Properties

Label 252.2.i
Level 252
Weight 2
Character orbit i
Rep. character \(\chi_{252}(25,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 16
Newform subspaces 2
Sturm bound 96
Trace bound 1

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Defining parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(252, [\chi])\).

Total New Old
Modular forms 108 16 92
Cusp forms 84 16 68
Eisenstein series 24 0 24

Trace form

\( 16q - 4q^{5} + q^{7} - 2q^{9} + O(q^{10}) \) \( 16q - 4q^{5} + q^{7} - 2q^{9} - 2q^{11} - q^{13} + 7q^{15} - 5q^{17} + 2q^{19} - 2q^{21} + 7q^{23} - 8q^{25} + 9q^{27} + 2q^{29} - 4q^{31} + 8q^{33} - 11q^{35} - q^{37} + 7q^{39} - 24q^{41} + 2q^{43} - 4q^{45} + 12q^{47} + 7q^{49} - 13q^{51} - 18q^{53} - 12q^{55} - 3q^{57} + 14q^{59} + 26q^{61} + 21q^{63} - 18q^{65} + 14q^{67} - 43q^{69} - 14q^{71} + 14q^{73} - 47q^{75} - 43q^{77} + 2q^{79} - 38q^{81} - 26q^{83} - 6q^{85} + 28q^{87} - 21q^{89} + 5q^{91} + 31q^{93} + 76q^{95} - q^{97} - 17q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(252, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
252.2.i.a \(2\) \(2.012\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(-2\) \(-5\) \(q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-3+\zeta_{6})q^{7}+\cdots\)
252.2.i.b \(14\) \(2.012\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(3\) \(-2\) \(6\) \(q+(-\beta _{1}+\beta _{3})q^{3}-\beta _{9}q^{5}+\beta _{10}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(252, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(252, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - 3 T + 7 T^{2} - 15 T^{3} + 34 T^{4} - 78 T^{5} + 144 T^{6} - 252 T^{7} + 432 T^{8} - 702 T^{9} + 918 T^{10} - 1215 T^{11} + 1701 T^{12} - 2187 T^{13} + 2187 T^{14} \))
$5$ (\( 1 + 2 T - T^{2} + 10 T^{3} + 25 T^{4} \))(\( 1 + 2 T - 11 T^{2} - 66 T^{3} - 25 T^{4} + 533 T^{5} + 1540 T^{6} - 603 T^{7} - 11032 T^{8} - 21691 T^{9} + 19925 T^{10} + 156945 T^{11} + 233805 T^{12} - 385788 T^{13} - 1886454 T^{14} - 1928940 T^{15} + 5845125 T^{16} + 19618125 T^{17} + 12453125 T^{18} - 67784375 T^{19} - 172375000 T^{20} - 47109375 T^{21} + 601562500 T^{22} + 1041015625 T^{23} - 244140625 T^{24} - 3222656250 T^{25} - 2685546875 T^{26} + 2441406250 T^{27} + 6103515625 T^{28} \))
$7$ (\( 1 + 5 T + 7 T^{2} \))(\( 1 - 6 T + 20 T^{2} - 77 T^{3} + 309 T^{4} - 961 T^{5} + 2706 T^{6} - 7572 T^{7} + 18942 T^{8} - 47089 T^{9} + 105987 T^{10} - 184877 T^{11} + 336140 T^{12} - 705894 T^{13} + 823543 T^{14} \))
$11$ (\( 1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4} \))(\( 1 - 2 T - 32 T^{2} + 42 T^{3} + 722 T^{4} - 614 T^{5} - 9113 T^{6} - 2733 T^{7} + 72959 T^{8} + 126076 T^{9} + 256457 T^{10} - 2291538 T^{11} - 13152543 T^{12} + 8654619 T^{13} + 213521742 T^{14} + 95200809 T^{15} - 1591457703 T^{16} - 3050037078 T^{17} + 3754786937 T^{18} + 20304665876 T^{19} + 129251318999 T^{20} - 53258438343 T^{21} - 1953452482553 T^{22} - 1447779882274 T^{23} + 18726820561922 T^{24} + 11983090165662 T^{25} - 100429708055072 T^{26} - 69045424287862 T^{27} + 379749833583241 T^{28} \))
