Properties

Label 1170.2.b
Level $1170$
Weight $2$
Character orbit 1170.b
Rep. character $\chi_{1170}(181,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $7$
Sturm bound $504$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(504\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1170, [\chi])\).

Total New Old
Modular forms 268 26 242
Cusp forms 236 26 210
Eisenstein series 32 0 32

Trace form

\( 26 q - 26 q^{4} + O(q^{10}) \) \( 26 q - 26 q^{4} - 2 q^{10} - 4 q^{13} + 26 q^{16} + 4 q^{17} - 26 q^{25} - 10 q^{26} - 20 q^{29} - 8 q^{35} + 24 q^{38} + 2 q^{40} - 4 q^{43} - 42 q^{49} + 4 q^{52} + 8 q^{53} + 20 q^{61} - 4 q^{62} - 26 q^{64} - 2 q^{65} - 4 q^{68} - 20 q^{74} + 64 q^{77} - 16 q^{79} - 16 q^{82} + 40 q^{91} - 8 q^{94} - 32 q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1170, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1170.2.b.a 1170.b 13.b $2$ $9.342$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-q^{4}+iq^{5}-iq^{8}-q^{10}+\cdots\)
1170.2.b.b 1170.b 13.b $2$ $9.342$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-q^{4}-iq^{5}-iq^{8}+q^{10}+\cdots\)
1170.2.b.c 1170.b 13.b $2$ $9.342$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}-q^{4}+iq^{5}+iq^{8}+q^{10}+\cdots\)
1170.2.b.d 1170.b 13.b $4$ $9.342$ \(\Q(i, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-q^{4}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1170.2.b.e 1170.b 13.b $4$ $9.342$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{2}-q^{4}-\zeta_{12}q^{5}-\zeta_{12}q^{8}+\cdots\)
1170.2.b.f 1170.b 13.b $4$ $9.342$ \(\Q(i, \sqrt{17})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-q^{4}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1170.2.b.g 1170.b 13.b $8$ $9.342$ 8.0.3057647616.6 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-q^{4}-\beta _{1}q^{5}-\beta _{7}q^{7}+\beta _{1}q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(234, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(390, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(585, [\chi])\)\(^{\oplus 2}\)