# Properties

 Label 350.3.b Level $350$ Weight $3$ Character orbit 350.b Rep. character $\chi_{350}(251,\cdot)$ Character field $\Q$ Dimension $24$ Newform subspaces $4$ Sturm bound $180$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 350.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$180$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$23$$

## Dimensions

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

Total New Old
Modular forms 132 24 108
Cusp forms 108 24 84
Eisenstein series 24 0 24

## Trace form

 $$24 q + 48 q^{4} + 4 q^{7} - 48 q^{9} + O(q^{10})$$ $$24 q + 48 q^{4} + 4 q^{7} - 48 q^{9} + 8 q^{14} + 96 q^{16} - 32 q^{18} + 8 q^{21} + 48 q^{22} + 72 q^{23} + 8 q^{28} - 64 q^{29} - 96 q^{36} - 136 q^{37} - 312 q^{39} - 80 q^{42} - 128 q^{43} + 32 q^{46} + 72 q^{49} - 296 q^{51} + 208 q^{53} + 16 q^{56} + 24 q^{57} + 224 q^{58} - 380 q^{63} + 192 q^{64} + 112 q^{67} + 312 q^{71} - 64 q^{72} + 256 q^{74} + 408 q^{77} - 320 q^{78} - 96 q^{79} - 88 q^{81} + 16 q^{84} - 160 q^{86} + 96 q^{88} + 208 q^{91} + 144 q^{92} + 824 q^{93} - 48 q^{98} + 392 q^{99} + O(q^{100})$$

## 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.b.a $4$ $9.537$ 4.0.7168.1 None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+\beta _{3}q^{2}+\beta _{1}q^{3}+2q^{4}+(-\beta _{1}+\beta _{2}+\cdots)q^{6}+\cdots$$
350.3.b.b $4$ $9.537$ 4.0.7168.1 None $$0$$ $$0$$ $$0$$ $$12$$ $$q-\beta _{3}q^{2}+\beta _{1}q^{3}+2q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots$$
350.3.b.c $8$ $9.537$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{2}-\beta _{5}q^{3}+2q^{4}+(-\beta _{4}+\beta _{7})q^{6}+\cdots$$
350.3.b.d $8$ $9.537$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q-\beta _{5}q^{2}-\beta _{1}q^{3}+2q^{4}-\beta _{4}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots$$

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

$$S_{3}^{\mathrm{old}}(350, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$