Properties

Label 38.2.a
Level 38
Weight 2
Character orbit a
Rep. character \(\chi_{38}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newforms 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: \(\Q\)
Newforms: \( 2 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(38))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19
38.2.a.a \(1\) \(0.303\) \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
38.2.a.b \(1\) \(0.303\) \(\Q\) None \(1\) \(-1\) \(-4\) \(3\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)

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

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