Properties

Label 90.3.b
Level $90$
Weight $3$
Character orbit 90.b
Rep. character $\chi_{90}(89,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $1$
Sturm bound $54$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 90.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(54\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(90, [\chi])\).

Total New Old
Modular forms 44 4 40
Cusp forms 28 4 24
Eisenstein series 16 0 16

Trace form

\( 4 q + 8 q^{4} + O(q^{10}) \) \( 4 q + 8 q^{4} + 20 q^{10} + 16 q^{16} - 96 q^{19} + 16 q^{31} + 8 q^{34} + 40 q^{40} - 224 q^{46} + 132 q^{49} - 160 q^{55} + 440 q^{61} + 32 q^{64} + 80 q^{70} - 192 q^{76} + 144 q^{79} + 20 q^{85} - 288 q^{91} + 160 q^{94} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.3.b.a 90.b 15.d $4$ $2.452$ \(\Q(\zeta_{8})\) None 90.3.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}-\zeta_{8}^{3})q^{2}+2q^{4}+5\zeta_{8}q^{5}-4\zeta_{8}^{2}q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(90, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)