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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 19
38.8.a.a 38.a 1.a $1$ $11.871$ \(\Q\) None \(-8\) \(77\) \(440\) \(951\) $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+77q^{3}+2^{6}q^{4}+440q^{5}+\cdots\)
38.8.a.b 38.a 1.a $2$ $11.871$ \(\Q(\sqrt{2737}) \) None \(-16\) \(-61\) \(175\) \(-2592\) $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-30-\beta )q^{3}+2^{6}q^{4}+(95+\cdots)q^{5}+\cdots\)
38.8.a.c 38.a 1.a $2$ $11.871$ \(\Q(\sqrt{17953}) \) None \(-16\) \(-11\) \(-69\) \(-348\) $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+(-5-\beta )q^{3}+2^{6}q^{4}+(-6^{2}+\cdots)q^{5}+\cdots\)
38.8.a.d 38.a 1.a $2$ $11.871$ \(\Q(\sqrt{633}) \) None \(16\) \(-69\) \(155\) \(-2238\) $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+(-33-3\beta )q^{3}+2^{6}q^{4}+(62+\cdots)q^{5}+\cdots\)
38.8.a.e 38.a 1.a $4$ $11.871$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32\) \(12\) \(-279\) \(2485\) $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+(3-\beta _{1})q^{3}+2^{6}q^{4}+(-70+\cdots)q^{5}+\cdots\)

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

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