# Properties

 Label 378.2.l Level 378 Weight 2 Character orbit l Rep. character $$\chi_{378}(143,\cdot)$$ Character field $$\Q(\zeta_{6})$$ Dimension 16 Newform subspaces 1 Sturm bound 144 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ = $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 378.l (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$63$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$1$$ Sturm bound: $$144$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(378, [\chi])$$.

Total New Old
Modular forms 168 16 152
Cusp forms 120 16 104
Eisenstein series 48 0 48

## Trace form

 $$16q - 16q^{4} + 2q^{7} + O(q^{10})$$ $$16q - 16q^{4} + 2q^{7} - 12q^{11} + 6q^{13} + 6q^{14} + 16q^{16} + 18q^{17} + 6q^{23} - 8q^{25} - 12q^{26} - 2q^{28} - 6q^{29} - 30q^{35} - 2q^{37} + 6q^{41} - 2q^{43} + 12q^{44} + 6q^{46} + 36q^{47} - 8q^{49} + 12q^{50} - 6q^{52} + 36q^{53} - 6q^{56} + 6q^{58} - 60q^{59} - 36q^{62} - 16q^{64} - 28q^{67} - 18q^{68} - 18q^{70} - 18q^{74} + 42q^{77} + 32q^{79} - 12q^{85} - 24q^{86} + 24q^{89} - 12q^{91} - 6q^{92} + 6q^{97} + 24q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(378, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
378.2.l.a $$16$$ $$3.018$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$2$$ $$q+(\beta _{1}+\beta _{7})q^{2}-q^{4}+(-\beta _{5}-\beta _{10}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(378, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(378, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{8}$$
$3$ 1
$5$ $$1 - 16 T^{2} + 24 T^{3} + 123 T^{4} - 330 T^{5} - 98 T^{6} + 2124 T^{7} - 4594 T^{8} + 102 T^{9} + 38946 T^{10} - 84396 T^{11} - 70916 T^{12} + 641196 T^{13} - 735151 T^{14} - 1444464 T^{15} + 6891966 T^{16} - 7222320 T^{17} - 18378775 T^{18} + 80149500 T^{19} - 44322500 T^{20} - 263737500 T^{21} + 608531250 T^{22} + 7968750 T^{23} - 1794531250 T^{24} + 4148437500 T^{25} - 957031250 T^{26} - 16113281250 T^{27} + 30029296875 T^{28} + 29296875000 T^{29} - 97656250000 T^{30} + 152587890625 T^{32}$$
$7$ $$1 - 2 T + 6 T^{2} + 8 T^{3} + 23 T^{4} - 210 T^{5} + 463 T^{6} - 1148 T^{7} - 1350 T^{8} - 8036 T^{9} + 22687 T^{10} - 72030 T^{11} + 55223 T^{12} + 134456 T^{13} + 705894 T^{14} - 1647086 T^{15} + 5764801 T^{16}$$
