↑

Properties

Label 1170.2.m
Level $1170$
Weight $2$
Character orbit 1170.m
Rep. character $\chi_{1170}(73,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $70$
Newform subspaces $9$
Sturm bound $504$
Trace bound $5$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.m (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 9 \)
Sturm bound: \(504\)
Trace bound: \(5\)
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 536 70 466
Cusp forms 472 70 402
Eisenstein series 64 0 64

Trace form

\( 70 q - 2 q^{2} + 70 q^{4} - 2 q^{8} + O(q^{10}) \) \( 70 q - 2 q^{2} + 70 q^{4} - 2 q^{8} + 4 q^{11} + 8 q^{13} + 70 q^{16} + 2 q^{17} + 20 q^{19} + 8 q^{23} + 6 q^{25} + 24 q^{31} - 2 q^{32} - 18 q^{34} - 12 q^{35} + 10 q^{41} - 8 q^{43} + 4 q^{44} + 4 q^{46} - 78 q^{49} + 18 q^{50} + 8 q^{52} + 30 q^{53} + 40 q^{55} - 12 q^{59} + 32 q^{61} + 12 q^{62} + 70 q^{64} + 46 q^{65} + 48 q^{67} + 2 q^{68} + 12 q^{70} + 16 q^{71} + 60 q^{73} + 20 q^{76} + 56 q^{77} + 26 q^{82} - 22 q^{85} + 6 q^{89} + 8 q^{92} - 24 q^{95} + 48 q^{97} + 50 q^{98} + O(q^{100}) \)

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.m.a 1170.m 65.k $2$ $9.342$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}+q^{4}+(-1+2i)q^{5}+2iq^{7}+\cdots\)
1170.2.m.b 1170.m 65.k $2$ $9.342$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+q^{4}+(1-2i)q^{5}+2iq^{7}+q^{8}+\cdots\)
1170.2.m.c 1170.m 65.k $2$ $9.342$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+q^{4}+(2+i)q^{5}+4iq^{7}+q^{8}+\cdots\)
1170.2.m.d 1170.m 65.k $4$ $9.342$ \(\Q(\zeta_{12})\) None \(-4\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}+q^{4}+(-2\zeta_{12}+\zeta_{12}^{2}+2\zeta_{12}^{3})q^{5}+\cdots\)
1170.2.m.e 1170.m 65.k $4$ $9.342$ \(\Q(i, \sqrt{11})\) None \(4\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+q^{4}+(-2-\beta _{3})q^{5}-3\beta _{2}q^{7}+\cdots\)
1170.2.m.f 1170.m 65.k $12$ $9.342$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+q^{4}+\beta _{11}q^{5}+(\beta _{2}-\beta _{3}-\beta _{7}+\cdots)q^{7}+\cdots\)
1170.2.m.g 1170.m 65.k $14$ $9.342$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-14\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}+q^{4}+\beta _{8}q^{5}+\beta _{9}q^{7}-q^{8}+\cdots\)
1170.2.m.h 1170.m 65.k $14$ $9.342$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(14\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+q^{4}-\beta _{8}q^{5}+\beta _{9}q^{7}+q^{8}+\cdots\)
1170.2.m.i 1170.m 65.k $16$ $9.342$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-16\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}+q^{4}+\beta _{11}q^{5}+(-\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)

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

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