Properties

Label 22.4.c
Level 22
Weight 4
Character orbit c
Rep. character \(\chi_{22}(3,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 12
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 22.c (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(22, [\chi])\).

Total New Old
Modular forms 44 12 32
Cusp forms 28 12 16
Eisenstein series 16 0 16

Trace form

\( 12q + 2q^{2} + 2q^{3} - 12q^{4} + 8q^{5} + 42q^{6} + 24q^{7} + 8q^{8} - 145q^{9} + O(q^{10}) \) \( 12q + 2q^{2} + 2q^{3} - 12q^{4} + 8q^{5} + 42q^{6} + 24q^{7} + 8q^{8} - 145q^{9} - 104q^{10} - 111q^{11} - 72q^{12} + 98q^{13} + 52q^{14} + 224q^{15} - 48q^{16} + 184q^{17} + 304q^{18} - 213q^{19} + 32q^{20} - 280q^{21} - 22q^{22} + 404q^{23} + 168q^{24} + 109q^{25} - 12q^{26} + 245q^{27} - 104q^{28} - 392q^{29} - 396q^{30} + 96q^{31} - 128q^{32} - 1155q^{33} - 516q^{34} - 662q^{35} - 40q^{36} - 12q^{37} - 548q^{38} - 448q^{39} + 224q^{40} + 708q^{41} + 1940q^{42} + 2602q^{43} + 1016q^{44} + 2404q^{45} - 424q^{46} - 818q^{47} + 32q^{48} - 661q^{49} + 166q^{50} - 1579q^{51} - 408q^{52} - 526q^{53} - 2288q^{54} + 26q^{55} + 128q^{56} - 769q^{57} - 916q^{58} - 955q^{59} - 1224q^{60} + 718q^{61} + 2020q^{62} + 1064q^{63} - 192q^{64} + 44q^{65} + 2656q^{66} + 490q^{67} + 736q^{68} - 554q^{69} + 24q^{70} - 1550q^{71} - 824q^{72} - 1640q^{73} - 220q^{74} - 3561q^{75} - 712q^{76} + 498q^{77} - 2320q^{78} + 748q^{79} - 512q^{80} + 3006q^{81} + 62q^{82} + 2829q^{83} + 2400q^{84} - 148q^{85} + 2186q^{86} + 3792q^{87} + 472q^{88} - 474q^{89} + 1640q^{90} + 4278q^{91} - 1624q^{92} + 7052q^{93} - 1040q^{94} - 720q^{95} - 128q^{96} - 3109q^{97} - 4776q^{98} - 7867q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(22, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
22.4.c.a \(4\) \(1.298\) \(\Q(\zeta_{10})\) None \(-2\) \(-1\) \(3\) \(25\) \(q-2\zeta_{10}q^{2}+(-3+3\zeta_{10}+8\zeta_{10}^{3})q^{3}+\cdots\)
22.4.c.b \(8\) \(1.298\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(4\) \(3\) \(5\) \(-1\) \(q-2\beta _{3}q^{2}+(1+\beta _{3}-\beta _{4}-\beta _{5})q^{3}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(22, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(22, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(11, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} \))(\( ( 1 - 2 T + 4 T^{2} - 8 T^{3} + 16 T^{4} )^{2} \))
$3$ (\( 1 + T + 49 T^{2} - 83 T^{3} + 1204 T^{4} - 2241 T^{5} + 35721 T^{6} + 19683 T^{7} + 531441 T^{8} \))(\( 1 - 3 T - 12 T^{2} + 158 T^{3} - 1228 T^{4} + 6473 T^{5} + 959 T^{6} - 225344 T^{7} + 1236232 T^{8} - 6084288 T^{9} + 699111 T^{10} + 127408059 T^{11} - 652609548 T^{12} + 2267127306 T^{13} - 4649045868 T^{14} - 31381059609 T^{15} + 282429536481 T^{16} \))
