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Properties

Label 370.2.h
Level $370$
Weight $2$
Character orbit 370.h
Rep. character $\chi_{370}(117,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $38$
Newform subspaces $5$
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.h (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 185 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 5 \)
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 122 38 84
Cusp forms 106 38 68
Eisenstein series 16 0 16

Trace form

\( 38 q - 2 q^{2} + 8 q^{3} + 38 q^{4} - 6 q^{5} - 2 q^{8} + O(q^{10}) \) \( 38 q - 2 q^{2} + 8 q^{3} + 38 q^{4} - 6 q^{5} - 2 q^{8} + 2 q^{10} + 8 q^{12} - 8 q^{13} + 4 q^{14} - 8 q^{15} + 38 q^{16} + 12 q^{19} - 6 q^{20} - 8 q^{23} + 10 q^{25} - 8 q^{26} - 40 q^{27} + 18 q^{29} - 16 q^{31} - 2 q^{32} - 8 q^{35} - 38 q^{37} - 8 q^{39} + 2 q^{40} + 16 q^{43} + 4 q^{45} - 8 q^{47} + 8 q^{48} - 10 q^{50} - 8 q^{52} - 6 q^{53} + 12 q^{55} + 4 q^{56} + 48 q^{57} - 18 q^{58} - 20 q^{59} - 8 q^{60} + 22 q^{61} - 40 q^{62} + 38 q^{64} + 24 q^{65} - 8 q^{66} + 16 q^{67} - 88 q^{69} + 16 q^{70} - 16 q^{71} + 2 q^{73} + 26 q^{74} - 72 q^{75} + 12 q^{76} - 40 q^{77} - 20 q^{78} - 32 q^{79} - 6 q^{80} + 10 q^{81} + 20 q^{83} - 16 q^{86} - 46 q^{89} - 40 q^{90} - 40 q^{91} - 8 q^{92} - 48 q^{93} + 36 q^{94} + 40 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.h.a 370.h 185.k $2$ $2.954$ \(\Q(\sqrt{-1}) \) None 370.2.g.b \(2\) \(0\) \(-2\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+q^{4}+(2 i-1)q^{5}+(2 i-2)q^{7}+\cdots\)
370.2.h.b 370.h 185.k $2$ $2.954$ \(\Q(\sqrt{-1}) \) None 370.2.g.a \(2\) \(2\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+(-i+1)q^{3}+q^{4}+(-2 i-1)q^{5}+\cdots\)
370.2.h.c 370.h 185.k $4$ $2.954$ \(\Q(\zeta_{8})\) None 370.2.g.c \(4\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+2\zeta_{8}q^{3}+q^{4}+(-\zeta_{8}-2\zeta_{8}^{3})q^{5}+\cdots\)
370.2.h.d 370.h 185.k $10$ $2.954$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None 370.2.g.d \(10\) \(2\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+(\beta _{1}-\beta _{4})q^{3}+q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
370.2.h.e 370.h 185.k $20$ $2.954$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 370.2.g.e \(-20\) \(4\) \(-4\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}-\beta _{4}q^{3}+q^{4}+\beta _{13}q^{5}+\beta _{4}q^{6}+\cdots\)

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

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