# 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

# Related objects

## 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

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