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Properties

Label 90.3.g
Level $90$
Weight $3$
Character orbit 90.g
Rep. character $\chi_{90}(37,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $10$
Newform subspaces $4$
Sturm bound $54$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(54\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 88 10 78
Cusp forms 56 10 46
Eisenstein series 32 0 32

Trace form

\( 10 q - 2 q^{2} + 20 q^{7} + 4 q^{8} + O(q^{10}) \) \( 10 q - 2 q^{2} + 20 q^{7} + 4 q^{8} + 22 q^{10} + 32 q^{11} + 6 q^{13} - 40 q^{16} - 58 q^{17} - 28 q^{20} - 16 q^{22} + 20 q^{23} + 14 q^{25} - 12 q^{26} - 40 q^{28} - 152 q^{31} + 8 q^{32} + 92 q^{35} - 126 q^{37} + 104 q^{38} - 4 q^{40} - 16 q^{41} + 108 q^{43} + 136 q^{46} - 60 q^{47} - 142 q^{50} + 12 q^{52} - 206 q^{53} + 144 q^{55} - 80 q^{56} - 8 q^{58} + 96 q^{61} + 184 q^{62} + 234 q^{65} + 284 q^{67} + 116 q^{68} + 184 q^{70} - 8 q^{71} - 62 q^{73} + 16 q^{76} + 160 q^{77} - 232 q^{82} - 228 q^{83} - 514 q^{85} - 264 q^{86} + 32 q^{88} + 120 q^{91} + 40 q^{92} + 136 q^{95} - 222 q^{97} + 206 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.3.g.a 90.g 5.c $2$ $2.452$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(-6\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(-3+4i)q^{5}+\cdots\)
90.3.g.b 90.g 5.c $2$ $2.452$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+5iq^{5}+(2+2i)q^{7}+\cdots\)
90.3.g.c 90.g 5.c $2$ $2.452$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(6\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+(3-4i)q^{5}+(8+\cdots)q^{7}+\cdots\)
90.3.g.d 90.g 5.c $4$ $2.452$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{1})q^{2}+2\beta _{1}q^{4}+(\beta _{1}-\beta _{3})q^{5}+\cdots\)

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

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