Properties

Label 37.2.f
Level $37$
Weight $2$
Character orbit 37.f
Rep. character $\chi_{37}(7,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $12$
Newform subspaces $2$
Sturm bound $6$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 37 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(37, [\chi])\).

Total New Old
Modular forms 24 24 0
Cusp forms 12 12 0
Eisenstein series 12 12 0

Trace form

\( 12 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 12 q^{6} - 9 q^{7} + 3 q^{8} - 3 q^{9} + O(q^{10}) \) \( 12 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 12 q^{6} - 9 q^{7} + 3 q^{8} - 3 q^{9} + 3 q^{10} + 9 q^{11} + 9 q^{12} - 21 q^{13} - 9 q^{14} + 9 q^{15} + 21 q^{16} + 6 q^{17} + 12 q^{18} - 6 q^{19} - 27 q^{20} + 18 q^{21} + 15 q^{22} + 15 q^{23} + 15 q^{24} - 12 q^{25} + 6 q^{26} - 3 q^{27} + 15 q^{28} - 6 q^{29} - 6 q^{30} - 12 q^{31} - 30 q^{32} - 24 q^{33} + 12 q^{34} + 12 q^{35} - 42 q^{36} - 18 q^{37} - 18 q^{38} - 12 q^{39} - 3 q^{40} + 15 q^{41} + 3 q^{42} - 24 q^{43} - 6 q^{44} + 24 q^{45} - 18 q^{46} + 24 q^{47} + 15 q^{48} + 9 q^{49} + 33 q^{50} - 33 q^{51} + 39 q^{52} + 45 q^{53} + 39 q^{54} + 24 q^{55} - 21 q^{56} + 33 q^{57} + 30 q^{58} + 30 q^{60} - 36 q^{61} - 12 q^{62} - 15 q^{63} - 9 q^{64} - 3 q^{65} - 15 q^{66} + 6 q^{67} + 18 q^{68} - 30 q^{69} - 66 q^{70} + 21 q^{71} - 3 q^{72} + 12 q^{73} - 54 q^{74} - 66 q^{75} + 15 q^{76} - 27 q^{77} - 9 q^{78} + 30 q^{79} - 36 q^{80} + 3 q^{81} - 27 q^{82} + 24 q^{83} - 18 q^{84} - 15 q^{85} + 12 q^{86} - 9 q^{87} + 6 q^{88} - 12 q^{89} + 27 q^{90} + 45 q^{91} + 18 q^{93} + 6 q^{94} - 3 q^{95} + 33 q^{96} - 9 q^{97} + 39 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(37, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
37.2.f.a 37.f 37.f $6$ $0.295$ \(\Q(\zeta_{18})\) None \(-6\) \(3\) \(-6\) \(3\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-1+\zeta_{18}^{5})q^{2}+(\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\cdots)q^{3}+\cdots\)
37.2.f.b 37.f 37.f $6$ $0.295$ \(\Q(\zeta_{18})\) None \(3\) \(-6\) \(6\) \(-12\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)