Properties

Label 90.2.e
Level $90$
Weight $2$
Character orbit 90.e
Rep. character $\chi_{90}(31,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $3$
Sturm bound $36$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(36\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(90, [\chi])\).

Total New Old
Modular forms 44 8 36
Cusp forms 28 8 20
Eisenstein series 16 0 16

Trace form

\( 8 q + 2 q^{2} + 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} - 4 q^{8} - 4 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{2} + 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{6} + 4 q^{7} - 4 q^{8} - 4 q^{9} - 6 q^{11} - 4 q^{12} + 4 q^{13} - 2 q^{14} + 4 q^{15} - 4 q^{16} - 12 q^{17} + 4 q^{18} + 4 q^{19} + 2 q^{20} + 4 q^{21} - 6 q^{22} - 12 q^{23} + 4 q^{24} - 4 q^{25} + 16 q^{26} - 16 q^{27} - 8 q^{28} - 6 q^{29} + 2 q^{30} + 4 q^{31} + 2 q^{32} - 6 q^{33} - 6 q^{34} - 8 q^{35} + 2 q^{36} - 8 q^{37} + 10 q^{38} + 28 q^{39} + 12 q^{41} + 20 q^{42} - 2 q^{43} + 12 q^{44} + 4 q^{45} + 12 q^{46} + 12 q^{47} + 2 q^{48} - 6 q^{49} + 2 q^{50} - 18 q^{51} + 4 q^{52} + 24 q^{53} - 8 q^{54} - 2 q^{56} - 26 q^{57} - 6 q^{59} + 4 q^{60} + 22 q^{61} - 8 q^{62} + 40 q^{63} + 8 q^{64} + 4 q^{65} - 36 q^{66} + 22 q^{67} + 6 q^{68} - 42 q^{69} - 6 q^{70} - 24 q^{71} + 10 q^{72} - 44 q^{73} - 20 q^{74} + 2 q^{75} - 2 q^{76} + 36 q^{77} - 4 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 12 q^{82} - 12 q^{83} + 10 q^{84} - 12 q^{85} - 14 q^{86} + 36 q^{87} - 6 q^{88} + 60 q^{89} - 16 q^{90} - 40 q^{91} - 12 q^{92} - 20 q^{93} - 6 q^{94} - 8 q^{95} - 2 q^{96} - 2 q^{97} - 36 q^{98} - 24 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.2.e.a 90.e 9.c $2$ $0.719$ \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.b 90.e 9.c $2$ $0.719$ \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.c 90.e 9.c $4$ $0.719$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(2\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(90, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)