# Properties

 Label 90.4.e Level $90$ Weight $4$ Character orbit 90.e Rep. character $\chi_{90}(31,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $5$ Sturm bound $72$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$90 = 2 \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 90.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$5$$ Sturm bound: $$72$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

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

Total New Old
Modular forms 116 24 92
Cusp forms 100 24 76
Eisenstein series 16 0 16

## Trace form

 $$24 q - 4 q^{2} + 2 q^{3} - 48 q^{4} - 10 q^{5} + 28 q^{6} - 24 q^{7} + 32 q^{8} + 68 q^{9} + O(q^{10})$$ $$24 q - 4 q^{2} + 2 q^{3} - 48 q^{4} - 10 q^{5} + 28 q^{6} - 24 q^{7} + 32 q^{8} + 68 q^{9} - 42 q^{11} - 16 q^{12} + 48 q^{13} - 44 q^{14} + 40 q^{15} - 192 q^{16} + 348 q^{17} - 296 q^{18} - 420 q^{19} - 40 q^{20} + 4 q^{21} + 36 q^{22} + 168 q^{23} + 64 q^{24} - 300 q^{25} + 160 q^{26} + 308 q^{27} + 192 q^{28} + 642 q^{29} + 140 q^{30} - 60 q^{31} - 64 q^{32} + 18 q^{33} - 180 q^{34} + 280 q^{35} - 280 q^{36} - 672 q^{37} - 668 q^{38} - 92 q^{39} + 1020 q^{41} + 152 q^{42} - 258 q^{43} + 336 q^{44} - 380 q^{45} - 504 q^{46} + 948 q^{47} + 32 q^{48} + 90 q^{49} - 100 q^{50} - 1446 q^{51} + 192 q^{52} + 1320 q^{53} - 632 q^{54} - 176 q^{56} + 1186 q^{57} + 1398 q^{59} + 160 q^{60} - 1698 q^{61} + 1360 q^{62} - 1544 q^{63} + 1536 q^{64} - 1160 q^{65} - 600 q^{66} - 1950 q^{67} - 696 q^{68} + 1230 q^{69} - 180 q^{70} - 1176 q^{71} - 80 q^{72} + 516 q^{73} - 536 q^{74} + 50 q^{75} + 840 q^{76} - 2520 q^{77} + 2408 q^{78} + 1776 q^{79} + 320 q^{80} + 2948 q^{81} + 2088 q^{82} - 552 q^{83} - 1400 q^{84} + 720 q^{85} + 1636 q^{86} - 3408 q^{87} + 144 q^{88} - 3972 q^{89} - 160 q^{90} - 4008 q^{91} + 672 q^{92} + 1528 q^{93} - 612 q^{94} - 1040 q^{95} - 704 q^{96} + 1038 q^{97} - 2232 q^{98} - 3168 q^{99} + O(q^{100})$$

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

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
90.4.e.a $2$ $5.310$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-9$$ $$5$$ $$16$$ $$q+(-2+2\zeta_{6})q^{2}+(-3-3\zeta_{6})q^{3}+\cdots$$
90.4.e.b $4$ $5.310$ $$\Q(\sqrt{-3}, \sqrt{-5})$$ None $$-4$$ $$12$$ $$10$$ $$-16$$ $$q+(-2+2\beta _{1})q^{2}+(4-2\beta _{1}+\beta _{2})q^{3}+\cdots$$
90.4.e.c $4$ $5.310$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$4$$ $$6$$ $$10$$ $$2$$ $$q+2\beta _{1}q^{2}+(3-3\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\cdots$$
90.4.e.d $6$ $5.310$ 6.0.41783472.1 None $$6$$ $$-9$$ $$-15$$ $$-3$$ $$q+(2+2\beta _{1})q^{2}+(-2-\beta _{4})q^{3}+4\beta _{1}q^{4}+\cdots$$
90.4.e.e $8$ $5.310$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-8$$ $$2$$ $$-20$$ $$-23$$ $$q+(-2-2\beta _{3})q^{2}-\beta _{1}q^{3}+4\beta _{3}q^{4}+\cdots$$

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

$$S_{4}^{\mathrm{old}}(90, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 2}$$