Properties

Label 1176.2.q
Level $1176$
Weight $2$
Character orbit 1176.q
Rep. character $\chi_{1176}(361,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $40$
Newform subspaces $15$
Sturm bound $448$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1176.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 15 \)
Sturm bound: \(448\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1176, [\chi])\).

Total New Old
Modular forms 512 40 472
Cusp forms 384 40 344
Eisenstein series 128 0 128

Trace form

\( 40 q - 2 q^{3} - 20 q^{9} + O(q^{10}) \) \( 40 q - 2 q^{3} - 20 q^{9} - 4 q^{11} - 12 q^{13} + 4 q^{15} - 4 q^{17} - 14 q^{19} + 12 q^{23} - 6 q^{25} + 4 q^{27} + 8 q^{29} + 10 q^{33} + 6 q^{37} + 2 q^{39} - 24 q^{41} - 4 q^{43} + 8 q^{51} - 12 q^{53} - 28 q^{55} - 20 q^{57} + 20 q^{61} - 24 q^{65} + 2 q^{67} - 8 q^{69} - 32 q^{71} - 10 q^{73} - 22 q^{75} + 28 q^{79} - 20 q^{81} - 40 q^{83} + 72 q^{85} - 2 q^{87} - 32 q^{89} - 22 q^{93} - 48 q^{95} + 12 q^{97} + 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1176, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1176.2.q.a 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1176.2.q.b 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1176.2.q.c 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{9}+4q^{13}+(-4+\cdots)q^{17}+\cdots\)
1176.2.q.d 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1176.2.q.e 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
1176.2.q.f 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}-\zeta_{6}q^{9}+6q^{13}+\cdots\)
1176.2.q.g 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}-\zeta_{6}q^{9}+(-3+\cdots)q^{11}+\cdots\)
1176.2.q.h 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{9}-4q^{13}+(4-4\zeta_{6})q^{17}+\cdots\)
1176.2.q.i 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+(-4+\cdots)q^{11}+\cdots\)
1176.2.q.j 1176.q 7.c $2$ $9.390$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{5}-\zeta_{6}q^{9}+2q^{13}+\cdots\)
1176.2.q.k 1176.q 7.c $4$ $9.390$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{3}+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots\)
1176.2.q.l 1176.q 7.c $4$ $9.390$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(-2\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(-1-\beta _{1}+\beta _{3})q^{5}+(-1+\cdots)q^{9}+\cdots\)
1176.2.q.m 1176.q 7.c $4$ $9.390$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(-2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{3}+(-\beta _{1}-2\beta _{2}-\beta _{3})q^{5}+\cdots\)
1176.2.q.n 1176.q 7.c $4$ $9.390$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{3}+(-\beta _{1}+2\beta _{2}-\beta _{3})q^{5}+\cdots\)
1176.2.q.o 1176.q 7.c $4$ $9.390$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{3}+(-\beta _{1}-2\beta _{2}-\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1176, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1176, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)