Properties

Label 49.2.a
Level 49
Weight 2
Character orbit a
Rep. character \(\chi_{49}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 9
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(9\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(49))\).

Total New Old
Modular forms 8 6 2
Cusp forms 1 1 0
Eisenstein series 7 5 2

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

\(7\)Dim.
\(-\)\(1\)

Trace form

\( q + q^{2} - q^{4} - 3q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} - q^{4} - 3q^{8} - 3q^{9} + 4q^{11} - q^{16} - 3q^{18} + 4q^{22} + 8q^{23} - 5q^{25} + 2q^{29} + 5q^{32} + 3q^{36} - 6q^{37} - 12q^{43} - 4q^{44} + 8q^{46} - 5q^{50} - 10q^{53} + 2q^{58} + 7q^{64} + 4q^{67} + 16q^{71} + 9q^{72} - 6q^{74} + 8q^{79} + 9q^{81} - 12q^{86} - 12q^{88} - 8q^{92} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7
49.2.a.a \(1\) \(0.391\) \(\Q\) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) \(-\) \(q+q^{2}-q^{4}-3q^{8}-3q^{9}+4q^{11}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 + 5 T^{2} \)
$7$ 1
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 + 17 T^{2} \)
$19$ \( 1 + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 - 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 6 T + 37 T^{2} \)
$41$ \( 1 + 41 T^{2} \)
$43$ \( 1 + 12 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 + 10 T + 53 T^{2} \)
$59$ \( 1 + 59 T^{2} \)
$61$ \( 1 + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 - 16 T + 71 T^{2} \)
$73$ \( 1 + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 + 97 T^{2} \)
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