# Properties

 Label 363.4.d Level $363$ Weight $4$ Character orbit 363.d Rep. character $\chi_{363}(362,\cdot)$ Character field $\Q$ Dimension $100$ Newform subspaces $5$ Sturm bound $176$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$363 = 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 363.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$33$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$176$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(363, [\chi])$$.

Total New Old
Modular forms 144 116 28
Cusp forms 120 100 20
Eisenstein series 24 16 8

## Trace form

 $$100 q + 4 q^{3} + 372 q^{4} + 36 q^{9} + O(q^{10})$$ $$100 q + 4 q^{3} + 372 q^{4} + 36 q^{9} + 102 q^{12} - 32 q^{15} + 1420 q^{16} - 2068 q^{25} - 608 q^{27} - 104 q^{31} - 908 q^{34} + 1382 q^{36} - 140 q^{37} + 220 q^{42} + 1542 q^{45} + 1494 q^{48} - 1044 q^{49} - 2844 q^{58} - 3612 q^{60} + 5136 q^{64} - 2140 q^{67} + 2760 q^{69} - 5356 q^{70} - 3666 q^{75} + 1372 q^{78} + 5732 q^{81} + 5808 q^{82} - 1608 q^{91} - 2336 q^{93} + 3100 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(363, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
363.4.d.a $4$ $21.418$ $$\Q(\sqrt{-2}, \sqrt{3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-8q^{4}+(-2\beta _{2}+\beta _{3})q^{7}+\cdots$$
363.4.d.b $4$ $21.418$ $$\Q(\sqrt{-2}, \sqrt{13})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}+(1-\beta _{1})q^{3}+5q^{4}+4\beta _{1}q^{5}+\cdots$$
363.4.d.c $12$ $21.418$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{7}q^{2}-\beta _{1}q^{3}+(8-\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots$$
363.4.d.d $40$ $21.418$ None $$0$$ $$-8$$ $$0$$ $$0$$
363.4.d.e $40$ $21.418$ None $$0$$ $$8$$ $$0$$ $$0$$

## Decomposition of $$S_{4}^{\mathrm{old}}(363, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(363, [\chi]) \cong$$