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Properties

Label 1170.2.i
Level $1170$
Weight $2$
Character orbit 1170.i
Rep. character $\chi_{1170}(451,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $44$
Newform subspaces $17$
Sturm bound $504$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 17 \)
Sturm bound: \(504\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\), \(11\), \(29\)

Dimensions

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

Total New Old
Modular forms 536 44 492
Cusp forms 472 44 428
Eisenstein series 64 0 64

Trace form

\( 44 q - 22 q^{4} + 8 q^{7} + O(q^{10}) \) \( 44 q - 22 q^{4} + 8 q^{7} + 2 q^{10} - 14 q^{11} - 4 q^{13} - 4 q^{14} - 22 q^{16} - 8 q^{17} - 6 q^{19} + 8 q^{22} - 8 q^{23} + 44 q^{25} - 8 q^{26} + 8 q^{28} + 20 q^{29} - 8 q^{31} + 16 q^{34} - 6 q^{35} + 4 q^{37} + 24 q^{38} - 4 q^{40} + 20 q^{41} + 16 q^{43} + 28 q^{44} + 16 q^{46} + 16 q^{47} - 12 q^{49} + 8 q^{52} - 16 q^{53} - 4 q^{55} + 2 q^{56} - 20 q^{58} + 16 q^{59} - 24 q^{61} + 44 q^{64} - 10 q^{65} - 16 q^{67} - 8 q^{68} + 56 q^{73} + 2 q^{74} - 6 q^{76} + 24 q^{77} + 104 q^{79} - 48 q^{83} - 12 q^{85} - 56 q^{86} + 8 q^{88} - 46 q^{89} + 106 q^{91} + 16 q^{92} - 2 q^{94} - 44 q^{97} - 32 q^{98} + O(q^{100}) \)

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.i.a 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}+2\zeta_{6}q^{7}+\cdots\)
1170.2.i.b 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}+4\zeta_{6}q^{7}+\cdots\)
1170.2.i.c 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}-3\zeta_{6}q^{7}+\cdots\)
1170.2.i.d 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}-2\zeta_{6}q^{7}+\cdots\)
1170.2.i.e 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}+q^{8}+\cdots\)
1170.2.i.f 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}+\zeta_{6}q^{7}+\cdots\)
1170.2.i.g 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}+5\zeta_{6}q^{7}+\cdots\)
1170.2.i.h 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}-q^{8}+(-1+\cdots)q^{10}+\cdots\)
1170.2.i.i 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}+3\zeta_{6}q^{7}+\cdots\)
1170.2.i.j 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}+3\zeta_{6}q^{7}+\cdots\)
1170.2.i.k 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}-2\zeta_{6}q^{7}+\cdots\)
1170.2.i.l 1170.i 13.c $2$ $9.342$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+q^{5}+4\zeta_{6}q^{7}+\cdots\)
1170.2.i.m 1170.i 13.c $4$ $9.342$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{12})q^{2}-\zeta_{12}q^{4}-q^{5}+(-2\zeta_{12}+\cdots)q^{7}+\cdots\)
1170.2.i.n 1170.i 13.c $4$ $9.342$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{12})q^{2}-\zeta_{12}q^{4}-q^{5}+(\zeta_{12}+\cdots)q^{7}+\cdots\)
1170.2.i.o 1170.i 13.c $4$ $9.342$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(2\) \(0\) \(-4\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{2})q^{2}-\beta _{2}q^{4}-q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1170.2.i.p 1170.i 13.c $4$ $9.342$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{12})q^{2}-\zeta_{12}q^{4}+q^{5}+(-2\zeta_{12}+\cdots)q^{7}+\cdots\)
1170.2.i.q 1170.i 13.c $4$ $9.342$ \(\Q(\sqrt{-3}, \sqrt{10})\) None \(2\) \(0\) \(4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+q^{5}-\beta _{2}q^{7}+\cdots\)

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

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