Properties

Label 50.2.b
Level 50
Weight 2
Character orbit b
Rep. character \(\chi_{50}(49,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 1
Sturm bound 15
Trace bound 0

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Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 50.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(15\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(50, [\chi])\).

Total New Old
Modular forms 14 2 12
Cusp forms 2 2 0
Eisenstein series 12 0 12

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 \(S_{2}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 - 5 T^{2} + 9 T^{4} \)
$5$ 1
$7$ \( 1 - 10 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 3 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( 1 - 25 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 5 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 10 T^{2} + 529 T^{4} \)
$29$ \( ( 1 + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 2 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 + 3 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 50 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 35 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 - 12 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 25 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 10 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 85 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 15 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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