$11$ $$1 + 12 T + 106 T^{2} + 696 T^{3} + 3735 T^{4} + 17808 T^{5} + 77432 T^{6} + 326142 T^{7} + 1351880 T^{8} + 5458080 T^{9} + 21295098 T^{10} + 79655676 T^{11} + 287397262 T^{12} + 1023482544 T^{13} + 3603683689 T^{14} + 12488798178 T^{15} + 42242848866 T^{16} + 137376779958 T^{17} + 436045726369 T^{18} + 1362255266064 T^{19} + 4207783312942 T^{20} + 12828626275476 T^{21} + 37725565107978 T^{22} + 106362538291680 T^{23} + 289787484046280 T^{24} + 769025775838122 T^{25} + 2008386661704632 T^{26} + 5080830230240688 T^{27} + 11722029987052935 T^{28} + 24027807652175976 T^{29} + 40253482359823546 T^{30} + 50126978032987812 T^{31} + 45949729863572161 T^{32}$$
$13$ $$1 - 6 T + 47 T^{2} - 210 T^{3} + 888 T^{4} - 2658 T^{5} + 9181 T^{6} - 12564 T^{7} + 21641 T^{8} + 214770 T^{9} - 1221396 T^{10} + 6263526 T^{11} - 19011605 T^{12} + 72302586 T^{13} - 179114935 T^{14} + 742674348 T^{15} - 2320143696 T^{16} + 9654766524 T^{17} - 30270424015 T^{18} + 158848781442 T^{19} - 542990450405 T^{20} + 2325603359118 T^{21} - 5895445205364 T^{22} + 13476498996090 T^{23} + 17653228533161 T^{24} - 133234930122372 T^{25} + 1265678813665669 T^{26} - 4763562327350346 T^{27} + 20688699588763128 T^{28} - 63603772384373130 T^{29} + 185056690127866583 T^{30} - 307115358084544542 T^{31} + 665416609183179841 T^{32}$$
$17$ $$1 - 18 T + 95 T^{2} + 42 T^{3} - 849 T^{4} - 7584 T^{5} + 29152 T^{6} + 139356 T^{7} - 154873 T^{8} - 2564070 T^{9} - 13186653 T^{10} + 89037906 T^{11} + 222290986 T^{12} - 933503742 T^{13} - 5890959001 T^{14} + 7644891330 T^{15} + 98422426836 T^{16} + 129963152610 T^{17} - 1702487151289 T^{18} - 4586303884446 T^{19} + 18565965441706 T^{20} + 126421094099442 T^{21} - 318293746666557 T^{22} - 1052137081279110 T^{23} - 1080356482159993 T^{24} + 16525932117115932 T^{25} + 58770254185889248 T^{26} - 259918061597088672 T^{27} - 494646279408067089 T^{28} + 415992277382049354 T^{29} + 15995893523143088255 T^{30} - 51523614927176684274 T^{31} + 48661191875666868481 T^{32}$$
$19$ $$1 + 80 T^{2} + 3483 T^{4} - 882 T^{5} + 107770 T^{6} - 26190 T^{7} + 2596658 T^{8} + 477594 T^{9} + 51326154 T^{10} + 58243698 T^{11} + 858124540 T^{12} + 2331197604 T^{13} + 13169269781 T^{14} + 60976736712 T^{15} + 222780924306 T^{16} + 1158557997528 T^{17} + 4754106390941 T^{18} + 15989684365836 T^{19} + 111831648177340 T^{20} + 144217162374102 T^{21} + 2414684133271674 T^{22} + 426907779315966 T^{23} + 44100504838916978 T^{24} - 8451190804832010 T^{25} + 660745010603213770 T^{26} - 102744408348229158 T^{27} + 7708975863107438763 T^{28} + 63920534862630729680 T^{30} +$$$$28\!\cdots\!81$$$$T^{32}$$
