Properties

Label 42.2.e
Level 42
Weight 2
Character orbit e
Rep. character \(\chi_{42}(25,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 4
Newform subspaces 2
Sturm bound 16
Trace bound 2

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Defining parameters

Level: \( N \) = \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 42.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

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

\( 4q - 2q^{4} - 4q^{5} - 4q^{6} + 6q^{7} - 2q^{9} + O(q^{10}) \) \( 4q - 2q^{4} - 4q^{5} - 4q^{6} + 6q^{7} - 2q^{9} + 2q^{10} - 8q^{11} - 8q^{13} + 8q^{14} + 4q^{15} - 2q^{16} + 4q^{17} - 4q^{19} + 8q^{20} + 4q^{21} + 4q^{22} + 4q^{23} + 2q^{24} - 4q^{26} - 6q^{28} + 8q^{29} + 4q^{30} - 2q^{31} + 2q^{33} - 8q^{34} - 8q^{35} + 4q^{36} - 4q^{37} - 12q^{38} + 4q^{39} + 2q^{40} - 6q^{42} - 16q^{43} - 8q^{44} - 4q^{45} + 4q^{46} + 12q^{47} - 2q^{49} - 16q^{50} - 4q^{51} + 4q^{52} + 12q^{53} + 2q^{54} + 28q^{55} - 4q^{56} - 24q^{57} + 14q^{58} + 8q^{59} - 2q^{60} + 16q^{61} + 8q^{62} + 4q^{64} + 12q^{65} + 8q^{66} + 12q^{67} + 4q^{68} + 8q^{69} - 2q^{70} - 8q^{71} - 12q^{73} + 12q^{74} - 8q^{75} + 8q^{76} - 28q^{77} + 8q^{78} - 2q^{79} - 4q^{80} - 2q^{81} - 32q^{83} - 8q^{84} - 8q^{85} - 12q^{86} - 14q^{87} - 2q^{88} - 4q^{90} - 20q^{91} - 8q^{92} + 4q^{93} + 4q^{95} + 2q^{96} + 12q^{97} + 24q^{98} + 16q^{99} + O(q^{100}) \)

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.e.a \(2\) \(0.335\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(-1\) \(1\) \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{3}-\zeta_{6}q^{4}+(-1+\cdots)q^{5}+\cdots\)
42.2.e.b \(2\) \(0.335\) \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(-3\) \(5\) \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}+(-3+\cdots)q^{5}+\cdots\)

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

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} \))(\( 1 - T + T^{2} \))
$3$ (\( 1 - T + T^{2} \))(\( 1 + T + T^{2} \))
$5$ (\( 1 + T - 4 T^{2} + 5 T^{3} + 25 T^{4} \))(\( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 - 5 T + 7 T^{2} \))
$11$ (\( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} \))(\( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \))
$13$ (\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 + 4 T + 13 T^{2} )^{2} \))
$17$ (\( 1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4} \))(\( 1 - 17 T^{2} + 289 T^{4} \))
$19$ (\( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \))(\( 1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 4 T - 7 T^{2} - 92 T^{3} + 529 T^{4} \))(\( 1 - 23 T^{2} + 529 T^{4} \))
$29$ (\( ( 1 + 5 T + 29 T^{2} )^{2} \))(\( ( 1 - 9 T + 29 T^{2} )^{2} \))
$31$ (\( 1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4} \))(\( 1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 4 T - 21 T^{2} - 148 T^{3} + 1369 T^{4} \))(\( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 + 41 T^{2} )^{2} \))
$43$ (\( ( 1 - 2 T + 43 T^{2} )^{2} \))(\( ( 1 + 10 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))(\( 1 - 6 T - 11 T^{2} - 282 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 9 T + 28 T^{2} - 477 T^{3} + 2809 T^{4} \))(\( 1 - 3 T - 44 T^{2} - 159 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 11 T + 62 T^{2} - 649 T^{3} + 3481 T^{4} \))(\( 1 + 3 T - 50 T^{2} + 177 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4} \))(\( 1 - 10 T + 39 T^{2} - 610 T^{3} + 3721 T^{4} \))
$67$ (\( 1 - 2 T - 63 T^{2} - 134 T^{3} + 4489 T^{4} \))(\( 1 - 10 T + 33 T^{2} - 670 T^{3} + 4489 T^{4} \))
$71$ (\( ( 1 - 2 T + 71 T^{2} )^{2} \))(\( ( 1 + 6 T + 71 T^{2} )^{2} \))
$73$ (\( ( 1 - 7 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 + 2 T - 69 T^{2} + 146 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 3 T - 70 T^{2} + 237 T^{3} + 6241 T^{4} \))(\( 1 - T - 78 T^{2} - 79 T^{3} + 6241 T^{4} \))
$83$ (\( ( 1 + 7 T + 83 T^{2} )^{2} \))(\( ( 1 + 9 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))
$97$ (\( ( 1 - 7 T + 97 T^{2} )^{2} \))(\( ( 1 + T + 97 T^{2} )^{2} \))
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