Properties

Label 6.27.b.a
Level $6$
Weight $27$
Character orbit 6.b
Analytic conductor $25.698$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 27 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.6975752438\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 11609466 x^{6} - 3416571600 x^{5} + 38618090622117 x^{4} + 19832313245955600 x^{3} - 25659229618759655524 x^{2} - 8410094250198512820000 x + 6059395276689883547331396\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{59}\cdot 3^{34}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 274365 - 33 \beta_{1} + \beta_{2} ) q^{3} -33554432 q^{4} + ( -69639 \beta_{1} + 22 \beta_{2} + \beta_{5} + \beta_{7} ) q^{5} + ( 1122361344 + 274362 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{6} + ( 22901407970 - 9860 \beta_{1} - 21956 \beta_{2} - 2 \beta_{3} + 29 \beta_{4} + 11 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{7} -33554432 \beta_{1} q^{8} + ( -606741412887 + 7169526 \beta_{1} + 359613 \beta_{2} - 216 \beta_{3} - 504 \beta_{4} - 1116 \beta_{5} + 657 \beta_{6} - 1431 \beta_{7} ) q^{9} +O(q^{10})\) \( q +\beta_{1} q^{2} +(274365 - 33 \beta_{1} + \beta_{2}) q^{3} -33554432 q^{4} +(-69639 \beta_{1} + 22 \beta_{2} + \beta_{5} + \beta_{7}) q^{5} +(1122361344 + 274362 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7}) q^{6} +(22901407970 - 9860 \beta_{1} - 21956 \beta_{2} - 2 \beta_{3} + 29 \beta_{4} + 11 \beta_{5} - 4 \beta_{6} + 2 \beta_{7}) q^{7} -33554432 \beta_{1} q^{8} +(-606741412887 + 7169526 \beta_{1} + 359613 \beta_{2} - 216 \beta_{3} - 504 \beta_{4} - 1116 \beta_{5} + 657 \beta_{6} - 1431 \beta_{7}) q^{9} +(2337034813440 - 419362 \beta_{1} - 934790 \beta_{2} - 4974 \beta_{3} - 372 \beta_{4} + 3192 \beta_{5} + 1420 \beta_{6} - 396 \beta_{7}) q^{10} +(-1596617584 \beta_{1} + 651454 \beta_{2} + 5431 \beta_{3} - 5431 \beta_{4} + 27759 \beta_{5} + 5431 \beta_{6} - 22156 \beta_{7}) q^{11} +(-9206161735680 + 1107296256 \beta_{1} - 33554432 \beta_{2}) q^{12} +(64053627818570 + 32598450 \beta_{1} + 72595610 \beta_{2} + 13840 \beta_{3} - 53260 \beta_{4} - 26980 \beta_{5} + 57870 \beta_{6} - 2920 \beta_{7}) q^{13} +(22839063090 \beta_{1} - 139131776 \beta_{2} + 6272 \beta_{3} - 6272 \beta_{4} - 322144 \beta_{5} + 6272 \beta_{6} + 109440 \beta_{7}) q^{14} +(-133053333194880 + 48302893170 \beta_{1} + 38535942 \beta_{2} + 1038978 \beta_{3} - 222891 \beta_{4} - 1477803 \beta_{5} + 58260 \beta_{6} + 863358 \beta_{7}) q^{15} +1125899906842624 q^{16} +(366717289196 \beta_{1} - 2691724624 \beta_{2} + 1299004 \beta_{3} - 1299004 \beta_{4} + 3415366 \beta_{5} + 1299004 \beta_{6} + 1760280 \beta_{7}) q^{17} +(-235160446648320 - 606330959589 \beta_{1} + 925481430 \beta_{2} + 6976638 \beta_{3} - 1008684 \beta_{4} - 482040 \beta_{5} - 1084716 \beta_{6} - 8195796 \beta_{7}) q^{18} +(1656312853216538 + 1447708210 \beta_{1} + 3217973480 \beta_{2} - 30410187 \beta_{3} - 11683083 \beta_{4} + 16379097 \beta_{5} + 13676897 \beta_{6} - 3118020 \beta_{7}) q^{19} +(2336697090048 \beta_{1} - 738197504 \beta_{2} - 33554432 \beta_{5} - 33554432 \beta_{7}) q^{20} +(-47010905052372198 - 10837987733541 \beta_{1} + 24253707506 \beta_{2} + 26481168 \beta_{3} + 7981092 \beta_{4} + 35842257 \beta_{5} - 20275920 \beta_{6} + 15956271 \beta_{7}) q^{21} +(53582839169433600 - 37581802218 \beta_{1} - 83696568926 \beta_{2} - 31350502 \beta_{3} + 54929980 \beta_{4} + 39210328 \beta_{5} - 63363780 \beta_{6} + 1746628 \beta_{7}) q^{22} +(40263550905184 \beta_{1} + 152612820884 \beta_{2} - 34457494 \beta_{3} + 34457494 \beta_{4} + 334124934 \beta_{5} - 34457494 \beta_{6} + 111850000 \beta_{7}) q^{23} +(-37660197396676608 - 9206061072384 \beta_{1} + 33554432 \beta_{2} - 100663296 \beta_{3} - 67108864 \beta_{4} - 134217728 \beta_{5} - 67108864 \beta_{6} + 67108864 \beta_{7}) q^{24} +(-702978947491586375 + 95529630110 \beta_{1} + 212684104350 \beta_{2} + 4709620 \beta_{3} - 589042540 \beta_{4} - 199487260 \beta_{5} - 310609850 \beta_{6} - 43283920 \beta_{7}) q^{25} +(64073546151690 \beta_{1} + 45528035840 \beta_{2} + 152076800 \beta_{3} - 152076800 \beta_{4} + 763797120 \beta_{5} + 152076800 \beta_{6} - 736064000 \beta_{7}) q^{26} +(1393067868534780165 - 23867937689157 \beta_{1} - 804520686717 \beta_{2} - 1814259141 \beta_{3} - 928049481 \beta_{4} + 2045259477 \beta_{5} + 453129471 \beta_{6} - 1868831352 \beta_{7}) q^{27} +(-768443736433623040 + 330846699520 \beta_{1} + 736721108992 \beta_{2} + 67108864 \beta_{3} - 973078528 \beta_{4} - 369098752 \beta_{5} + 134217728 \beta_{6} - 67108864 \beta_{7}) q^{28} +(529990798319691 \beta_{1} - 2599632946094 \beta_{2} + 2417669112 \beta_{3} - 2417669112 \beta_{4} + 6522956647 \beta_{5} + 2417669112 \beta_{6} - 5470285541 \beta_{7}) q^{29} +(-1620179753160130560 - 131903767223658 \beta_{1} + 2562744263298 \beta_{2} - 1095973254 \beta_{3} - 4484974212 \beta_{4} + 10182611736 \beta_{5} + 1815389820 \beta_{6} + 2162355588 \beta_{7}) q^{30} +(-3910551161750446222 + 628331034280 \beta_{1} + 1396582116540 \beta_{2} - 7766306080 \beta_{3} - 12963039635 \beta_{4} + 856524175 \beta_{5} - 7130439670 \beta_{6} - 1535507090 \beta_{7}) q^{31} +1125899906842624 \beta_{1} q^{32} +(-3391887528816099840 + 5618485203201882 \beta_{1} + 2993273692971 \beta_{2} + 7965603540 \beta_{3} - 10189212348 \beta_{4} - 9531325086 \beta_{5} - 12347908437 \beta_{6} - 25417713813 \beta_{7}) q^{33} +(-12345551731179454464 - 2892374838040 \beta_{1} - 6446967949000 \beta_{2} - 26323628968 \beta_{3} - 10219878512 \beta_{4} + 14142459808 \beta_{5} - 4801807472 \beta_{6} - 2706926480 \beta_{7}) q^{34} +(-6931139219443928 \beta_{1} + 33299885825444 \beta_{2} + 2321048450 \beta_{3} - 2321048450 \beta_{4} + 135917634852 \beta_{5} + 2321048450 \beta_{6} + 2391813652 \beta_{7}) q^{35} +(20358863480300765184 - 240569372639232 \beta_{1} - 12066609954816 \beta_{2} + 7247757312 \beta_{3} + 16911433728 \beta_{4} + 37446746112 \beta_{5} - 22045261824 \beta_{6} + 48016392192 \beta_{7}) q^{36} +(-\)\(11\!