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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 $2$

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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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(38))\) 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
38.6.a.a 38.a 1.a $1$ $6.095$ \(\Q\) None \(-4\) \(-6\) \(31\) \(-27\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-6q^{3}+2^{4}q^{4}+31q^{5}+24q^{6}+\cdots\)
38.6.a.b 38.a 1.a $1$ $6.095$ \(\Q\) None \(4\) \(-14\) \(-45\) \(-121\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-14q^{3}+2^{4}q^{4}-45q^{5}-56q^{6}+\cdots\)
38.6.a.c 38.a 1.a $2$ $6.095$ \(\Q(\sqrt{1441}) \) None \(-8\) \(3\) \(-45\) \(114\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(2-\beta )q^{3}+2^{4}q^{4}+(-21+\cdots)q^{5}+\cdots\)
38.6.a.d 38.a 1.a $3$ $6.095$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(12\) \(13\) \(81\) \(228\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(4+\beta _{1})q^{3}+2^{4}q^{4}+(3^{3}-\beta _{1}+\cdots)q^{5}+\cdots\)

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

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