# Properties

 Label 39.6.b Level $39$ Weight $6$ Character orbit 39.b Rep. character $\chi_{39}(25,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $2$ Sturm bound $28$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 39.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$28$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(39, [\chi])$$.

Total New Old
Modular forms 26 10 16
Cusp forms 22 10 12
Eisenstein series 4 0 4

## Trace form

 $$10 q - 18 q^{3} - 96 q^{4} + 810 q^{9} + O(q^{10})$$ $$10 q - 18 q^{3} - 96 q^{4} + 810 q^{9} - 524 q^{10} + 1260 q^{12} - 578 q^{13} + 4416 q^{14} - 268 q^{16} + 468 q^{17} - 11324 q^{22} + 2112 q^{23} - 14718 q^{25} + 19032 q^{26} - 1458 q^{27} - 21156 q^{29} + 2916 q^{30} - 13728 q^{35} - 7776 q^{36} - 5328 q^{38} + 10818 q^{39} + 88292 q^{40} - 43848 q^{42} + 4376 q^{43} - 27504 q^{48} + 53614 q^{49} + 2700 q^{51} + 56436 q^{52} + 83916 q^{53} - 119360 q^{55} - 84192 q^{56} - 116644 q^{61} + 49944 q^{62} + 40936 q^{64} + 67632 q^{65} + 61452 q^{66} + 8856 q^{68} + 24624 q^{69} - 145944 q^{74} + 145926 q^{75} + 39192 q^{77} - 78624 q^{78} - 238384 q^{79} + 65610 q^{81} - 123380 q^{82} + 119124 q^{87} + 74372 q^{88} - 42444 q^{90} - 226248 q^{91} + 172440 q^{92} - 230972 q^{94} - 487296 q^{95} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(39, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
39.6.b.a $4$ $6.255$ 4.0.31066572.1 None $$0$$ $$36$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+9q^{3}+(5+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots$$
39.6.b.b $6$ $6.255$ $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ None $$0$$ $$-54$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-9q^{3}+(-20+\beta _{4})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(39, [\chi])$$ into lower level spaces

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