# Properties

 Label 78.4.b Level $78$ Weight $4$ Character orbit 78.b Rep. character $\chi_{78}(25,\cdot)$ Character field $\Q$ Dimension $6$ Newform subspaces $2$ Sturm bound $56$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 78.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$56$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

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

Total New Old
Modular forms 46 6 40
Cusp forms 38 6 32
Eisenstein series 8 0 8

## Trace form

 $$6 q + 6 q^{3} - 24 q^{4} + 54 q^{9} + O(q^{10})$$ $$6 q + 6 q^{3} - 24 q^{4} + 54 q^{9} + 104 q^{10} - 24 q^{12} + 110 q^{13} + 56 q^{14} + 96 q^{16} - 188 q^{17} - 144 q^{22} + 184 q^{23} - 314 q^{25} + 8 q^{26} + 54 q^{27} + 164 q^{29} + 120 q^{30} - 1448 q^{35} - 216 q^{36} + 456 q^{38} - 138 q^{39} - 416 q^{40} - 168 q^{42} + 760 q^{43} + 96 q^{48} - 782 q^{49} - 12 q^{51} - 440 q^{52} + 876 q^{53} + 96 q^{55} - 224 q^{56} + 2156 q^{61} + 1928 q^{62} - 384 q^{64} + 1240 q^{65} - 1152 q^{66} + 752 q^{68} + 1896 q^{69} - 704 q^{74} - 1674 q^{75} - 2064 q^{77} - 600 q^{78} - 2096 q^{79} + 486 q^{81} + 40 q^{82} + 2532 q^{87} + 576 q^{88} + 936 q^{90} + 1720 q^{91} - 736 q^{92} - 288 q^{94} - 696 q^{95} + O(q^{100})$$

## 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.b.a $2$ $4.602$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-6$$ $$0$$ $$0$$ $$q+iq^{2}-3q^{3}-4q^{4}-4iq^{5}-3iq^{6}+\cdots$$
78.4.b.b $4$ $4.602$ $$\Q(i, \sqrt{17})$$ None $$0$$ $$12$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+3q^{3}-4q^{4}+(4\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$

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

$$S_{4}^{\mathrm{old}}(78, [\chi]) \simeq$$ $$S_{4}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$