\cdots\!50\)\( - 11611658316594 \beta_{1} - 25882110001034 \beta_{2} - 118755690088 \beta_{3} - 25817031716 \beta_{4} + 70564782820 \beta_{5} + 7754584026 \beta_{6} - 10709090504 \beta_{7}) q^{37} +(1667929635813386 \beta_{1} + 26094439413632 \beta_{2} + 8788591488 \beta_{3} - 8788591488 \beta_{4} - 99213470368 \beta_{5} + 8788591488 \beta_{6} - 284473141120 \beta_{7}) q^{38} +(\)\(19\!\cdots\!30\)\( + 1186780357340070 \beta_{1} + 45722150222750 \beta_{2} - 32782374270 \beta_{3} - 84166767900 \beta_{4} - 50381454870 \beta_{5} + 83104471530 \beta_{6} - 136183767480 \beta_{7}) q^{39} +(-78417875729205166080 + 14071453712384 \beta_{1} + 31366347489280 \beta_{2} + 166899744768 \beta_{3} + 12482248704 \beta_{4} - 107105746944 \beta_{5} - 47647293440 \beta_{6} + 13287555072 \beta_{7}) q^{40} +(25932996017943686 \beta_{1} - 302468435398652 \beta_{2} + 5415530608 \beta_{3} - 5415530608 \beta_{4} - 295270983690 \beta_{5} + 5415530608 \beta_{6} + 751301057174 \beta_{7}) q^{41} +(\)\(36\!\cdots\!20\)\( - 47039155817155116 \beta_{1} - 63101544738098 \beta_{2} - 103614591594 \beta_{3} + 143512281316 \beta_{4} + 277775562728 \beta_{5} + 21106284004 \beta_{6} + 294318052892 \beta_{7}) q^{42} +(-\)\(64\!\cdots\!30\)\( + 169402011565818 \beta_{1} + 377559573601848 \beta_{2} + 1429242254361 \beta_{3} + 499581536877 \beta_{4} - 786300990615 \beta_{5} + 245372885853 \beta_{6} + 142875836388 \beta_{7}) q^{43} +(53573596152332288 \beta_{1} - 21859168944128 \beta_{2} - 182234120192 \beta_{3} + 182234120192 \beta_{4} - 931437477888 \beta_{5} - 182234120192 \beta_{6} + 743431995392 \beta_{7}) q^{44} +(\)\(25\!\cdots\!40\)\( - 121072521953842029 \beta_{1} - 483762011075424 \beta_{2} + 95293640376 \beta_{3} + 830401757928 \beta_{4} - 1796992295805 \beta_{5} + 157516158870 \beta_{6} + 330568322157 \beta_{7}) q^{45} +(-\)\(13\!\cdots\!56\)\( - 224191855864620 \beta_{1} - 499288113331780 \beta_{2} - 511957469492 \beta_{3} + 748729897352 \beta_{4} + 590881612112 \beta_{5} + 343987018632 \beta_{6} + 17538698360 \beta_{7}) q^{46} +(-253065657106040720 \beta_{1} + 255357378892440 \beta_{2} - 206540468980 \beta_{3} + 206540468980 \beta_{4} - 2665231388240 \beta_{5} - 206540468980 \beta_{6} - 1935643792600 \beta_{7}) q^{47} +(\)\(30\!\cdots\!60\)\( - 37154696925806592 \beta_{1} + 1125899906842624 \beta_{2}) q^{48} +(-\)\(37\!