Properties

Label 21.8.a
Level 21
Weight 8
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 4
Sturm bound 21
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(21\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(21))\).

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

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(3\)

Trace form

\( 8q + 2q^{2} + 770q^{4} - 776q^{5} - 108q^{6} + 686q^{7} + 1854q^{8} + 5832q^{9} + O(q^{10}) \) \( 8q + 2q^{2} + 770q^{4} - 776q^{5} - 108q^{6} + 686q^{7} + 1854q^{8} + 5832q^{9} + 3852q^{10} + 1432q^{11} - 15336q^{12} + 12112q^{13} - 8918q^{14} - 216q^{15} + 92882q^{16} - 40968q^{17} + 1458q^{18} - 97424q^{19} + 572q^{20} + 18522q^{21} - 68400q^{22} + 134808q^{23} - 54756q^{24} + 138632q^{25} - 651436q^{26} + 107702q^{28} - 439328q^{29} + 367848q^{30} + 442432q^{31} + 523654q^{32} + 190296q^{33} - 935460q^{34} - 58996q^{35} + 561330q^{36} - 518624q^{37} + 444368q^{38} - 30672q^{39} + 1448676q^{40} + 1751048q^{41} + 148176q^{42} + 271216q^{43} - 3091024q^{44} - 565704q^{45} + 37512q^{46} + 1356144q^{47} - 1884816q^{48} + 941192q^{49} - 2099378q^{50} + 2017224q^{51} - 696572q^{52} - 1085904q^{53} - 78732q^{54} - 3055728q^{55} - 3825822q^{56} + 2204928q^{57} - 2019924q^{58} - 1151088q^{59} - 4629744q^{60} + 5183200q^{61} + 12950208q^{62} + 500094q^{63} + 8451050q^{64} + 3820624q^{65} - 4485672q^{66} - 6863648q^{67} - 6501060q^{68} - 4491288q^{69} - 1140132q^{70} - 3994680q^{71} + 1351566q^{72} - 10351520q^{73} + 12127020q^{74} + 3748896q^{75} + 832192q^{76} + 2118368q^{77} + 1252800q^{78} + 13872064q^{79} + 28655804q^{80} + 4251528q^{81} - 21988356q^{82} - 9046368q^{83} + 3556224q^{84} - 16470384q^{85} - 4645208q^{86} + 271296q^{87} + 14529696q^{88} - 353816q^{89} + 2808108q^{90} + 9888004q^{91} - 15951096q^{92} - 10010736q^{93} + 7381056q^{94} - 21731744q^{95} - 16331436q^{96} - 5445920q^{97} + 235298q^{98} + 1043928q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(21))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
21.8.a.a \(1\) \(6.560\) \(\Q\) None \(2\) \(27\) \(-278\) \(-343\) \(-\) \(+\) \(q+2q^{2}+3^{3}q^{3}-124q^{4}-278q^{5}+\cdots\)
21.8.a.b \(2\) \(6.560\) \(\Q(\sqrt{1065}) \) None \(-9\) \(-54\) \(-360\) \(686\) \(+\) \(-\) \(q+(-4-\beta )q^{2}-3^{3}q^{3}+(154+9\beta )q^{4}+\cdots\)
21.8.a.c \(2\) \(6.560\) \(\Q(\sqrt{67}) \) None \(12\) \(-54\) \(-24\) \(-686\) \(+\) \(+\) \(q+(6+\beta )q^{2}-3^{3}q^{3}+(176+12\beta )q^{4}+\cdots\)
21.8.a.d \(3\) \(6.560\) 3.3.2910828.1 None \(-3\) \(81\) \(-114\) \(1029\) \(-\) \(-\) \(q+(-1+\beta _{1})q^{2}+3^{3}q^{3}+(75+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 128 T^{2} \))(\( 1 + 9 T + 10 T^{2} + 1152 T^{3} + 16384 T^{4} \))(\( 1 - 12 T + 24 T^{2} - 1536 T^{3} + 16384 T^{4} \))(\( 1 + 3 T + 84 T^{2} - 24 T^{3} + 10752 T^{4} + 49152 T^{5} + 2097152 T^{6} \))
$3$ (\( 1 - 27 T \))(\( ( 1 + 27 T )^{2} \))(\( ( 1 + 27 T )^{2} \))(\( ( 1 - 27 T )^{3} \))
$5$ (\( 1 + 278 T + 78125 T^{2} \))(\( 1 + 360 T + 82150 T^{2} + 28125000 T^{3} + 6103515625 T^{4} \))(\( 1 + 24 T + 139242 T^{2} + 1875000 T^{3} + 6103515625 T^{4} \))(\( 1 + 114 T + 53895 T^{2} + 29447700 T^{3} + 4210546875 T^{4} + 695800781250 T^{5} + 476837158203125 T^{6} \))
