# Properties

 Label 400.2.l Level $400$ Weight $2$ Character orbit 400.l Rep. character $\chi_{400}(101,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $70$ Newform subspaces $9$ Sturm bound $120$ Trace bound $6$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.l (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$16$$ Character field: $$\Q(i)$$ Newform subspaces: $$9$$ Sturm bound: $$120$$ Trace bound: $$6$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(400, [\chi])$$.

Total New Old
Modular forms 132 82 50
Cusp forms 108 70 38
Eisenstein series 24 12 12

## Trace form

 $$70q + 2q^{2} + 2q^{3} + 4q^{4} - 4q^{8} + O(q^{10})$$ $$70q + 2q^{2} + 2q^{3} + 4q^{4} - 4q^{8} - 2q^{11} + 16q^{12} + 2q^{13} + 24q^{14} - 20q^{16} + 4q^{17} - 2q^{18} - 6q^{19} - 12q^{21} + 20q^{22} - 20q^{24} - 12q^{26} - 16q^{27} - 4q^{28} + 10q^{29} - 8q^{32} + 4q^{33} - 40q^{34} + 16q^{36} + 10q^{37} - 8q^{38} - 60q^{42} - 18q^{43} - 44q^{44} - 32q^{46} + 24q^{47} + 32q^{48} - 30q^{49} + 28q^{51} - 52q^{52} - 6q^{53} - 44q^{54} - 52q^{56} + 24q^{58} + 30q^{59} + 10q^{61} - 8q^{62} - 44q^{63} + 76q^{64} + 28q^{66} - 30q^{67} + 48q^{68} + 36q^{69} + 36q^{72} + 36q^{74} + 12q^{76} - 20q^{77} + 20q^{78} - 34q^{81} + 76q^{82} - 38q^{83} + 156q^{84} - 28q^{86} - 8q^{88} - 60q^{91} + 76q^{92} + 32q^{93} - 56q^{94} - 20q^{96} + 4q^{97} - 54q^{98} + 18q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(400, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
400.2.l.a $$2$$ $$3.194$$ $$\Q(\sqrt{-1})$$ None $$-2$$ $$-2$$ $$0$$ $$0$$ $$q+(-1+i)q^{2}+(-1+i)q^{3}-2iq^{4}+\cdots$$
400.2.l.b $$2$$ $$3.194$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$2$$ $$0$$ $$0$$ $$q+(1-i)q^{2}+(1-i)q^{3}-2iq^{4}-2iq^{6}+\cdots$$
400.2.l.c $$2$$ $$3.194$$ $$\Q(\sqrt{-1})$$ None $$2$$ $$2$$ $$0$$ $$0$$ $$q+(1+i)q^{2}+(1-i)q^{3}+2iq^{4}+2q^{6}+\cdots$$
400.2.l.d $$4$$ $$3.194$$ $$\Q(i, \sqrt{11})$$ None $$-4$$ $$-2$$ $$0$$ $$0$$ $$q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots$$
400.2.l.e $$4$$ $$3.194$$ $$\Q(i, \sqrt{11})$$ None $$4$$ $$2$$ $$0$$ $$0$$ $$q+(1+\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots$$
400.2.l.f $$12$$ $$3.194$$ 12.0.$$\cdots$$.1 None $$-4$$ $$-2$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots$$
400.2.l.g $$12$$ $$3.194$$ 12.0.$$\cdots$$.1 None $$4$$ $$2$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots$$
400.2.l.h $$16$$ $$3.194$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{8}q^{2}+(\beta _{4}+\beta _{11})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots$$
400.2.l.i $$16$$ $$3.194$$ 16.0.$$\cdots$$.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{10}q^{2}+(-\beta _{2}+\beta _{9}+\beta _{12}-\beta _{14}+\cdots)q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(400, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(400, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$