Properties

Label 50.2.a
Level 50
Weight 2
Character orbit a
Rep. character \(\chi_{50}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 15
Trace bound 2

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Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 50.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(15\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(50))\).

Total New Old
Modular forms 13 2 11
Cusp forms 2 2 0
Eisenstein series 11 0 11

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

\(2\)\(5\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2q + 2q^{4} - 2q^{6} - 4q^{9} + O(q^{10}) \) \( 2q + 2q^{4} - 2q^{6} - 4q^{9} - 6q^{11} - 4q^{14} + 2q^{16} + 10q^{19} + 4q^{21} - 2q^{24} + 8q^{26} + 4q^{31} + 6q^{34} - 4q^{36} - 8q^{39} - 6q^{41} - 6q^{44} - 12q^{46} - 6q^{49} - 6q^{51} + 10q^{54} - 4q^{56} + 4q^{61} + 2q^{64} + 6q^{66} + 12q^{69} + 24q^{71} - 4q^{74} + 10q^{76} - 20q^{79} + 2q^{81} + 4q^{84} + 8q^{86} + 30q^{89} - 16q^{91} - 24q^{94} - 2q^{96} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(50))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
50.2.a.a \(1\) \(0.399\) \(\Q\) None \(-1\) \(1\) \(0\) \(2\) \(+\) \(-\) \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
50.2.a.b \(1\) \(0.399\) \(\Q\) None \(1\) \(-1\) \(0\) \(-2\) \(-\) \(+\) \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)

Hecke Characteristic Polynomials

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