$5$ (\( 1 - 3 T - 121 T^{2} + 123 T^{3} + 15376 T^{4} + 15375 T^{5} - 1890625 T^{6} - 5859375 T^{7} + 244140625 T^{8} \))(\( 1 - 5 T - 104 T^{2} - 1630 T^{3} + 10006 T^{4} + 283885 T^{5} + 2835571 T^{6} - 31638350 T^{7} - 387953844 T^{8} - 3954793750 T^{9} + 44305796875 T^{10} + 554462890625 T^{11} + 2442871093750 T^{12} - 49743652343750 T^{13} - 396728515625000 T^{14} - 2384185791015625 T^{15} + 59604644775390625 T^{16} \))
$7$ (\( 1 - 25 T + 117 T^{2} - 6065 T^{3} + 231284 T^{4} - 2080295 T^{5} + 13764933 T^{6} - 1008840175 T^{7} + 13841287201 T^{8} \))(\( 1 + T + 12 T^{2} - 1104 T^{3} - 87888 T^{4} - 922401 T^{5} + 28062801 T^{6} + 211635072 T^{7} + 13215739812 T^{8} + 72590829696 T^{9} + 3301560474849 T^{10} - 37222207450407 T^{11} - 1216483049521488 T^{12} - 5241307906977072 T^{13} + 19540963174925388 T^{14} + 558545864083284007 T^{15} + \)\(19\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 - 44 T + 726 T^{2} - 58564 T^{3} + 1771561 T^{4} \))(\( 1 + 155 T + 13111 T^{2} + 755095 T^{3} + 31999176 T^{4} + 1005031445 T^{5} + 23226936271 T^{6} + 365481892105 T^{7} + 3138428376721 T^{8} \))
$13$ (\( 1 - 91 T + 1299 T^{2} + 43003 T^{3} - 510136 T^{4} + 94477591 T^{5} + 6270024891 T^{6} - 965009442943 T^{7} + 23298085122481 T^{8} \))(\( 1 - 7 T + 1930 T^{2} + 79384 T^{3} + 10224596 T^{4} + 326819897 T^{5} + 10009625561 T^{6} + 1235524128730 T^{7} + 47709276128944 T^{8} + 2714446510819810 T^{9} + 48314550744464849 T^{10} + 3465761392820424581 T^{11} + \)\(23\!\cdots\!76\)\( T^{12} + \)\(40\!\cdots\!88\)\( T^{13} + \)\(21\!\cdots\!70\)\( T^{14} - \)\(17\!\cdots\!91\)\( T^{15} + \)\(54\!\cdots\!61\)\( T^{16} \))
$17$ (\( 1 - 23 T + 6011 T^{2} - 260829 T^{3} + 20906864 T^{4} - 1281452877 T^{5} + 145090927259 T^{6} - 2727521159431 T^{7} + 582622237229761 T^{8} \))(\( 1 - 161 T + 5020 T^{2} + 211188 T^{3} + 21130156 T^{4} - 1323457999 T^{5} - 265862283281 T^{6} + 13237803594360 T^{7} + 277440196391544 T^{8} + 65037329059090680 T^{9} - 6417269207192683889 T^{10} - \)\(15\!\cdots\!03\)\( T^{11} + \)\(12\!\cdots\!16\)\( T^{12} + \)\(60\!\cdots\!84\)\( T^{13} + \)\(70\!\cdots\!80\)\( T^{14} - \)\(11\!\cdots\!37\)\( T^{15} + \)\(33\!\cdots\!21\)\( T^{16} \))
$19$ (\( 1 - 59 T - 5223 T^{2} + 483473 T^{3} + 10576100 T^{4} + 3316141307 T^{5} - 245720636463 T^{6} - 19038574168961 T^{7} + 2213314919066161 T^{8} \))(\( 1 + 272 T + 33201 T^{2} + 3039584 T^{3} + 292957015 T^{4} + 21884364992 T^{5} + 909384182595 T^{6} + 32108296725040 T^{7} + 2715838419478696 T^{8} + 220230807237049360 T^{9} + 42782780037646641195 T^{10} + \)\(70\!\cdots\!68\)\( T^{11} + \)\(64\!\cdots\!15\)\( T^{12} + \)\(46\!\cdots\!16\)\( T^{13} + \)\(34\!\cdots\!41\)\( T^{14} + \)\(19\!\cdots\!68\)\( T^{15} + \)\(48\!\cdots\!21\)\( T^{16} \))
$23$ (\( ( 1 + 112 T + 26750 T^{2} + 1362704 T^{3} + 148035889 T^{4} )^{2} \))(\( ( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 97855798574 T^{5} + 9150986514424 T^{6} - 565561935699382 T^{7} + 21914624432020321 T^{8} )^{2} \))
