Properties

Label 7.6
Level 7
Weight 6
Dimension 7
Nonzero newspaces 2
Newform subspaces 3
Sturm bound 24
Trace bound 1

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Defining parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 3 \)
Sturm bound: \(24\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(7))\).

Total New Old
Modular forms 13 11 2
Cusp forms 7 7 0
Eisenstein series 6 4 2

Trace form

\( 7q - 3q^{2} - 12q^{3} + 61q^{4} - 36q^{5} - 222q^{6} - 119q^{7} - 177q^{8} + 891q^{9} + O(q^{10}) \) \( 7q - 3q^{2} - 12q^{3} + 61q^{4} - 36q^{5} - 222q^{6} - 119q^{7} - 177q^{8} + 891q^{9} + 1542q^{10} + 204q^{11} - 2310q^{12} - 2338q^{13} - 1743q^{14} + 912q^{15} + 3601q^{16} + 2424q^{17} + 3795q^{18} - 3004q^{19} - 4704q^{20} - 1134q^{21} + 180q^{22} + 3924q^{23} + 1794q^{24} - 5099q^{25} + 2268q^{26} + 3432q^{27} - 539q^{28} - 3990q^{29} - 19866q^{30} - 11212q^{31} - 753q^{32} + 20166q^{33} + 37998q^{34} + 29358q^{35} + 5241q^{36} - 15256q^{37} - 34368q^{38} - 20160q^{39} + 7440q^{40} - 17010q^{41} - 20034q^{42} - 17500q^{43} - 20988q^{44} + 6570q^{45} + 12462q^{46} + 43980q^{47} + 43710q^{48} + 12103q^{49} + 40173q^{50} + 32088q^{51} + 23324q^{52} + 8688q^{53} - 111054q^{54} - 102048q^{55} - 40593q^{56} - 23868q^{57} + 61986q^{58} + 19140q^{59} + 41244q^{60} + 3380q^{61} - 6984q^{62} + 24885q^{63} - 18239q^{64} + 50064q^{65} + 125190q^{66} - 65932q^{67} - 137550q^{68} - 118620q^{69} - 130494q^{70} + 34440q^{71} + 58143q^{72} + 136040q^{73} + 194784q^{74} + 75948q^{75} + 58226q^{76} + 73374q^{77} - 37632q^{78} + 74036q^{79} - 218832q^{80} - 157671q^{81} + 53550q^{82} - 111636q^{83} + 14994q^{84} - 47640q^{85} - 345624q^{86} - 191040q^{87} - 103680q^{88} + 60360q^{89} + 328836q^{90} + 125342q^{91} + 254376q^{92} - 23982q^{93} - 31806q^{94} + 227892q^{95} + 239106q^{96} + 380534q^{97} + 96285q^{98} - 161532q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(7))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
7.6.a \(\chi_{7}(1, \cdot)\) 7.6.a.a 1 1
7.6.a.b 2
7.6.c \(\chi_{7}(2, \cdot)\) 7.6.c.a 4 2