$7$ (\( 1 + 343 T \))(\( ( 1 - 343 T )^{2} \))(\( ( 1 + 343 T )^{2} \))(\( ( 1 - 343 T )^{3} \))
$11$ (\( 1 + 4496 T + 19487171 T^{2} \))(\( 1 + 4932 T + 19797958 T^{2} + 96110727372 T^{3} + 379749833583241 T^{4} \))(\( 1 - 2124 T + 19090986 T^{2} - 41390751204 T^{3} + 379749833583241 T^{4} \))(\( 1 - 8736 T + 72843645 T^{2} - 319375577040 T^{3} + 1419516566378295 T^{4} - 3317494546183193376 T^{5} + \)\(74\!\cdots\!11\)\( T^{6} \))
$13$ (\( 1 + 7274 T + 62748517 T^{2} \))(\( 1 - 7708 T + 118266510 T^{2} - 483665569036 T^{3} + 3937376385699289 T^{4} \))(\( 1 + 1084 T + 103561806 T^{2} + 68019392428 T^{3} + 3937376385699289 T^{4} \))(\( 1 - 12762 T + 102114315 T^{2} - 845823244124 T^{3} + 6407521830720855 T^{4} - 50248797434294326218 T^{5} + \)\(24\!\cdots\!13\)\( T^{6} \))
$17$ (\( 1 - 11382 T + 410338673 T^{2} \))(\( 1 + 28584 T + 942631150 T^{2} + 11729120629032 T^{3} + 168377826559400929 T^{4} \))(\( 1 + 29256 T + 533114098 T^{2} + 12004868217288 T^{3} + 168377826559400929 T^{4} \))(\( 1 - 5490 T + 315344595 T^{2} + 7961596230492 T^{3} + 129398082650022435 T^{4} - \)\(92\!\cdots\!10\)\( T^{5} + \)\(69\!\cdots\!17\)\( T^{6} \))
$19$ (\( 1 + 15884 T + 893871739 T^{2} \))(\( 1 + 63728 T + 2569797414 T^{2} + 56964658182992 T^{3} + 799006685782884121 T^{4} \))(\( 1 + 25816 T + 1627717254 T^{2} + 23076192814024 T^{3} + 799006685782884121 T^{4} \))(\( 1 - 8004 T + 1277977953 T^{2} - 19746518206232 T^{3} + 1142348375251770267 T^{4} - \)\(63\!\cdots\!84\)\( T^{5} + \)\(71\!\cdots\!19\)\( T^{6} \))
$23$ (\( 1 - 86100 T + 3404825447 T^{2} \))(\( 1 - 82260 T + 8202169294 T^{2} - 280080941270220 T^{3} + 11592836324538749809 T^{4} \))(\( 1 - 68316 T + 7976265490 T^{2} - 232604055237252 T^{3} + 11592836324538749809 T^{4} \))(\( 1 + 101868 T + 10443822969 T^{2} + 590778772265544 T^{3} + 35559394208814292143 T^{4} + \)\(11\!\cdots\!12\)\( T^{5} + \)\(39\!\cdots\!23\)\( T^{6} \))
$29$ (\( 1 - 40702 T + 17249876309 T^{2} \))(\( 1 + 435996 T + 80862098782 T^{2} + 7520877071218764 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))(\( 1 - 211308 T + 35807598606 T^{2} - 3645036863102172 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))(\( 1 + 255342 T + 69145430955 T^{2} + 9017630182097076 T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(75\!\cdots\!02\)\( T^{5} + \)\(51\!\cdots\!29\)\( T^{6} \))
$31$ (\( 1 + 44760 T + 27512614111 T^{2} \))(\( 1 + 29240 T + 27798421182 T^{2} + 804468836605640 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( 1 - 435840 T + 98660711870 T^{2} - 11991097734138240 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( 1 - 80592 T + 42262036701 T^{2} - 1959662063128160 T^{3} + \)\(11\!\cdots\!11\)\( T^{4} - \)\(61\!\cdots\!32\)\( T^{5} + \)\(20\!\cdots\!31\)\( T^{6} \))
