# Properties

 Label 39.4.b Level $39$ Weight $4$ Character orbit 39.b Rep. character $\chi_{39}(25,\cdot)$ Character field $\Q$ Dimension $8$ Newform subspaces $2$ Sturm bound $18$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$18$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 16 8 8
Cusp forms 12 8 4
Eisenstein series 4 0 4

## Trace form

 $$8 q - 40 q^{4} + 72 q^{9} + O(q^{10})$$ $$8 q - 40 q^{4} + 72 q^{9} - 68 q^{10} + 36 q^{12} - 84 q^{13} - 264 q^{14} + 372 q^{16} + 312 q^{17} + 244 q^{22} - 168 q^{23} - 64 q^{25} - 432 q^{26} - 432 q^{29} - 324 q^{30} + 984 q^{35} - 360 q^{36} + 24 q^{38} + 180 q^{39} - 316 q^{40} + 1296 q^{42} - 224 q^{43} - 1008 q^{48} - 904 q^{49} - 360 q^{51} + 460 q^{52} + 120 q^{53} + 256 q^{55} + 1728 q^{56} - 1992 q^{61} + 2592 q^{62} + 264 q^{64} + 1272 q^{65} + 1548 q^{66} - 5784 q^{68} - 1224 q^{69} + 840 q^{74} + 72 q^{75} + 600 q^{77} - 936 q^{78} + 4608 q^{79} + 648 q^{81} - 3116 q^{82} - 1008 q^{87} + 3428 q^{88} - 612 q^{90} - 384 q^{91} - 7560 q^{92} + 4996 q^{94} + 1512 q^{95} + O(q^{100})$$

## 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.b.a $4$ $2.301$ 4.0.5054412.1 None $$0$$ $$-12$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-3q^{3}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}+\cdots$$
39.4.b.b $4$ $2.301$ 4.0.1362828.1 None $$0$$ $$12$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+3q^{3}+(-4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(39, [\chi])$$ into lower level spaces

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