Properties

Label 912.6.a
Level $912$
Weight $6$
Character orbit 912.a
Rep. character $\chi_{912}(1,\cdot)$
Character field $\Q$
Dimension $90$
Newform subspaces $27$
Sturm bound $960$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 27 \)
Sturm bound: \(960\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(912))\).

Total New Old
Modular forms 812 90 722
Cusp forms 788 90 698
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.
\(+\)\(+\)\(+\)\(+\)\(12\)
\(+\)\(+\)\(-\)\(-\)\(10\)
\(+\)\(-\)\(+\)\(-\)\(13\)
\(+\)\(-\)\(-\)\(+\)\(9\)
\(-\)\(+\)\(+\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(+\)\(11\)
\(-\)\(-\)\(+\)\(+\)\(11\)
\(-\)\(-\)\(-\)\(-\)\(12\)
Plus space\(+\)\(43\)
Minus space\(-\)\(47\)

Trace form

\( 90 q - 76 q^{5} + 196 q^{7} + 7290 q^{9} + O(q^{10}) \) \( 90 q - 76 q^{5} + 196 q^{7} + 7290 q^{9} - 1208 q^{11} + 244 q^{13} - 404 q^{17} - 2166 q^{19} + 5500 q^{23} + 62094 q^{25} + 420 q^{29} - 9432 q^{33} + 16788 q^{35} + 25060 q^{37} + 12168 q^{39} + 2476 q^{41} + 10540 q^{43} - 6156 q^{45} - 5016 q^{47} + 197698 q^{49} + 89700 q^{53} - 6636 q^{55} + 118984 q^{59} - 18316 q^{61} + 15876 q^{63} - 86904 q^{65} + 12232 q^{67} - 15448 q^{71} + 31748 q^{73} - 96624 q^{75} + 14896 q^{77} + 132976 q^{79} + 590490 q^{81} - 104172 q^{83} + 100968 q^{85} + 90828 q^{87} + 175580 q^{89} - 229384 q^{91} - 146492 q^{97} - 97848 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\) 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
912.6.a.a 912.a 1.a $1$ $146.270$ \(\Q\) None \(0\) \(-9\) \(-98\) \(-240\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}-98q^{5}-240q^{7}+3^{4}q^{9}+\cdots\)
912.6.a.b 912.a 1.a $1$ $146.270$ \(\Q\) None \(0\) \(-9\) \(-91\) \(33\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}-91q^{5}+33q^{7}+3^{4}q^{9}+91q^{11}+\cdots\)
912.6.a.c 912.a 1.a $1$ $146.270$ \(\Q\) None \(0\) \(-9\) \(-54\) \(-104\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}-54q^{5}-104q^{7}+3^{4}q^{9}+\cdots\)
912.6.a.d 912.a 1.a $1$ $146.270$ \(\Q\) None \(0\) \(-9\) \(6\) \(176\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+6q^{5}+176q^{7}+3^{4}q^{9}+496q^{11}+\cdots\)
912.6.a.e 912.a 1.a $1$ $146.270$ \(\Q\) None \(0\) \(-9\) \(81\) \(247\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+3^{4}q^{5}+247q^{7}+3^{4}q^{9}+\cdots\)
912.6.a.f 912.a 1.a $1$ $146.270$ \(\Q\) None \(0\) \(9\) \(21\) \(143\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+21q^{5}+143q^{7}+3^{4}q^{9}+\cdots\)
912.6.a.g 912.a 1.a $2$ $146.270$ \(\Q(\sqrt{17}) \) None \(0\) \(-18\) \(-87\) \(251\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-41-5\beta )q^{5}+(11^{2}+9\beta )q^{7}+\cdots\)
912.6.a.h 912.a 1.a $2$ $146.270$ \(\Q(\sqrt{4089}) \) None \(0\) \(-18\) \(-49\) \(105\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-24-\beta )q^{5}+(50+5\beta )q^{7}+\cdots\)
912.6.a.i 912.a 1.a $2$ $146.270$ \(\Q(\sqrt{201}) \) None \(0\) \(18\) \(-13\) \(33\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-4-5\beta )q^{5}+(22-11\beta )q^{7}+\cdots\)
912.6.a.j 912.a 1.a $2$ $146.270$ \(\Q(\sqrt{2441}) \) None \(0\) \(18\) \(-5\) \(105\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-3-\beta )q^{5}+(53+\beta )q^{7}+\cdots\)
912.6.a.k 912.a 1.a $3$ $146.270$ 3.3.286833.1 None \(0\) \(-27\) \(-5\) \(33\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+(13+\cdots)q^{7}+\cdots\)
