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Properties

Label 38.2.c
Level $38$
Weight $2$
Character orbit 38.c
Rep. character $\chi_{38}(7,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $6$
Newform subspaces $2$
Sturm bound $10$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(38, [\chi])\).

Total New Old
Modular forms 14 6 8
Cusp forms 6 6 0
Eisenstein series 8 0 8

Trace form

\( 6 q - q^{2} - q^{3} - 3 q^{4} - 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{8} - 6 q^{9} + O(q^{10}) \) \( 6 q - q^{2} - q^{3} - 3 q^{4} - 2 q^{5} + q^{6} - 4 q^{7} + 2 q^{8} - 6 q^{9} - 2 q^{10} - 2 q^{11} + 2 q^{12} - 6 q^{13} - 6 q^{14} + 14 q^{15} - 3 q^{16} + 6 q^{17} + 20 q^{18} + 5 q^{19} + 4 q^{20} - 10 q^{21} + 7 q^{22} + 4 q^{23} + q^{24} - q^{25} + 4 q^{26} - 10 q^{27} + 2 q^{28} + 2 q^{29} - 28 q^{30} - 8 q^{31} - q^{32} + 11 q^{33} - 6 q^{34} + 12 q^{35} - 6 q^{36} - 8 q^{37} - 8 q^{38} + 4 q^{39} - 2 q^{40} - 19 q^{41} - 18 q^{42} - 8 q^{43} + q^{44} + 16 q^{45} + 16 q^{46} - 14 q^{47} - q^{48} + 22 q^{49} + 22 q^{50} + 6 q^{51} - 6 q^{52} - 2 q^{53} - 5 q^{54} - 10 q^{55} + 12 q^{56} + 36 q^{57} - 4 q^{58} + 9 q^{59} + 14 q^{60} + 18 q^{61} + 8 q^{62} - 16 q^{63} + 6 q^{64} + 8 q^{65} + 17 q^{66} + 11 q^{67} - 12 q^{68} - 40 q^{69} + 12 q^{70} + 22 q^{71} - 10 q^{72} - 13 q^{73} - 16 q^{74} - 66 q^{75} - 13 q^{76} - 60 q^{77} + 2 q^{78} + 12 q^{79} - 2 q^{80} + 9 q^{81} - q^{82} + 6 q^{83} + 20 q^{84} - 16 q^{86} + 28 q^{87} - 14 q^{88} - 6 q^{89} - 8 q^{90} + 4 q^{91} + 4 q^{92} + 12 q^{93} + 28 q^{94} - 28 q^{95} - 2 q^{96} + q^{97} + 7 q^{98} + 22 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(38, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
38.2.c.a 38.c 19.c $2$ $0.303$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
38.2.c.b 38.c 19.c $4$ $0.303$ \(\Q(\sqrt{-3}, \sqrt{7})\) None \(-2\) \(0\) \(-2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots\)