$13$ (\( 1 + 3 T - 4 T^{2} + 39 T^{3} + 169 T^{4} \))(\( 1 - 2 T - 25 T^{2} - 20 T^{3} + 172 T^{4} + 1281 T^{5} + 1882 T^{6} - 1142 T^{7} - 27931 T^{8} - 309997 T^{9} + 9092 T^{10} + 1859069 T^{11} + 4530175 T^{12} + 5185603 T^{13} - 73974588 T^{14} + 67412839 T^{15} + 765599575 T^{16} + 4084374593 T^{17} + 259676612 T^{18} - 115099716121 T^{19} - 134817602179 T^{20} - 71658806414 T^{21} + 1535205216922 T^{22} + 13584363696813 T^{23} + 23711660598028 T^{24} - 35843207880740 T^{25} - 582452128062025 T^{26} - 605750213184506 T^{27} + 3937376385699289 T^{28} \))
$17$ (\( 1 + 7 T + 32 T^{2} + 119 T^{3} + 289 T^{4} \))(\( 1 - 2 T - 65 T^{2} + 210 T^{3} + 2087 T^{4} - 9143 T^{5} - 37340 T^{6} + 240381 T^{7} + 293834 T^{8} - 4176065 T^{9} + 3382763 T^{10} + 47662233 T^{11} - 157347285 T^{12} - 273515658 T^{13} + 3170983122 T^{14} - 4649766186 T^{15} - 45473365365 T^{16} + 234164550729 T^{17} + 282531748523 T^{18} - 5929415122705 T^{19} + 7092438449546 T^{20} + 98637620554413 T^{21} - 260474782846940 T^{22} - 1084248954812071 T^{23} + 4207379270237063 T^{24} + 7197098224602930 T^{25} - 37870445419934465 T^{26} - 19809156065811874 T^{27} + 168377826559400929 T^{28} \))
$19$ (\( 1 + 5 T + 6 T^{2} + 95 T^{3} + 361 T^{4} \))(\( 1 - 7 T - 54 T^{2} + 381 T^{3} + 1875 T^{4} - 9873 T^{5} - 65652 T^{6} + 221430 T^{7} + 1870425 T^{8} - 4319703 T^{9} - 46476858 T^{10} + 61637031 T^{11} + 1073881146 T^{12} - 457871775 T^{13} - 21789737442 T^{14} - 8699563725 T^{15} + 387671093706 T^{16} + 422768395629 T^{17} - 6056910611418 T^{18} - 10696012278597 T^{19} + 87995791969425 T^{20} + 197930019166770 T^{21} - 1115004880767732 T^{22} - 3185895640172067 T^{23} + 11495749233376875 T^{24} + 44382788640221439 T^{25} - 119519005629572694 T^{26} - 294370884235799413 T^{27} + 799006685782884121 T^{28} \))
$23$ (\( 1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4} \))(\( 1 - 11 T - 8 T^{2} + 333 T^{3} + 401 T^{4} - 3407 T^{5} - 42005 T^{6} + 36987 T^{7} + 1121822 T^{8} + 103015 T^{9} - 29774440 T^{10} + 52947609 T^{11} + 527481093 T^{12} - 1421356596 T^{13} - 4817982924 T^{14} - 32691201708 T^{15} + 279037498197 T^{16} + 644213558703 T^{17} - 8332109064040 T^{18} + 663039874145 T^{19} + 166069917069758 T^{20} + 125934278808189 T^{21} - 3289452936728405 T^{22} - 6136527117604441 T^{23} + 16612030996673249 T^{24} + 317285649385337691 T^{25} - 175316995456162568 T^{26} - 5544399981301141213 T^{27} + 11592836324538749809 T^{28} \))
