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Properties

Label 90.2.c
Level $90$
Weight $2$
Character orbit 90.c
Rep. character $\chi_{90}(19,\cdot)$
Character field $\Q$
Dimension $2$
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.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
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 26 2 24
Cusp forms 10 2 8
Eisenstein series 16 0 16

Trace form

\( 2 q - 2 q^{4} + 4 q^{5} + O(q^{10}) \) \( 2 q - 2 q^{4} + 4 q^{5} - 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{20} + 6 q^{25} + 12 q^{26} - 16 q^{31} + 4 q^{34} - 4 q^{35} + 2 q^{40} - 4 q^{41} + 4 q^{44} + 8 q^{46} + 6 q^{49} - 8 q^{50} - 8 q^{55} + 4 q^{56} + 20 q^{59} + 4 q^{61} - 2 q^{64} + 12 q^{65} - 8 q^{70} - 24 q^{71} - 4 q^{74} + 4 q^{80} + 4 q^{85} - 8 q^{86} - 20 q^{89} + 24 q^{91} - 16 q^{94} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.2.c.a 90.c 5.b $2$ $0.719$ \(\Q(\sqrt{-1}) \) None 30.2.c.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+i q^{2}-q^{4}+(i+2)q^{5}+2 i q^{7}+\cdots\)

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

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