Properties

Label 7.6.c
Level 7
Weight 6
Character orbit c
Rep. character \(\chi_{7}(2,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 4
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(7, [\chi])\).

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

Trace form

\( 4q - 2q^{2} + 8q^{3} - 12q^{4} + 38q^{5} - 164q^{6} - 168q^{7} + 192q^{8} + 380q^{9} + O(q^{10}) \) \( 4q - 2q^{2} + 8q^{3} - 12q^{4} + 38q^{5} - 164q^{6} - 168q^{7} + 192q^{8} + 380q^{9} + 778q^{10} - 424q^{11} + 196q^{12} - 1848q^{13} - 2674q^{14} + 1784q^{15} + 2064q^{16} + 2346q^{17} - 212q^{18} + 360q^{19} - 3416q^{20} - 1526q^{21} + 4252q^{22} + 12q^{23} - 1392q^{24} - 1872q^{25} - 1148q^{26} + 5744q^{27} + 2548q^{28} - 14104q^{29} - 5258q^{30} - 3548q^{31} + 8096q^{32} + 3398q^{33} + 14844q^{34} + 27496q^{35} - 2192q^{36} - 11090q^{37} - 20138q^{38} - 1624q^{39} - 15936q^{40} + 7000q^{41} - 3472q^{42} - 25360q^{43} - 5948q^{44} - 1300q^{45} + 5118q^{46} + 22956q^{47} + 22432q^{48} + 4900q^{49} + 59984q^{50} + 384q^{51} + 1400q^{52} - 3042q^{53} - 32546q^{54} - 50152q^{55} - 57792q^{56} - 38116q^{57} + 58852q^{58} + 65808q^{59} - 14084q^{60} + 42486q^{61} - 98724q^{62} - 4760q^{63} + 70912q^{64} + 3164q^{65} + 25894q^{66} - 42312q^{67} - 5460q^{68} + 10308q^{69} - 113050q^{70} - 4416q^{71} + 32448q^{72} + 50506q^{73} + 47370q^{74} + 35608q^{75} + 77672q^{76} + 65338q^{77} + 55048q^{78} - 9004q^{79} - 68816q^{80} - 51178q^{81} - 67732q^{82} - 208656q^{83} + 44492q^{84} - 106212q^{85} - 86776q^{86} - 80008q^{87} + 20496q^{88} + 26666q^{89} + 261304q^{90} + 135632q^{91} - 20568q^{92} - 38718q^{93} - 98034q^{94} + 198140q^{95} - 54880q^{96} + 418264q^{97} + 98686q^{98} - 133888q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(7, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
7.6.c.a \(4\) \(1.123\) \(\Q(\sqrt{-3}, \sqrt{37})\) None \(-2\) \(8\) \(38\) \(-168\) \(q+(-\beta _{1}-\beta _{2})q^{2}+(4-4\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T - 24 T^{2} - 72 T^{3} - 368 T^{4} - 2304 T^{5} - 24576 T^{6} + 65536 T^{7} + 1048576 T^{8} \)
$3$ \( 1 - 8 T - 401 T^{2} + 168 T^{3} + 141624 T^{4} + 40824 T^{5} - 23678649 T^{6} - 114791256 T^{7} + 3486784401 T^{8} \)
$5$ \( 1 - 38 T - 1467 T^{2} + 126882 T^{3} - 5804204 T^{4} + 396506250 T^{5} - 14326171875 T^{6} - 1159667968750 T^{7} + 95367431640625 T^{8} \)
$7$ \( 1 + 168 T + 11662 T^{2} + 2823576 T^{3} + 282475249 T^{4} \)
$11$ \( 1 + 424 T - 167697 T^{2} + 10757304 T^{3} + 65846956552 T^{4} + 1732474566504 T^{5} - 4349628293313897 T^{6} + 1771153223832236024 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( ( 1 + 924 T + 927022 T^{2} + 343074732 T^{3} + 137858491849 T^{4} )^{2} \)
$17$ \( 1 - 2346 T + 1932761 T^{2} - 1715491386 T^{3} + 2921236022964 T^{4} - 2435752452851802 T^{5} + 3896434387025709689 T^{6} - \)\(67\!\cdots\!78\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \)
$19$ \( 1 - 360 T - 2016025 T^{2} + 1010366280 T^{3} - 1848262047576 T^{4} + 2501766935541720 T^{5} - 12360382852383261025 T^{6} - \)\(54\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \)
