Properties

Label 38.4.a
Level $38$
Weight $4$
Character orbit 38.a
Rep. character $\chi_{38}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $3$
Sturm bound $20$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(38))\).

Total New Old
Modular forms 17 5 12
Cusp forms 13 5 8
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.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(1\)

Trace form

\( 5 q - 2 q^{2} + 8 q^{3} + 20 q^{4} - 8 q^{5} + 20 q^{6} + 8 q^{7} - 8 q^{8} + 35 q^{9} + O(q^{10}) \) \( 5 q - 2 q^{2} + 8 q^{3} + 20 q^{4} - 8 q^{5} + 20 q^{6} + 8 q^{7} - 8 q^{8} + 35 q^{9} - 20 q^{10} + 50 q^{11} + 32 q^{12} - 22 q^{13} - 88 q^{14} - 304 q^{15} + 80 q^{16} - 62 q^{17} + 22 q^{18} + 19 q^{19} - 32 q^{20} - 48 q^{21} - 168 q^{22} - 154 q^{23} + 80 q^{24} + 229 q^{25} + 112 q^{26} + 368 q^{27} + 32 q^{28} - 226 q^{29} + 8 q^{30} + 228 q^{31} - 32 q^{32} + 28 q^{33} - 196 q^{34} + 906 q^{35} + 140 q^{36} + 346 q^{37} + 114 q^{38} + 42 q^{39} - 80 q^{40} - 1098 q^{41} - 812 q^{42} + 906 q^{43} + 200 q^{44} - 884 q^{45} + 600 q^{46} - 10 q^{47} + 128 q^{48} + 1705 q^{49} + 18 q^{50} - 1924 q^{51} - 88 q^{52} - 1254 q^{53} + 20 q^{54} - 850 q^{55} - 352 q^{56} + 114 q^{57} + 464 q^{58} - 416 q^{59} - 1216 q^{60} - 380 q^{61} + 392 q^{62} + 278 q^{63} + 320 q^{64} + 272 q^{65} - 216 q^{66} - 572 q^{67} - 248 q^{68} + 2020 q^{69} + 264 q^{70} + 1248 q^{71} + 88 q^{72} + 14 q^{73} + 76 q^{74} - 84 q^{75} + 76 q^{76} - 1298 q^{77} + 2120 q^{78} + 756 q^{79} - 128 q^{80} - 139 q^{81} + 1996 q^{82} + 1544 q^{83} - 192 q^{84} + 2358 q^{85} + 896 q^{86} + 2074 q^{87} - 672 q^{88} - 1706 q^{89} - 2588 q^{90} - 688 q^{91} - 616 q^{92} - 2140 q^{93} - 1536 q^{94} - 532 q^{95} + 320 q^{96} + 854 q^{97} - 3170 q^{98} - 826 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(38))\) 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
38.4.a.a 38.a 1.a $1$ $2.242$ \(\Q\) None \(-2\) \(-2\) \(-9\) \(-31\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}-9q^{5}+4q^{6}+\cdots\)
38.4.a.b 38.a 1.a $2$ $2.242$ \(\Q(\sqrt{177}) \) None \(-4\) \(1\) \(10\) \(57\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta )q^{3}+4q^{4}+(4+2\beta )q^{5}+\cdots\)
38.4.a.c 38.a 1.a $2$ $2.242$ \(\Q(\sqrt{73}) \) None \(4\) \(9\) \(-9\) \(-18\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(5-\beta )q^{3}+4q^{4}+(-6+3\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(38))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(38)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)