Properties

Label 21.10.a
Level 21
Weight 10
Character orbit a
Rep. character \(\chi_{21}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 4
Sturm bound 26
Trace bound 2

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Defining parameters

Level: \( N \) = \( 21 = 3 \cdot 7 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 21.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(26\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(21))\).

Total New Old
Modular forms 26 8 18
Cusp forms 22 8 14
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.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(3\)
Minus space\(-\)\(5\)

Trace form

\( 8q + 2q^{2} + 2050q^{4} + 4552q^{5} - 6156q^{6} - 4802q^{7} + 9390q^{8} + 52488q^{9} + O(q^{10}) \) \( 8q + 2q^{2} + 2050q^{4} + 4552q^{5} - 6156q^{6} - 4802q^{7} + 9390q^{8} + 52488q^{9} + 16108q^{10} - 78140q^{11} + 115992q^{12} + 279176q^{13} + 24010q^{14} - 3564q^{15} + 214546q^{16} - 378912q^{17} + 13122q^{18} + 586184q^{19} + 1411100q^{20} - 388962q^{21} + 2396416q^{22} + 725724q^{23} - 1495908q^{24} - 2577248q^{25} + 957428q^{26} - 4057690q^{28} + 8576440q^{29} + 3227688q^{30} - 8356584q^{31} - 45299498q^{32} + 5409504q^{33} + 28069404q^{34} - 6204184q^{35} + 13450050q^{36} - 37232136q^{37} - 30025936q^{38} - 14262480q^{39} + 52787556q^{40} + 13951376q^{41} - 6223392q^{42} + 50242720q^{43} - 42073024q^{44} + 29865672q^{45} - 98920632q^{46} + 62344392q^{47} + 92730096q^{48} + 46118408q^{49} + 73994302q^{50} + 23031540q^{51} + 231982372q^{52} - 217936680q^{53} - 40389516q^{54} + 130559768q^{55} + 63083874q^{56} + 5276664q^{57} - 443463044q^{58} + 69912216q^{59} - 29254608q^{60} - 85792600q^{61} - 308107200q^{62} - 31505922q^{63} - 56923702q^{64} + 246963088q^{65} + 35066520q^{66} + 179326184q^{67} - 853640772q^{68} + 547698672q^{69} + 291106844q^{70} + 151882212q^{71} + 61607790q^{72} + 781362720q^{73} - 758102628q^{74} - 638703792q^{75} - 219853280q^{76} + 467349848q^{77} - 408004128q^{78} + 75820360q^{79} - 225047332q^{80} + 344373768q^{81} - 1050294052q^{82} - 167217024q^{83} - 298722816q^{84} - 2054101752q^{85} + 3203864920q^{86} - 472571496q^{87} + 1195436784q^{88} + 473386912q^{89} + 105684588q^{90} - 720175148q^{91} + 3191894664q^{92} + 89676072q^{93} + 130819488q^{94} + 1028816560q^{95} - 374635692q^{96} + 2488240800q^{97} + 11529602q^{98} - 512676540q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(21))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7
21.10.a.a \(1\) \(10.816\) \(\Q\) None \(-24\) \(81\) \(-144\) \(2401\) \(-\) \(-\) \(q-24q^{2}+3^{4}q^{3}+2^{6}q^{4}-12^{2}q^{5}+\cdots\)
21.10.a.b \(2\) \(10.816\) \(\Q(\sqrt{2353}) \) None \(9\) \(-162\) \(1170\) \(-4802\) \(+\) \(+\) \(q+(5-\beta )q^{2}-3^{4}q^{3}+(101-9\beta )q^{4}+\cdots\)
21.10.a.c \(2\) \(10.816\) \(\Q(\sqrt{345}) \) None \(30\) \(-162\) \(1128\) \(4802\) \(+\) \(-\) \(q+(15-\beta )q^{2}-3^{4}q^{3}+(58-30\beta )q^{4}+\cdots\)
21.10.a.d \(3\) \(10.816\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-13\) \(243\) \(2398\) \(-7203\) \(-\) \(+\) \(q+(-4-\beta _{1})q^{2}+3^{4}q^{3}+(555+7\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 24 T + 512 T^{2} \))(\( 1 - 9 T + 456 T^{2} - 4608 T^{3} + 262144 T^{4} \))(\( 1 - 30 T + 904 T^{2} - 15360 T^{3} + 262144 T^{4} \))(\( 1 + 13 T + 14 T^{2} + 2328 T^{3} + 7168 T^{4} + 3407872 T^{5} + 134217728 T^{6} \))
