Properties

Label 63.2.s
Level 63
Weight 2
Character orbit s
Rep. character \(\chi_{63}(47,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 12
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 63.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(63, [\chi])\).

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

Trace form

\( 12q - 3q^{2} - 3q^{3} + 5q^{4} - 6q^{5} - 6q^{6} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 12q - 3q^{2} - 3q^{3} + 5q^{4} - 6q^{5} - 6q^{6} - 2q^{7} + 3q^{9} - 6q^{10} - 3q^{12} + 3q^{13} + 3q^{14} + 6q^{15} - q^{16} - 9q^{17} - 27q^{18} - 6q^{19} - 6q^{20} + 24q^{21} + 2q^{22} + 24q^{24} - 6q^{25} + 6q^{26} + 27q^{27} - 2q^{28} - 24q^{29} - 18q^{30} - 15q^{31} + 39q^{32} - 6q^{33} - 6q^{34} - 12q^{36} - q^{37} + 54q^{38} + 18q^{39} - 6q^{41} - 21q^{42} + 2q^{43} + 27q^{44} + 6q^{45} - 4q^{46} + 15q^{47} - 15q^{48} - 12q^{49} - 9q^{50} - 24q^{51} - 24q^{53} + 36q^{54} - 54q^{56} - 27q^{57} + 2q^{58} - 18q^{59} + 6q^{60} + 36q^{61} + 24q^{62} + 6q^{63} + 4q^{64} + 6q^{65} - 33q^{66} - 6q^{67} - 48q^{68} - 12q^{69} - 18q^{70} + 27q^{72} - 6q^{73} - 33q^{75} + 48q^{77} + 15q^{78} + 12q^{79} - 45q^{80} - 57q^{81} - 30q^{83} + 69q^{84} + 9q^{85} + 39q^{87} + 22q^{88} + 27q^{89} + 51q^{90} - 15q^{91} + 30q^{92} + 12q^{93} - 3q^{94} + 27q^{95} + 12q^{96} + 3q^{97} + 21q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(63, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
63.2.s.a \(2\) \(0.503\) \(\Q(\sqrt{-3}) \) None \(-3\) \(-3\) \(-6\) \(-5\) \(q+(-1-\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
63.2.s.b \(10\) \(0.503\) 10.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(3\) \(q+(\beta _{4}+\beta _{7}+\beta _{8})q^{2}+(-\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4} \))(\( 1 + 3 T^{2} + 4 T^{4} - 6 T^{5} - 2 T^{6} + 3 T^{7} - 14 T^{8} + 39 T^{9} + T^{10} + 78 T^{11} - 56 T^{12} + 24 T^{13} - 32 T^{14} - 192 T^{15} + 256 T^{16} + 768 T^{18} + 1024 T^{20} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 9 T^{3} + 12 T^{4} + 9 T^{5} + 36 T^{6} - 81 T^{7} + 243 T^{10} \))
$5$ (\( ( 1 + 3 T + 5 T^{2} )^{2} \))(\( ( 1 + 16 T^{2} - 6 T^{3} + 127 T^{4} - 51 T^{5} + 635 T^{6} - 150 T^{7} + 2000 T^{8} + 3125 T^{10} )^{2} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 - 3 T + 16 T^{2} - 62 T^{3} + 220 T^{4} - 473 T^{5} + 1540 T^{6} - 3038 T^{7} + 5488 T^{8} - 7203 T^{9} + 16807 T^{10} \))
$11$ (\( 1 - 19 T^{2} + 121 T^{4} \))(\( 1 - 66 T^{2} + 2257 T^{4} - 51461 T^{6} + 855052 T^{8} - 10752323 T^{10} + 103461292 T^{12} - 753440501 T^{14} + 3998413177 T^{16} - 14147686146 T^{18} + 25937424601 T^{20} \))
