Properties

Label 42.2.f
Level 42
Weight 2
Character orbit f
Rep. character \(\chi_{42}(5,\cdot)\)
Character field \(\Q(\zeta_{6})\)
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.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 21 \)
Character field: \(\Q(\zeta_{6})\)
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 24 4 20
Cusp forms 8 4 4
Eisenstein series 16 0 16

Trace form

\( 4q + 2q^{4} - 10q^{7} - 6q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 10q^{7} - 6q^{9} - 6q^{10} + 12q^{15} - 2q^{16} + 12q^{19} + 12q^{22} + 6q^{24} + 4q^{25} - 2q^{28} - 6q^{31} - 18q^{33} - 12q^{36} + 4q^{37} - 6q^{40} - 6q^{42} - 32q^{43} - 12q^{46} + 22q^{49} + 12q^{51} + 12q^{52} + 18q^{54} + 6q^{58} + 6q^{60} + 24q^{63} - 4q^{64} - 4q^{67} + 18q^{70} + 24q^{73} - 24q^{78} + 2q^{79} - 18q^{81} - 24q^{82} - 24q^{85} - 18q^{87} + 6q^{88} - 12q^{91} - 24q^{94} + 6q^{96} + O(q^{100}) \)

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.f.a \(4\) \(0.335\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-10\) \(q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 + 3 T^{2} + 9 T^{4} \)
$5$ \( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} \)
$7$ \( ( 1 + 5 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 13 T^{2} + 48 T^{4} + 1573 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 22 T^{2} + 195 T^{4} - 6358 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + T + 19 T^{2} )^{2} \)
$23$ \( 1 + 10 T^{2} - 429 T^{4} + 5290 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 49 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 2 T - 33 T^{2} - 74 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 + 34 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 8 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 46 T^{2} - 93 T^{4} - 101614 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 115 T^{2} + 9744 T^{4} - 400315 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 61 T^{2} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 2 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 12 T + 121 T^{2} - 876 T^{3} + 5329 T^{4} )^{2} \)
$79$ \( ( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 91 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 70 T^{2} - 3021 T^{4} - 554470 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 19 T + 97 T^{2} )^{2}( 1 + 19 T + 97 T^{2} )^{2} \)
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