# Properties

 Label 338.8.b Level $338$ Weight $8$ Character orbit 338.b Rep. character $\chi_{338}(337,\cdot)$ Character field $\Q$ Dimension $88$ Newform subspaces $11$ Sturm bound $364$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 338.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$364$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(338, [\chi])$$.

Total New Old
Modular forms 332 88 244
Cusp forms 304 88 216
Eisenstein series 28 0 28

## Trace form

 $$88 q - 54 q^{3} - 5632 q^{4} + 57482 q^{9} + O(q^{10})$$ $$88 q - 54 q^{3} - 5632 q^{4} + 57482 q^{9} + 1136 q^{10} + 3456 q^{12} - 5600 q^{14} + 360448 q^{16} + 17252 q^{17} - 13264 q^{22} - 130592 q^{23} - 1165546 q^{25} + 79248 q^{27} + 90702 q^{29} + 585568 q^{30} - 1181932 q^{35} - 3678848 q^{36} - 779920 q^{38} - 72704 q^{40} - 1832224 q^{42} - 2981042 q^{43} - 221184 q^{48} - 9548636 q^{49} - 1932476 q^{51} + 4648578 q^{53} - 5048840 q^{55} + 358400 q^{56} - 14202614 q^{61} + 1860416 q^{62} - 23068672 q^{64} + 7062592 q^{66} - 1104128 q^{68} + 344652 q^{69} - 20903216 q^{74} + 22539702 q^{75} + 5373020 q^{77} + 32332684 q^{79} - 5403568 q^{81} - 17895712 q^{82} - 62233460 q^{87} + 848896 q^{88} + 22099440 q^{90} + 8357888 q^{92} - 13879040 q^{94} - 4853268 q^{95} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(338, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.8.b.a $2$ $105.586$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-174$$ $$0$$ $$0$$ $$q-8iq^{2}-87q^{3}-2^{6}q^{4}-321iq^{5}+\cdots$$
338.8.b.b $2$ $105.586$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-78$$ $$0$$ $$0$$ $$q-8iq^{2}-39q^{3}-2^{6}q^{4}+385iq^{5}+\cdots$$
338.8.b.c $2$ $105.586$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-54$$ $$0$$ $$0$$ $$q+8iq^{2}-3^{3}q^{3}-2^{6}q^{4}-245iq^{5}+\cdots$$
338.8.b.d $2$ $105.586$ $$\Q(\sqrt{-1})$$ None $$0$$ $$24$$ $$0$$ $$0$$ $$q-4iq^{2}+12q^{3}-2^{6}q^{4}-105iq^{5}+\cdots$$
338.8.b.e $4$ $105.586$ $$\Q(i, \sqrt{105})$$ None $$0$$ $$-24$$ $$0$$ $$0$$ $$q-8\beta _{1}q^{2}+(-6+7\beta _{3})q^{3}-2^{6}q^{4}+\cdots$$
338.8.b.f $4$ $105.586$ $$\Q(i, \sqrt{2305})$$ None $$0$$ $$174$$ $$0$$ $$0$$ $$q+4\beta _{2}q^{2}+(43+\beta _{3})q^{3}-2^{6}q^{4}+(5\beta _{1}+\cdots)q^{5}+\cdots$$
338.8.b.g $6$ $105.586$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+8\beta _{2}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+(2\beta _{1}-110\beta _{2}+\cdots)q^{5}+\cdots$$
338.8.b.h $8$ $105.586$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-8\beta _{2}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$$
338.8.b.i $16$ $105.586$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{10}q^{2}+\beta _{2}q^{3}-2^{6}q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots$$
338.8.b.j $18$ $105.586$ $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ None $$0$$ $$-138$$ $$0$$ $$0$$ $$q+8\beta _{3}q^{2}+(-7-\beta _{1}-\beta _{2})q^{3}-2^{6}q^{4}+\cdots$$
338.8.b.k $24$ $105.586$ None $$0$$ $$216$$ $$0$$ $$0$$

## Decomposition of $$S_{8}^{\mathrm{old}}(338, [\chi])$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(338, [\chi]) \simeq$$ $$S_{8}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(169, [\chi])$$$$^{\oplus 2}$$