$29$ (\( 1 + 425 T + 71171 T^{2} + 10071255 T^{3} + 1728423176 T^{4} + 245627838195 T^{5} + 42334170578891 T^{6} + 6165537039744325 T^{7} + 353814783205469041 T^{8} \))(\( 1 - 33 T + 14176 T^{2} + 2091144 T^{3} + 419909220 T^{4} + 81926257017 T^{5} + 7198320043835 T^{6} + 2719607629962300 T^{7} + 63828590693142176 T^{8} + 66328510487150534700 T^{9} + \)\(42\!\cdots\!35\)\( T^{10} + \)\(11\!\cdots\!73\)\( T^{11} + \)\(14\!\cdots\!20\)\( T^{12} + \)\(18\!\cdots\!56\)\( T^{13} + \)\(29\!\cdots\!36\)\( T^{14} - \)\(16\!\cdots\!57\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \))
$31$ (\( 1 + 227 T + 63923 T^{2} + 11938579 T^{3} + 2979456040 T^{4} + 355662206989 T^{5} + 56731897800563 T^{6} + 6001794230472317 T^{7} + 787662783788549761 T^{8} \))(\( 1 - 323 T - 4098 T^{2} + 5799192 T^{3} + 305184598 T^{4} - 93424183073 T^{5} + 36115996851965 T^{6} - 3925905573337100 T^{7} - 564631402757797204 T^{8} - \)\(11\!\cdots\!00\)\( T^{9} + \)\(32\!\cdots\!65\)\( T^{10} - \)\(24\!\cdots\!83\)\( T^{11} + \)\(24\!\cdots\!78\)\( T^{12} + \)\(13\!\cdots\!92\)\( T^{13} - \)\(28\!\cdots\!18\)\( T^{14} - \)\(67\!\cdots\!13\)\( T^{15} + \)\(62\!\cdots\!21\)\( T^{16} \))
$37$ (\( 1 + 61 T - 8617 T^{2} + 10210715 T^{3} + 2977938736 T^{4} + 517203346895 T^{5} - 22108864466353 T^{6} + 7927666127499697 T^{7} + 6582952005840035281 T^{8} \))(\( 1 - 49 T - 7868 T^{2} + 4260746 T^{3} - 566286138 T^{4} + 1042572020189 T^{5} + 91349107260671 T^{6} - 14416097444880138 T^{7} + 5705343181934634812 T^{8} - \)\(73\!\cdots\!14\)\( T^{9} + \)\(23\!\cdots\!39\)\( T^{10} + \)\(13\!\cdots\!53\)\( T^{11} - \)\(37\!\cdots\!78\)\( T^{12} + \)\(14\!\cdots\!78\)\( T^{13} - \)\(13\!\cdots\!72\)\( T^{14} - \)\(41\!\cdots\!13\)\( T^{15} + \)\(43\!\cdots\!61\)\( T^{16} \))
$41$ (\( 1 - 347 T - 5917 T^{2} - 2642769 T^{3} + 5378594000 T^{4} - 182142282249 T^{5} - 28106366793997 T^{6} - 113601531234704467 T^{7} + 22563490300366186081 T^{8} \))(\( 1 - 361 T - 19140 T^{2} + 12862580 T^{3} + 3853225260 T^{4} - 605378813703 T^{5} - 283363657590617 T^{6} + 99783171614693080 T^{7} - 22290482644766813160 T^{8} + \)\(68\!\cdots\!80\)\( T^{9} - \)\(13\!\cdots\!97\)\( T^{10} - \)\(19\!\cdots\!83\)\( T^{11} + \)\(86\!\cdots\!60\)\( T^{12} + \)\(20\!\cdots\!80\)\( T^{13} - \)\(20\!\cdots\!40\)\( T^{14} - \)\(26\!\cdots\!01\)\( T^{15} + \)\(50\!\cdots\!61\)\( T^{16} \))
$43$ (\( ( 1 - 580 T + 237334 T^{2} - 46114060 T^{3} + 6321363049 T^{4} )^{2} \))(\( ( 1 - 721 T + 420117 T^{2} - 154459221 T^{3} + 51447883420 T^{4} - 12280589284047 T^{5} + 2655712080056733 T^{6} - 362369273206463803 T^{7} + 39959630797262576401 T^{8} )^{2} \))
$47$ (\( 1 - 251 T - 62767 T^{2} + 35545485 T^{3} - 2090662324 T^{4} + 3690438889155 T^{5} - 676579008555343 T^{6} - 280901748748794517 T^{7} + \)\(11\!\cdots\!41\)\( T^{8} \))(\( 1 + 1069 T + 301070 T^{2} - 43368042 T^{3} - 24642745634 T^{4} + 6223420821981 T^{5} + 4635059307315839 T^{6} + 746453682627128500 T^{7} + 29698413240018564184 T^{8} + \)\(77\!\cdots\!00\)\( T^{9} + \)\(49\!\cdots\!31\)\( T^{10} + \)\(69\!\cdots\!27\)\( T^{11} - \)\(28\!\cdots\!94\)\( T^{12} - \)\(52\!\cdots\!06\)\( T^{13} + \)\(37\!\cdots\!30\)\( T^{14} + \)\(13\!\cdots\!43\)\( T^{15} + \)\(13\!\cdots\!81\)\( T^{16} \))
