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

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

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
370.2.h.a \(2\) \(2.954\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(-2\) \(-4\) \(q+q^{2}+q^{4}+(-1+2i)q^{5}+(-2+2i)q^{7}+\cdots\)
370.2.h.b \(2\) \(2.954\) \(\Q(\sqrt{-1}) \) None \(2\) \(2\) \(-2\) \(2\) \(q+q^{2}+(1-i)q^{3}+q^{4}+(-1-2i)q^{5}+\cdots\)
370.2.h.c \(4\) \(2.954\) \(\Q(\zeta_{8})\) None \(4\) \(0\) \(0\) \(8\) \(q+q^{2}+2\zeta_{8}q^{3}+q^{4}+(-\zeta_{8}-2\zeta_{8}^{3})q^{5}+\cdots\)
370.2.h.d \(10\) \(2.954\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(10\) \(2\) \(2\) \(-4\) \(q+q^{2}+(\beta _{1}-\beta _{4})q^{3}+q^{4}+(-\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
370.2.h.e \(20\) \(2.954\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-20\) \(4\) \(-4\) \(-2\) \(q-q^{2}-\beta _{4}q^{3}+q^{4}+\beta _{13}q^{5}+\beta _{4}q^{6}+\cdots\)

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

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