# Properties

 Label 400.2.u Level $400$ Weight $2$ Character orbit 400.u Rep. character $\chi_{400}(81,\cdot)$ Character field $\Q(\zeta_{5})$ Dimension $56$ Newform subspaces $7$ Sturm bound $120$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 400.u (of order $$5$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$25$$ Character field: $$\Q(\zeta_{5})$$ Newform subspaces: $$7$$ Sturm bound: $$120$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

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

Total New Old
Modular forms 264 64 200
Cusp forms 216 56 160
Eisenstein series 48 8 40

## Trace form

 $$56 q + 3 q^{3} - 5 q^{5} + 8 q^{7} - 15 q^{9} + O(q^{10})$$ $$56 q + 3 q^{3} - 5 q^{5} + 8 q^{7} - 15 q^{9} + 9 q^{11} - 5 q^{13} + 7 q^{15} - q^{17} + 9 q^{19} - 12 q^{21} - 15 q^{23} - 7 q^{25} + 9 q^{27} - 5 q^{29} - 3 q^{31} - 5 q^{33} + 24 q^{35} + 12 q^{37} - 11 q^{39} - q^{41} - 8 q^{43} - q^{45} + 3 q^{47} + 16 q^{49} - 46 q^{51} + 9 q^{55} - 18 q^{57} + 21 q^{59} - 13 q^{61} + 54 q^{63} - 26 q^{65} + 45 q^{67} + 3 q^{69} - 11 q^{71} - q^{73} + 3 q^{75} - 24 q^{77} - 7 q^{79} + 21 q^{81} - 35 q^{83} + 12 q^{85} - 67 q^{87} - 18 q^{89} + 36 q^{91} + 62 q^{93} + 3 q^{95} - 13 q^{97} - 88 q^{99} + O(q^{100})$$

## 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.u.a $4$ $3.194$ $$\Q(\zeta_{10})$$ None $$0$$ $$-5$$ $$-5$$ $$-6$$ $$q+(-2+2\zeta_{10}+\zeta_{10}^{3})q^{3}+(-2+\zeta_{10}+\cdots)q^{5}+\cdots$$
400.2.u.b $4$ $3.194$ $$\Q(\zeta_{10})$$ None $$0$$ $$1$$ $$-5$$ $$2$$ $$q+\zeta_{10}^{3}q^{3}+(-2+\zeta_{10}-2\zeta_{10}^{2}+\cdots)q^{5}+\cdots$$
400.2.u.c $4$ $3.194$ $$\Q(\zeta_{10})$$ None $$0$$ $$1$$ $$5$$ $$12$$ $$q+(1-\zeta_{10}-2\zeta_{10}^{3})q^{3}+(2-2\zeta_{10}+\cdots)q^{5}+\cdots$$
400.2.u.d $8$ $3.194$ 8.0.58140625.2 None $$0$$ $$3$$ $$0$$ $$-4$$ $$q+(\beta _{2}+\beta _{3}+\beta _{5}+\beta _{6})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$
400.2.u.e $8$ $3.194$ 8.0.58140625.2 None $$0$$ $$6$$ $$5$$ $$-4$$ $$q+(1-\beta _{3})q^{3}+(-\beta _{4}+\beta _{6})q^{5}+(-1+\cdots)q^{7}+\cdots$$
400.2.u.f $12$ $3.194$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-2$$ $$-4$$ $$2$$ $$q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{2}+\beta _{3}-2\beta _{5}+\cdots)q^{5}+\cdots$$
400.2.u.g $16$ $3.194$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$-1$$ $$-1$$ $$6$$ $$q-\beta _{1}q^{3}+(-1+\beta _{2}-\beta _{4}+\beta _{5}+\beta _{14}+\cdots)q^{5}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(400, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$