Properties

Label 42.3.g
Level $42$
Weight $3$
Character orbit 42.g
Rep. character $\chi_{42}(19,\cdot)$
Character field $\Q(\zeta_{6})$
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.g (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
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 40 4 36
Cusp forms 24 4 20
Eisenstein series 16 0 16

Trace form

\( 4q + 6q^{3} - 4q^{4} + 12q^{5} - 10q^{7} + 6q^{9} + O(q^{10}) \) \( 4q + 6q^{3} - 4q^{4} + 12q^{5} - 10q^{7} + 6q^{9} - 24q^{10} - 12q^{11} - 12q^{12} + 24q^{14} + 24q^{15} - 8q^{16} - 48q^{17} - 42q^{19} + 24q^{23} + 22q^{25} + 96q^{26} + 40q^{28} - 24q^{30} + 102q^{31} - 36q^{33} - 108q^{35} - 24q^{36} + 22q^{37} + 24q^{38} + 6q^{39} + 48q^{40} + 24q^{42} + 28q^{43} - 24q^{44} + 36q^{45} - 72q^{46} - 132q^{47} - 2q^{49} - 192q^{50} - 48q^{51} + 12q^{52} + 120q^{53} - 48q^{56} - 84q^{57} - 96q^{58} - 24q^{59} - 24q^{60} - 72q^{61} + 30q^{63} + 32q^{64} + 180q^{65} + 110q^{67} + 96q^{68} + 96q^{70} + 312q^{71} - 66q^{73} - 48q^{74} + 66q^{75} + 120q^{77} + 192q^{78} - 10q^{79} - 48q^{80} - 18q^{81} + 48q^{82} + 60q^{84} - 288q^{85} - 24q^{86} - 72q^{89} - 222q^{91} - 96q^{92} + 102q^{93} - 24q^{94} - 132q^{95} - 72q^{99} + O(q^{100}) \)

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.g.a \(4\) \(1.144\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(6\) \(12\) \(-10\) \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{3}+2\beta _{2}q^{4}+(2+\cdots)q^{5}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} + 4 T^{4} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )^{2} \)
$5$ \( 1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 11400 T^{5} + 53750 T^{6} - 187500 T^{7} + 390625 T^{8} \)
$7$ \( 1 + 10 T + 51 T^{2} + 490 T^{3} + 2401 T^{4} \)
$11$ \( ( 1 + 6 T - 85 T^{2} + 726 T^{3} + 14641 T^{4} )^{2} \)
$13$ \( 1 + 98 T^{2} + 54915 T^{4} + 2798978 T^{6} + 815730721 T^{8} \)
$17$ \( 1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 10348512 T^{5} + 126450794 T^{6} + 1158603312 T^{7} + 6975757441 T^{8} \)
$19$ \( 1 + 42 T + 1433 T^{2} + 35490 T^{3} + 795972 T^{4} + 12811890 T^{5} + 186749993 T^{6} + 1975927002 T^{7} + 16983563041 T^{8} \)
$23$ \( 1 - 24 T + 22 T^{2} + 12096 T^{3} - 277629 T^{4} + 6398784 T^{5} + 6156502 T^{6} - 3552861336 T^{7} + 78310985281 T^{8} \)
$29$ \( ( 1 + 530 T^{2} + 707281 T^{4} )^{2} \)
$31$ \( 1 - 102 T + 6041 T^{2} - 262446 T^{3} + 9029556 T^{4} - 252210606 T^{5} + 5578990361 T^{6} - 90525375462 T^{7} + 852891037441 T^{8} \)
$37$ \( 1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 5029706 T^{5} - 3911374007 T^{6} - 56445980998 T^{7} + 3512479453921 T^{8} \)
$41$ \( 1 - 2476 T^{2} + 6405414 T^{4} - 6996584236 T^{6} + 7984925229121 T^{8} \)
$43$ \( ( 1 - 14 T + 3675 T^{2} - 25886 T^{3} + 3418801 T^{4} )^{2} \)
$47$ \( 1 + 132 T + 11654 T^{2} + 771672 T^{3} + 42125907 T^{4} + 1704623448 T^{5} + 56867802374 T^{6} + 1422856423428 T^{7} + 23811286661761 T^{8} \)
$53$ \( 1 - 120 T + 5830 T^{2} - 354240 T^{3} + 25104819 T^{4} - 995060160 T^{5} + 46001504230 T^{6} - 2659723335480 T^{7} + 62259690411361 T^{8} \)
$59$ \( 1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 487395696 T^{5} + 73019217386 T^{6} + 1012332807384 T^{7} + 146830437604321 T^{8} \)
$61$ \( 1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 2006660880 T^{5} + 127630962338 T^{6} + 3709466953992 T^{7} + 191707312997281 T^{8} \)
$67$ \( 1 - 110 T + 3625 T^{2} + 55330 T^{3} - 2642396 T^{4} + 248376370 T^{5} + 73047813625 T^{6} - 9950422038590 T^{7} + 406067677556641 T^{8} \)
$71$ \( ( 1 - 156 T + 12638 T^{2} - 786396 T^{3} + 25411681 T^{4} )^{2} \)
$73$ \( 1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 499785594 T^{5} + 81588146393 T^{6} + 9988058935074 T^{7} + 806460091894081 T^{8} \)
$79$ \( 1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 542155670 T^{5} - 143920549295 T^{6} + 2430874555210 T^{7} + 1517108809906561 T^{8} \)
$83$ \( 1 - 9628 T^{2} + 107694438 T^{4} - 456928714588 T^{6} + 2252292232139041 T^{8} \)
$89$ \( ( 1 + 36 T + 8353 T^{2} + 285156 T^{3} + 62742241 T^{4} )^{2} \)
$97$ \( 1 - 36580 T^{2} + 511416774 T^{4} - 3238401098980 T^{6} + 7837433594376961 T^{8} \)
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