# Properties

 Label 650.6.b Level $650$ Weight $6$ Character orbit 650.b Rep. character $\chi_{650}(599,\cdot)$ Character field $\Q$ Dimension $90$ Newform subspaces $15$ Sturm bound $630$ Trace bound $9$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$650 = 2 \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 650.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$15$$ Sturm bound: $$630$$ Trace bound: $$9$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(650, [\chi])$$.

Total New Old
Modular forms 538 90 448
Cusp forms 514 90 424
Eisenstein series 24 0 24

## Trace form

 $$90 q - 1440 q^{4} - 304 q^{6} - 6282 q^{9} + O(q^{10})$$ $$90 q - 1440 q^{4} - 304 q^{6} - 6282 q^{9} + 148 q^{11} + 160 q^{14} + 23040 q^{16} + 1444 q^{19} + 14368 q^{21} + 4864 q^{24} - 4056 q^{26} - 9792 q^{29} + 7728 q^{31} + 9248 q^{34} + 100512 q^{36} - 20376 q^{41} - 2368 q^{44} + 47936 q^{46} - 256722 q^{49} + 29208 q^{51} + 15856 q^{54} - 2560 q^{56} + 122096 q^{59} - 237040 q^{61} - 368640 q^{64} + 282128 q^{66} + 35248 q^{69} - 184880 q^{71} - 15360 q^{74} - 23104 q^{76} - 39608 q^{79} + 394314 q^{81} - 229888 q^{84} + 212320 q^{86} - 110112 q^{89} - 181844 q^{91} + 192800 q^{94} - 77824 q^{96} - 1134344 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(650, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
650.6.b.a $2$ $104.249$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}-2^{4}q^{4}+170iq^{7}-2^{6}iq^{8}+\cdots$$
650.6.b.b $4$ $104.249$ $$\Q(i, \sqrt{235})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{1}q^{2}+(-13\beta _{1}+\beta _{3})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.c $4$ $104.249$ $$\Q(i, \sqrt{14})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{1}q^{2}+(8\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.d $4$ $104.249$ $$\Q(i, \sqrt{2785})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{2}q^{2}+(\beta _{1}-4\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.e $4$ $104.249$ $$\Q(i, \sqrt{19})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{1}q^{2}+(-3\beta _{1}+3\beta _{3})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.f $4$ $104.249$ $$\Q(i, \sqrt{145})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{1}q^{2}+(-3\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.g $4$ $104.249$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{1}q^{2}+(2\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.h $4$ $104.249$ $$\Q(i, \sqrt{849})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{2}q^{2}+(\beta _{1}-4\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.i $6$ $104.249$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{2}q^{2}+(\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.j $6$ $104.249$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{1}q^{2}+(7\beta _{1}+\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.k $8$ $104.249$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{2}q^{2}+(\beta _{1}+5\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.l $8$ $104.249$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{1}q^{2}+(-5\beta _{1}-\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.m $10$ $104.249$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{5}q^{2}+(\beta _{1}-2\beta _{5})q^{3}-2^{4}q^{4}+\cdots$$
650.6.b.n $10$ $104.249$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{6}q^{2}+\beta _{1}q^{3}-2^{4}q^{4}-4\beta _{2}q^{6}+\cdots$$
650.6.b.o $12$ $104.249$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{7}q^{2}+(\beta _{1}+2\beta _{7})q^{3}-2^{4}q^{4}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(650, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(650, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(65, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(130, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(325, [\chi])$$$$^{\oplus 2}$$