# Properties

 Label 650.2.b Level $650$ Weight $2$ Character orbit 650.b Rep. character $\chi_{650}(599,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $9$ Sturm bound $210$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 650.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$210$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(650, [\chi])$$.

Total New Old
Modular forms 118 18 100
Cusp forms 94 18 76
Eisenstein series 24 0 24

## Trace form

 $$18 q - 18 q^{4} + 4 q^{6} - 18 q^{9} + O(q^{10})$$ $$18 q - 18 q^{4} + 4 q^{6} - 18 q^{9} - 12 q^{11} - 8 q^{14} + 18 q^{16} + 4 q^{19} - 16 q^{21} - 4 q^{24} - 6 q^{26} - 16 q^{29} - 16 q^{31} + 8 q^{34} + 18 q^{36} + 8 q^{41} + 12 q^{44} - 26 q^{49} - 8 q^{51} - 4 q^{54} + 8 q^{56} + 32 q^{59} - 18 q^{64} - 28 q^{66} - 48 q^{71} - 32 q^{74} - 4 q^{76} + 40 q^{79} - 30 q^{81} + 16 q^{84} + 8 q^{86} + 64 q^{89} + 12 q^{91} - 40 q^{94} + 4 q^{96} + 104 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(650, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
650.2.b.a $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+3iq^{3}-q^{4}-3q^{6}+iq^{7}+\cdots$$
650.2.b.b $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+2iq^{3}-q^{4}-2q^{6}+5iq^{7}+\cdots$$
650.2.b.c $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}-4iq^{7}+\cdots$$
650.2.b.d $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}+iq^{7}-iq^{8}+\cdots$$
650.2.b.e $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-q^{4}-iq^{8}+3q^{9}-iq^{13}+\cdots$$
650.2.b.f $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-2iq^{3}-q^{4}+2q^{6}+4iq^{7}+\cdots$$
650.2.b.g $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-2iq^{3}-q^{4}+2q^{6}-4iq^{7}+\cdots$$
650.2.b.h $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-2iq^{3}-q^{4}+2q^{6}+iq^{7}+\cdots$$
650.2.b.i $2$ $5.190$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}-3iq^{3}-q^{4}+3q^{6}-iq^{8}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(650, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(650, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 2}$$