# Properties

 Label 338.4.c Level $338$ Weight $4$ Character orbit 338.c Rep. character $\chi_{338}(191,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $78$ Newform subspaces $16$ Sturm bound $182$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.c (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$16$$ Sturm bound: $$182$$ Trace bound: $$5$$ 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 302 78 224
Cusp forms 246 78 168
Eisenstein series 56 0 56

## Trace form

 $$78 q + 2 q^{2} - 6 q^{3} - 156 q^{4} + 10 q^{5} - 50 q^{7} - 16 q^{8} - 337 q^{9} + O(q^{10})$$ $$78 q + 2 q^{2} - 6 q^{3} - 156 q^{4} + 10 q^{5} - 50 q^{7} - 16 q^{8} - 337 q^{9} - 18 q^{10} + 18 q^{11} + 48 q^{12} + 200 q^{14} - 104 q^{15} - 624 q^{16} - 105 q^{17} - 308 q^{18} + 78 q^{19} - 20 q^{20} + 52 q^{21} + 12 q^{22} - 242 q^{23} + 2704 q^{25} + 672 q^{27} - 200 q^{28} - 157 q^{29} - 268 q^{30} - 352 q^{31} + 32 q^{32} - 806 q^{33} + 628 q^{34} + 318 q^{35} - 1348 q^{36} - 279 q^{37} + 40 q^{38} + 144 q^{40} + 437 q^{41} + 20 q^{42} + 258 q^{43} - 144 q^{44} + 83 q^{45} + 440 q^{46} - 1224 q^{47} - 96 q^{48} - 2135 q^{49} + 8 q^{50} - 1880 q^{51} - 514 q^{53} - 936 q^{54} + 528 q^{55} - 400 q^{56} + 676 q^{57} - 370 q^{58} + 102 q^{59} + 832 q^{60} + 683 q^{61} + 1528 q^{62} - 912 q^{63} + 4992 q^{64} + 3904 q^{66} + 710 q^{67} - 420 q^{68} - 142 q^{69} - 1104 q^{70} - 490 q^{71} + 616 q^{72} + 786 q^{73} - 2554 q^{74} + 840 q^{75} + 312 q^{76} + 204 q^{77} - 4288 q^{79} - 80 q^{80} - 4687 q^{81} - 170 q^{82} + 4992 q^{83} - 104 q^{84} - 359 q^{85} + 1168 q^{86} + 826 q^{87} + 48 q^{88} - 1912 q^{89} - 2772 q^{90} + 1936 q^{92} + 2496 q^{93} + 12 q^{94} + 484 q^{95} - 976 q^{97} + 1506 q^{98} - 4384 q^{99} + O(q^{100})$$

## 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.c.a $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-4$$ $$-36$$ $$-20$$ $$q+(-2+2\zeta_{6})q^{2}+(-4+4\zeta_{6})q^{3}+\cdots$$
338.4.c.b $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-3$$ $$-22$$ $$19$$ $$q+(-2+2\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots$$
338.4.c.c $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$1$$ $$34$$ $$35$$ $$q+(-2+2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
338.4.c.d $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$3$$ $$-4$$ $$-5$$ $$q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
338.4.c.e $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-4$$ $$36$$ $$20$$ $$q+(2-2\zeta_{6})q^{2}+(-4+4\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
338.4.c.f $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-3$$ $$22$$ $$-19$$ $$q+(2-2\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
338.4.c.g $2$ $19.943$ $$\Q(\sqrt{-3})$$ None $$2$$ $$1$$ $$-34$$ $$-35$$ $$q+(2-2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
338.4.c.h $4$ $19.943$ $$\Q(\sqrt{-3}, \sqrt{217})$$ None $$-4$$ $$3$$ $$38$$ $$-7$$ $$q-2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots$$
338.4.c.i $4$ $19.943$ $$\Q(\sqrt{-3}, \sqrt{217})$$ None $$4$$ $$3$$ $$-38$$ $$7$$ $$q+2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots$$
338.4.c.j $4$ $19.943$ $$\Q(\sqrt{-3}, \sqrt{217})$$ None $$4$$ $$3$$ $$14$$ $$-45$$ $$q+2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots$$
338.4.c.k $6$ $19.943$ 6.0.64827.1 None $$-6$$ $$12$$ $$-24$$ $$-27$$ $$q+(-2+2\beta _{5})q^{2}+(5-3\beta _{4}-5\beta _{5})q^{3}+\cdots$$
338.4.c.l $6$ $19.943$ 6.0.64827.1 None $$6$$ $$12$$ $$24$$ $$27$$ $$q+(2-2\beta _{5})q^{2}+(5-3\beta _{4}-5\beta _{5})q^{3}+\cdots$$
338.4.c.m $8$ $19.943$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$-8$$ $$-6$$ $$-8$$ $$-20$$ $$q+(-2-2\beta _{2})q^{2}+(-2-\beta _{2}+\beta _{3}+\cdots)q^{3}+\cdots$$
338.4.c.n $8$ $19.943$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$8$$ $$-6$$ $$8$$ $$20$$ $$q+(2+2\beta _{2})q^{2}+(-2-\beta _{2}+\beta _{3}+\beta _{7})q^{3}+\cdots$$
338.4.c.o $12$ $19.943$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-12$$ $$-9$$ $$36$$ $$-25$$ $$q+(-2+2\beta _{1})q^{2}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$$
338.4.c.p $12$ $19.943$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$12$$ $$-9$$ $$-36$$ $$25$$ $$q+(2-2\beta _{1})q^{2}+(-1+2\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots$$

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

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