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 $3$

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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

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