Properties

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

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Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 38.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(38))\).

Total New Old
Modular forms 7 2 5
Cusp forms 4 2 2
Eisenstein series 3 0 3

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.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2 q + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{7} - 4 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{7} - 4 q^{9} - 4 q^{10} - 4 q^{11} + 4 q^{13} + 4 q^{14} + 4 q^{15} + 2 q^{16} + 6 q^{17} - 4 q^{20} - 4 q^{21} + 8 q^{22} + 2 q^{23} - 2 q^{24} + 6 q^{25} - 6 q^{26} + 2 q^{28} + 4 q^{29} + 4 q^{30} - 12 q^{31} - 8 q^{33} - 12 q^{35} - 4 q^{36} - 2 q^{38} + 6 q^{39} - 4 q^{40} - 8 q^{41} - 2 q^{42} + 12 q^{43} - 4 q^{44} + 8 q^{45} - 4 q^{46} + 8 q^{47} - 4 q^{49} + 16 q^{50} + 4 q^{52} - 4 q^{53} + 10 q^{54} - 8 q^{55} + 4 q^{56} + 2 q^{57} - 14 q^{58} + 24 q^{59} + 4 q^{60} - 8 q^{61} - 4 q^{62} - 4 q^{63} + 2 q^{64} + 4 q^{65} + 4 q^{66} + 8 q^{67} + 6 q^{68} + 4 q^{69} - 12 q^{70} - 4 q^{71} + 2 q^{73} - 4 q^{74} - 16 q^{75} + 12 q^{77} - 4 q^{78} - 20 q^{79} - 4 q^{80} + 2 q^{81} - 8 q^{82} - 12 q^{83} - 4 q^{84} - 12 q^{85} - 4 q^{86} + 14 q^{87} + 8 q^{88} - 12 q^{89} + 8 q^{90} - 8 q^{91} + 2 q^{92} + 4 q^{93} + 8 q^{94} + 4 q^{95} - 2 q^{96} - 12 q^{97} + 8 q^{98} + 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(38))\) 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
38.2.a.a 38.a 1.a $1$ $0.303$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
38.2.a.b 38.a 1.a $1$ $0.303$ \(\Q\) None \(1\) \(-1\) \(-4\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(38))\) into lower level spaces

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