$29$ (\( 1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4} \))(\( 1 - T - 89 T^{2} + 606 T^{3} + 3413 T^{4} - 45595 T^{5} + 49603 T^{6} + 1643802 T^{7} - 7893892 T^{8} - 19552444 T^{9} + 271585946 T^{10} - 420585531 T^{11} - 3441402105 T^{12} + 10956432363 T^{13} + 10513309734 T^{14} + 317736538527 T^{15} - 2894219170305 T^{16} - 10257660515559 T^{17} + 192087579472826 T^{18} - 401043092198156 T^{19} - 4695471055055332 T^{20} + 28355381176486818 T^{21} + 24813722822104483 T^{22} - 661453320769747055 T^{23} + 1435873787253586013 T^{24} + 7393508918017732374 T^{25} - 31489515705286744649 T^{26} - 10260628712958602189 T^{27} + \)\(29\!\cdots\!81\)\( T^{28} \))
$31$ (\( ( 1 + 3 T + 31 T^{2} )^{2} \))(\( ( 1 - T + 86 T^{2} - 194 T^{3} + 4879 T^{4} - 9670 T^{5} + 191454 T^{6} - 403758 T^{7} + 5935074 T^{8} - 9292870 T^{9} + 145350289 T^{10} - 179163074 T^{11} + 2462106986 T^{12} - 887503681 T^{13} + 27512614111 T^{14} )^{2} \))
$37$ (\( ( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} ) \))(\( 1 - 10 T - 84 T^{2} + 1296 T^{3} + 1134 T^{4} - 66630 T^{5} + 77382 T^{6} + 1851174 T^{7} - 1251993 T^{8} - 21837810 T^{9} - 268575252 T^{10} - 774884982 T^{11} + 27666538119 T^{12} + 26159905206 T^{13} - 1381876923558 T^{14} + 967916492622 T^{15} + 37875490684911 T^{16} - 39250248993246 T^{17} - 503353262863572 T^{18} - 1514320157614170 T^{19} - 3212271503983137 T^{20} + 175735422719804142 T^{21} + 271802685103314822 T^{22} - 8659350722545980510 T^{23} + 5452934678321840766 T^{24} + \)\(23\!\cdots\!48\)\( T^{25} - \)\(55\!\cdots\!04\)\( T^{26} - \)\(24\!\cdots\!70\)\( T^{27} + \)\(90\!\cdots\!89\)\( T^{28} \))
$41$ (\( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} \))(\( 1 + 33 T + 463 T^{2} + 3882 T^{3} + 26359 T^{4} + 177381 T^{5} + 987377 T^{6} + 3338436 T^{7} - 1489480 T^{8} - 146370792 T^{9} - 1532374696 T^{10} - 10553354379 T^{11} - 66691501599 T^{12} - 460808734581 T^{13} - 3110830401306 T^{14} - 18893158117821 T^{15} - 112108414187919 T^{16} - 727347737155059 T^{17} - 4330124653343656 T^{18} - 16957963898481192 T^{19} - 7075185264884680 T^{20} + 650174679078190116 T^{21} + 7884131517953805617 T^{22} + 58071334904735196141 T^{23} + \)\(35\!\cdots\!59\)\( T^{24} + \)\(21\!\cdots\!62\)\( T^{25} + \)\(10\!\cdots\!03\)\( T^{26} + \)\(30\!\cdots\!93\)\( T^{27} + \)\(37\!\cdots\!61\)\( T^{28} \))
$43$ (\( ( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} ) \))(\( 1 - 7 T - 222 T^{2} + 1221 T^{3} + 30003 T^{4} - 121065 T^{5} - 2964828 T^{6} + 8400318 T^{7} + 232132089 T^{8} - 430114695 T^{9} - 14993117802 T^{10} + 15328887375 T^{11} + 817956723570 T^{12} - 259275458991 T^{13} - 38016795252930 T^{14} - 11148844736613 T^{15} + 1512401981880930 T^{16} + 1218753848524125 T^{17} - 51258486134595402 T^{18} - 63230491623369885 T^{19} + 1467391209891779361 T^{20} + 2283362771617132026 T^{21} - 34653503452639217628 T^{22} - 60846374564133897795 T^{23} + \)\(64\!\cdots\!47\)\( T^{24} + \)\(11\!\cdots\!47\)\( T^{25} - \)\(88\!\cdots\!22\)\( T^{26} - \)\(12\!\cdots\!01\)\( T^{27} + \)\(73\!\cdots\!49\)\( T^{28} \))
