# Properties

 Label 402.2.d Level 402 Weight 2 Character orbit d Rep. character $$\chi_{402}(401,\cdot)$$ Character field $$\Q$$ Dimension 24 Newform subspaces 2 Sturm bound 136 Trace bound 2

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## Defining parameters

 Level: $$N$$ = $$402 = 2 \cdot 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 402.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$201$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$136$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(402, [\chi])$$.

Total New Old
Modular forms 72 24 48
Cusp forms 64 24 40
Eisenstein series 8 0 8

## Trace form

 $$24q + 24q^{4} - 2q^{6} + 10q^{9} + O(q^{10})$$ $$24q + 24q^{4} - 2q^{6} + 10q^{9} - 8q^{15} + 24q^{16} - 40q^{19} - 12q^{22} - 2q^{24} + 48q^{25} + 10q^{36} - 44q^{37} + 12q^{39} - 4q^{49} + 10q^{54} - 16q^{55} - 8q^{60} + 24q^{64} - 16q^{67} - 40q^{73} - 40q^{76} - 22q^{81} - 28q^{82} - 12q^{88} + 4q^{90} - 16q^{91} - 24q^{93} - 2q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(402, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
402.2.d.a $$12$$ $$3.210$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-12$$ $$1$$ $$0$$ $$0$$ $$q-q^{2}+\beta _{8}q^{3}+q^{4}-\beta _{7}q^{5}-\beta _{8}q^{6}+\cdots$$
402.2.d.b $$12$$ $$3.210$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$12$$ $$-1$$ $$0$$ $$0$$ $$q+q^{2}-\beta _{8}q^{3}+q^{4}+\beta _{7}q^{5}-\beta _{8}q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(402, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(402, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(201, [\chi])$$$$^{\oplus 2}$$