$3$ (\( 1 - 81 T \))(\( ( 1 + 81 T )^{2} \))(\( ( 1 + 81 T )^{2} \))(\( ( 1 - 81 T )^{3} \))
$5$ (\( 1 + 144 T + 1953125 T^{2} \))(\( 1 - 1170 T + 1366050 T^{2} - 2285156250 T^{3} + 3814697265625 T^{4} \))(\( 1 - 1128 T + 2533846 T^{2} - 2203125000 T^{3} + 3814697265625 T^{4} \))(\( 1 - 2398 T + 7454315 T^{2} - 9553653300 T^{3} + 14559208984375 T^{4} - 9147644042968750 T^{5} + 7450580596923828125 T^{6} \))
$7$ (\( 1 - 2401 T \))(\( ( 1 + 2401 T )^{2} \))(\( ( 1 - 2401 T )^{2} \))(\( ( 1 + 2401 T )^{3} \))
$11$ (\( 1 + 15030 T + 2357947691 T^{2} \))(\( 1 + 145746 T + 9848423886 T^{2} + 343661444172486 T^{3} + 5559917313492231481 T^{4} \))(\( 1 - 73284 T + 5040255766 T^{2} - 172799838587244 T^{3} + 5559917313492231481 T^{4} \))(\( 1 - 9352 T + 5750954321 T^{2} - 44520233946288 T^{3} + 13560449462248422811 T^{4} - \)\(51\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!71\)\( T^{6} \))
$13$ (\( 1 + 151486 T + 10604499373 T^{2} \))(\( 1 - 86528 T + 23073911094 T^{2} - 917586121746944 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - 141100 T + 26090120766 T^{2} - 1496294861530300 T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \))(\( 1 - 203034 T + 40057043091 T^{2} - 4399977434379676 T^{3} + \)\(42\!\cdots\!43\)\( T^{4} - \)\(22\!\cdots\!86\)\( T^{5} + \)\(11\!\cdots\!17\)\( T^{6} \))
$17$ (\( 1 + 350448 T + 118587876497 T^{2} \))(\( 1 + 229842 T - 51722753798 T^{2} + 27256474709823474 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))(\( 1 + 101784 T + 163890278158 T^{2} + 12070348421370648 T^{3} + \)\(14\!\cdots\!09\)\( T^{4} \))(\( 1 - 303162 T + 141948419919 T^{2} - 83158795428499212 T^{3} + \)\(16\!\cdots\!43\)\( T^{4} - \)\(42\!\cdots\!58\)\( T^{5} + \)\(16\!\cdots\!73\)\( T^{6} \))
$19$ (\( 1 + 691108 T + 322687697779 T^{2} \))(\( 1 + 221224 T + 657561618294 T^{2} + 71386263253461496 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))(\( 1 - 481744 T + 700509251862 T^{2} - 155452862278846576 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} \))(\( 1 - 1016772 T + 562195142985 T^{2} - 271724066942712280 T^{3} + \)\(18\!\cdots\!15\)\( T^{4} - \)\(10\!\cdots\!52\)\( T^{5} + \)\(33\!\cdots\!39\)\( T^{6} \))
$23$ (\( 1 - 892458 T + 1801152661463 T^{2} \))(\( 1 + 2035782 T + 4618931326270 T^{2} + 3666754167458469066 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 + 982212 T + 684471671662 T^{2} + 1769113757920896156 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 2851260 T + 3029091102789 T^{2} - 2159954408299106760 T^{3} + \)\(54\!\cdots\!07\)\( T^{4} - \)\(92\!\cdots\!40\)\( T^{5} + \)\(58\!\cdots\!47\)\( T^{6} \))
$29$ (\( 1 - 1648518 T + 14507145975869 T^{2} \))(\( 1 - 9756252 T + 52470669069246 T^{2} - \)\(14\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( 1 + 2550924 T + 23659017143182 T^{2} + 37006626841347652956 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( 1 + 277406 T + 21834047921699 T^{2} - 27292035365754403404 T^{3} + \)\(31\!\cdots\!31\)\( T^{4} + \)\(58\!\cdots\!66\)\( T^{5} + \)\(30\!\cdots\!09\)\( T^{6} \))
$31$ (\( 1 + 3734296 T + 26439622160671 T^{2} \))(\( 1 - 204000 T + 30790423515710 T^{2} - 5393682920776884000 T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( 1 + 4935848 T + 45703928191038 T^{2} + \)\(13\!\cdots\!08\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} \))(\( 1 - 109560 T + 3751113600093 T^{2} + \)\(11\!\cdots\!80\)\( T^{3} + \)\(99\!\cdots\!03\)\( T^{4} - \)\(76\!\cdots\!60\)\( T^{5} + \)\(18\!\cdots\!11\)\( T^{6} \))
