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Properties

Label 90.2.f
Level $90$
Weight $2$
Character orbit 90.f
Rep. character $\chi_{90}(17,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $4$
Newform subspaces $1$
Sturm bound $36$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 90.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(36\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 52 4 48
Cusp forms 20 4 16
Eisenstein series 32 0 32

Trace form

\( 4 q + 8 q^{7} + O(q^{10}) \) \( 4 q + 8 q^{7} - 8 q^{10} - 12 q^{13} - 4 q^{16} + 8 q^{22} - 16 q^{25} + 8 q^{28} + 16 q^{31} - 12 q^{37} - 4 q^{40} + 16 q^{46} + 12 q^{52} + 24 q^{55} + 4 q^{58} - 16 q^{67} - 8 q^{70} + 4 q^{73} - 32 q^{76} - 28 q^{82} + 32 q^{85} + 8 q^{88} - 48 q^{91} - 12 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.f.a 90.f 15.e $4$ $0.719$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}+(\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(90, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)