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Properties

Label 90.4.c
Level $90$
Weight $4$
Character orbit 90.c
Rep. character $\chi_{90}(19,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $3$
Sturm bound $72$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(72\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(90, [\chi])\).

Total New Old
Modular forms 62 8 54
Cusp forms 46 8 38
Eisenstein series 16 0 16

Trace form

\( 8 q - 32 q^{4} + 6 q^{5} + O(q^{10}) \) \( 8 q - 32 q^{4} + 6 q^{5} + 4 q^{10} - 84 q^{11} + 96 q^{14} + 128 q^{16} - 264 q^{19} - 24 q^{20} + 108 q^{25} - 264 q^{26} + 612 q^{29} + 224 q^{31} - 152 q^{34} - 564 q^{35} - 16 q^{40} - 72 q^{41} + 336 q^{44} + 488 q^{46} - 600 q^{49} + 576 q^{50} + 1552 q^{55} - 384 q^{56} - 780 q^{59} - 1448 q^{61} - 512 q^{64} - 948 q^{65} - 1448 q^{70} + 3336 q^{71} - 72 q^{74} + 1056 q^{76} + 3600 q^{79} + 96 q^{80} - 548 q^{85} - 2616 q^{86} - 2856 q^{89} - 5544 q^{91} - 424 q^{94} + 696 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(90, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.4.c.a 90.c 5.b $2$ $5.310$ \(\Q(\sqrt{-1}) \) None 30.4.c.a \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2 i q^{2}-4 q^{4}+(11 i-2)q^{5}+2 i q^{7}+\cdots\)
90.4.c.b 90.c 5.b $2$ $5.310$ \(\Q(\sqrt{-1}) \) None 10.4.b.a \(0\) \(0\) \(10\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-4 q^{4}+(-5\beta+5)q^{5}-13\beta q^{7}+\cdots\)
90.4.c.c 90.c 5.b $4$ $5.310$ \(\Q(i, \sqrt{31})\) None 90.4.c.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-4q^{4}-\beta _{1}q^{5}-\beta _{3}q^{7}+4\beta _{2}q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(90, [\chi])\) into lower level spaces

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