Properties

Label 42.2.d
Level 42
Weight 2
Character orbit d
Rep. character \(\chi_{42}(41,\cdot)\)
Character field \(\Q\)
Dimension 4
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.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(42, [\chi])\).

Total New Old
Modular forms 12 4 8
Cusp forms 4 4 0
Eisenstein series 8 0 8

Trace form

\( 4q - 4q^{4} - 4q^{7} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{7} - 12q^{15} + 4q^{16} + 12q^{18} + 12q^{21} + 4q^{25} + 4q^{28} - 12q^{30} - 8q^{37} - 12q^{39} - 12q^{42} + 16q^{43} - 24q^{46} - 20q^{49} + 24q^{51} + 12q^{57} + 24q^{58} + 12q^{60} - 4q^{64} + 32q^{67} + 24q^{70} - 12q^{72} + 12q^{78} - 40q^{79} - 36q^{81} - 12q^{84} - 48q^{85} + 24q^{91} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
42.2.d.a \(4\) \(0.335\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(-4\) \(q-\beta _{2}q^{2}+\beta _{1}q^{3}-q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( ( 1 + 4 T^{2} + 25 T^{4} )^{2} \)
$7$ \( ( 1 + 2 T + 7 T^{2} )^{2} \)
$11$ \( ( 1 - 11 T^{2} )^{4} \)
$13$ \( ( 1 - 20 T^{2} + 169 T^{4} )^{2} \)
$17$ \( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 32 T^{2} + 361 T^{4} )^{2} \)
$23$ \( ( 1 - 10 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 - 22 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} )^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{4} \)
$41$ \( ( 1 + 58 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 4 T + 43 T^{2} )^{4} \)
$47$ \( ( 1 + 70 T^{2} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 - 70 T^{2} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 - 32 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 28 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{4} \)
$71$ \( ( 1 - 71 T^{2} )^{4} \)
$73$ \( ( 1 - 14 T + 73 T^{2} )^{2}( 1 + 14 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 10 T + 79 T^{2} )^{4} \)
$83$ \( ( 1 + 160 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 + 89 T^{2} )^{4} \)
$97$ \( ( 1 - 170 T^{2} + 9409 T^{4} )^{2} \)
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