$47$ (\( ( 1 - 3 T + 47 T^{2} )^{2} \))(\( ( 1 - 3 T + 224 T^{2} - 711 T^{3} + 24846 T^{4} - 72810 T^{5} + 1742359 T^{6} - 4337544 T^{7} + 81890873 T^{8} - 160837290 T^{9} + 2579586258 T^{10} - 3469453191 T^{11} + 51373281568 T^{12} - 32337645987 T^{13} + 506623120463 T^{14} )^{2} \))
$53$ (\( 1 + 3 T - 44 T^{2} + 159 T^{3} + 2809 T^{4} \))(\( 1 + 15 T - 44 T^{2} - 1563 T^{3} + 1621 T^{4} + 132831 T^{5} + 280796 T^{6} - 6875916 T^{7} - 32544895 T^{8} + 334617081 T^{9} + 3332374934 T^{10} - 9404405181 T^{11} - 219667861866 T^{12} + 268474558113 T^{13} + 14089335458730 T^{14} + 14229151579989 T^{15} - 617047023981594 T^{16} - 1400099630131737 T^{17} + 26294041101603254 T^{18} + 139935355155015933 T^{19} - 721336805685386455 T^{20} - 8077215121783465692 T^{21} + 17482272028748523356 T^{22} + \)\(43\!\cdots\!23\)\( T^{23} + \)\(28\!\cdots\!29\)\( T^{24} - \)\(14\!\cdots\!11\)\( T^{25} - \)\(21\!\cdots\!04\)\( T^{26} + \)\(39\!\cdots\!95\)\( T^{27} + \)\(13\!\cdots\!69\)\( T^{28} \))
$59$ (\( ( 1 + 7 T + 59 T^{2} )^{2} \))(\( ( 1 - 14 T + 237 T^{2} - 2625 T^{3} + 28687 T^{4} - 249069 T^{5} + 2197811 T^{6} - 16839260 T^{7} + 129670849 T^{8} - 867009189 T^{9} + 5891707373 T^{10} - 31808072625 T^{11} + 169437058863 T^{12} - 590527470974 T^{13} + 2488651484819 T^{14} )^{2} \))
$61$ (\( ( 1 - 3 T + 61 T^{2} )^{2} \))(\( ( 1 - 10 T + 377 T^{2} - 2981 T^{3} + 62533 T^{4} - 402217 T^{5} + 6040473 T^{6} - 31437684 T^{7} + 368468853 T^{8} - 1496649457 T^{9} + 14193802873 T^{10} - 41274452021 T^{11} + 318412805477 T^{12} - 515203743610 T^{13} + 3142742836021 T^{14} )^{2} \))
$67$ (\( ( 1 - 13 T + 67 T^{2} )^{2} \))(\( ( 1 + 6 T + 344 T^{2} + 1114 T^{3} + 48744 T^{4} + 55499 T^{5} + 4192269 T^{6} + 1055346 T^{7} + 280882023 T^{8} + 249135011 T^{9} + 14660391672 T^{10} + 22448348794 T^{11} + 464443036808 T^{12} + 542750293014 T^{13} + 6060711605323 T^{14} )^{2} \))
$71$ (\( ( 1 + 8 T + 71 T^{2} )^{2} \))(\( ( 1 - T + 381 T^{2} - 417 T^{3} + 66850 T^{4} - 68550 T^{5} + 7142942 T^{6} - 6246700 T^{7} + 507148882 T^{8} - 345560550 T^{9} + 23926350350 T^{10} - 10596670977 T^{11} + 687411382731 T^{12} - 128100283921 T^{13} + 9095120158391 T^{14} )^{2} \))
$73$ (\( ( 1 - 10 T + 73 T^{2} )( 1 + 17 T + 73 T^{2} ) \))(\( 1 - 21 T - 107 T^{2} + 4532 T^{3} + 8593 T^{4} - 574123 T^{5} - 1508435 T^{6} + 52395764 T^{7} + 342478696 T^{8} - 4227610868 T^{9} - 46283154524 T^{10} + 253115011813 T^{11} + 4673586532751 T^{12} - 7744045394141 T^{13} - 369527474872486 T^{14} - 565315313772293 T^{15} + 24905542633030079 T^{16} + 98466042550457821 T^{17} - 1314360176412792284 T^{18} - 8764139996708872724 T^{19} + 51828748479625639144 T^{20} + \)\(57\!\cdots\!08\)\( T^{21} - \)\(12\!\cdots\!35\)\( T^{22} - \)\(33\!\cdots\!99\)\( T^{23} + \)\(36\!\cdots\!57\)\( T^{24} + \)\(14\!\cdots\!64\)\( T^{25} - \)\(24\!\cdots\!47\)\( T^{26} - \)\(35\!\cdots\!93\)\( T^{27} + \)\(12\!\cdots\!09\)\( T^{28} \))