$23$ \( 1 - 12 T - 12696421 T^{2} + 2113452 T^{3} + 119775324752184 T^{4} + 13602901986036 T^{5} - \)\(52\!\cdots\!29\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( ( 1 + 7052 T + 35324974 T^{2} + 144644622748 T^{3} + 420707233300201 T^{4} )^{2} \)
$31$ \( 1 + 3548 T - 28901749 T^{2} - 55945747452 T^{3} + 541403754912104 T^{4} - 1601679251611173252 T^{5} - \)\(23\!\cdots\!49\)\( T^{6} + \)\(83\!\cdots\!48\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$37$ \( 1 + 11090 T - 23355139 T^{2} + 84897554250 T^{3} + 8079277710681572 T^{4} + 5887132351317167250 T^{5} - \)\(11\!\cdots\!11\)\( T^{6} + \)\(36\!\cdots\!70\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \)
$41$ \( ( 1 - 3500 T + 206898214 T^{2} - 405496703500 T^{3} + 13422659310152401 T^{4} )^{2} \)
$43$ \( ( 1 + 12680 T + 267378054 T^{2} + 1864067057240 T^{3} + 21611482313284249 T^{4} )^{2} \)
$47$ \( 1 - 22956 T + 35452763 T^{2} - 753763910004 T^{3} + 68138105036084328 T^{4} - \)\(17\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} - \)\(27\!\cdots\!08\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( 1 + 3042 T - 707161075 T^{2} - 364967439174 T^{3} + 334492868775797220 T^{4} - \)\(15\!\cdots\!82\)\( T^{5} - \)\(12\!\cdots\!75\)\( T^{6} + \)\(22\!\cdots\!94\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \)
$59$ \( 1 - 65808 T + 1828603607 T^{2} - 70562013287472 T^{3} + 2653216336714668312 T^{4} - \)\(50\!\cdots\!28\)\( T^{5} + \)\(93\!\cdots\!07\)\( T^{6} - \)\(24\!\cdots\!92\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 - 42486 T + 303064037 T^{2} + 7953228077298 T^{3} + 18102385363935684 T^{4} + \)\(67\!\cdots\!98\)\( T^{5} + \)\(21\!\cdots\!37\)\( T^{6} - \)\(25\!\cdots\!86\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 + 42312 T - 1326272449 T^{2} + 17615652522648 T^{3} + 5473083141138317592 T^{4} + \)\(23\!\cdots\!36\)\( T^{5} - \)\(24\!\cdots\!01\)\( T^{6} + \)\(10\!\cdots\!16\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 + 2208 T + 3433192846 T^{2} + 3983738407008 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( 1 - 50506 T - 222184951 T^{2} + 69349899662694 T^{3} - 1895976700185808828 T^{4} + \)\(14\!\cdots\!42\)\( T^{5} - \)\(95\!\cdots\!99\)\( T^{6} - \)\(44\!\cdots\!42\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$79$ \( 1 + 9004 T - 5095520573 T^{2} - 8801591961836 T^{3} + 17079371584250245336 T^{4} - \)\(27\!\cdots\!64\)\( T^{5} - \)\(48\!\cdots\!73\)\( T^{6} + \)\(26\!\cdots\!96\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \)
$83$ \( ( 1 + 104328 T + 9837878230 T^{2} + 410952232202904 T^{3} + 15516041187205853449 T^{4} )^{2} \)
$89$ \( 1 - 26666 T - 7396026999 T^{2} + 81625061802438 T^{3} + 30572703729886615780 T^{4} + \)\(45\!\cdots\!62\)\( T^{5} - \)\(23\!\cdots\!99\)\( T^{6} - \)\(46\!\cdots\!34\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \)
$97$ \( ( 1 - 209132 T + 28107307478 T^{2} - 1795887642626924 T^{3} + 73742412689492826049 T^{4} )^{2} \)
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