# Properties

 Label 38.8.a Level $38$ Weight $8$ Character orbit 38.a Rep. character $\chi_{38}(1,\cdot)$ Character field $\Q$ Dimension $11$ Newform subspaces $5$ Sturm bound $40$ Trace bound $3$

# Learn more about

## Defining parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 38.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$40$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(38))$$.

Total New Old
Modular forms 37 11 26
Cusp forms 33 11 22
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.
$$+$$$$+$$$$+$$$$3$$
$$+$$$$-$$$$-$$$$2$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$7$$
Minus space$$-$$$$4$$

## Trace form

 $$11q + 8q^{2} - 52q^{3} + 704q^{4} + 422q^{5} - 496q^{6} - 1742q^{7} + 512q^{8} + 17597q^{9} + O(q^{10})$$ $$11q + 8q^{2} - 52q^{3} + 704q^{4} + 422q^{5} - 496q^{6} - 1742q^{7} + 512q^{8} + 17597q^{9} - 5360q^{10} - 2656q^{11} - 3328q^{12} - 13710q^{13} + 17888q^{14} + 14396q^{15} + 45056q^{16} + 7896q^{17} + 24616q^{18} + 6859q^{19} + 27008q^{20} + 143388q^{21} + 52832q^{22} - 147286q^{23} - 31744q^{24} + 481849q^{25} - 18656q^{26} - 191080q^{27} - 111488q^{28} + 229594q^{29} - 245152q^{30} - 7148q^{31} + 32768q^{32} + 122272q^{33} - 213104q^{34} - 530124q^{35} + 1126208q^{36} - 1027866q^{37} + 164616q^{38} - 1620546q^{39} - 343040q^{40} - 278006q^{41} + 559312q^{42} + 614464q^{43} - 169984q^{44} - 1635854q^{45} - 88288q^{46} - 1921600q^{47} - 212992q^{48} + 2499061q^{49} + 504888q^{50} + 1993400q^{51} - 877440q^{52} - 1438894q^{53} - 3196720q^{54} - 79000q^{55} + 1144832q^{56} + 370386q^{57} - 3994912q^{58} - 2570676q^{59} + 921344q^{60} - 3286670q^{61} + 7807968q^{62} + 12567356q^{63} + 2883584q^{64} - 4343408q^{65} + 5952864q^{66} + 7302740q^{67} + 505344q^{68} - 3802148q^{69} - 15502176q^{70} + 6003140q^{71} + 1575424q^{72} + 10005364q^{73} + 8300304q^{74} - 4118364q^{75} + 438976q^{76} + 3133152q^{77} - 790240q^{78} + 1989692q^{79} + 1728512q^{80} + 14974883q^{81} - 9223920q^{82} - 24955968q^{83} + 9176832q^{84} + 18133548q^{85} - 19710336q^{86} + 8284678q^{87} + 3381248q^{88} + 12483990q^{89} - 19958288q^{90} - 15873176q^{91} - 9426304q^{92} - 60880612q^{93} + 22559168q^{94} - 7668362q^{95} - 2031616q^{96} - 4839490q^{97} - 12249592q^{98} - 67805044q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(38))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 19
38.8.a.a $$1$$ $$11.871$$ $$\Q$$ None $$-8$$ $$77$$ $$440$$ $$951$$ $$+$$ $$+$$ $$q-8q^{2}+77q^{3}+2^{6}q^{4}+440q^{5}+\cdots$$
38.8.a.b $$2$$ $$11.871$$ $$\Q(\sqrt{2737})$$ None $$-16$$ $$-61$$ $$175$$ $$-2592$$ $$+$$ $$+$$ $$q-8q^{2}+(-30-\beta )q^{3}+2^{6}q^{4}+(95+\cdots)q^{5}+\cdots$$
38.8.a.c $$2$$ $$11.871$$ $$\Q(\sqrt{17953})$$ None $$-16$$ $$-11$$ $$-69$$ $$-348$$ $$+$$ $$-$$ $$q-8q^{2}+(-5-\beta )q^{3}+2^{6}q^{4}+(-6^{2}+\cdots)q^{5}+\cdots$$
38.8.a.d $$2$$ $$11.871$$ $$\Q(\sqrt{633})$$ None $$16$$ $$-69$$ $$155$$ $$-2238$$ $$-$$ $$+$$ $$q+8q^{2}+(-33-3\beta )q^{3}+2^{6}q^{4}+(62+\cdots)q^{5}+\cdots$$
38.8.a.e $$4$$ $$11.871$$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$32$$ $$12$$ $$-279$$ $$2485$$ $$-$$ $$-$$ $$q+8q^{2}+(3-\beta _{1})q^{3}+2^{6}q^{4}+(-70+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_0(38))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_0(38)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 2}$$