# Properties

 Label 350.3.w Level $350$ Weight $3$ Character orbit 350.w Rep. character $\chi_{350}(23,\cdot)$ Character field $\Q(\zeta_{60})$ Dimension $640$ Newform subspaces $2$ Sturm bound $180$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 350.w (of order $$60$$ and degree $$16$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$175$$ Character field: $$\Q(\zeta_{60})$$ Newform subspaces: $$2$$ Sturm bound: $$180$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 1984 640 1344
Cusp forms 1856 640 1216
Eisenstein series 128 0 128

## Trace form

 $$640 q - 4 q^{5} + 4 q^{7} + O(q^{10})$$ $$640 q - 4 q^{5} + 4 q^{7} + 16 q^{10} + 72 q^{15} - 320 q^{16} + 188 q^{17} + 32 q^{18} - 64 q^{22} - 32 q^{23} + 4 q^{25} + 288 q^{27} - 16 q^{28} + 400 q^{29} - 144 q^{30} + 44 q^{33} - 452 q^{35} - 960 q^{36} + 28 q^{37} - 32 q^{38} + 400 q^{39} - 272 q^{42} + 200 q^{43} - 388 q^{45} - 220 q^{47} + 192 q^{50} - 524 q^{53} + 8 q^{55} - 488 q^{57} - 160 q^{58} + 400 q^{59} - 32 q^{60} + 1104 q^{62} + 1356 q^{63} - 20 q^{65} + 80 q^{67} - 176 q^{68} + 1400 q^{69} + 176 q^{70} + 64 q^{72} + 268 q^{73} - 304 q^{75} - 1156 q^{77} + 64 q^{78} + 16 q^{80} - 720 q^{81} + 512 q^{82} - 1952 q^{83} - 600 q^{84} + 128 q^{85} - 388 q^{87} + 96 q^{88} + 1500 q^{89} - 640 q^{90} + 288 q^{92} + 372 q^{93} + 668 q^{95} + 448 q^{97} + 224 q^{98} + O(q^{100})$$

## 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.w.a $320$ $9.537$ None $$-40$$ $$0$$ $$-6$$ $$2$$
350.3.w.b $320$ $9.537$ None $$40$$ $$0$$ $$2$$ $$2$$

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

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