Properties

Label 370.2.r
Level $370$
Weight $2$
Character orbit 370.r
Rep. character $\chi_{370}(23,\cdot)$
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.r (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

\( 76 q - 4 q^{3} + 38 q^{4} - 4 q^{5} + O(q^{10}) \) \( 76 q - 4 q^{3} + 38 q^{4} - 4 q^{5} - 2 q^{10} + 4 q^{12} - 8 q^{14} - 38 q^{16} + 2 q^{17} - 10 q^{18} + 12 q^{19} + 4 q^{20} + 16 q^{22} - 8 q^{25} + 8 q^{26} - 64 q^{27} + 18 q^{29} - 24 q^{30} + 16 q^{31} - 12 q^{33} + 16 q^{35} + 36 q^{37} + 64 q^{39} - 16 q^{40} - 18 q^{41} - 28 q^{42} + 12 q^{44} + 18 q^{45} + 8 q^{47} + 8 q^{48} - 72 q^{49} + 32 q^{50} - 18 q^{53} + 12 q^{55} - 4 q^{56} - 120 q^{57} + 24 q^{58} - 20 q^{59} - 4 q^{61} + 20 q^{62} - 76 q^{64} - 12 q^{65} + 8 q^{66} + 100 q^{67} + 4 q^{68} + 80 q^{69} + 8 q^{70} - 8 q^{71} + 10 q^{72} - 4 q^{73} - 52 q^{74} - 120 q^{75} - 24 q^{76} + 44 q^{77} - 20 q^{78} + 16 q^{79} + 8 q^{80} + 62 q^{81} - 48 q^{82} - 32 q^{83} - 8 q^{86} + 12 q^{87} + 32 q^{88} - 58 q^{89} + 40 q^{90} - 20 q^{91} - 12 q^{92} + 36 q^{94} - 44 q^{95} - 56 q^{97} - 10 q^{98} + O(q^{100}) \)

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.r.a 370.r 185.u $4$ $2.954$ \(\Q(\zeta_{12})\) None \(0\) \(2\) \(2\) \(4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
370.2.r.b 370.r 185.u $4$ $2.954$ \(\Q(\zeta_{12})\) None \(0\) \(6\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
370.2.r.c 370.r 185.u $8$ $2.954$ \(\Q(\zeta_{24})\) None \(0\) \(8\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\zeta_{24}^{2}+\zeta_{24}^{6})q^{2}+(1+\zeta_{24}^{2}+\cdots)q^{3}+\cdots\)
370.2.r.d 370.r 185.u $12$ $2.954$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(6\) \(-8\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{8}q^{2}+(1-\beta _{1}-\beta _{3}+\beta _{4}+\beta _{8}+\cdots)q^{3}+\cdots\)
370.2.r.e 370.r 185.u $16$ $2.954$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(-16\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\beta _{5}-\beta _{13})q^{2}+(-1-\beta _{5}+\beta _{13}+\cdots)q^{3}+\cdots\)
370.2.r.f 370.r 185.u $32$ $2.954$ None \(0\) \(-10\) \(6\) \(2\) $\mathrm{SU}(2)[C_{12}]$

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

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