$13$ (\( ( 1 - 2 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} ) \))(\( 1 - 6 T + 68 T^{2} - 336 T^{3} + 2292 T^{4} - 9399 T^{5} + 51837 T^{6} - 187401 T^{7} + 909867 T^{8} - 3004662 T^{9} + 13054461 T^{10} - 39060606 T^{11} + 153767523 T^{12} - 411719997 T^{13} + 1480516557 T^{14} - 3489782907 T^{15} + 11063046228 T^{16} - 21083501712 T^{17} + 55469689028 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \))
$17$ (\( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \))(\( 1 + 12 T + 26 T^{2} - 36 T^{3} + 1143 T^{4} + 5247 T^{5} - 21540 T^{6} - 73476 T^{7} + 337539 T^{8} - 599625 T^{9} - 13374333 T^{10} - 10193625 T^{11} + 97548771 T^{12} - 360987588 T^{13} - 1799042340 T^{14} + 7449989679 T^{15} + 27589241367 T^{16} - 14772192228 T^{17} + 181369693466 T^{18} + 1423054517964 T^{19} + 2015993900449 T^{20} \))
$19$ (\( ( 1 + T + 19 T^{2} )( 1 + 8 T + 19 T^{2} ) \))(\( 1 - 3 T + 71 T^{2} - 204 T^{3} + 2646 T^{4} - 8547 T^{5} + 74607 T^{6} - 258954 T^{7} + 1743726 T^{8} - 5989488 T^{9} + 35261703 T^{10} - 113800272 T^{11} + 629485086 T^{12} - 1776165486 T^{13} + 9722858847 T^{14} - 21163218153 T^{15} + 124483401126 T^{16} - 182349834756 T^{17} + 1205832975911 T^{18} - 968063093337 T^{19} + 6131066257801 T^{20} \))
$23$ (\( 1 - 19 T^{2} + 529 T^{4} \))(\( 1 - 165 T^{2} + 13144 T^{4} - 668429 T^{6} + 24050461 T^{8} - 639494549 T^{10} + 12722693869 T^{12} - 187053839789 T^{14} + 1945783725016 T^{16} - 12921312571365 T^{18} + 41426511213649 T^{20} \))
$29$ (\( 1 + 9 T + 56 T^{2} + 261 T^{3} + 841 T^{4} \))(\( 1 + 15 T + 150 T^{2} + 1125 T^{3} + 6691 T^{4} + 30108 T^{5} + 81631 T^{6} - 232971 T^{7} - 5137202 T^{8} - 44535417 T^{9} - 275752187 T^{10} - 1291527093 T^{11} - 4320386882 T^{12} - 5681929719 T^{13} + 57736055311 T^{14} + 617549674092 T^{15} + 3979962840811 T^{16} + 19406110847625 T^{17} + 75036961944150 T^{18} + 217607189638035 T^{19} + 420707233300201 T^{20} \))
$31$ (\( 1 + 6 T + 43 T^{2} + 186 T^{3} + 961 T^{4} \))(\( 1 + 9 T + 149 T^{2} + 1098 T^{3} + 10878 T^{4} + 60723 T^{5} + 461409 T^{6} + 2027286 T^{7} + 13421802 T^{8} + 50078664 T^{9} + 374531595 T^{10} + 1552438584 T^{11} + 12898351722 T^{12} + 60394877226 T^{13} + 426120901089 T^{14} + 1738447936173 T^{15} + 9654265041918 T^{16} + 30208850293878 T^{17} + 127080764578709 T^{18} + 237956599446039 T^{19} + 819628286980801 T^{20} \))
$37$ (\( 1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4} \))(\( 1 - 6 T - 97 T^{2} + 194 T^{3} + 7179 T^{4} + 3556 T^{5} - 323794 T^{6} - 533292 T^{7} + 10739317 T^{8} + 10946526 T^{9} - 345629139 T^{10} + 405021462 T^{11} + 14702124973 T^{12} - 27012839676 T^{13} - 606842086834 T^{14} + 246587111092 T^{15} + 18419349890211 T^{16} + 18416784163802 T^{17} - 340710507030337 T^{18} - 779770438770462 T^{19} + 4808584372417849 T^{20} \))
$41$ (\( 1 - 3 T - 32 T^{2} - 123 T^{3} + 1681 T^{4} \))(\( 1 + 9 T - 34 T^{2} - 747 T^{3} - 2085 T^{4} + 20394 T^{5} + 110775 T^{6} - 623979 T^{7} - 5992218 T^{8} + 18494757 T^{9} + 381591615 T^{10} + 758285037 T^{11} - 10072918458 T^{12} - 43005256659 T^{13} + 313023674775 T^{14} + 2362771363194 T^{15} - 9903967342485 T^{16} - 145481442589107 T^{17} - 271487457790114 T^{18} + 2946437409545649 T^{19} + 13422659310152401 T^{20} \))
