Properties

Label 370.2.q
Level $370$
Weight $2$
Character orbit 370.q
Rep. character $\chi_{370}(97,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $76$
Newform subspaces $6$
Sturm bound $114$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.q (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 185 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 6 \)
Sturm bound: \(114\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(370, [\chi])\).

Total New Old
Modular forms 244 76 168
Cusp forms 212 76 136
Eisenstein series 32 0 32

Trace form

\( 76 q + 2 q^{2} + 4 q^{3} - 38 q^{4} + 6 q^{5} - 4 q^{8} + O(q^{10}) \) \( 76 q + 2 q^{2} + 4 q^{3} - 38 q^{4} + 6 q^{5} - 4 q^{8} - 2 q^{10} + 4 q^{12} + 8 q^{13} + 8 q^{14} + 8 q^{15} - 38 q^{16} - 12 q^{19} + 8 q^{23} + 2 q^{25} + 8 q^{26} + 64 q^{27} - 18 q^{29} - 24 q^{30} + 16 q^{31} + 2 q^{32} + 12 q^{33} - 16 q^{35} + 14 q^{37} - 64 q^{39} - 14 q^{40} - 18 q^{41} - 84 q^{42} - 16 q^{43} - 12 q^{44} + 8 q^{45} + 8 q^{47} - 8 q^{48} + 72 q^{49} - 20 q^{50} + 8 q^{52} - 42 q^{53} - 12 q^{55} - 4 q^{56} + 24 q^{57} + 6 q^{58} + 20 q^{59} + 8 q^{60} - 4 q^{61} - 20 q^{62} + 76 q^{64} + 12 q^{65} + 8 q^{66} + 68 q^{67} - 80 q^{69} + 8 q^{70} - 8 q^{71} + 4 q^{73} + 52 q^{74} - 120 q^{75} - 24 q^{76} - 44 q^{77} + 20 q^{78} - 16 q^{79} - 6 q^{80} + 62 q^{81} - 8 q^{83} - 8 q^{86} - 36 q^{87} + 58 q^{89} + 40 q^{90} - 20 q^{91} - 4 q^{92} + 72 q^{93} - 36 q^{94} + 68 q^{95} - 24 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(370, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
370.2.q.a 370.q 185.p $4$ $2.954$ \(\Q(\zeta_{12})\) None 370.2.q.a \(-2\) \(4\) \(-4\) \(4\) $\mathrm{SU}(2)[C_{12}]$ \(q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
370.2.q.b 370.q 185.p $4$ $2.954$ \(\Q(\zeta_{12})\) None 370.2.q.b \(-2\) \(6\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{12}]$ \(q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}+\zeta_{12}^{2}-2\zeta_{12}^{3})q^{3}+\cdots\)
370.2.q.c 370.q 185.p $8$ $2.954$ \(\Q(\zeta_{24})\) None 370.2.q.c \(4\) \(4\) \(4\) \(12\) $\mathrm{SU}(2)[C_{12}]$ \(q+\zeta_{24}^{4}q^{2}+(\zeta_{24}^{4}+\zeta_{24}^{5}+\zeta_{24}^{6}+\cdots)q^{3}+\cdots\)
370.2.q.d 370.q 185.p $12$ $2.954$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 370.2.q.d \(-6\) \(0\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1-\beta _{7})q^{2}+(1-\beta _{1}-\beta _{3}+2\beta _{4}+\cdots)q^{3}+\cdots\)
370.2.q.e 370.q 185.p $16$ $2.954$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 370.2.q.e \(-8\) \(-8\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1-\beta _{14})q^{2}+(-1+\beta _{2}+\beta _{13}+\cdots)q^{3}+\cdots\)
370.2.q.f 370.q 185.p $32$ $2.954$ None 370.2.q.f \(16\) \(-2\) \(6\) \(-10\) $\mathrm{SU}(2)[C_{12}]$

Decomposition of \(S_{2}^{\mathrm{old}}(370, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(370, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(185, [\chi])\)\(^{\oplus 2}\)