$53$ (\( 1 + 245 T + 135263 T^{2} + 62167475 T^{3} + 40936657664 T^{4} + 9255307175575 T^{5} + 2998017979391927 T^{6} + 808442079991522585 T^{7} + \)\(49\!\cdots\!41\)\( T^{8} \))(\( 1 + 281 T - 343802 T^{2} - 60864958 T^{3} + 33213391926 T^{4} + 4932564267181 T^{5} + 8874412003121445 T^{6} + 107096647673733916 T^{7} - \)\(25\!\cdots\!56\)\( T^{8} + \)\(15\!\cdots\!32\)\( T^{9} + \)\(19\!\cdots\!05\)\( T^{10} + \)\(16\!\cdots\!73\)\( T^{11} + \)\(16\!\cdots\!66\)\( T^{12} - \)\(44\!\cdots\!06\)\( T^{13} - \)\(37\!\cdots\!78\)\( T^{14} + \)\(45\!\cdots\!93\)\( T^{15} + \)\(24\!\cdots\!81\)\( T^{16} \))
$59$ (\( 1 + 827 T + 139925 T^{2} + 42419807 T^{3} + 50592146204 T^{4} + 8712137541853 T^{5} + 5902111169716925 T^{6} + 7164297542027634553 T^{7} + \)\(17\!\cdots\!81\)\( T^{8} \))(\( 1 + 128 T - 87119 T^{2} - 50674984 T^{3} + 39669673335 T^{4} + 496570131448 T^{5} - 4049042036547885 T^{6} - 22790010341236160 T^{7} + \)\(27\!\cdots\!36\)\( T^{8} - \)\(46\!\cdots\!40\)\( T^{9} - \)\(17\!\cdots\!85\)\( T^{10} + \)\(43\!\cdots\!72\)\( T^{11} + \)\(70\!\cdots\!35\)\( T^{12} - \)\(18\!\cdots\!16\)\( T^{13} - \)\(65\!\cdots\!99\)\( T^{14} + \)\(19\!\cdots\!52\)\( T^{15} + \)\(31\!\cdots\!61\)\( T^{16} \))
$61$ (\( 1 - 1335 T + 583559 T^{2} - 108492345 T^{3} + 24155367616 T^{4} - 24625700960445 T^{5} + 30065178141730799 T^{6} - 15611685033933578235 T^{7} + \)\(26\!\cdots\!21\)\( T^{8} \))(\( 1 + 617 T - 60798 T^{2} - 112995618 T^{3} + 22253617738 T^{4} + 7295748837377 T^{5} - 13214872371636415 T^{6} + 3429112403997460700 T^{7} + \)\(68\!\cdots\!36\)\( T^{8} + \)\(77\!\cdots\!00\)\( T^{9} - \)\(68\!\cdots\!15\)\( T^{10} + \)\(85\!\cdots\!57\)\( T^{11} + \)\(59\!\cdots\!98\)\( T^{12} - \)\(68\!\cdots\!18\)\( T^{13} - \)\(83\!\cdots\!38\)\( T^{14} + \)\(19\!\cdots\!37\)\( T^{15} + \)\(70\!\cdots\!41\)\( T^{16} \))
$67$ (\( ( 1 + 44 T + 585190 T^{2} + 13233572 T^{3} + 90458382169 T^{4} )^{2} \))(\( ( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} + \)\(81\!\cdots\!61\)\( T^{8} )^{2} \))
$71$ (\( 1 + 1665 T + 687799 T^{2} - 442234575 T^{3} - 563174800784 T^{4} - 158280618972825 T^{5} + 88107247180579879 T^{6} + 76337753696217636615 T^{7} + \)\(16\!\cdots\!41\)\( T^{8} \))(\( 1 - 115 T - 138428 T^{2} + 42960000 T^{3} + 77925723838 T^{4} - 71653925274035 T^{5} + 12191505393571699 T^{6} + 3285417296443548450 T^{7} + \)\(11\!\cdots\!80\)\( T^{8} + \)\(11\!\cdots\!50\)\( T^{9} + \)\(15\!\cdots\!79\)\( T^{10} - \)\(32\!\cdots\!85\)\( T^{11} + \)\(12\!\cdots\!58\)\( T^{12} + \)\(25\!\cdots\!00\)\( T^{13} - \)\(29\!\cdots\!08\)\( T^{14} - \)\(86\!\cdots\!65\)\( T^{15} + \)\(26\!\cdots\!81\)\( T^{16} \))
