# Properties

 Label 338.4.e Level $338$ Weight $4$ Character orbit 338.e Rep. character $\chi_{338}(23,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $76$ Newform subspaces $9$ Sturm bound $182$ Trace bound $10$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.e (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$9$$ Sturm bound: $$182$$ Trace bound: $$10$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(338, [\chi])$$.

Total New Old
Modular forms 300 76 224
Cusp forms 244 76 168
Eisenstein series 56 0 56

## Trace form

 $$76 q + 6 q^{3} + 152 q^{4} - 18 q^{7} - 356 q^{9} + O(q^{10})$$ $$76 q + 6 q^{3} + 152 q^{4} - 18 q^{7} - 356 q^{9} + 8 q^{10} + 18 q^{11} + 48 q^{12} - 40 q^{14} + 192 q^{15} - 608 q^{16} - 110 q^{17} - 594 q^{19} - 72 q^{20} + 100 q^{22} + 318 q^{23} - 2048 q^{25} - 192 q^{27} - 72 q^{28} + 22 q^{29} - 244 q^{30} + 42 q^{33} + 190 q^{35} + 1424 q^{36} + 768 q^{37} + 824 q^{38} + 64 q^{40} + 564 q^{41} + 820 q^{42} - 38 q^{43} - 630 q^{45} - 576 q^{46} + 96 q^{48} + 2362 q^{49} - 1968 q^{50} - 1384 q^{51} - 384 q^{53} - 648 q^{54} - 1488 q^{55} - 80 q^{56} + 1848 q^{58} + 1110 q^{59} - 1046 q^{61} - 1824 q^{62} + 1980 q^{63} - 4864 q^{64} + 3392 q^{66} - 510 q^{67} + 440 q^{68} + 2730 q^{69} + 1470 q^{71} - 576 q^{72} + 376 q^{74} + 1320 q^{75} - 2376 q^{76} - 3708 q^{77} - 2024 q^{79} - 288 q^{80} - 6302 q^{81} - 440 q^{82} + 2136 q^{84} + 2898 q^{85} - 2410 q^{87} - 400 q^{88} + 4434 q^{89} + 4536 q^{90} + 2544 q^{92} - 3108 q^{93} + 2908 q^{94} + 580 q^{95} - 1854 q^{97} - 4272 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(338, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.4.e.a $4$ $19.943$ $$\Q(\zeta_{12})$$ None $$0$$ $$-8$$ $$0$$ $$0$$ $$q+\zeta_{12}q^{2}+(-4+4\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots$$
338.4.e.b $4$ $19.943$ $$\Q(\zeta_{12})$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+2\zeta_{12}q^{2}+(-3+3\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots$$
338.4.e.c $4$ $19.943$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q+2\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots$$
338.4.e.d $4$ $19.943$ $$\Q(\zeta_{12})$$ None $$0$$ $$6$$ $$0$$ $$0$$ $$q+2\zeta_{12}q^{2}+(3-3\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots$$
338.4.e.e $8$ $19.943$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$-6$$ $$0$$ $$-18$$ $$q+\beta _{3}q^{2}+(-2+2\beta _{1}+\beta _{4}-\beta _{7})q^{3}+\cdots$$
338.4.e.f $8$ $19.943$ 8.0.$$\cdots$$.49 None $$0$$ $$6$$ $$0$$ $$0$$ $$q-2\beta _{6}q^{2}+(2-\beta _{2}+\beta _{4}+\beta _{5})q^{3}+\cdots$$
338.4.e.g $8$ $19.943$ 8.0.$$\cdots$$.49 None $$0$$ $$6$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}+(1-2\beta _{2}-\beta _{4}-\beta _{6})q^{3}+\cdots$$
338.4.e.h $12$ $19.943$ 12.0.$$\cdots$$.1 None $$0$$ $$24$$ $$0$$ $$0$$ $$q+2\beta _{6}q^{2}+(2-3\beta _{4}-2\beta _{7}-3\beta _{9}+\cdots)q^{3}+\cdots$$
338.4.e.i $24$ $19.943$ None $$0$$ $$-18$$ $$0$$ $$0$$

## Decomposition of $$S_{4}^{\mathrm{old}}(338, [\chi])$$ into lower level spaces

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