Properties

Label 42.2.a
Level 42
Weight 2
Character orbit a
Rep. character \(\chi_{42}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 16
Trace bound 0

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 42.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(42))\).

Total New Old
Modular forms 12 1 11
Cusp forms 5 1 4
Eisenstein series 7 0 7

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

\(2\)\(3\)\(7\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} + 6q^{13} - q^{14} + 2q^{15} + q^{16} + 2q^{17} + q^{18} - 4q^{19} - 2q^{20} + q^{21} - 4q^{22} + 8q^{23} - q^{24} - q^{25} + 6q^{26} - q^{27} - q^{28} - 2q^{29} + 2q^{30} + q^{32} + 4q^{33} + 2q^{34} + 2q^{35} + q^{36} - 10q^{37} - 4q^{38} - 6q^{39} - 2q^{40} - 6q^{41} + q^{42} - 4q^{43} - 4q^{44} - 2q^{45} + 8q^{46} - q^{48} + q^{49} - q^{50} - 2q^{51} + 6q^{52} + 6q^{53} - q^{54} + 8q^{55} - q^{56} + 4q^{57} - 2q^{58} + 4q^{59} + 2q^{60} + 6q^{61} - q^{63} + q^{64} - 12q^{65} + 4q^{66} + 4q^{67} + 2q^{68} - 8q^{69} + 2q^{70} + 8q^{71} + q^{72} + 10q^{73} - 10q^{74} + q^{75} - 4q^{76} + 4q^{77} - 6q^{78} - 2q^{80} + q^{81} - 6q^{82} - 4q^{83} + q^{84} - 4q^{85} - 4q^{86} + 2q^{87} - 4q^{88} - 6q^{89} - 2q^{90} - 6q^{91} + 8q^{92} + 8q^{95} - q^{96} - 14q^{97} + q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(42))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
42.2.a.a \(1\) \(0.335\) \(\Q\) None \(1\) \(-1\) \(-2\) \(-1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(42)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T \)
$3$ \( 1 + T \)
$5$ \( 1 + 2 T + 5 T^{2} \)
$7$ \( 1 + T \)
$11$ \( 1 + 4 T + 11 T^{2} \)
$13$ \( 1 - 6 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 - 8 T + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 31 T^{2} \)
$37$ \( 1 + 10 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 - 4 T + 59 T^{2} \)
$61$ \( 1 - 6 T + 61 T^{2} \)
$67$ \( 1 - 4 T + 67 T^{2} \)
$71$ \( 1 - 8 T + 71 T^{2} \)
$73$ \( 1 - 10 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 4 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 14 T + 97 T^{2} \)
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