# Properties

 Label 690.2.d Level $690$ Weight $2$ Character orbit 690.d Rep. character $\chi_{690}(139,\cdot)$ Character field $\Q$ Dimension $20$ Newform subspaces $4$ Sturm bound $288$ Trace bound $14$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$288$$ Trace bound: $$14$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(690, [\chi])$$.

Total New Old
Modular forms 152 20 132
Cusp forms 136 20 116
Eisenstein series 16 0 16

## Trace form

 $$20 q - 20 q^{4} - 4 q^{6} - 20 q^{9} + O(q^{10})$$ $$20 q - 20 q^{4} - 4 q^{6} - 20 q^{9} + 4 q^{10} + 4 q^{15} + 20 q^{16} - 16 q^{19} + 16 q^{21} + 4 q^{24} - 4 q^{25} + 16 q^{26} - 24 q^{29} + 16 q^{31} - 16 q^{34} + 24 q^{35} + 20 q^{36} - 16 q^{39} - 4 q^{40} - 24 q^{41} - 8 q^{46} - 20 q^{49} - 8 q^{50} + 4 q^{54} + 24 q^{55} + 40 q^{59} - 4 q^{60} - 20 q^{64} + 16 q^{65} + 8 q^{69} - 8 q^{70} - 56 q^{71} + 16 q^{75} + 16 q^{76} - 24 q^{79} + 20 q^{81} - 16 q^{84} + 32 q^{86} + 16 q^{89} - 4 q^{90} - 16 q^{91} + 32 q^{94} - 40 q^{95} - 4 q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(690, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
690.2.d.a $4$ $5.510$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{3}q^{5}+q^{6}+\cdots$$
690.2.d.b $4$ $5.510$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{2}-\zeta_{8}^{2}q^{3}-q^{4}+(2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots$$
690.2.d.c $6$ $5.510$ 6.0.350464.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{3}q^{5}-q^{6}+\cdots$$
690.2.d.d $6$ $5.510$ 6.0.5161984.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{4}q^{5}-q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(690, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(690, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(115, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(230, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(345, [\chi])$$$$^{\oplus 2}$$