$73$ (\( 1 + 153 T - 378613 T^{2} - 19574565 T^{3} + 149317325056 T^{4} - 7614838552605 T^{5} - 57297105417957157 T^{6} + 9007352766364990689 T^{7} + \)\(22\!\cdots\!21\)\( T^{8} \))(\( 1 + 1487 T + 142728 T^{2} - 1045087272 T^{3} - 569519703768 T^{4} + 366060407928453 T^{5} + 393291607153115139 T^{6} - 68019158899540158384 T^{7} - \)\(19\!\cdots\!28\)\( T^{8} - \)\(26\!\cdots\!28\)\( T^{9} + \)\(59\!\cdots\!71\)\( T^{10} + \)\(21\!\cdots\!89\)\( T^{11} - \)\(13\!\cdots\!28\)\( T^{12} - \)\(93\!\cdots\!04\)\( T^{13} + \)\(49\!\cdots\!32\)\( T^{14} + \)\(20\!\cdots\!51\)\( T^{15} + \)\(52\!\cdots\!41\)\( T^{16} \))
$79$ (\( 1 - 677 T + 1031235 T^{2} - 490717477 T^{3} + 653512317224 T^{4} - 241942854142603 T^{5} + 250680292194198435 T^{6} - 81139530480232601963 T^{7} + \)\(59\!\cdots\!41\)\( T^{8} \))(\( 1 - 71 T - 500222 T^{2} - 73013624 T^{3} + 350790961998 T^{4} + 97550067497959 T^{5} - 161731343227399015 T^{6} + 8534351363848547400 T^{7} + \)\(59\!\cdots\!76\)\( T^{8} + \)\(42\!\cdots\!00\)\( T^{9} - \)\(39\!\cdots\!15\)\( T^{10} + \)\(11\!\cdots\!21\)\( T^{11} + \)\(20\!\cdots\!18\)\( T^{12} - \)\(21\!\cdots\!76\)\( T^{13} - \)\(71\!\cdots\!42\)\( T^{14} - \)\(50\!\cdots\!09\)\( T^{15} + \)\(34\!\cdots\!81\)\( T^{16} \))
$83$ (\( 1 - 887 T + 364607 T^{2} - 628939715 T^{3} + 778142659496 T^{4} - 359619552820705 T^{5} + 119204748712950983 T^{6} - \)\(16\!\cdots\!61\)\( T^{7} + \)\(10\!\cdots\!61\)\( T^{8} \))(\( 1 - 1942 T + 875395 T^{2} + 857035884 T^{3} - 790010610449 T^{4} - 618864428088728 T^{5} + 845503237658086801 T^{6} + \)\(17\!\cdots\!30\)\( T^{7} - \)\(59\!\cdots\!96\)\( T^{8} + \)\(10\!\cdots\!10\)\( T^{9} + \)\(27\!\cdots\!69\)\( T^{10} - \)\(11\!\cdots\!84\)\( T^{11} - \)\(84\!\cdots\!89\)\( T^{12} + \)\(52\!\cdots\!88\)\( T^{13} + \)\(30\!\cdots\!55\)\( T^{14} - \)\(38\!\cdots\!86\)\( T^{15} + \)\(11\!\cdots\!21\)\( T^{16} \))
$89$ (\( ( 1 - 864 T + 492062 T^{2} - 609093216 T^{3} + 496981290961 T^{4} )^{2} \))(\( ( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1349696810654843 T^{5} + 1196181784307576095 T^{6} + \)\(38\!\cdots\!09\)\( T^{7} + \)\(24\!\cdots\!21\)\( T^{8} )^{2} \))
$97$ (\( 1 - 2019 T + 872763 T^{2} + 94786015 T^{3} + 88862307216 T^{4} + 86508636668095 T^{5} + 726987145937848827 T^{6} - \)\(15\!\cdots\!23\)\( T^{7} + \)\(69\!\cdots\!41\)\( T^{8} \))(\( 1 + 5128 T + 10343177 T^{2} + 9500759764 T^{3} + 1479496697751 T^{4} - 4959875190278012 T^{5} - 1475957592224572125 T^{6} + \)\(96\!\cdots\!12\)\( T^{7} + \)\(15\!\cdots\!64\)\( T^{8} + \)\(88\!\cdots\!76\)\( T^{9} - \)\(12\!\cdots\!25\)\( T^{10} - \)\(37\!\cdots\!04\)\( T^{11} + \)\(10\!\cdots\!91\)\( T^{12} + \)\(60\!\cdots\!52\)\( T^{13} + \)\(59\!\cdots\!53\)\( T^{14} + \)\(27\!\cdots\!16\)\( T^{15} + \)\(48\!\cdots\!81\)\( T^{16} \))
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