$37$ (\( 1 + 11471902 T + 129961739795077 T^{2} \))(\( 1 + 13959816 T + 306736999959926 T^{2} + \)\(18\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))(\( 1 + 16256516 T + 258501173397198 T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!29\)\( T^{4} \))(\( 1 - 4456098 T + 206200212680379 T^{2} - \)\(81\!\cdots\!68\)\( T^{3} + \)\(26\!\cdots\!83\)\( T^{4} - \)\(75\!\cdots\!42\)\( T^{5} + \)\(21\!\cdots\!33\)\( T^{6} \))
$41$ (\( 1 - 13985724 T + 327381934393961 T^{2} \))(\( 1 + 42362550 T + 1063812640967922 T^{2} + \)\(13\!\cdots\!50\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 - 48707856 T + 1234544246122606 T^{2} - \)\(15\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} \))(\( 1 + 6379654 T + 487977636599255 T^{2} + \)\(68\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!55\)\( T^{4} + \)\(68\!\cdots\!34\)\( T^{5} + \)\(35\!\cdots\!81\)\( T^{6} \))
$43$ (\( 1 - 16794524 T + 502592611936843 T^{2} \))(\( 1 + 4763912 T + 345063686509734 T^{2} + \)\(23\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))(\( 1 - 7989640 T + 1011320095256166 T^{2} - \)\(40\!\cdots\!20\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \))(\( 1 - 30222468 T + 156063845146257 T^{2} + \)\(10\!\cdots\!16\)\( T^{3} + \)\(78\!\cdots\!51\)\( T^{4} - \)\(76\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!07\)\( T^{6} \))
$47$ (\( 1 + 14012052 T + 1119130473102767 T^{2} \))(\( 1 + 48278484 T + 2529195823681630 T^{2} + \)\(54\!\cdots\!28\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 - 85572408 T + 3865344002776030 T^{2} - \)\(95\!\cdots\!36\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \))(\( 1 - 39062520 T + 3179627012062221 T^{2} - \)\(73\!\cdots\!80\)\( T^{3} + \)\(35\!\cdots\!07\)\( T^{4} - \)\(48\!\cdots\!80\)\( T^{5} + \)\(14\!\cdots\!63\)\( T^{6} \))
$53$ (\( 1 + 97439910 T + 3299763591802133 T^{2} \))(\( 1 + 108980352 T + 6796561315809670 T^{2} + \)\(35\!\cdots\!16\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))(\( 1 - 26565324 T + 1785888321799630 T^{2} - \)\(87\!\cdots\!92\)\( T^{3} + \)\(10\!\cdots\!89\)\( T^{4} \))(\( 1 + 38081742 T + 6630432600967467 T^{2} + \)\(12\!\cdots\!36\)\( T^{3} + \)\(21\!\cdots\!11\)\( T^{4} + \)\(41\!\cdots\!38\)\( T^{5} + \)\(35\!\cdots\!37\)\( T^{6} \))
$59$ (\( 1 - 110798304 T + 8662995818654939 T^{2} \))(\( 1 + 188376804 T + 23346971643078934 T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( 1 - 115200960 T + 19706779116773158 T^{2} - \)\(99\!\cdots\!40\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( 1 - 32289756 T + 18651472724075649 T^{2} - \)\(60\!\cdots\!16\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} - \)\(24\!\cdots\!76\)\( T^{5} + \)\(65\!\cdots\!19\)\( T^{6} \))
$61$ (\( 1 + 93816682 T + 11694146092834141 T^{2} \))(\( 1 - 19722092 T + 14767573710118686 T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))(\( 1 + 142820204 T + 14471165594380686 T^{2} + \)\(16\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} \))(\( 1 - 131122194 T + 33623318056565955 T^{2} - \)\(30\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!55\)\( T^{4} - \)\(17\!\cdots\!14\)\( T^{5} + \)\(15\!\cdots\!21\)\( T^{6} \))