912.6.a.l 912.a 1.a $3$ $146.270$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(0\) \(-27\) \(135\) \(-125\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+(45-\beta _{1})q^{5}+(-42-\beta _{2})q^{7}+\cdots\)
912.6.a.m 912.a 1.a $3$ $146.270$ 3.3.616092.1 None \(0\) \(-27\) \(206\) \(-186\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+(69+\beta _{1})q^{5}+(-61+\beta _{1}+\cdots)q^{7}+\cdots\)
912.6.a.n 912.a 1.a $3$ $146.270$ 3.3.9153.1 None \(0\) \(27\) \(-9\) \(-141\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-3-3\beta _{1}-2\beta _{2})q^{5}+(-47+\cdots)q^{7}+\cdots\)
912.6.a.o 912.a 1.a $3$ $146.270$ 3.3.2922585.1 None \(0\) \(27\) \(63\) \(-125\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+(21+\beta _{1}-\beta _{2})q^{5}+(-40+\cdots)q^{7}+\cdots\)
912.6.a.p 912.a 1.a $4$ $146.270$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(36\) \(-84\) \(-54\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-21+\beta _{2})q^{5}+(-13-\beta _{1}+\cdots)q^{7}+\cdots\)
912.6.a.q 912.a 1.a $4$ $146.270$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(36\) \(-20\) \(-70\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-5+\beta _{2})q^{5}+(-19+4\beta _{1}+\cdots)q^{7}+\cdots\)
912.6.a.r 912.a 1.a $4$ $146.270$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(36\) \(-8\) \(142\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-3-\beta _{1}-3\beta _{3})q^{5}+(33+\cdots)q^{7}+\cdots\)
912.6.a.s 912.a 1.a $4$ $146.270$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(36\) \(117\) \(33\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(29-\beta _{2})q^{5}+(8-\beta _{1})q^{7}+\cdots\)
912.6.a.t 912.a 1.a $5$ $146.270$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-45\) \(-59\) \(149\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-12-\beta _{1})q^{5}+(30-\beta _{1}+\cdots)q^{7}+\cdots\)
912.6.a.u 912.a 1.a $5$ $146.270$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-45\) \(-54\) \(-70\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-11+\beta _{2})q^{5}+(-14-\beta _{4})q^{7}+\cdots\)
912.6.a.v 912.a 1.a $5$ $146.270$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-45\) \(-6\) \(-54\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-1-\beta _{1})q^{5}+(-11+\beta _{1}+\cdots)q^{7}+\cdots\)
912.6.a.w 912.a 1.a $5$ $146.270$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-45\) \(66\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-9q^{3}+(13+\beta _{1})q^{5}-\beta _{2}q^{7}+3^{4}q^{9}+\cdots\)
912.6.a.x 912.a 1.a $5$ $146.270$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(45\) \(-90\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-18+\beta _{2}-\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
912.6.a.y 912.a 1.a $6$ $146.270$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(54\) \(-65\) \(149\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+(-11-\beta _{2})q^{5}+(5^{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
912.6.a.z 912.a 1.a $7$ $146.270$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-63\) \(-29\) \(-119\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-9q^{3}+(-4-\beta _{1})q^{5}+(-17+\beta _{3}+\cdots)q^{7}+\cdots\)
912.6.a.ba 912.a 1.a $7$ $146.270$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(63\) \(55\) \(-119\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+9q^{3}+(8-\beta _{1})q^{5}+(-17-\beta _{1}-\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(912))\) into lower level spaces

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