$37$ (\( 1 + 580962 T + 94931877133 T^{2} \))(\( 1 + 709556 T + 313296440190 T^{2} + 67359483010982948 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))(\( 1 + 28428 T + 188976571454 T^{2} + 2698723403136924 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))(\( 1 - 800322 T + 410431030179 T^{2} - 137487387839318444 T^{3} + \)\(38\!\cdots\!07\)\( T^{4} - \)\(72\!\cdots\!58\)\( T^{5} + \)\(85\!\cdots\!37\)\( T^{6} \))
$41$ (\( 1 + 171658 T + 194754273881 T^{2} \))(\( 1 + 25056 T - 26592839954 T^{2} + 4879763086362336 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( 1 - 749760 T + 517210006962 T^{2} - 146018964385018560 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( 1 - 1198002 T + 590806308363 T^{2} - 222007935431949924 T^{3} + \)\(11\!\cdots\!03\)\( T^{4} - \)\(45\!\cdots\!22\)\( T^{5} + \)\(73\!\cdots\!41\)\( T^{6} \))
$43$ (\( 1 + 741148 T + 271818611107 T^{2} \))(\( 1 - 496216 T + 567160908438 T^{2} - 134880743929071112 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))(\( 1 - 397096 T + 472051884246 T^{2} - 107938083196145272 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))(\( 1 - 119052 T + 562409914905 T^{2} - 95535036097769864 T^{3} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(87\!\cdots\!48\)\( T^{5} + \)\(20\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 - 1071720 T + 506623120463 T^{2} \))(\( 1 + 1575000 T + 1564490669086 T^{2} + 797931414729225000 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))(\( 1 - 840168 T + 744642372910 T^{2} - 425648533873157784 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))(\( 1 - 1019256 T + 1392634189821 T^{2} - 808085371369677072 T^{3} + \)\(70\!\cdots\!23\)\( T^{4} - \)\(26\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!47\)\( T^{6} \))
$53$ (\( 1 + 1721778 T + 1174711139837 T^{2} \))(\( 1 - 2057436 T + 3149566808638 T^{2} - 2416892988701677932 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))(\( 1 + 246684 T + 2085795052990 T^{2} + 289782442819550508 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))(\( 1 + 1174878 T + 3974773460979 T^{2} + 2816856603947973204 T^{3} + \)\(46\!\cdots\!23\)\( T^{4} + \)\(16\!\cdots\!82\)\( T^{5} + \)\(16\!\cdots\!53\)\( T^{6} \))
$59$ (\( 1 + 1557012 T + 2488651484819 T^{2} \))(\( 1 + 1101024 T + 4521593481622 T^{2} + 2740065012421354656 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))(\( 1 - 2199504 T + 4631611593574 T^{2} - 5473798895465329776 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))(\( 1 + 692556 T + 5232348867513 T^{2} + 3135752992577101128 T^{3} + \)\(13\!\cdots\!47\)\( T^{4} + \)\(42\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \))
$61$ (\( 1 - 2597998 T + 3142742836021 T^{2} \))(\( 1 - 28996 T + 5887677435486 T^{2} - 91126971273264916 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))(\( 1 + 1951108 T + 6790017439086 T^{2} + 6131830689303261268 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))(\( 1 - 4507314 T + 12758183492811 T^{2} - 24537301510123677836 T^{3} + \)\(40\!\cdots\!31\)\( T^{4} - \)\(44\!\cdots\!74\)\( T^{5} + \)\(31\!\cdots\!61\)\( T^{6} \))
