# Properties

 Label 350.3.i Level $350$ Weight $3$ Character orbit 350.i Rep. character $\chi_{350}(199,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $48$ Newform subspaces $3$ Sturm bound $180$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 350.i (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$35$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$180$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(350, [\chi])$$.

Total New Old
Modular forms 264 48 216
Cusp forms 216 48 168
Eisenstein series 48 0 48

## Trace form

 $$48 q + 48 q^{4} - 96 q^{9} + O(q^{10})$$ $$48 q + 48 q^{4} - 96 q^{9} + 4 q^{11} - 64 q^{14} - 96 q^{16} + 84 q^{19} + 16 q^{21} + 48 q^{24} + 96 q^{26} + 192 q^{29} - 48 q^{31} - 384 q^{36} - 264 q^{39} - 8 q^{44} + 56 q^{46} - 232 q^{49} + 32 q^{51} + 216 q^{54} - 112 q^{56} + 504 q^{59} + 264 q^{61} - 384 q^{64} - 480 q^{66} + 568 q^{71} - 256 q^{74} - 176 q^{79} - 320 q^{81} - 152 q^{84} + 256 q^{86} + 924 q^{89} + 1252 q^{91} + 720 q^{94} + 96 q^{96} - 72 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(350, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.3.i.a $8$ $9.537$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{5}q^{2}+(\zeta_{24}-2\zeta_{24}^{3}-2\zeta_{24}^{5}+\cdots)q^{3}+\cdots$$
350.3.i.b $16$ $9.537$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{4}+\beta _{11})q^{2}+(-\beta _{6}+\beta _{12})q^{3}+\cdots$$
350.3.i.c $24$ $9.537$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{3}^{\mathrm{old}}(350, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(350, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$