$67$ (\( 1 + 122446456 T + 27206534396294947 T^{2} \))(\( 1 - 70274396 T - 15240809102766810 T^{2} - \)\(19\!\cdots\!12\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))(\( 1 - 27521392 T + 42449494594232310 T^{2} - \)\(74\!\cdots\!24\)\( T^{3} + \)\(74\!\cdots\!09\)\( T^{4} \))(\( 1 - 203976852 T + 63042478662006489 T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + \)\(17\!\cdots\!83\)\( T^{4} - \)\(15\!\cdots\!68\)\( T^{5} + \)\(20\!\cdots\!23\)\( T^{6} \))
$71$ (\( 1 - 206197398 T + 45848500718449031 T^{2} \))(\( 1 + 382044186 T + 120425517984233086 T^{2} + \)\(17\!\cdots\!66\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( 1 - 38070180 T - 34252321852949618 T^{2} - \)\(17\!\cdots\!80\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} \))(\( 1 - 289658820 T + 132124660051052373 T^{2} - \)\(26\!\cdots\!40\)\( T^{3} + \)\(60\!\cdots\!63\)\( T^{4} - \)\(60\!\cdots\!20\)\( T^{5} + \)\(96\!\cdots\!91\)\( T^{6} \))
$73$ (\( 1 - 250337558 T + 58871586708267913 T^{2} \))(\( 1 - 191785896 T + 124167221244272702 T^{2} - \)\(11\!\cdots\!48\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))(\( 1 + 2095316 T + 100469358493934310 T^{2} + \)\(12\!\cdots\!08\)\( T^{3} + \)\(34\!\cdots\!69\)\( T^{4} \))(\( 1 - 341334582 T + 133520714771776167 T^{2} - \)\(35\!\cdots\!96\)\( T^{3} + \)\(78\!\cdots\!71\)\( T^{4} - \)\(11\!\cdots\!58\)\( T^{5} + \)\(20\!\cdots\!97\)\( T^{6} \))
$79$ (\( 1 + 38314852 T + 119851595982618319 T^{2} \))(\( 1 + 72592148 T + 239185581484447902 T^{2} + \)\(87\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))(\( 1 - 435097048 T + 140284960617569694 T^{2} - \)\(52\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} \))(\( 1 + 248369688 T + 280918275520438125 T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!75\)\( T^{4} + \)\(35\!\cdots\!68\)\( T^{5} + \)\(17\!\cdots\!59\)\( T^{6} \))
$83$ (\( 1 + 514086924 T + 186940255267540403 T^{2} \))(\( 1 - 187994232 T + 258704825037511030 T^{2} - \)\(35\!\cdots\!96\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))(\( 1 - 264288744 T - 19361594307718730 T^{2} - \)\(49\!\cdots\!32\)\( T^{3} + \)\(34\!\cdots\!09\)\( T^{4} \))(\( 1 + 105413076 T + 344029310499723081 T^{2} + \)\(42\!\cdots\!04\)\( T^{3} + \)\(64\!\cdots\!43\)\( T^{4} + \)\(36\!\cdots\!84\)\( T^{5} + \)\(65\!\cdots\!27\)\( T^{6} \))
$89$ (\( 1 + 1061294916 T + 350356403707485209 T^{2} \))(\( 1 - 42930954 T + 403096805893580370 T^{2} - \)\(15\!\cdots\!86\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))(\( 1 - 642673776 T + 754914610596232462 T^{2} - \)\(22\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} \))(\( 1 - 849077098 T + 1099697730140654375 T^{2} - \)\(58\!\cdots\!40\)\( T^{3} + \)\(38\!\cdots\!75\)\( T^{4} - \)\(10\!\cdots\!38\)\( T^{5} + \)\(43\!\cdots\!29\)\( T^{6} \))
$97$ (\( 1 + 73841578 T + 760231058654565217 T^{2} \))(\( 1 - 1726854096 T + 1929285565356657566 T^{2} - \)\(13\!\cdots\!32\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 - 345361228 T + 1029300318136190550 T^{2} - \)\(26\!\cdots\!76\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \))(\( 1 - 489867054 T + 498400224610481103 T^{2} - \)\(51\!\cdots\!08\)\( T^{3} + \)\(37\!\cdots\!51\)\( T^{4} - \)\(28\!\cdots\!06\)\( T^{5} + \)\(43\!\cdots\!13\)\( T^{6} \))
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