# Properties

 Label 690.2.m Level $690$ Weight $2$ Character orbit 690.m Rep. character $\chi_{690}(31,\cdot)$ Character field $\Q(\zeta_{11})$ Dimension $160$ Newform subspaces $8$ Sturm bound $288$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.m (of order $$11$$ and degree $$10$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$23$$ Character field: $$\Q(\zeta_{11})$$ Newform subspaces: $$8$$ Sturm bound: $$288$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

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

Total New Old
Modular forms 1520 160 1360
Cusp forms 1360 160 1200
Eisenstein series 160 0 160

## Trace form

 $$160 q - 16 q^{4} - 16 q^{9} + O(q^{10})$$ $$160 q - 16 q^{4} - 16 q^{9} - 4 q^{10} - 24 q^{11} - 16 q^{13} - 8 q^{14} - 16 q^{16} + 56 q^{17} + 64 q^{19} + 80 q^{22} - 24 q^{23} - 16 q^{25} + 64 q^{26} + 64 q^{29} - 4 q^{30} + 40 q^{31} - 8 q^{33} - 24 q^{34} + 28 q^{35} - 16 q^{36} - 40 q^{37} - 32 q^{38} + 36 q^{39} - 4 q^{40} + 40 q^{41} - 24 q^{42} + 64 q^{43} - 24 q^{44} - 24 q^{46} - 48 q^{47} - 64 q^{49} + 56 q^{51} - 16 q^{52} + 56 q^{53} + 20 q^{55} - 8 q^{56} - 8 q^{57} + 24 q^{58} - 20 q^{59} + 40 q^{61} - 16 q^{62} - 16 q^{64} - 24 q^{65} - 24 q^{66} + 32 q^{67} - 32 q^{68} - 24 q^{69} - 104 q^{71} - 24 q^{73} - 8 q^{74} - 24 q^{76} - 48 q^{77} - 16 q^{78} + 104 q^{79} - 16 q^{81} + 56 q^{82} + 8 q^{83} - 8 q^{85} - 32 q^{86} - 16 q^{87} - 8 q^{88} - 4 q^{90} - 128 q^{91} - 24 q^{92} - 16 q^{93} - 16 q^{94} - 32 q^{95} + 8 q^{97} - 32 q^{98} - 24 q^{99} + O(q^{100})$$

## 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.m.a $10$ $5.510$ $$\Q(\zeta_{22})$$ None $$-1$$ $$-1$$ $$1$$ $$0$$ $$q+\zeta_{22}^{4}q^{2}+(-1+\zeta_{22}-\zeta_{22}^{2}+\zeta_{22}^{3}+\cdots)q^{3}+\cdots$$
690.2.m.b $10$ $5.510$ $$\Q(\zeta_{22})$$ None $$1$$ $$-1$$ $$-1$$ $$0$$ $$q-\zeta_{22}^{4}q^{2}+(-1+\zeta_{22}-\zeta_{22}^{2}+\zeta_{22}^{3}+\cdots)q^{3}+\cdots$$
690.2.m.c $20$ $5.510$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-2$$ $$2$$ $$-2$$ $$2$$ $$q+\beta _{16}q^{2}-\beta _{14}q^{3}-\beta _{12}q^{4}+\beta _{11}q^{5}+\cdots$$
690.2.m.d $20$ $5.510$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-2$$ $$2$$ $$2$$ $$2$$ $$q+\beta _{7}q^{2}-\beta _{12}q^{3}+\beta _{10}q^{4}-\beta _{6}q^{5}+\cdots$$
690.2.m.e $20$ $5.510$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$2$$ $$2$$ $$-2$$ $$-2$$ $$q-\beta _{13}q^{2}-\beta _{16}q^{3}-\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots$$
690.2.m.f $20$ $5.510$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$2$$ $$2$$ $$2$$ $$-2$$ $$q+\beta _{8}q^{2}+(1+\beta _{5}+\beta _{6}-\beta _{7}-\beta _{8}+\cdots)q^{3}+\cdots$$
690.2.m.g $30$ $5.510$ None $$-3$$ $$-3$$ $$-3$$ $$-8$$
690.2.m.h $30$ $5.510$ None $$3$$ $$-3$$ $$3$$ $$8$$

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

$$S_{2}^{\mathrm{old}}(690, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(23, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(46, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(69, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(115, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(138, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(230, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(345, [\chi])$$$$^{\oplus 2}$$