$79$ (\( ( 1 + 9 T + 79 T^{2} )^{2} \))(\( ( 1 - 10 T + 326 T^{2} - 3770 T^{3} + 58966 T^{4} - 604927 T^{5} + 7130289 T^{6} - 58615338 T^{7} + 563292831 T^{8} - 3775349407 T^{9} + 29072537674 T^{10} - 146841805370 T^{11} + 1003120386074 T^{12} - 2430874555210 T^{13} + 19203908986159 T^{14} )^{2} \))
$83$ (\( 1 + T - 82 T^{2} + 83 T^{3} + 6889 T^{4} \))(\( 1 + 25 T + 157 T^{2} - 750 T^{3} - 6622 T^{4} + 69964 T^{5} + 1466905 T^{6} + 20424981 T^{7} + 74872112 T^{8} - 997051343 T^{9} + 7203133487 T^{10} + 269189728764 T^{11} + 1307717205141 T^{12} - 2467734607815 T^{13} - 39825488981322 T^{14} - 204821972448645 T^{15} + 9008863826216349 T^{16} + 153919187440781268 T^{17} + 341848621231895327 T^{18} - 3927425763234733549 T^{19} + 24478716252205585328 T^{20} + \)\(55\!\cdots\!87\)\( T^{21} + \)\(33\!\cdots\!05\)\( T^{22} + \)\(13\!\cdots\!92\)\( T^{23} - \)\(10\!\cdots\!78\)\( T^{24} - \)\(96\!\cdots\!50\)\( T^{25} + \)\(16\!\cdots\!77\)\( T^{26} + \)\(22\!\cdots\!75\)\( T^{27} + \)\(73\!\cdots\!29\)\( T^{28} \))
$89$ (\( 1 + 15 T + 136 T^{2} + 1335 T^{3} + 7921 T^{4} \))(\( 1 + 6 T - 395 T^{2} - 1770 T^{3} + 83413 T^{4} + 244017 T^{5} - 12791578 T^{6} - 19742595 T^{7} + 1607888462 T^{8} + 933579267 T^{9} - 173735706451 T^{10} - 11863751967 T^{11} + 16888438123377 T^{12} - 583089996186 T^{13} - 1538438146646466 T^{14} - 51895009660554 T^{15} + 133773318375269217 T^{16} - 8363577360424023 T^{17} - 10900567564453896691 T^{18} + 5213162127281843883 T^{19} + \)\(79\!\cdots\!82\)\( T^{20} - \)\(87\!\cdots\!55\)\( T^{21} - \)\(50\!\cdots\!18\)\( T^{22} + \)\(85\!\cdots\!53\)\( T^{23} + \)\(26\!\cdots\!13\)\( T^{24} - \)\(49\!\cdots\!30\)\( T^{25} - \)\(97\!\cdots\!95\)\( T^{26} + \)\(13\!\cdots\!14\)\( T^{27} + \)\(19\!\cdots\!41\)\( T^{28} \))
$97$ (\( 1 - 17 T + 192 T^{2} - 1649 T^{3} + 9409 T^{4} \))(\( 1 + 18 T - 332 T^{2} - 7282 T^{3} + 70792 T^{4} + 1694642 T^{5} - 11251793 T^{6} - 262778647 T^{7} + 1667779429 T^{8} + 30411590872 T^{9} - 218639545013 T^{10} - 2436064838528 T^{11} + 25890765026051 T^{12} + 94489464293929 T^{13} - 2645836565169718 T^{14} + 9165478036511113 T^{15} + 243606208130113859 T^{16} - 2223330604373865344 T^{17} - 19356001718168025653 T^{18} + \)\(26\!\cdots\!04\)\( T^{19} + \)\(13\!\cdots\!41\)\( T^{20} - \)\(21\!\cdots\!11\)\( T^{21} - \)\(88\!\cdots\!73\)\( T^{22} + \)\(12\!\cdots\!14\)\( T^{23} + \)\(52\!\cdots\!08\)\( T^{24} - \)\(52\!\cdots\!46\)\( T^{25} - \)\(23\!\cdots\!12\)\( T^{26} + \)\(12\!\cdots\!86\)\( T^{27} + \)\(65\!\cdots\!69\)\( T^{28} \))
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