$23$ $$1 - 6 T + 130 T^{2} - 708 T^{3} + 8559 T^{4} - 45012 T^{5} + 399452 T^{6} - 2115900 T^{7} + 15122540 T^{8} - 81277422 T^{9} + 486352566 T^{10} - 2645846004 T^{11} + 13727139058 T^{12} - 75456395706 T^{13} + 352786829089 T^{14} - 1928667059838 T^{15} + 8415100540206 T^{16} - 44359342376274 T^{17} + 186624232588081 T^{18} - 918077966554902 T^{19} + 3841416321129778 T^{20} - 17029572406923372 T^{21} + 71997634475241174 T^{22} - 276735434692157634 T^{23} + 1184261007351333740 T^{24} - 3811058916389561700 T^{25} + 16547902757314520348 T^{26} - 42887872823221682124 T^{27} +$$$$18\!\cdots\!39$$$$T^{28} -$$$$35\!\cdots\!64$$$$T^{29} +$$$$15\!\cdots\!70$$$$T^{30} -$$$$15\!\cdots\!42$$$$T^{31} +$$$$61\!\cdots\!61$$$$T^{32}$$
$29$ $$1 + 6 T + 196 T^{2} + 1104 T^{3} + 19989 T^{4} + 108858 T^{5} + 1435478 T^{6} + 7594740 T^{7} + 81322766 T^{8} + 416245488 T^{9} + 3844091772 T^{10} + 18872529852 T^{11} + 155981060614 T^{12} + 727660406082 T^{13} + 5509975202215 T^{14} + 24244157784798 T^{15} + 170557455776958 T^{16} + 703080575759142 T^{17} + 4633889145062815 T^{18} + 17746909643933898 T^{19} + 110322440532130534 T^{20} + 387097271801319948 T^{21} + 2286555434049814812 T^{22} + 7180183182179343792 T^{23} + 40681421983566770126 T^{24} +$$$$11\!\cdots\!60$$$$T^{25} +$$$$60\!\cdots\!78$$$$T^{26} +$$$$13\!\cdots\!82$$$$T^{27} +$$$$70\!\cdots\!49$$$$T^{28} +$$$$11\!\cdots\!56$$$$T^{29} +$$$$58\!\cdots\!76$$$$T^{30} +$$$$51\!\cdots\!94$$$$T^{31} +$$$$25\!\cdots\!21$$$$T^{32}$$
$31$ $$1 - 292 T^{2} + 41022 T^{4} - 3694910 T^{6} + 240356405 T^{8} - 12117919608 T^{10} + 500572972675 T^{12} - 17863857948586 T^{14} + 576631021225626 T^{16} - 17167167488591146 T^{18} + 462289652297788675 T^{20} - 10754698258162077048 T^{22} +$$$$20\!\cdots\!05$$$$T^{24} -$$$$30\!\cdots\!10$$$$T^{26} +$$$$32\!\cdots\!42$$$$T^{28} -$$$$22\!\cdots\!32$$$$T^{30} +$$$$72\!\cdots\!81$$$$T^{32}$$
$37$ $$1 + 2 T - 116 T^{2} - 656 T^{3} + 3854 T^{4} + 48538 T^{5} + 98460 T^{6} - 747702 T^{7} - 5251023 T^{8} - 50514234 T^{9} - 417738792 T^{10} + 801423570 T^{11} + 29966075382 T^{12} + 116541321036 T^{13} - 322717550496 T^{14} - 3343790405826 T^{15} - 14327838321804 T^{16} - 123720245015562 T^{17} - 441800326629024 T^{18} + 5903167534436508 T^{19} + 56161249804004502 T^{20} + 55573881576866490 T^{21} - 1071803450698157928 T^{22} - 4795411055555611122 T^{23} - 18444110399566611183 T^{24} - 97172652768258663054 T^{25} +$$$$47\!\cdots\!40$$$$T^{26} +$$$$86\!\cdots\!94$$$$T^{27} +$$$$25\!\cdots\!74$$$$T^{28} -$$$$15\!\cdots\!32$$$$T^{29} -$$$$10\!\cdots\!24$$$$T^{30} +$$$$66\!\cdots\!86$$$$T^{31} +$$$$12\!\cdots\!41$$$$T^{32}$$
