# Properties

 Label 400.3.p Level $400$ Weight $3$ Character orbit 400.p Rep. character $\chi_{400}(193,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $34$ Newform subspaces $12$ Sturm bound $180$ Trace bound $11$

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## Defining parameters

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 400.p (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$12$$ Sturm bound: $$180$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(400, [\chi])$$.

Total New Old
Modular forms 276 38 238
Cusp forms 204 34 170
Eisenstein series 72 4 68

## Trace form

 $$34 q - 2 q^{3} - 2 q^{7} + O(q^{10})$$ $$34 q - 2 q^{3} - 2 q^{7} + 4 q^{11} + 14 q^{13} - 2 q^{17} - 36 q^{21} + 46 q^{23} + 112 q^{27} + 132 q^{31} + 68 q^{33} + 22 q^{37} - 36 q^{41} - 66 q^{43} - 242 q^{47} - 572 q^{51} - 26 q^{53} + 16 q^{57} + 60 q^{61} + 222 q^{63} + 334 q^{67} + 836 q^{71} - 170 q^{73} + 100 q^{77} - 134 q^{81} - 274 q^{83} - 496 q^{87} - 188 q^{91} + 116 q^{93} + 22 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(400, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.3.p.a $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-6$$ $$0$$ $$6$$ $$q+(-3+3i)q^{3}+(3+3i)q^{7}-9iq^{9}+\cdots$$
400.3.p.b $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-4$$ $$0$$ $$4$$ $$q+(-2+2i)q^{3}+(2+2i)q^{7}+iq^{9}+\cdots$$
400.3.p.c $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$-14$$ $$q+(-1+i)q^{3}+(-7-7i)q^{7}+7iq^{9}+\cdots$$
400.3.p.d $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$-14$$ $$q+(1-i)q^{3}+(-7-7i)q^{7}+7iq^{9}+\cdots$$
400.3.p.e $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$-6$$ $$q+(1-i)q^{3}+(-3-3i)q^{7}+7iq^{9}+\cdots$$
400.3.p.f $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$14$$ $$q+(1-i)q^{3}+(7+7i)q^{7}+7iq^{9}+4q^{11}+\cdots$$
400.3.p.g $2$ $10.899$ $$\Q(\sqrt{-1})$$ None $$0$$ $$6$$ $$0$$ $$-6$$ $$q+(3-3i)q^{3}+(-3-3i)q^{7}-9iq^{9}+\cdots$$
400.3.p.h $4$ $10.899$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$-8$$ $$0$$ $$16$$ $$q+(-2+2\beta _{2}+\beta _{3})q^{3}+(4+4\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots$$
400.3.p.i $4$ $10.899$ $$\Q(i, \sqrt{41})$$ None $$0$$ $$-2$$ $$0$$ $$14$$ $$q+(-1+\beta _{2})q^{3}+(3-3\beta _{1}-\beta _{3})q^{7}+\cdots$$
400.3.p.j $4$ $10.899$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-4\beta _{3}q^{7}-6\beta _{2}q^{9}+3q^{11}+\cdots$$
400.3.p.k $4$ $10.899$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}-4\beta _{3}q^{7}+18\beta _{2}q^{9}+15q^{11}+\cdots$$
400.3.p.l $4$ $10.899$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$8$$ $$0$$ $$-16$$ $$q+(2-2\beta _{2}+\beta _{3})q^{3}+(-4+4\beta _{1}-4\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(400, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(400, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 2}$$