# Properties

 Label 350.6.c Level $350$ Weight $6$ Character orbit 350.c Rep. character $\chi_{350}(99,\cdot)$ Character field $\Q$ Dimension $44$ Newform subspaces $14$ Sturm bound $360$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 350.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$360$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(350, [\chi])$$.

Total New Old
Modular forms 312 44 268
Cusp forms 288 44 244
Eisenstein series 24 0 24

## Trace form

 $$44 q - 704 q^{4} - 3432 q^{9} + O(q^{10})$$ $$44 q - 704 q^{4} - 3432 q^{9} - 1244 q^{11} + 784 q^{14} + 11264 q^{16} - 7024 q^{19} - 3724 q^{21} - 9552 q^{26} + 2912 q^{29} + 27416 q^{31} + 880 q^{34} + 54912 q^{36} - 39352 q^{39} + 43148 q^{41} + 19904 q^{44} - 45376 q^{46} - 105644 q^{49} - 83068 q^{51} - 71856 q^{54} - 12544 q^{56} + 172012 q^{59} + 106548 q^{61} - 180224 q^{64} + 110480 q^{66} + 401744 q^{69} - 86952 q^{71} + 121472 q^{74} + 112384 q^{76} + 53320 q^{79} + 222036 q^{81} + 59584 q^{84} + 22240 q^{86} - 113964 q^{89} - 97804 q^{91} - 34560 q^{94} - 835056 q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(350, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.6.c.a $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}+17iq^{3}-2^{4}q^{4}-68q^{6}+\cdots$$
350.6.c.b $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}+11iq^{3}-2^{4}q^{4}-44q^{6}+\cdots$$
350.6.c.c $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}+11iq^{3}-2^{4}q^{4}-44q^{6}+\cdots$$
350.6.c.d $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{2}+10iq^{3}-2^{4}q^{4}-40q^{6}+\cdots$$
350.6.c.e $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4iq^{2}+3iq^{3}-2^{4}q^{4}+12q^{6}+\cdots$$
350.6.c.f $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4iq^{2}+8iq^{3}-2^{4}q^{4}+2^{5}q^{6}+\cdots$$
350.6.c.g $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4iq^{2}+9iq^{3}-2^{4}q^{4}+6^{2}q^{6}+\cdots$$
350.6.c.h $2$ $56.134$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4iq^{2}+23iq^{3}-2^{4}q^{4}+92q^{6}+\cdots$$
350.6.c.i $4$ $56.134$ $$\Q(i, \sqrt{79})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{1}q^{2}+(-10\beta _{1}+\beta _{3})q^{3}-2^{4}q^{4}+\cdots$$
350.6.c.j $4$ $56.134$ $$\Q(i, \sqrt{3369})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{2}q^{2}+(\beta _{1}-\beta _{2})q^{3}-2^{4}q^{4}+(8+\cdots)q^{6}+\cdots$$
350.6.c.k $4$ $56.134$ $$\Q(i, \sqrt{1129})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{2}q^{2}+(\beta _{1}-2\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$
350.6.c.l $4$ $56.134$ $$\Q(i, \sqrt{79})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4\beta _{1}q^{2}+(-4\beta _{1}-\beta _{3})q^{3}-2^{4}q^{4}+\cdots$$
350.6.c.m $6$ $56.134$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{2}q^{2}-\beta _{4}q^{3}-2^{4}q^{4}-4\beta _{1}q^{6}+\cdots$$
350.6.c.n $6$ $56.134$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\beta _{2}q^{2}+(\beta _{1}+3\beta _{2})q^{3}-2^{4}q^{4}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(350, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(350, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(175, [\chi])$$$$^{\oplus 2}$$