$41$ $$1 - 6 T - 223 T^{2} + 1686 T^{3} + 25980 T^{4} - 231654 T^{5} - 1971341 T^{6} + 20408106 T^{7} + 109216031 T^{8} - 1268388768 T^{9} - 4836103872 T^{10} + 57438167556 T^{11} + 192545389345 T^{12} - 1824668193534 T^{13} - 7677147470143 T^{14} + 27964729912410 T^{15} + 313424888729076 T^{16} + 1146553926408810 T^{17} - 12905284897310383 T^{18} - 125757956566556814 T^{19} + 544087251940916545 T^{20} + 6654567885439614756 T^{21} - 22971997512303721152 T^{22} -$$$$24\!\cdots\!08$$$$T^{23} +$$$$87\!\cdots\!51$$$$T^{24} +$$$$66\!\cdots\!66$$$$T^{25} -$$$$26\!\cdots\!41$$$$T^{26} -$$$$12\!\cdots\!14$$$$T^{27} +$$$$58\!\cdots\!80$$$$T^{28} +$$$$15\!\cdots\!06$$$$T^{29} -$$$$84\!\cdots\!03$$$$T^{30} -$$$$93\!\cdots\!06$$$$T^{31} +$$$$63\!\cdots\!41$$$$T^{32}$$
$43$ $$1 + 2 T - 209 T^{2} - 602 T^{3} + 20774 T^{4} + 72052 T^{5} - 1457073 T^{6} - 4933350 T^{7} + 91147851 T^{8} + 235722522 T^{9} - 5466637218 T^{10} - 9144255228 T^{11} + 301177025103 T^{12} + 290829011802 T^{13} - 14773623661707 T^{14} - 4808085837138 T^{15} + 658882369500660 T^{16} - 206747690996934 T^{17} - 27316430150496243 T^{18} + 23122942241341614 T^{19} + 1029664314599161503 T^{20} - 1344282723462890004 T^{21} - 34556598512153357682 T^{22} + 64073768536679251854 T^{23} +$$$$10\!\cdots\!51$$$$T^{24} -$$$$24\!\cdots\!50$$$$T^{25} -$$$$31\!\cdots\!77$$$$T^{26} +$$$$66\!\cdots\!64$$$$T^{27} +$$$$83\!\cdots\!74$$$$T^{28} -$$$$10\!\cdots\!86$$$$T^{29} -$$$$15\!\cdots\!41$$$$T^{30} +$$$$63\!\cdots\!14$$$$T^{31} +$$$$13\!\cdots\!01$$$$T^{32}$$
$47$ $$( 1 - 18 T + 379 T^{2} - 4272 T^{3} + 53788 T^{4} - 467070 T^{5} + 4528654 T^{6} - 32587152 T^{7} + 257763508 T^{8} - 1531596144 T^{9} + 10003796686 T^{10} - 48492608610 T^{11} + 262468281628 T^{12} - 979761869904 T^{13} + 4085322609691 T^{14} - 9119216168334 T^{15} + 23811286661761 T^{16} )^{2}$$
$53$ $$1 - 36 T + 766 T^{2} - 12024 T^{3} + 153999 T^{4} - 1662408 T^{5} + 15312632 T^{6} - 121481640 T^{7} + 835010600 T^{8} - 5038052724 T^{9} + 28358007246 T^{10} - 182105031078 T^{11} + 1622318702806 T^{12} - 17289317216124 T^{13} + 177226593270661 T^{14} - 1605696832118286 T^{15} + 12594715934262750 T^{16} - 85101932102269158 T^{17} + 497829500497286749 T^{18} - 2573981679184892748 T^{19} + 12800874900435389686 T^{20} - 76155503249444531454 T^{21} +$$$$62\!\cdots\!34$$$$T^{22} -$$$$59\!\cdots\!88$$$$T^{23} +$$$$51\!\cdots\!00$$$$T^{24} -$$$$40\!\cdots\!20$$$$T^{25} +$$$$26\!\cdots\!68$$$$T^{26} -$$$$15\!\cdots\!76$$$$T^{27} +$$$$75\!\cdots\!59$$$$T^{28} -$$$$31\!\cdots\!52$$$$T^{29} +$$$$10\!\cdots\!54$$$$T^{30} -$$$$26\!\cdots\!52$$$$T^{31} +$$$$38\!\cdots\!21$$$$T^{32}$$
