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Properties

Label 38.2.a
Level 38
Weight 2
Character orbit 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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(38))\) into newform subspaces

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))
$13$ (\( 1 - 5 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))
$19$ (\( 1 - T \))(\( 1 + T \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))
$29$ (\( 1 - 9 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))
$53$ (\( 1 + 3 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 - 15 T + 59 T^{2} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))
$67$ (\( 1 - 5 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))
$73$ (\( 1 + 7 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))
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