# Properties

 Label 13.4.e Level $13$ Weight $4$ Character orbit 13.e Rep. character $\chi_{13}(4,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $4$ Newform subspaces $2$ Sturm bound $4$ Trace bound $2$

# Related objects

## Defining parameters

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

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(13, [\chi])$$.

Total New Old
Modular forms 8 8 0
Cusp forms 4 4 0
Eisenstein series 4 4 0

## Trace form

 $$4 q - 3 q^{2} + 5 q^{3} - q^{4} - 48 q^{6} + 15 q^{7} + q^{9} + O(q^{10})$$ $$4 q - 3 q^{2} + 5 q^{3} - q^{4} - 48 q^{6} + 15 q^{7} + q^{9} + 51 q^{10} - 15 q^{11} + 76 q^{12} + 65 q^{13} - 204 q^{14} + 174 q^{15} + 79 q^{16} - 144 q^{17} - 351 q^{19} - 111 q^{20} + 102 q^{22} + 21 q^{23} + 246 q^{24} + 110 q^{25} + 351 q^{26} - 130 q^{27} + 276 q^{28} + 210 q^{29} - 330 q^{30} - 603 q^{32} - 321 q^{33} + 336 q^{35} + 203 q^{36} - 318 q^{37} + 216 q^{38} + 325 q^{39} - 462 q^{40} - 210 q^{41} - 498 q^{42} - 19 q^{43} + 597 q^{45} + 576 q^{46} - 562 q^{48} + 13 q^{49} + 768 q^{50} + 90 q^{51} - 494 q^{52} + 1038 q^{53} - 510 q^{54} + 336 q^{55} + 624 q^{56} + 9 q^{58} - 525 q^{59} - 128 q^{61} - 522 q^{62} - 1410 q^{63} - 742 q^{64} - 1209 q^{65} + 996 q^{66} - 1077 q^{67} - 477 q^{68} + 555 q^{69} + 2841 q^{71} + 1425 q^{72} - 111 q^{74} - 713 q^{75} + 378 q^{76} - 1398 q^{77} - 858 q^{78} + 64 q^{79} + 1923 q^{80} + 418 q^{81} + 1833 q^{82} + 852 q^{84} + 297 q^{85} - 201 q^{87} + 624 q^{88} - 1161 q^{89} - 1974 q^{90} - 741 q^{91} - 1236 q^{92} - 1422 q^{93} - 1710 q^{94} - 1026 q^{95} + 2487 q^{97} - 1437 q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(13, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.4.e.a $2$ $0.767$ $$\Q(\sqrt{-3})$$ None $$-6$$ $$7$$ $$0$$ $$39$$ $$q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots$$
13.4.e.b $2$ $0.767$ $$\Q(\sqrt{-3})$$ None $$3$$ $$-2$$ $$0$$ $$-24$$ $$q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots$$