-1-\beta )q^{7}+(-4+\beta )q^{13}+\cdots\)
8712.2.a.bk \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(0\) \(2\) \(-\) \(-\) \(-\) \(q-\beta q^{5}+(1+\beta )q^{7}+(4-\beta )q^{13}-q^{17}+\cdots\)
8712.2.a.bl \(2\) \(69.566\) \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(1\) \(-1\) \(-\) \(-\) \(-\) \(q+\beta q^{5}+(-1+\beta )q^{7}+(-2+\beta )q^{13}+\cdots\)
8712.2.a.bm \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(1\) \(-1\) \(+\) \(-\) \(-\) \(q+\beta q^{5}+(-1+\beta )q^{7}+(4-3\beta )q^{13}+\cdots\)
8712.2.a.bn \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(1\) \(0\) \(+\) \(-\) \(-\) \(q+\beta q^{5}+(1-2\beta )q^{7}+(-3+2\beta )q^{13}+\cdots\)
8712.2.a.bo \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(1\) \(0\) \(-\) \(-\) \(+\) \(q+\beta q^{5}+(-1+2\beta )q^{7}+(3-2\beta )q^{13}+\cdots\)
8712.2.a.bp \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(1\) \(1\) \(-\) \(-\) \(+\) \(q+\beta q^{5}+(1-\beta )q^{7}+(-4+3\beta )q^{13}+\cdots\)
8712.2.a.bq \(2\) \(69.566\) \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(1\) \(1\) \(+\) \(-\) \(-\) \(q+\beta q^{5}+(1-\beta )q^{7}+(2-\beta )q^{13}+(-4+\cdots)q^{17}+\cdots\)
8712.2.a.br \(2\) \(69.566\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(4\) \(-2\) \(-\) \(-\) \(+\) \(q+(2+\beta )q^{5}+(-1-\beta )q^{7}+(-1-2\beta )q^{13}+\cdots\)
8712.2.a.bs \(2\) \(69.566\) \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(4\) \(2\) \(+\) \(-\) \(+\) \(q+(2+\beta )q^{5}+(1+\beta )q^{7}+(1+2\beta )q^{13}+\cdots\)
8712.2.a.bt \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(5\) \(-4\) \(-\) \(-\) \(-\) \(q+(3-\beta )q^{5}+(-1-2\beta )q^{7}+(-1+2\beta )q^{13}+\cdots\)
8712.2.a.bu \(2\) \(69.566\) \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(5\) \(4\) \(+\) \(-\) \(+\) \(q+(3-\beta )q^{5}+(1+2\beta )q^{7}+(1-2\beta )q^{13}+\cdots\)
8712.2.a.bv \(3\) \(69.566\) 3.3.1436.1 None \(0\) \(0\) \(0\) \(-1\) \(-\) \(+\) \(-\) \(q+\beta _{1}q^{5}-\beta _{2}q^{7}+(1+\beta _{2})q^{13}+(-2+\cdots)q^{17}+\cdots\)
8712.2.a.bw \(3\) \(69.566\) 3.3.1436.1 None \(0\) \(0\) \(0\) \(-1\) \(+\) \(+\) \(-\) \(q-\beta _{1}q^{5}-\beta _{2}q^{7}+(1+\beta _{2})q^{13}+(2+\cdots)q^{17}+\cdots\)
8712.2.a.bx \(3\) \(69.566\) 3.3.1436.1 None \(0\) \(0\) \(0\) \(1\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{5}+\beta _{2}q^{7}+(-1-\beta _{2})q^{13}+\cdots\)
8712.2.a.by \(3\) \(69.566\) 3.3.1436.1 None \(0\) \(0\) \(0\) \(1\) \(+\) \(+\) \(-\) \(q+\beta _{1}q^{5}+\beta _{2}q^{7}+(-1-\beta _{2})q^{13}+\cdots\)
8712.2.a.bz \(4\) \(69.566\) 4.4.13625.1 None \(0\) \(0\) \(-2\) \(-5\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{5}+(-2+\beta _{2}-2\beta _{3})q^{7}+(\beta _{1}+\cdots)q^{13}+\cdots\)
8712.2.a.ca \(4\) \(69.566\) 4.4.46224.1 None \(0\) \(0\) \(-2\) \(-4\) \(-\) \(-\) \(+\) \(q-\beta _{1}q^{5}+(-1-\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
8712.2.a.cb \(4\) \(69.566\) 4.4.46224.1 None \(0\) \(0\) \(-2\) \(4\) \(+\) \(-\) \(+\) \(q-\beta _{1}q^{5}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+(3+\cdots)q^{13}+\cdots\)
8712.2.a.cc \(4\) \(69.566\) 4.4.13625.1 None \(0\) \(0\) \(-2\) \(5\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{5}+(2-\beta _{2}+2\beta _{3})q^{7}+(-\beta _{1}+\cdots)q^{13}+\cdots\)
8712.2.a.cd \(4\) \(69.566\) 4.4.5225.1 None \(0\) \(0\) \(-1\) \(-1\) \(-\) \(-\) \(-\) \(q+(-1+\beta _{1}+\beta _{3})q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)
8712.2.a.ce \(4\) \(69.566\) 4.4.5225.1 None \(0\) \(0\) \(-1\) \(1\) \(+\) \(-\) \(+\) \(q+(-1+\beta _{1}+\beta _{3})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
8712.2.a.cf \(4\) \(69.566\) \(\Q(\sqrt{3}, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{5}+(-\beta _{1}+\beta _{2})q^{7}-\beta _{1}q^{13}+\cdots\)
8712.2.a.cg \(4\) \(69.566\) \(\Q(\sqrt{3}, \sqrt{19})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(+\) \(q+\beta _{1}q^{5}+(\beta _{1}-\beta _{2})q^{7}+\beta _{1}q^{13}+(2+\cdots)q^{17}+\cdots\)
8712.2.a.ch \(6\) \(69.566\) 6.6.62158000.1 None \(0\) \(0\) \(-5\) \(-1\) \(+\) \(+\) \(+\) \(q+(-1+\beta _{2})q^{5}+(\beta _{1}+\beta _{5})q^{7}+(-1+\cdots)q^{13}+\cdots\)
8712.2.a.ci \(6\) \(69.566\) 6.6.62158000.1 None \(0\) \(0\) \(-5\) \(1\) \(-\) \(+\) \(-\) \(q+(-1+\beta _{2})q^{5}+(-\beta _{1}-\beta _{5})q^{7}+(1+\cdots)q^{13}+\cdots\)
8712.2.a.cj \(6\) \(69.566\) 6.6.62158000.1 None \(0\) \(0\) \(5\) \(-1\) \(-\) \(+\) \(+\) \(q+(1-\beta _{2})q^{5}+(\beta _{1}+\beta _{5})q^{7}+(-1+\beta _{2}+\cdots)q^{13}+\cdots\)
8712.2.a.ck \(6\) \(69.566\) 6.6.62158000.1 None \(0\) \(0\) \(5\) \(1\) \(+\) \(+\) \(-\) \(q+(1-\beta _{2})q^{5}+(-\beta _{1}-\beta _{5})q^{7}+(1-\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8712))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8712)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 24}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(264))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(363))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(396))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(484))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(726))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(792))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(968))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1089))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1452))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2178))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2904))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4356))\)\(^{\oplus 2}\)