Properties

Label 90.5.g
Level $90$
Weight $5$
Character orbit 90.g
Rep. character $\chi_{90}(37,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $20$
Newform subspaces $6$
Sturm bound $90$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(90\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(90, [\chi])\).

Total New Old
Modular forms 160 20 140
Cusp forms 128 20 108
Eisenstein series 32 0 32

Trace form

\( 20 q - 60 q^{5} - 140 q^{7} + 32 q^{10} - 456 q^{11} - 300 q^{13} - 1280 q^{16} + 780 q^{17} + 192 q^{20} - 1280 q^{22} - 1860 q^{23} - 2876 q^{25} - 1344 q^{26} + 1120 q^{28} + 3656 q^{31} + 5052 q^{35}+ \cdots - 42240 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{5}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.5.g.a 90.g 5.c $2$ $9.303$ \(\Q(\sqrt{-1}) \) None 10.5.c.b \(-4\) \(0\) \(30\) \(-38\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2 i-2)q^{2}+8 i q^{4}+(20 i+15)q^{5}+\cdots\)
90.5.g.b 90.g 5.c $2$ $9.303$ \(\Q(\sqrt{-1}) \) None 10.5.c.a \(4\) \(0\) \(30\) \(58\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2 i+2)q^{2}+8 i q^{4}+(-20 i+15)q^{5}+\cdots\)
90.5.g.c 90.g 5.c $4$ $9.303$ \(\Q(i, \sqrt{6})\) None 30.5.f.b \(-8\) \(0\) \(-36\) \(68\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2\beta _{2})q^{2}-8\beta _{2}q^{4}+(-9-11\beta _{1}+\cdots)q^{5}+\cdots\)
90.5.g.d 90.g 5.c $4$ $9.303$ \(\Q(i, \sqrt{26})\) None 90.5.g.d \(-8\) \(0\) \(-24\) \(-100\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-2\beta _{2})q^{2}+8\beta _{2}q^{4}+(-6-2\beta _{1}+\cdots)q^{5}+\cdots\)
90.5.g.e 90.g 5.c $4$ $9.303$ \(\Q(i, \sqrt{6})\) None 30.5.f.a \(8\) \(0\) \(-84\) \(-28\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2-2\beta _{2})q^{2}-8\beta _{2}q^{4}+(-21+\beta _{1}+\cdots)q^{5}+\cdots\)
90.5.g.f 90.g 5.c $4$ $9.303$ \(\Q(i, \sqrt{26})\) None 90.5.g.d \(8\) \(0\) \(24\) \(-100\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2\beta _{2})q^{2}+8\beta _{2}q^{4}+(6-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(90, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 2}\)