$67$ (\( 1 + 963548 T + 6060711605323 T^{2} \))(\( 1 + 4480784 T + 16777543143750 T^{2} + 27156739589745613232 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))(\( 1 - 1532048 T + 530136191190 T^{2} - 9285301093511891504 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))(\( 1 + 2951364 T + 20040859741281 T^{2} + 35350616522201858008 T^{3} + \)\(12\!\cdots\!63\)\( T^{4} + \)\(10\!\cdots\!56\)\( T^{5} + \)\(22\!\cdots\!67\)\( T^{6} \))
$71$ (\( 1 + 4063380 T + 9095120158391 T^{2} \))(\( 1 - 54540 T + 17803030672942 T^{2} - 496047853438645140 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))(\( 1 - 2024004 T + 14260375775986 T^{2} - 18408559581064017564 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))(\( 1 + 2009844 T + 4152749057385 T^{2} - 20203620844773313800 T^{3} + \)\(37\!\cdots\!35\)\( T^{4} + \)\(16\!\cdots\!64\)\( T^{5} + \)\(75\!\cdots\!71\)\( T^{6} \))
$73$ (\( 1 + 5370222 T + 11047398519097 T^{2} \))(\( 1 - 666604 T - 310686963642 T^{2} - 7364240042424136588 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))(\( 1 + 1709028 T + 11306816102198 T^{2} + 18880313396295307716 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))(\( 1 + 3938874 T + 30868256285319 T^{2} + 84184052509551167404 T^{3} + \)\(34\!\cdots\!43\)\( T^{4} + \)\(48\!\cdots\!66\)\( T^{5} + \)\(13\!\cdots\!73\)\( T^{6} \))
$79$ (\( 1 - 4094936 T + 19203908986159 T^{2} \))(\( 1 - 2322952 T + 38986658128734 T^{2} - 44609758787216021368 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))(\( 1 - 1048168 T + 11630316806382 T^{2} - 20128922874204306712 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))(\( 1 - 6406008 T + 65508630875421 T^{2} - \)\(24\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!39\)\( T^{4} - \)\(23\!\cdots\!48\)\( T^{5} + \)\(70\!\cdots\!79\)\( T^{6} \))
$83$ (\( 1 + 1343124 T + 27136050989627 T^{2} \))(\( 1 + 7384392 T + 61089776413510 T^{2} + \)\(20\!\cdots\!84\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))(\( 1 + 4894296 T + 47206286808070 T^{2} + \)\(13\!\cdots\!92\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))(\( 1 - 4575444 T + 60865744807905 T^{2} - \)\(16\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!35\)\( T^{4} - \)\(33\!\cdots\!76\)\( T^{5} + \)\(19\!\cdots\!83\)\( T^{6} \))
$89$ (\( 1 - 9081574 T + 44231334895529 T^{2} \))(\( 1 - 1784448 T + 26626978018894 T^{2} - 78928517091656932992 T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( 1 + 60864 T + 81562257471570 T^{2} + 2692095967081477056 T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( 1 + 11158974 T + 168392983416699 T^{2} + \)\(10\!\cdots\!12\)\( T^{3} + \)\(74\!\cdots\!71\)\( T^{4} + \)\(21\!\cdots\!34\)\( T^{5} + \)\(86\!\cdots\!89\)\( T^{6} \))
$97$ (\( 1 - 6487914 T + 80798284478113 T^{2} \))(\( 1 - 16266412 T + 223770781220502 T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))(\( 1 + 26046852 T + 325711749428294 T^{2} + \)\(21\!\cdots\!76\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))(\( 1 + 2153394 T + 159967477341855 T^{2} + \)\(41\!\cdots\!32\)\( T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(14\!\cdots\!86\)\( T^{5} + \)\(52\!\cdots\!97\)\( T^{6} \))
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