# Properties

 Label 38.6.a Level $38$ Weight $6$ Character orbit 38.a Rep. character $\chi_{38}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $4$ Sturm bound $30$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 38.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$30$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(38))$$.

Total New Old
Modular forms 27 7 20
Cusp forms 23 7 16
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$19$$FrickeDim.
$$+$$$$+$$$$+$$$$1$$
$$+$$$$-$$$$-$$$$2$$
$$-$$$$+$$$$-$$$$3$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$2$$
Minus space$$-$$$$5$$

## Trace form

 $$7q + 4q^{2} - 4q^{3} + 112q^{4} + 22q^{5} + 8q^{6} + 194q^{7} + 64q^{8} + 221q^{9} + O(q^{10})$$ $$7q + 4q^{2} - 4q^{3} + 112q^{4} + 22q^{5} + 8q^{6} + 194q^{7} + 64q^{8} + 221q^{9} + 200q^{10} + 320q^{11} - 64q^{12} + 1338q^{13} + 80q^{14} + 1868q^{15} + 1792q^{16} - 1316q^{17} + 628q^{18} - 361q^{19} + 352q^{20} - 2940q^{21} - 1424q^{22} - 3430q^{23} + 128q^{24} - 5899q^{25} - 2144q^{26} + 3464q^{27} + 3104q^{28} - 5566q^{29} - 7792q^{30} - 11164q^{31} + 1024q^{32} - 6800q^{33} + 3080q^{34} - 7788q^{35} + 3536q^{36} + 10038q^{37} - 4332q^{38} + 5046q^{39} + 3200q^{40} + 25722q^{41} + 8632q^{42} - 18208q^{43} + 5120q^{44} + 10810q^{45} + 688q^{46} - 2056q^{47} - 1024q^{48} - 2563q^{49} - 16260q^{50} + 47816q^{51} + 21408q^{52} + 53370q^{53} - 22072q^{54} + 60424q^{55} + 1280q^{56} - 6498q^{57} - 25376q^{58} - 31028q^{59} + 29888q^{60} + 102938q^{61} - 46192q^{62} + 22580q^{63} + 28672q^{64} - 108904q^{65} - 56400q^{66} - 29516q^{67} - 21056q^{68} - 121196q^{69} + 65232q^{70} + 5892q^{71} + 10048q^{72} - 115600q^{73} + 20104q^{74} - 136620q^{75} - 5776q^{76} - 7592q^{77} - 2800q^{78} + 202780q^{79} + 5632q^{80} - 350161q^{81} - 39768q^{82} + 79616q^{83} - 47040q^{84} - 101964q^{85} - 83968q^{86} - 214610q^{87} - 22784q^{88} + 202750q^{89} + 85720q^{90} + 481736q^{91} - 54880q^{92} + 289700q^{93} + 61312q^{94} - 72922q^{95} + 2048q^{96} + 16390q^{97} + 243076q^{98} + 324236q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(38))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 19
38.6.a.a $$1$$ $$6.095$$ $$\Q$$ None $$-4$$ $$-6$$ $$31$$ $$-27$$ $$+$$ $$+$$ $$q-4q^{2}-6q^{3}+2^{4}q^{4}+31q^{5}+24q^{6}+\cdots$$
38.6.a.b $$1$$ $$6.095$$ $$\Q$$ None $$4$$ $$-14$$ $$-45$$ $$-121$$ $$-$$ $$-$$ $$q+4q^{2}-14q^{3}+2^{4}q^{4}-45q^{5}-56q^{6}+\cdots$$
38.6.a.c $$2$$ $$6.095$$ $$\Q(\sqrt{1441})$$ None $$-8$$ $$3$$ $$-45$$ $$114$$ $$+$$ $$-$$ $$q-4q^{2}+(2-\beta )q^{3}+2^{4}q^{4}+(-21+\cdots)q^{5}+\cdots$$
38.6.a.d $$3$$ $$6.095$$ $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ None $$12$$ $$13$$ $$81$$ $$228$$ $$-$$ $$+$$ $$q+4q^{2}+(4+\beta _{1})q^{3}+2^{4}q^{4}+(3^{3}-\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(38))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(38)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 2}$$