# Properties

 Label 78.6.b Level $78$ Weight $6$ Character orbit 78.b Rep. character $\chi_{78}(25,\cdot)$ Character field $\Q$ Dimension $14$ Newform subspaces $2$ Sturm bound $84$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$78 = 2 \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$84$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(78, [\chi])$$.

Total New Old
Modular forms 74 14 60
Cusp forms 66 14 52
Eisenstein series 8 0 8

## Trace form

 $$14 q + 18 q^{3} - 224 q^{4} + 1134 q^{9} + O(q^{10})$$ $$14 q + 18 q^{3} - 224 q^{4} + 1134 q^{9} + 240 q^{10} - 288 q^{12} + 478 q^{13} - 1936 q^{14} + 3584 q^{16} - 236 q^{17} + 4320 q^{22} - 2072 q^{23} - 9410 q^{25} - 592 q^{26} + 1458 q^{27} + 15428 q^{29} - 3600 q^{30} + 33256 q^{35} - 18144 q^{36} - 13872 q^{38} - 5238 q^{39} - 3840 q^{40} + 7056 q^{42} - 7256 q^{43} + 4608 q^{48} - 62518 q^{49} + 12924 q^{51} - 7648 q^{52} - 32820 q^{53} + 130080 q^{55} + 30976 q^{56} + 55148 q^{61} - 51568 q^{62} - 57344 q^{64} - 210056 q^{65} + 62208 q^{66} + 3776 q^{68} - 11160 q^{69} + 11392 q^{74} - 97614 q^{75} + 129936 q^{77} + 19152 q^{78} - 13360 q^{79} + 91854 q^{81} - 15312 q^{82} - 199332 q^{87} - 69120 q^{88} + 19440 q^{90} + 76872 q^{91} + 33152 q^{92} - 96960 q^{94} + 198936 q^{95} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(78, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.6.b.a $6$ $12.510$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$-54$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-9q^{3}-2^{4}q^{4}+(3\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots$$
78.6.b.b $8$ $12.510$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$72$$ $$0$$ $$0$$ $$q+\beta _{4}q^{2}+9q^{3}-2^{4}q^{4}+(-\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(78, [\chi])$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(78, [\chi]) \simeq$$ $$S_{6}^{\mathrm{new}}(13, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$