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

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

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
90.2.e.a \(2\) \(0.719\) \(\Q(\sqrt{-3}) \) None \(-1\) \(3\) \(1\) \(1\) \(q+(-1+\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.b \(2\) \(0.719\) \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(-1\) \(4\) \(q+(1-\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
90.2.e.c \(4\) \(0.719\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(2\) \(2\) \(-1\) \(q+\beta _{2}q^{2}+(1-\beta _{1}+\beta _{3})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)

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

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