Properties

Label 4680.2.a
Level $4680$
Weight $2$
Character orbit 4680.a
Rep. character $\chi_{4680}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $38$
Sturm bound $2016$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 4680 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4680.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 38 \)
Sturm bound: \(2016\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4680))\).

Total New Old
Modular forms 1040 60 980
Cusp forms 977 60 917
Eisenstein series 63 0 63

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

\(2\)\(3\)\(5\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(3\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(27\)
Minus space\(-\)\(33\)

Trace form

\( 60 q + 2 q^{5} + O(q^{10}) \) \( 60 q + 2 q^{5} + 4 q^{11} - 2 q^{13} - 16 q^{17} - 12 q^{19} + 60 q^{25} - 16 q^{29} + 12 q^{37} - 16 q^{41} - 12 q^{43} + 60 q^{49} + 4 q^{53} - 4 q^{59} + 16 q^{61} + 4 q^{65} - 8 q^{67} + 16 q^{71} + 36 q^{73} - 8 q^{77} - 16 q^{83} - 4 q^{85} - 48 q^{89} - 8 q^{95} + 60 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4680))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5 13
4680.2.a.a 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(-5\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-5q^{7}-q^{11}+q^{13}+3q^{17}+\cdots\)
4680.2.a.b 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}-3q^{11}+q^{13}+3q^{17}+\cdots\)
4680.2.a.c 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}+3q^{11}-q^{13}+q^{17}+\cdots\)
4680.2.a.d 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+q^{13}+4q^{17}+6q^{19}+\cdots\)
4680.2.a.e 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{11}+q^{13}-2q^{17}+2q^{19}+\cdots\)
4680.2.a.f 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{13}+6q^{17}-8q^{19}-6q^{23}+\cdots\)
4680.2.a.g 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{11}+q^{13}-2q^{17}-4q^{19}+\cdots\)
4680.2.a.h 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-4q^{11}-q^{13}-8q^{17}+\cdots\)
4680.2.a.i 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-4q^{11}-q^{13}+2q^{19}+\cdots\)
4680.2.a.j 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(3\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+3q^{7}+5q^{11}-q^{13}-3q^{17}+\cdots\)
4680.2.a.k 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}-4q^{11}+q^{13}-6q^{17}+\cdots\)
4680.2.a.l 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+q^{13}-2q^{17}+q^{25}+\cdots\)
4680.2.a.m 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(-4\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{7}-4q^{11}+q^{13}+6q^{17}+\cdots\)
4680.2.a.n 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(-3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}+3q^{11}+q^{13}-3q^{17}+\cdots\)
4680.2.a.o 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}+5q^{11}+q^{13}-3q^{17}+\cdots\)
4680.2.a.p 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{11}+q^{13}+2q^{17}-4q^{19}+\cdots\)
4680.2.a.q 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{13}-2q^{17}-4q^{23}+q^{25}+\cdots\)
4680.2.a.r 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{13}-6q^{17}-8q^{19}+6q^{23}+\cdots\)
4680.2.a.s 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{13}+6q^{17}+4q^{19}+q^{25}+\cdots\)
4680.2.a.t 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{11}-q^{13}+6q^{17}+4q^{19}+\cdots\)
4680.2.a.u 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}+4q^{11}-q^{13}+8q^{17}+\cdots\)
4680.2.a.v 4680.a 1.a $1$ $37.370$ \(\Q\) None \(0\) \(0\) \(1\) \(3\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+3q^{7}-5q^{11}-q^{13}+3q^{17}+\cdots\)
4680.2.a.w 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-2\) \(-4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-3\beta q^{11}-q^{13}+(2+2\beta )q^{17}+\cdots\)
4680.2.a.x 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-2\) \(-4\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-q^{13}+(2-\beta )q^{17}+2q^{19}+\cdots\)
4680.2.a.y 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-2\) \(-1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-\beta q^{7}+(4-\beta )q^{11}+q^{13}+(2+\cdots)q^{17}+\cdots\)
4680.2.a.z 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{6}) \) None \(0\) \(0\) \(-2\) \(4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}+(2-\beta )q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4680.2.a.ba 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(-4\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-q^{13}+(-2-\beta )q^{17}+\cdots\)
4680.2.a.bb 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{41}) \) None \(0\) \(0\) \(2\) \(-1\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-\beta q^{7}+(2+\beta )q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4680.2.a.bc 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+(-3-\beta )q^{11}-q^{13}-2q^{17}+\cdots\)
4680.2.a.bd 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(0\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2\beta q^{7}+(-1-\beta )q^{11}+q^{13}+\cdots\)
4680.2.a.be 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(0\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2\beta q^{7}+(3+\beta )q^{11}+q^{13}+\cdots\)
4680.2.a.bf 4680.a 1.a $2$ $37.370$ \(\Q(\sqrt{33}) \) None \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}+\beta q^{7}+\beta q^{11}-q^{13}+(-2+\cdots)q^{17}+\cdots\)
4680.2.a.bg 4680.a 1.a $3$ $37.370$ 3.3.1016.1 None \(0\) \(0\) \(-3\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-\beta _{2}q^{7}+(2-\beta _{1})q^{11}-q^{13}+\cdots\)
4680.2.a.bh 4680.a 1.a $3$ $37.370$ 3.3.940.1 None \(0\) \(0\) \(-3\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+\beta _{2}q^{7}+(-2-\beta _{1})q^{11}-q^{13}+\cdots\)
4680.2.a.bi 4680.a 1.a $3$ $37.370$ 3.3.621.1 None \(0\) \(0\) \(-3\) \(3\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+(1+\beta _{2})q^{7}+(-1+\beta _{1}+\beta _{2})q^{11}+\cdots\)
4680.2.a.bj 4680.a 1.a $3$ $37.370$ 3.3.1016.1 None \(0\) \(0\) \(3\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-\beta _{2}q^{7}+(-2+\beta _{1})q^{11}-q^{13}+\cdots\)
4680.2.a.bk 4680.a 1.a $3$ $37.370$ 3.3.621.1 None \(0\) \(0\) \(3\) \(3\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(1+\beta _{2})q^{7}+(1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
4680.2.a.bl 4680.a 1.a $3$ $37.370$ 3.3.1849.1 None \(0\) \(0\) \(3\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+(2-\beta _{1})q^{7}+(-1-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4680))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4680)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(585))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(936))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1170))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1560))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2340))\)\(^{\oplus 2}\)