$43$ (\( 1 + T - 42 T^{2} + 43 T^{3} + 1849 T^{4} \))(\( 1 - 3 T - 79 T^{2} + 1100 T^{3} + 1674 T^{4} - 79931 T^{5} + 324899 T^{6} + 3229902 T^{7} - 28512986 T^{8} - 52724394 T^{9} + 1438527201 T^{10} - 2267148942 T^{11} - 52720511114 T^{12} + 256799818314 T^{13} + 1110765026099 T^{14} - 11750531857433 T^{15} + 10581961744026 T^{16} + 299000472217700 T^{17} - 923367821930479 T^{18} - 1507777835810529 T^{19} + 21611482313284249 T^{20} \))
$47$ (\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 - 15 T - 49 T^{2} + 1068 T^{3} + 9486 T^{4} - 83265 T^{5} - 760914 T^{6} + 2887821 T^{7} + 57371082 T^{8} - 59131839 T^{9} - 3026317959 T^{10} - 2779196433 T^{11} + 126732720138 T^{12} + 299822239683 T^{13} - 3713017588434 T^{14} - 19096412007855 T^{15} + 102251636610894 T^{16} + 541073492654484 T^{17} - 1166753046426289 T^{18} - 16786957096541505 T^{19} + 52599132235830049 T^{20} \))
$53$ (\( 1 + 15 T + 128 T^{2} + 795 T^{3} + 2809 T^{4} \))(\( 1 + 9 T + 237 T^{2} + 1890 T^{3} + 28720 T^{4} + 208689 T^{5} + 2523571 T^{6} + 16852668 T^{7} + 178203742 T^{8} + 1088604978 T^{9} + 10361882797 T^{10} + 57696063834 T^{11} + 500574311278 T^{12} + 2508974653836 T^{13} + 19912189027651 T^{14} + 87272799238677 T^{15} + 636560451624880 T^{16} + 2220204054291930 T^{17} + 14755546627492557 T^{18} + 29697872326219197 T^{19} + 174887470365513049 T^{20} \))
$59$ (\( 1 - 59 T^{2} + 3481 T^{4} \))(\( 1 + 18 T - 55 T^{2} - 1536 T^{3} + 19971 T^{4} + 205494 T^{5} - 1764945 T^{6} - 8798931 T^{7} + 181121100 T^{8} + 308804295 T^{9} - 11121159681 T^{10} + 18219453405 T^{11} + 630482549100 T^{12} - 1807115649849 T^{13} - 21386475710145 T^{14} + 146912653898706 T^{15} + 842387437344411 T^{16} - 3822568680681984 T^{17} - 8075674068237655 T^{18} + 155933924735788902 T^{19} + 511116753300641401 T^{20} \))
$61$ (\( 1 - 24 T + 253 T^{2} - 1464 T^{3} + 3721 T^{4} \))(\( 1 - 12 T + 251 T^{2} - 2436 T^{3} + 34779 T^{4} - 347730 T^{5} + 3716049 T^{6} - 34819755 T^{7} + 301038894 T^{8} - 2709237273 T^{9} + 20488848807 T^{10} - 165263473653 T^{11} + 1120165724574 T^{12} - 7903422809655 T^{13} + 51451823602209 T^{14} - 293691471746730 T^{15} + 1791827099901219 T^{16} - 7655721548547156 T^{17} + 48118535562317531 T^{18} - 140329753114009692 T^{19} + 713342911662882601 T^{20} \))
$67$ (\( 1 - 4 T - 51 T^{2} - 268 T^{3} + 4489 T^{4} \))(\( 1 + 10 T - 182 T^{2} - 2448 T^{3} + 18867 T^{4} + 319605 T^{5} - 772530 T^{6} - 23213154 T^{7} - 12219093 T^{8} + 697872289 T^{9} + 3674653819 T^{10} + 46757443363 T^{11} - 54851508477 T^{12} - 6981657836502 T^{13} - 15567345506130 T^{14} + 431506734822735 T^{15} + 1706678296382523 T^{16} - 14836622009830704 T^{17} - 73904317315308662 T^{18} + 272065343962949470 T^{19} + 1822837804551761449 T^{20} \))
