# Properties

 Label 78.4.k Level $78$ Weight $4$ Character orbit 78.k Rep. character $\chi_{78}(11,\cdot)$ Character field $\Q(\zeta_{12})$ Dimension $56$ Newform subspaces $1$ Sturm bound $56$ Trace bound $0$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 78.k (of order $$12$$ and degree $$4$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$39$$ Character field: $$\Q(\zeta_{12})$$ Newform subspaces: $$1$$ Sturm bound: $$56$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(78, [\chi])$$.

Total New Old
Modular forms 184 56 128
Cusp forms 152 56 96
Eisenstein series 32 0 32

## Trace form

 $$56 q - 92 q^{7} + O(q^{10})$$ $$56 q - 92 q^{7} + 144 q^{10} + 168 q^{13} - 168 q^{15} + 448 q^{16} + 96 q^{18} - 716 q^{19} - 144 q^{21} - 72 q^{27} + 368 q^{28} + 816 q^{30} + 140 q^{31} + 156 q^{33} + 240 q^{34} - 768 q^{36} - 1804 q^{37} - 912 q^{39} - 576 q^{42} - 1092 q^{43} + 492 q^{45} + 48 q^{46} + 2196 q^{49} - 16 q^{52} - 1584 q^{54} + 792 q^{55} + 1020 q^{57} - 1056 q^{58} + 384 q^{60} - 120 q^{61} + 8148 q^{63} - 96 q^{66} + 160 q^{67} + 3516 q^{69} - 192 q^{70} + 576 q^{72} - 4628 q^{73} - 7020 q^{75} - 2864 q^{76} - 5232 q^{78} + 6816 q^{79} - 4224 q^{81} + 864 q^{82} + 1056 q^{84} - 1656 q^{85} - 1008 q^{87} + 8044 q^{91} + 3912 q^{93} + 2256 q^{94} + 4220 q^{97} + 8892 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(78, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.4.k.a $56$ $4.602$ None $$0$$ $$0$$ $$0$$ $$-92$$

## Decomposition of $$S_{4}^{\mathrm{old}}(78, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(78, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$