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 10 T + 32 T^{2} \))(\( 1 - 9 T + 70 T^{2} - 288 T^{3} + 1024 T^{4} \))(\( 1 + 2 T - 24 T^{2} - 72 T^{3} - 368 T^{4} - 2304 T^{5} - 24576 T^{6} + 65536 T^{7} + 1048576 T^{8} \))
$3$ (\( 1 + 14 T + 243 T^{2} \))(\( 1 + 6 T - 18 T^{2} + 1458 T^{3} + 59049 T^{4} \))(\( 1 - 8 T - 401 T^{2} + 168 T^{3} + 141624 T^{4} + 40824 T^{5} - 23678649 T^{6} - 114791256 T^{7} + 3486784401 T^{8} \))
$5$ (\( 1 + 56 T + 3125 T^{2} \))(\( 1 + 18 T + 4906 T^{2} + 56250 T^{3} + 9765625 T^{4} \))(\( 1 - 38 T - 1467 T^{2} + 126882 T^{3} - 5804204 T^{4} + 396506250 T^{5} - 14326171875 T^{6} - 1159667968750 T^{7} + 95367431640625 T^{8} \))
$7$ (\( 1 + 49 T \))(\( ( 1 - 49 T )^{2} \))(\( 1 + 168 T + 11662 T^{2} + 2823576 T^{3} + 282475249 T^{4} \))
$11$ (\( 1 - 232 T + 161051 T^{2} \))(\( 1 - 396 T + 142198 T^{2} - 63776196 T^{3} + 25937424601 T^{4} \))(\( 1 + 424 T - 167697 T^{2} + 10757304 T^{3} + 65846956552 T^{4} + 1732474566504 T^{5} - 4349628293313897 T^{6} + 1771153223832236024 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \))
$13$ (\( 1 + 140 T + 371293 T^{2} \))(\( 1 + 350 T + 546978 T^{2} + 129952550 T^{3} + 137858491849 T^{4} \))(\( ( 1 + 924 T + 927022 T^{2} + 343074732 T^{3} + 137858491849 T^{4} )^{2} \))
$17$ (\( 1 + 1722 T + 1419857 T^{2} \))(\( 1 - 1800 T + 3567406 T^{2} - 2555742600 T^{3} + 2015993900449 T^{4} \))(\( 1 - 2346 T + 1932761 T^{2} - 1715491386 T^{3} + 2921236022964 T^{4} - 2435752452851802 T^{5} + 3896434387025709689 T^{6} - \)\(67\!\cdots\!78\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} \))
$19$ (\( 1 + 98 T + 2476099 T^{2} \))(\( 1 + 3266 T + 7614270 T^{2} + 8086939334 T^{3} + 6131066257801 T^{4} \))(\( 1 - 360 T - 2016025 T^{2} + 1010366280 T^{3} - 1848262047576 T^{4} + 2501766935541720 T^{5} - 12360382852383261025 T^{6} - \)\(54\!\cdots\!40\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} \))
$23$ (\( 1 - 1824 T + 6436343 T^{2} \))(\( 1 - 2088 T + 9365230 T^{2} - 13439084184 T^{3} + 41426511213649 T^{4} \))(\( 1 - 12 T - 12696421 T^{2} + 2113452 T^{3} + 119775324752184 T^{4} + 13602901986036 T^{5} - \)\(52\!\cdots\!29\)\( T^{6} - \)\(31\!\cdots\!84\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \))
$29$ (\( 1 - 3418 T + 20511149 T^{2} \))(\( 1 - 6696 T + 51326470 T^{2} - 137342653704 T^{3} + 420707233300201 T^{4} \))(\( ( 1 + 7052 T + 35324974 T^{2} + 144644622748 T^{3} + 420707233300201 T^{4} )^{2} \))
$31$ (\( 1 + 7644 T + 28629151 T^{2} \))(\( 1 + 20 T + 53103102 T^{2} + 572583020 T^{3} + 819628286980801 T^{4} \))(\( 1 + 3548 T - 28901749 T^{2} - 55945747452 T^{3} + 541403754912104 T^{4} - 1601679251611173252 T^{5} - \)\(23\!\cdots\!49\)\( T^{6} + \)\(83\!\cdots\!48\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \))
$37$ (\( 1 + 10398 T + 69343957 T^{2} \))(\( 1 - 6232 T + 144242070 T^{2} - 432151540024 T^{3} + 4808584372417849 T^{4} \))(\( 1 + 11090 T - 23355139 T^{2} + 84897554250 T^{3} + 8079277710681572 T^{4} + 5887132351317167250 T^{5} - \)\(11\!\cdots\!11\)\( T^{6} + \)\(36\!\cdots\!70\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} \))
$41$ (\( 1 + 17962 T + 115856201 T^{2} \))(\( 1 + 6048 T + 223864366 T^{2} + 700698303648 T^{3} + 13422659310152401 T^{4} \))(\( ( 1 - 3500 T + 206898214 T^{2} - 405496703500 T^{3} + 13422659310152401 T^{4} )^{2} \))
$43$ (\( 1 - 10880 T + 147008443 T^{2} \))(\( 1 + 3020 T - 30383466 T^{2} + 443965497860 T^{3} + 21611482313284249 T^{4} \))(\( ( 1 + 12680 T + 267378054 T^{2} + 1864067057240 T^{3} + 21611482313284249 T^{4} )^{2} \))
