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Properties

Label 1170.2.j
Level $1170$
Weight $2$
Character orbit 1170.j
Rep. character $\chi_{1170}(391,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $96$
Newform subspaces $14$
Sturm bound $504$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(504\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 520 96 424
Cusp forms 488 96 392
Eisenstein series 32 0 32

Trace form

\( 96 q - 4 q^{2} - 4 q^{3} - 48 q^{4} - 4 q^{5} + 4 q^{6} + 8 q^{8} - 8 q^{9} + O(q^{10}) \) \( 96 q - 4 q^{2} - 4 q^{3} - 48 q^{4} - 4 q^{5} + 4 q^{6} + 8 q^{8} - 8 q^{9} + 4 q^{11} + 8 q^{12} + 4 q^{14} - 48 q^{16} - 8 q^{17} + 8 q^{18} - 24 q^{19} - 4 q^{20} - 32 q^{21} + 12 q^{22} + 24 q^{23} - 8 q^{24} - 48 q^{25} - 16 q^{27} + 4 q^{29} - 4 q^{30} - 4 q^{32} + 52 q^{33} + 12 q^{34} + 4 q^{36} + 4 q^{38} - 40 q^{41} + 12 q^{43} - 8 q^{44} + 8 q^{45} - 24 q^{46} - 4 q^{48} - 36 q^{49} - 4 q^{50} + 20 q^{51} - 64 q^{53} - 8 q^{54} + 4 q^{56} + 28 q^{57} + 44 q^{59} + 12 q^{61} - 96 q^{63} + 96 q^{64} - 8 q^{65} - 16 q^{66} + 12 q^{67} + 4 q^{68} + 76 q^{69} + 12 q^{70} + 128 q^{71} - 4 q^{72} - 24 q^{73} + 8 q^{74} - 4 q^{75} + 12 q^{76} - 56 q^{77} + 8 q^{80} + 40 q^{81} - 24 q^{82} - 20 q^{84} - 4 q^{86} + 24 q^{87} + 12 q^{88} + 8 q^{89} + 16 q^{90} + 24 q^{92} + 8 q^{93} + 12 q^{94} + 4 q^{96} + 12 q^{97} + 72 q^{98} + 32 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.j.a 1170.j 9.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(-3\) \(1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
1170.2.j.b 1170.j 9.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(1\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
1170.2.j.c 1170.j 9.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(1\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
1170.2.j.d 1170.j 9.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(1\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
1170.2.j.e 1170.j 9.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-1\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
1170.2.j.f 1170.j 9.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
1170.2.j.g 1170.j 9.c $4$ $9.342$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{12}q^{2}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
1170.2.j.h 1170.j 9.c $8$ $9.342$ 8.0.856615824.2 None \(-4\) \(-1\) \(4\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}-\beta _{1}q^{4}+\beta _{1}q^{5}+\cdots\)
1170.2.j.i 1170.j 9.c $10$ $9.342$ 10.0.\(\cdots\).1 None \(-5\) \(-1\) \(5\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{2})q^{2}+(-\beta _{1}-\beta _{4})q^{3}-\beta _{2}q^{4}+\cdots\)
1170.2.j.j 1170.j 9.c $10$ $9.342$ 10.0.\(\cdots\).1 None \(5\) \(1\) \(-5\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{6}q^{2}-\beta _{2}q^{3}+(-1+\beta _{6})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1170.2.j.k 1170.j 9.c $12$ $9.342$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-6\) \(-2\) \(-6\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{8})q^{2}-\beta _{4}q^{3}-\beta _{8}q^{4}-\beta _{8}q^{5}+\cdots\)
1170.2.j.l 1170.j 9.c $12$ $9.342$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(6\) \(0\) \(6\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{5})q^{2}-\beta _{4}q^{3}+\beta _{5}q^{4}-\beta _{5}q^{5}+\cdots\)
1170.2.j.m 1170.j 9.c $12$ $9.342$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(6\) \(1\) \(-6\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{5}q^{2}+\beta _{1}q^{3}+(-1+\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1170.2.j.n 1170.j 9.c $16$ $9.342$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-8\) \(-2\) \(-8\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{5})q^{2}+(-\beta _{1}-\beta _{7})q^{3}-\beta _{5}q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(117, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(234, [\chi])\)\(^{\oplus 2}\)