$59$ $$( 1 + 30 T + 700 T^{2} + 11136 T^{3} + 154015 T^{4} + 1732056 T^{5} + 17715805 T^{6} + 156334302 T^{7} + 1281973738 T^{8} + 9223723818 T^{9} + 61668717205 T^{10} + 355727929224 T^{11} + 1866255354415 T^{12} + 7961396993664 T^{13} + 29526373548700 T^{14} + 74659544544570 T^{15} + 146830437604321 T^{16} )^{2}$$
$61$ $$1 - 472 T^{2} + 107634 T^{4} - 15742946 T^{6} + 1661711189 T^{8} - 136358974992 T^{10} + 9312516091879 T^{12} - 573351459745642 T^{14} + 34616863644604218 T^{16} - 2133440781713533882 T^{18} +$$$$12\!\cdots\!39$$$$T^{20} -$$$$70\!\cdots\!12$$$$T^{22} +$$$$31\!\cdots\!09$$$$T^{24} -$$$$11\!\cdots\!46$$$$T^{26} +$$$$28\!\cdots\!14$$$$T^{28} -$$$$46\!\cdots\!52$$$$T^{30} +$$$$36\!\cdots\!61$$$$T^{32}$$
$67$ $$( 1 + 14 T + 435 T^{2} + 5596 T^{3} + 90428 T^{4} + 1008078 T^{5} + 11375926 T^{6} + 106594220 T^{7} + 933924096 T^{8} + 7141812740 T^{9} + 51066531814 T^{10} + 303192563514 T^{11} + 1822225569788 T^{12} + 7555300098772 T^{13} + 39349396243515 T^{14} + 84849962474522 T^{15} + 406067677556641 T^{16} )^{2}$$
$71$ $$1 - 650 T^{2} + 199389 T^{4} - 38930632 T^{6} + 5566228364 T^{8} - 640511863116 T^{10} + 63163988645884 T^{12} - 5475157404521894 T^{14} + 416179213677825948 T^{16} - 27600268476194867654 T^{18} +$$$$16\!\cdots\!04$$$$T^{20} -$$$$82\!\cdots\!36$$$$T^{22} +$$$$35\!\cdots\!04$$$$T^{24} -$$$$12\!\cdots\!32$$$$T^{26} +$$$$32\!\cdots\!49$$$$T^{28} -$$$$53\!\cdots\!50$$$$T^{30} +$$$$41\!\cdots\!21$$$$T^{32}$$
$73$ $$1 + 434 T^{2} + 101253 T^{4} + 70380 T^{5} + 16361620 T^{6} + 29833866 T^{7} + 2027081504 T^{8} + 6817588704 T^{9} + 204710173188 T^{10} + 1064159524062 T^{11} + 17706469884892 T^{12} + 124668495857988 T^{13} + 1383495353685575 T^{14} + 11427374402737884 T^{15} + 102528286806790386 T^{16} + 834198331399865532 T^{17} + 7372646739790429175 T^{18} + 48498164253186917796 T^{19} +$$$$50\!\cdots\!72$$$$T^{20} +$$$$22\!\cdots\!66$$$$T^{21} +$$$$30\!\cdots\!32$$$$T^{22} +$$$$75\!\cdots\!88$$$$T^{23} +$$$$16\!\cdots\!24$$$$T^{24} +$$$$17\!\cdots\!58$$$$T^{25} +$$$$70\!\cdots\!80$$$$T^{26} +$$$$22\!\cdots\!60$$$$T^{27} +$$$$23\!\cdots\!13$$$$T^{28} +$$$$52\!\cdots\!06$$$$T^{30} +$$$$65\!\cdots\!61$$$$T^{32}$$
$79$ $$( 1 - 16 T + 483 T^{2} - 6542 T^{3} + 113126 T^{4} - 1278384 T^{5} + 16380706 T^{6} - 152801626 T^{7} + 1573503228 T^{8} - 12071328454 T^{9} + 102231986146 T^{10} - 630293168976 T^{11} + 4406266863206 T^{12} - 20130102962258 T^{13} + 117411241016643 T^{14} - 307262543778544 T^{15} + 1517108809906561 T^{16} )^{2}$$
