Properties

Label 37.2.a
Level 37
Weight 2
Character orbit a
Rep. character \(\chi_{37}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 6
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(37))\).

Total New Old
Modular forms 3 3 0
Cusp forms 2 2 0
Eisenstein series 1 1 0

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(37\)Dim.
\(+\)\(1\)
\(-\)\(1\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 37
37.2.a.a \(1\) \(0.295\) \(\Q\) None \(-2\) \(-3\) \(-2\) \(-1\) \(+\) \(q-2q^{2}-3q^{3}+2q^{4}-2q^{5}+6q^{6}+\cdots\)
37.2.a.b \(1\) \(0.295\) \(\Q\) None \(0\) \(1\) \(0\) \(-1\) \(-\) \(q+q^{3}-2q^{4}-q^{7}-2q^{9}+3q^{11}+\cdots\)

Hecke characteristic polynomials

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