$71$ (\( 1 - 130 T^{2} + 5041 T^{4} \))(\( 1 - 351 T^{2} + 63037 T^{4} - 7800935 T^{6} + 747809113 T^{8} - 58386380555 T^{10} + 3769705738633 T^{12} - 198234871721735 T^{14} + 8075057597528077 T^{16} - 226659489467262111 T^{18} + 3255243551009881201 T^{20} \))
$73$ (\( 1 + 9 T + 100 T^{2} + 657 T^{3} + 5329 T^{4} \))(\( 1 - 3 T + 260 T^{2} - 771 T^{3} + 36639 T^{4} - 155724 T^{5} + 3663555 T^{6} - 21043473 T^{7} + 293486934 T^{8} - 2074196103 T^{9} + 21799556757 T^{10} - 151416315519 T^{11} + 1563991871286 T^{12} - 8186268736041 T^{13} + 104038517806755 T^{14} - 322827000748332 T^{15} + 5544734717002671 T^{16} - 8517544258223787 T^{17} + 209679623892461060 T^{18} - 176614760124803739 T^{19} + 4297625829703557649 T^{20} \))
$79$ (\( 1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4} \))(\( 1 - 20 T + 46 T^{2} - 144 T^{3} + 21153 T^{4} - 101181 T^{5} - 106944 T^{6} - 9264000 T^{7} + 7962453 T^{8} + 230795113 T^{9} + 4723714795 T^{10} + 18232813927 T^{11} + 49693669173 T^{12} - 4567513296000 T^{13} - 4165477462464 T^{14} - 311339643507219 T^{15} + 5142028946635713 T^{16} - 2765362894006896 T^{17} + 69787005255701806 T^{18} - 2397031919652366380 T^{19} + 9468276082626847201 T^{20} \))
$83$ (\( 1 + 15 T + 142 T^{2} + 1245 T^{3} + 6889 T^{4} \))(\( 1 + 15 T - 136 T^{2} - 1773 T^{3} + 22674 T^{4} + 93717 T^{5} - 3687774 T^{6} - 10067337 T^{7} + 346135869 T^{8} + 496605294 T^{9} - 27460905396 T^{10} + 41218239402 T^{11} + 2384530001541 T^{12} - 5756372421219 T^{13} - 175015562267454 T^{14} + 369155071940031 T^{15} + 7413046025768706 T^{16} - 48112218404608671 T^{17} - 306311743570909576 T^{18} + 2804103829013106045 T^{19} + 15516041187205853449 T^{20} \))
$89$ (\( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} \))(\( 1 - 24 T + 44 T^{2} + 2592 T^{3} - 1287 T^{4} - 278721 T^{5} - 235110 T^{6} + 13705920 T^{7} + 186157425 T^{8} - 904992183 T^{9} - 14602879521 T^{10} - 80544304287 T^{11} + 1474552963425 T^{12} + 9662248716480 T^{13} - 14751328281510 T^{14} - 1556394633684729 T^{15} - 639614921466807 T^{16} + 114647620049211168 T^{17} + 173209907450891564 T^{18} - 8408553688979645016 T^{19} + 31181719929966183601 T^{20} \))
$97$ (\( 1 + 3 T + 100 T^{2} + 291 T^{3} + 9409 T^{4} \))(\( 1 - 6 T + 311 T^{2} - 1794 T^{3} + 51903 T^{4} - 385032 T^{5} + 6010353 T^{6} - 63309837 T^{7} + 574248354 T^{8} - 8264282925 T^{9} + 54719955099 T^{10} - 801635443725 T^{11} + 5403102762786 T^{12} - 57781178864301 T^{13} + 532092229646193 T^{14} - 3306400793833224 T^{15} + 43233745971829887 T^{16} - 144952122353734722 T^{17} + 2437441847851234871 T^{18} - 4561386351927391302 T^{19} + 73742412689492826049 T^{20} \))
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