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Properties

Label 370.2.d
Level $370$
Weight $2$
Character orbit 370.d
Rep. character $\chi_{370}(221,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $3$
Sturm bound $114$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 37 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(114\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

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

Total New Old
Modular forms 62 10 52
Cusp forms 54 10 44
Eisenstein series 8 0 8

Trace form

\( 10 q - 4 q^{3} - 10 q^{4} + 16 q^{7} + 2 q^{9} + O(q^{10}) \) \( 10 q - 4 q^{3} - 10 q^{4} + 16 q^{7} + 2 q^{9} + 2 q^{10} - 4 q^{11} + 4 q^{12} + 10 q^{16} - 16 q^{21} - 10 q^{25} - 4 q^{26} - 16 q^{27} - 16 q^{28} - 4 q^{30} + 40 q^{33} - 24 q^{34} - 2 q^{36} - 20 q^{37} + 8 q^{38} - 2 q^{40} + 8 q^{41} + 4 q^{44} + 8 q^{46} - 8 q^{47} - 4 q^{48} + 22 q^{49} + 8 q^{53} - 20 q^{58} - 28 q^{62} + 40 q^{63} - 10 q^{64} - 12 q^{65} + 44 q^{67} + 4 q^{70} - 40 q^{71} + 12 q^{73} + 2 q^{74} + 4 q^{75} + 16 q^{77} + 16 q^{78} + 42 q^{81} - 68 q^{83} + 16 q^{84} + 4 q^{85} + 28 q^{86} + 26 q^{90} - 24 q^{95} - 52 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
370.2.d.a 370.d 37.b $2$ $2.954$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-q^{4}+iq^{5}-2q^{7}-iq^{8}+\cdots\)
370.2.d.b 370.d 37.b $2$ $2.954$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-q^{4}+iq^{5}+5q^{7}-iq^{8}+\cdots\)
370.2.d.c 370.d 37.b $6$ $2.954$ 6.0.399424.1 None \(0\) \(-4\) \(0\) \(10\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-1-\beta _{1}-\beta _{3})q^{3}-q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(370, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(37, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(74, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(185, [\chi])\)\(^{\oplus 2}\)