$47$ (\( 1 - 9324 T + 229345007 T^{2} \))(\( 1 - 11700 T + 292735582 T^{2} - 2683336581900 T^{3} + 52599132235830049 T^{4} \))(\( 1 - 22956 T + 35452763 T^{2} - 753763910004 T^{3} + 68138105036084328 T^{4} - \)\(17\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} - \)\(27\!\cdots\!08\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \))
$53$ (\( 1 - 2262 T + 418195493 T^{2} \))(\( 1 - 9468 T + 858185230 T^{2} - 3959474927724 T^{3} + 174887470365513049 T^{4} \))(\( 1 + 3042 T - 707161075 T^{2} - 364967439174 T^{3} + 334492868775797220 T^{4} - \)\(15\!\cdots\!82\)\( T^{5} - \)\(12\!\cdots\!75\)\( T^{6} + \)\(22\!\cdots\!94\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} \))
$59$ (\( 1 + 2730 T + 714924299 T^{2} \))(\( 1 + 43938 T + 1852599934 T^{2} + 31412343849462 T^{3} + 511116753300641401 T^{4} \))(\( 1 - 65808 T + 1828603607 T^{2} - 70562013287472 T^{3} + 2653216336714668312 T^{4} - \)\(50\!\cdots\!28\)\( T^{5} + \)\(93\!\cdots\!07\)\( T^{6} - \)\(24\!\cdots\!92\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \))
$61$ (\( 1 - 25648 T + 844596301 T^{2} \))(\( 1 + 64754 T + 2408321418 T^{2} + 54690988874954 T^{3} + 713342911662882601 T^{4} \))(\( 1 - 42486 T + 303064037 T^{2} + 7953228077298 T^{3} + 18102385363935684 T^{4} + \)\(67\!\cdots\!98\)\( T^{5} + \)\(21\!\cdots\!37\)\( T^{6} - \)\(25\!\cdots\!86\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \))
$67$ (\( 1 + 48404 T + 1350125107 T^{2} \))(\( 1 - 24784 T + 2799959190 T^{2} - 33461500651888 T^{3} + 1822837804551761449 T^{4} \))(\( 1 + 42312 T - 1326272449 T^{2} + 17615652522648 T^{3} + 5473083141138317592 T^{4} + \)\(23\!\cdots\!36\)\( T^{5} - \)\(24\!\cdots\!01\)\( T^{6} + \)\(10\!\cdots\!16\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \))
$71$ (\( 1 + 58560 T + 1804229351 T^{2} \))(\( 1 - 97416 T + 5729557966 T^{2} - 175760806457016 T^{3} + 3255243551009881201 T^{4} \))(\( ( 1 + 2208 T + 3433192846 T^{2} + 3983738407008 T^{3} + 3255243551009881201 T^{4} )^{2} \))
$73$ (\( 1 - 68082 T + 2073071593 T^{2} \))(\( 1 - 17452 T + 3828622374 T^{2} - 36179245441036 T^{3} + 4297625829703557649 T^{4} \))(\( 1 - 50506 T - 222184951 T^{2} + 69349899662694 T^{3} - 1895976700185808828 T^{4} + \)\(14\!\cdots\!42\)\( T^{5} - \)\(95\!\cdots\!99\)\( T^{6} - \)\(44\!\cdots\!42\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \))
$79$ (\( 1 - 31784 T + 3077056399 T^{2} \))(\( 1 - 51256 T + 3645565854 T^{2} - 157717602787144 T^{3} + 9468276082626847201 T^{4} \))(\( 1 + 9004 T - 5095520573 T^{2} - 8801591961836 T^{3} + 17079371584250245336 T^{4} - \)\(27\!\cdots\!64\)\( T^{5} - \)\(48\!\cdots\!73\)\( T^{6} + \)\(26\!\cdots\!96\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \))
$83$ (\( 1 + 20538 T + 3939040643 T^{2} \))(\( 1 - 117558 T + 7798161502 T^{2} - 463065739909794 T^{3} + 15516041187205853449 T^{4} \))(\( ( 1 + 104328 T + 9837878230 T^{2} + 410952232202904 T^{3} + 15516041187205853449 T^{4} )^{2} \))
$89$ (\( 1 + 50582 T + 5584059449 T^{2} \))(\( 1 - 84276 T + 5915697430 T^{2} - 470602194123924 T^{3} + 31181719929966183601 T^{4} \))(\( 1 - 26666 T - 7396026999 T^{2} + 81625061802438 T^{3} + 30572703729886615780 T^{4} + \)\(45\!\cdots\!62\)\( T^{5} - \)\(23\!\cdots\!99\)\( T^{6} - \)\(46\!\cdots\!34\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} \))
$97$ (\( 1 + 58506 T + 8587340257 T^{2} \))(\( 1 - 20776 T + 16174049358 T^{2} - 178410581179432 T^{3} + 73742412689492826049 T^{4} \))(\( ( 1 - 209132 T + 28107307478 T^{2} - 1795887642626924 T^{3} + 73742412689492826049 T^{4} )^{2} \))
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