Properties

Label 42.3.b
Level $42$
Weight $3$
Character orbit 42.b
Rep. character $\chi_{42}(29,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $1$
Sturm bound $24$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 42.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(42, [\chi])\).

Total New Old
Modular forms 20 4 16
Cusp forms 12 4 8
Eisenstein series 8 0 8

Trace form

\( 4q - 8q^{4} - 8q^{6} + 20q^{9} + O(q^{10}) \) \( 4q - 8q^{4} - 8q^{6} + 20q^{9} - 16q^{10} + 40q^{13} - 16q^{15} + 16q^{16} - 64q^{19} - 28q^{21} + 40q^{22} + 16q^{24} + 12q^{25} + 56q^{30} - 128q^{31} + 40q^{33} + 16q^{34} - 40q^{36} + 80q^{37} - 112q^{39} + 32q^{40} + 80q^{43} + 112q^{45} - 8q^{46} + 28q^{49} + 16q^{51} - 80q^{52} - 152q^{54} - 32q^{55} - 160q^{58} + 32q^{60} - 56q^{61} - 32q^{64} + 112q^{66} + 240q^{67} - 8q^{69} - 56q^{70} + 120q^{73} + 224q^{75} + 128q^{76} - 80q^{78} - 128q^{79} - 124q^{81} + 240q^{82} + 56q^{84} - 248q^{85} - 160q^{87} - 80q^{88} - 80q^{90} + 112q^{91} - 280q^{93} + 48q^{94} - 32q^{96} - 360q^{97} + 224q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
42.3.b.a \(4\) \(1.144\) \(\Q(\sqrt{-2}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}-2q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 2 T^{2} )^{2} \)
$3$ \( 1 - 10 T^{2} + 81 T^{4} \)
$5$ \( 1 - 56 T^{2} + 1586 T^{4} - 35000 T^{6} + 390625 T^{8} \)
$7$ \( ( 1 - 7 T^{2} )^{2} \)
$11$ \( 1 - 272 T^{2} + 36578 T^{4} - 3982352 T^{6} + 214358881 T^{8} \)
$13$ \( ( 1 - 20 T + 326 T^{2} - 3380 T^{3} + 28561 T^{4} )^{2} \)
$17$ \( 1 - 440 T^{2} + 204242 T^{4} - 36749240 T^{6} + 6975757441 T^{8} \)
$19$ \( ( 1 + 16 T + 361 T^{2} )^{4} \)
$23$ \( 1 + 688 T^{2} + 666818 T^{4} + 192530608 T^{6} + 78310985281 T^{8} \)
$29$ \( 1 - 1652 T^{2} + 1917638 T^{4} - 1168428212 T^{6} + 500246412961 T^{8} \)
$31$ \( ( 1 + 64 T + 2246 T^{2} + 61504 T^{3} + 923521 T^{4} )^{2} \)
$37$ \( ( 1 - 20 T + 1369 T^{2} )^{4} \)
$41$ \( 1 - 856 T^{2} - 2330094 T^{4} - 2418851416 T^{6} + 7984925229121 T^{8} \)
$43$ \( ( 1 - 40 T + 3090 T^{2} - 73960 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( ( 1 - 4346 T^{2} + 4879681 T^{4} )^{2} \)
$53$ \( ( 1 - 3026 T^{2} + 7890481 T^{4} )^{2} \)
$59$ \( 1 - 10532 T^{2} + 49098278 T^{4} - 127620046052 T^{6} + 146830437604321 T^{8} \)
$61$ \( ( 1 + 28 T + 4838 T^{2} + 104188 T^{3} + 13845841 T^{4} )^{2} \)
$67$ \( ( 1 - 120 T + 12466 T^{2} - 538680 T^{3} + 20151121 T^{4} )^{2} \)
$71$ \( 1 - 12176 T^{2} + 74167106 T^{4} - 309412627856 T^{6} + 645753531245761 T^{8} \)
$73$ \( ( 1 - 60 T + 9766 T^{2} - 319740 T^{3} + 28398241 T^{4} )^{2} \)
$79$ \( ( 1 + 64 T + 10706 T^{2} + 399424 T^{3} + 38950081 T^{4} )^{2} \)
$83$ \( 1 - 6356 T^{2} - 6983674 T^{4} - 301645088276 T^{6} + 2252292232139041 T^{8} \)
$89$ \( 1 - 27832 T^{2} + 318232338 T^{4} - 1746242051512 T^{6} + 3936588805702081 T^{8} \)
$97$ \( ( 1 + 180 T + 26470 T^{2} + 1693620 T^{3} + 88529281 T^{4} )^{2} \)
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