# Properties

 Label 38.4.e Level $38$ Weight $4$ Character orbit 38.e Rep. character $\chi_{38}(5,\cdot)$ Character field $\Q(\zeta_{9})$ Dimension $30$ Newform subspaces $2$ Sturm bound $20$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 38.e (of order $$9$$ and degree $$6$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{9})$$ Newform subspaces: $$2$$ Sturm bound: $$20$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(38, [\chi])$$.

Total New Old
Modular forms 102 30 72
Cusp forms 78 30 48
Eisenstein series 24 0 24

## Trace form

 $$30 q - 3 q^{3} - 30 q^{6} - 12 q^{7} + 24 q^{8} + 15 q^{9} + O(q^{10})$$ $$30 q - 3 q^{3} - 30 q^{6} - 12 q^{7} + 24 q^{8} + 15 q^{9} - 84 q^{11} + 72 q^{12} + 138 q^{13} + 24 q^{14} + 240 q^{15} - 42 q^{17} - 276 q^{18} - 834 q^{19} - 192 q^{20} - 486 q^{21} + 24 q^{22} + 132 q^{23} - 120 q^{24} + 876 q^{25} + 228 q^{26} + 867 q^{27} + 408 q^{28} + 828 q^{29} + 30 q^{31} - 1245 q^{33} + 360 q^{34} - 2130 q^{35} + 60 q^{36} + 60 q^{37} - 474 q^{38} + 564 q^{39} - 63 q^{41} + 72 q^{42} - 516 q^{43} + 768 q^{44} + 2790 q^{45} + 1200 q^{46} + 1932 q^{47} + 96 q^{48} + 555 q^{49} + 714 q^{50} + 345 q^{51} - 816 q^{52} - 882 q^{53} - 3438 q^{54} - 1296 q^{55} - 864 q^{56} - 5484 q^{57} - 1296 q^{58} - 4197 q^{59} + 240 q^{60} - 990 q^{61} - 1764 q^{62} + 1578 q^{63} - 960 q^{64} + 2250 q^{65} + 3162 q^{66} + 2385 q^{67} - 540 q^{68} + 3456 q^{69} + 3720 q^{70} + 4650 q^{71} + 1104 q^{72} - 378 q^{73} - 396 q^{74} - 6300 q^{75} - 408 q^{76} + 4404 q^{77} + 4884 q^{78} + 216 q^{79} + 5205 q^{81} + 714 q^{82} + 1308 q^{83} + 1296 q^{84} - 2028 q^{85} - 2208 q^{86} - 1614 q^{87} + 528 q^{88} - 1980 q^{89} - 7224 q^{90} - 516 q^{91} - 2784 q^{92} - 7734 q^{93} + 168 q^{94} + 1434 q^{95} - 453 q^{97} - 576 q^{98} - 561 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(38, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
38.4.e.a $12$ $2.242$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$-9$$ $$0$$ $$21$$ $$q-2\beta _{3}q^{2}+(-1+2\beta _{2}-\beta _{4}+2\beta _{6}+\cdots)q^{3}+\cdots$$
38.4.e.b $18$ $2.242$ $$\mathbb{Q}[x]/(x^{18} + \cdots)$$ None $$0$$ $$6$$ $$0$$ $$-33$$ $$q-2\beta _{8}q^{2}+(\beta _{3}+\beta _{4}+\beta _{6}+\beta _{9})q^{3}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(38, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(38, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 2}$$