# Properties

 Label 385.2.i Level $385$ Weight $2$ Character orbit 385.i Rep. character $\chi_{385}(221,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $56$ Newform subspaces $4$ Sturm bound $96$ Trace bound $1$

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## Defining parameters

 Level: $$N$$ $$=$$ $$385 = 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 385.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$4$$ Sturm bound: $$96$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(385, [\chi])$$.

Total New Old
Modular forms 104 56 48
Cusp forms 88 56 32
Eisenstein series 16 0 16

## Trace form

 $$56q + 4q^{3} - 32q^{4} + 4q^{5} + 8q^{6} - 4q^{7} - 36q^{9} + O(q^{10})$$ $$56q + 4q^{3} - 32q^{4} + 4q^{5} + 8q^{6} - 4q^{7} - 36q^{9} - 12q^{12} + 32q^{13} + 4q^{14} - 44q^{16} - 20q^{18} + 4q^{19} - 16q^{20} - 20q^{21} + 28q^{24} - 28q^{25} - 12q^{26} - 32q^{27} + 20q^{28} + 32q^{29} - 4q^{30} - 24q^{31} - 40q^{32} + 8q^{33} + 24q^{34} + 4q^{35} + 160q^{36} - 8q^{37} + 44q^{38} - 12q^{39} - 8q^{41} + 48q^{42} - 24q^{43} - 4q^{44} - 16q^{46} + 12q^{47} - 32q^{48} + 24q^{49} - 8q^{51} - 40q^{52} + 12q^{53} - 48q^{54} - 16q^{55} - 84q^{56} - 8q^{57} - 20q^{58} - 4q^{59} - 4q^{61} + 56q^{62} - 28q^{63} + 128q^{64} - 20q^{66} + 8q^{67} + 16q^{68} - 24q^{69} + 4q^{70} - 24q^{71} + 16q^{72} + 4q^{73} + 40q^{74} + 4q^{75} + 8q^{76} + 4q^{77} - 136q^{78} + 16q^{79} + 20q^{80} - 52q^{81} + 12q^{82} + 112q^{83} + 12q^{84} + 8q^{85} - 64q^{86} + 48q^{87} - 40q^{89} + 72q^{90} + 12q^{91} + 36q^{93} - 12q^{94} + 56q^{96} - 120q^{97} + 52q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(385, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
385.2.i.a $$8$$ $$3.074$$ 8.0.310217769.2 None $$3$$ $$3$$ $$-4$$ $$1$$ $$q+(1+\beta _{1}+\beta _{2}-\beta _{5}-\beta _{7})q^{2}+(-\beta _{2}+\cdots)q^{3}+\cdots$$
385.2.i.b $$12$$ $$3.074$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$3$$ $$-1$$ $$6$$ $$-3$$ $$q+(-\beta _{1}+\beta _{5})q^{2}-\beta _{9}q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$$
385.2.i.c $$16$$ $$3.074$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$-3$$ $$-1$$ $$-8$$ $$-1$$ $$q-\beta _{1}q^{2}+(-\beta _{4}+\beta _{7})q^{3}+(-\beta _{8}+\beta _{11}+\cdots)q^{4}+\cdots$$
385.2.i.d $$20$$ $$3.074$$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-3$$ $$3$$ $$10$$ $$-1$$ $$q-\beta _{1}q^{2}+(-\beta _{10}+\beta _{17})q^{3}+(-\beta _{3}+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(385, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(385, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(77, [\chi])$$$$^{\oplus 2}$$