$83$ $$1 - 487 T^{2} - 312 T^{3} + 123774 T^{4} + 132990 T^{5} - 22183883 T^{6} - 29138634 T^{7} + 3170469341 T^{8} + 4110229572 T^{9} - 386467088226 T^{10} - 393115428402 T^{11} + 41656592194789 T^{12} + 25282823866380 T^{13} - 4030130568645907 T^{14} - 780079655467782 T^{15} + 351851707607703156 T^{16} - 64746611403825906 T^{17} - 27763569487401653323 T^{18} + 14456390010085821060 T^{19} +$$$$19\!\cdots\!69$$$$T^{20} -$$$$15\!\cdots\!86$$$$T^{21} -$$$$12\!\cdots\!94$$$$T^{22} +$$$$11\!\cdots\!44$$$$T^{23} +$$$$71\!\cdots\!81$$$$T^{24} -$$$$54\!\cdots\!02$$$$T^{25} -$$$$34\!\cdots\!67$$$$T^{26} +$$$$17\!\cdots\!30$$$$T^{27} +$$$$13\!\cdots\!14$$$$T^{28} -$$$$27\!\cdots\!56$$$$T^{29} -$$$$35\!\cdots\!23$$$$T^{30} +$$$$50\!\cdots\!81$$$$T^{32}$$
$89$ $$1 - 24 T - 10 T^{2} + 3636 T^{3} + 1197 T^{4} - 543420 T^{5} + 1792912 T^{6} + 58468350 T^{7} - 544062388 T^{8} - 3510821196 T^{9} + 61278252588 T^{10} + 296095488138 T^{11} - 6613804791944 T^{12} - 16297764313308 T^{13} + 597103648531739 T^{14} - 218011613471172 T^{15} - 39896652829458150 T^{16} - 19403033598934308 T^{17} + 4729658000019904619 T^{18} - 11489418610188427452 T^{19} -$$$$41\!\cdots\!04$$$$T^{20} +$$$$16\!\cdots\!62$$$$T^{21} +$$$$30\!\cdots\!68$$$$T^{22} -$$$$15\!\cdots\!84$$$$T^{23} -$$$$21\!\cdots\!28$$$$T^{24} +$$$$20\!\cdots\!50$$$$T^{25} +$$$$55\!\cdots\!12$$$$T^{26} -$$$$15\!\cdots\!80$$$$T^{27} +$$$$29\!\cdots\!37$$$$T^{28} +$$$$79\!\cdots\!84$$$$T^{29} -$$$$19\!\cdots\!10$$$$T^{30} -$$$$41\!\cdots\!76$$$$T^{31} +$$$$15\!\cdots\!61$$$$T^{32}$$
$97$ $$1 - 6 T + 395 T^{2} - 2298 T^{3} + 68379 T^{4} - 571872 T^{5} + 8922340 T^{6} - 109039644 T^{7} + 1233764039 T^{8} - 14418445794 T^{9} + 171203907675 T^{10} - 1605167799666 T^{11} + 21002714173786 T^{12} - 194363495836338 T^{13} + 2133943680044999 T^{14} - 22669145285986746 T^{15} + 200280216945639852 T^{16} - 2198907092740714362 T^{17} + 20078276085543395591 T^{18} -$$$$17\!\cdots\!74$$$$T^{19} +$$$$18\!\cdots\!66$$$$T^{20} -$$$$13\!\cdots\!62$$$$T^{21} +$$$$14\!\cdots\!75$$$$T^{22} -$$$$11\!\cdots\!22$$$$T^{23} +$$$$96\!\cdots\!79$$$$T^{24} -$$$$82\!\cdots\!48$$$$T^{25} +$$$$65\!\cdots\!60$$$$T^{26} -$$$$40\!\cdots\!16$$$$T^{27} +$$$$47\!\cdots\!39$$$$T^{28} -$$$$15\!\cdots\!46$$$$T^{29} +$$$$25\!\cdots\!55$$$$T^{30} -$$$$37\!\cdots\!58$$$$T^{31} +$$$$61\!\cdots\!21$$$$T^{32}$$