# Properties

 Label 39.4.f Level $39$ Weight $4$ Character orbit 39.f Rep. character $\chi_{39}(5,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $24$ Newform subspaces $2$ Sturm bound $18$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.f (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$39$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$18$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(39, [\chi])$$.

Total New Old
Modular forms 32 32 0
Cusp forms 24 24 0
Eisenstein series 8 8 0

## Trace form

 $$24 q - 4 q^{3} + 44 q^{6} + 4 q^{7} - 4 q^{9} + O(q^{10})$$ $$24 q - 4 q^{3} + 44 q^{6} + 4 q^{7} - 4 q^{9} - 76 q^{13} - 76 q^{15} - 272 q^{16} + 296 q^{18} + 148 q^{19} - 208 q^{21} - 224 q^{22} + 36 q^{24} - 592 q^{27} + 264 q^{28} - 84 q^{31} + 872 q^{33} + 816 q^{34} - 1440 q^{37} + 368 q^{39} + 3288 q^{40} + 124 q^{42} + 260 q^{45} - 1560 q^{46} - 1084 q^{48} - 2336 q^{52} - 232 q^{54} - 872 q^{55} + 1028 q^{57} - 1352 q^{58} - 1064 q^{60} + 1960 q^{61} - 652 q^{63} - 7664 q^{66} + 844 q^{67} + 1192 q^{70} + 6984 q^{72} - 416 q^{73} + 2712 q^{76} + 728 q^{78} + 6544 q^{79} + 3116 q^{81} + 5204 q^{84} - 8304 q^{85} + 3136 q^{87} + 4036 q^{91} - 4156 q^{93} - 6056 q^{94} - 5956 q^{96} - 5232 q^{97} + 1700 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(39, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
39.4.f.a $4$ $2.301$ $$\Q(\zeta_{12})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-40$$ $$q-\zeta_{12}^{3}q^{3}+8\zeta_{12}q^{4}+(-10+10\zeta_{12}+\cdots)q^{7}+\cdots$$
39.4.f.b $20$ $2.301$ $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ None $$0$$ $$-4$$ $$0$$ $$44$$ $$q+\beta _{8}q^{2}-\beta _{5}q^{3}+(-6\beta _{4}-\beta _{14})q^{4}+\cdots$$