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Properties

Label 370.2.n
Level $370$
Weight $2$
Character orbit 370.n
Rep. character $\chi_{370}(269,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $36$
Newform subspaces $6$
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.n (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 185 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
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 124 36 88
Cusp forms 108 36 72
Eisenstein series 16 0 16

Trace form

\( 36 q + 18 q^{4} + 4 q^{5} + 18 q^{9} + O(q^{10}) \) \( 36 q + 18 q^{4} + 4 q^{5} + 18 q^{9} + 4 q^{10} - 12 q^{11} - 8 q^{14} - 18 q^{16} + 14 q^{19} - 4 q^{20} + 16 q^{21} - 10 q^{25} - 12 q^{26} + 16 q^{29} - 8 q^{30} - 32 q^{31} - 12 q^{34} + 8 q^{35} + 36 q^{36} - 20 q^{39} + 2 q^{40} + 4 q^{41} - 6 q^{44} + 64 q^{45} - 6 q^{46} - 6 q^{49} - 32 q^{50} - 8 q^{51} - 12 q^{55} - 4 q^{56} + 46 q^{59} - 24 q^{61} - 36 q^{64} - 30 q^{65} + 8 q^{66} - 52 q^{69} + 12 q^{70} + 36 q^{71} - 2 q^{74} + 16 q^{75} - 14 q^{76} + 12 q^{79} - 8 q^{80} - 34 q^{81} + 32 q^{84} - 24 q^{85} + 28 q^{86} - 30 q^{89} - 14 q^{90} - 6 q^{94} + 24 q^{95} - 94 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
370.2.n.a 370.n 185.n $4$ $2.954$ \(\Q(\zeta_{12})\) None 370.2.n.a \(0\) \(0\) \(-8\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-2+\zeta_{12}^{3})q^{5}+\cdots\)
370.2.n.b 370.n 185.n $4$ $2.954$ \(\Q(\zeta_{12})\) None 370.2.n.b \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(-\zeta_{12}+2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
370.2.n.c 370.n 185.n $4$ $2.954$ \(\Q(\zeta_{12})\) None 370.2.n.a \(0\) \(0\) \(4\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+(2-\zeta_{12}-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
370.2.n.d 370.n 185.n $4$ $2.954$ \(\Q(\zeta_{12})\) None 370.2.n.d \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(3\zeta_{12}-3\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
370.2.n.e 370.n 185.n $8$ $2.954$ 8.0.303595776.1 None 370.2.n.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}-2\beta _{5}q^{3}-\beta _{4}q^{4}+(\beta _{1}+\beta _{5}+\cdots)q^{5}+\cdots\)
370.2.n.f 370.n 185.n $12$ $2.954$ 12.0.\(\cdots\).1 None 370.2.n.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(\beta _{5}-\beta _{6}-\beta _{7}+\beta _{9})q^{3}+\cdots\)

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

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