Properties

Label 912.4.a
Level $912$
Weight $4$
Character orbit 912.a
Rep. character $\chi_{912}(1,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $23$
Sturm bound $640$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(640\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(912))\).

Total New Old
Modular forms 492 54 438
Cusp forms 468 54 414
Eisenstein series 24 0 24

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

\(2\)\(3\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(5\)
\(+\)\(-\)\(-\)$+$\(9\)
\(-\)\(+\)\(+\)$-$\(6\)
\(-\)\(+\)\(-\)$+$\(7\)
\(-\)\(-\)\(+\)$+$\(7\)
\(-\)\(-\)\(-\)$-$\(6\)
Plus space\(+\)\(29\)
Minus space\(-\)\(25\)

Trace form

\( 54 q - 4 q^{5} - 28 q^{7} + 486 q^{9} + O(q^{10}) \) \( 54 q - 4 q^{5} - 28 q^{7} + 486 q^{9} + 40 q^{11} + 92 q^{13} + 52 q^{17} + 114 q^{19} + 412 q^{23} + 1138 q^{25} + 684 q^{29} + 24 q^{33} - 348 q^{35} - 116 q^{37} - 312 q^{39} - 236 q^{41} + 892 q^{43} - 36 q^{45} + 744 q^{47} + 3054 q^{49} - 180 q^{53} - 1804 q^{55} - 1496 q^{59} + 156 q^{61} - 252 q^{63} + 696 q^{65} - 408 q^{67} + 1256 q^{71} + 1724 q^{73} + 1104 q^{75} - 1904 q^{77} - 1680 q^{79} + 4374 q^{81} - 2076 q^{83} + 1400 q^{85} - 1044 q^{87} - 220 q^{89} - 104 q^{91} - 164 q^{97} + 360 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(912))\) 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 19
912.4.a.a 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(-3\) \(-12\) \(20\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-12q^{5}+20q^{7}+9q^{9}+4q^{11}+\cdots\)
912.4.a.b 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(-3\) \(-7\) \(15\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-7q^{5}+15q^{7}+9q^{9}+7^{2}q^{11}+\cdots\)
912.4.a.c 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(-3\) \(-3\) \(17\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-3q^{5}+17q^{7}+9q^{9}+19q^{11}+\cdots\)
912.4.a.d 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(-3\) \(12\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+12q^{5}-4q^{7}+9q^{9}-8q^{11}+\cdots\)
912.4.a.e 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(3\) \(-19\) \(-9\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-19q^{5}-9q^{7}+9q^{9}+13q^{11}+\cdots\)
912.4.a.f 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(3\) \(-11\) \(15\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-11q^{5}+15q^{7}+9q^{9}+29q^{11}+\cdots\)
912.4.a.g 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(3\) \(-7\) \(-21\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-7q^{5}-21q^{7}+9q^{9}+37q^{11}+\cdots\)
912.4.a.h 912.a 1.a $1$ $53.810$ \(\Q\) None \(0\) \(3\) \(4\) \(12\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+4q^{5}+12q^{7}+9q^{9}-40q^{11}+\cdots\)
912.4.a.i 912.a 1.a $2$ $53.810$ \(\Q(\sqrt{97}) \) None \(0\) \(-6\) \(-13\) \(-3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-6-\beta )q^{5}-3\beta q^{7}+9q^{9}+\cdots\)
912.4.a.j 912.a 1.a $2$ $53.810$ \(\Q(\sqrt{273}) \) None \(0\) \(-6\) \(11\) \(-9\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(6-\beta )q^{5}+(-4-\beta )q^{7}+9q^{9}+\cdots\)
912.4.a.k 912.a 1.a $2$ $53.810$ \(\Q(\sqrt{105}) \) None \(0\) \(6\) \(-9\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-4-\beta )q^{5}+(-2+\beta )q^{7}+\cdots\)
912.4.a.l 912.a 1.a $2$ $53.810$ \(\Q(\sqrt{897}) \) None \(0\) \(6\) \(3\) \(17\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(1+\beta )q^{5}+(9-\beta )q^{7}+9q^{9}+\cdots\)
912.4.a.m 912.a 1.a $2$ $53.810$ \(\Q(\sqrt{17}) \) None \(0\) \(6\) \(18\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(9+\beta )q^{5}+(-2+2\beta )q^{7}+\cdots\)
912.4.a.n 912.a 1.a $2$ $53.810$ \(\Q(\sqrt{33}) \) None \(0\) \(6\) \(22\) \(-36\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(11-\beta )q^{5}+(-18+2\beta )q^{7}+\cdots\)
912.4.a.o 912.a 1.a $3$ $53.810$ 3.3.226425.1 None \(0\) \(-9\) \(31\) \(-9\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(10+\beta _{1})q^{5}+(-3-\beta _{2})q^{7}+\cdots\)
912.4.a.p 912.a 1.a $3$ $53.810$ 3.3.2700.1 None \(0\) \(9\) \(-12\) \(18\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-4+\beta _{1}+2\beta _{2})q^{5}+(6-\beta _{1}+\cdots)q^{7}+\cdots\)
912.4.a.q 912.a 1.a $3$ $53.810$ 3.3.24665.1 None \(0\) \(9\) \(-1\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+\beta _{1}q^{5}+(-3-2\beta _{1}+3\beta _{2})q^{7}+\cdots\)
912.4.a.r 912.a 1.a $4$ $53.810$ 4.4.4914253.1 None \(0\) \(-12\) \(-16\) \(14\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-4+\beta _{2}+\beta _{3})q^{5}+(4+\beta _{1}+\cdots)q^{7}+\cdots\)
912.4.a.s 912.a 1.a $4$ $53.810$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(-8\) \(-10\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2+\beta _{1})q^{5}+(-3-\beta _{1}+\cdots)q^{7}+\cdots\)
912.4.a.t 912.a 1.a $4$ $53.810$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(-6\) \(-38\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-2-\beta _{3})q^{5}+(-10+2\beta _{2}+\cdots)q^{7}+\cdots\)
912.4.a.u 912.a 1.a $4$ $53.810$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-12\) \(9\) \(-7\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(2+\beta _{2})q^{5}+(-2+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
912.4.a.v 912.a 1.a $4$ $53.810$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(12\) \(4\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(1-\beta _{1})q^{5}+(3-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
912.4.a.w 912.a 1.a $5$ $53.810$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(15\) \(6\) \(-10\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(1+\beta _{1})q^{5}+(-2+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(912))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(912)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 2}\)