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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