\cdots\!05\)\( - 637313426908830 \beta_{1} - 1419554078682110 \beta_{2} - 2424907769956 \beta_{3} + 1564567227916 \beta_{4} + 2138127589276 \beta_{5} + 1006544862666 \beta_{6} - 63728929040 \beta_{7}) q^{49} +(-700990645451755015 \beta_{1} + 4428944434705920 \beta_{2} - 1360805747200 \beta_{3} + 1360805747200 \beta_{4} + 5552786776960 \beta_{5} - 1360805747200 \beta_{6} + 2070750824960 \beta_{7}) q^{50} +(\)\(69\!\cdots\!48\)\( + 566270076411040392 \beta_{1} - 520988870482344 \beta_{2} + 5490887688468 \beta_{3} + 933493982112 \beta_{4} - 352786000302 \beta_{5} - 916615025676 \beta_{6} + 3321248954868 \beta_{7}) q^{51} +(-\)\(21\!\cdots\!40\)\( - 1093822473830400 \beta_{1} - 2435904459243520 \beta_{2} - 464393338880 \beta_{3} + 1787109048320 \beta_{4} + 905298575360 \beta_{5} - 1941794979840 \beta_{6} + 97978941440 \beta_{7}) q^{52} +(597689948819279661 \beta_{1} - 9917423690514674 \beta_{2} + 2519546084064 \beta_{3} - 2519546084064 \beta_{4} - 7793354135339 \beta_{5} + 2519546084064 \beta_{6} + 5216323226005 \beta_{7}) q^{53} +(\)\(78\!\cdots\!00\)\( + 1393207536167442468 \beta_{1} + 313176651187077 \beta_{2} - 2104177448079 \beta_{3} + 3580341099894 \beta_{4} - 11561928060276 \beta_{5} - 6663699900810 \beta_{6} - 9244193477238 \beta_{7}) q^{54} +(-\)\(95\!\cdots\!40\)\( + 8211387086666972 \beta_{1} + 18285753569310240 \beta_{2} + 5319755169994 \beta_{3} - 21923851867768 \beta_{4} - 10854454069252 \beta_{5} + 3342379161430 \beta_{6} - 1229933088724 \beta_{7}) q^{55} +(-766351789397114880 \beta_{1} + 4668487716831232 \beta_{2} - 210453397504 \beta_{3} + 210453397504 \beta_{4} + 10809358942208 \beta_{5} - 210453397504 \beta_{6} - 3672197038080 \beta_{7}) q^{56} +(\)\(83\!\cdots\!10\)\( + 1828160916379364364 \beta_{1} + 1615265461430939 \beta_{2} - 14445140742576 \beta_{3} - 35756694499392 \beta_{4} + 54456136060050 \beta_{5} + 21116557516707 \beta_{6} + 1611224469477 \beta_{7}) q^{57} +(-\)\(17\!\cdots\!80\)\( - 9564988375933798 \beta_{1} - 21304951534988690 \beta_{2} - 19605796147530 \beta_{3} + 2201032851876 \beta_{4} + 13804208382312 \beta_{5} - 21642182863196 \beta_{6} - 1289241725604 \beta_{7}) q^{58} +(3119727491569881848 \beta_{1} - 27593866317394690 \beta_{2} + 27575697365527 \beta_{3} - 27575697365527 \beta_{4} + 65052614301169 \beta_{5} + 27575697365527 \beta_{6} - 80122391070288 \beta_{7}) q^{59} +(\)\(44\!\cdots\!60\)\( - 1620776144276029440 \beta_{1} - 1293051645394944 \beta_{2} - 34862316650496 \beta_{3} + 7478980902912 \beta_{4} + 49586840272896 \beta_{5} - 1954881208320 \beta_{6} - 28969487302656 \beta_{7}) q^{60} +(\)\(29\!\cdots\!54\)\( + 6162066410275910 \beta_{1} + 13719263677074030 \beta_{2} - 4149935619416 \beta_{3} - 30051442672084 \beta_{4} - 7250523811084 \beta_{5} - 7602063368654 \beta_{6} - 2533435429000 \beta_{7}) q^{61} +(-3864133109939918462 \beta_{1} + 103380707086945920 \beta_{2} - 37344848547200 \beta_{3} + 37344848547200 \beta_{4} + 58488146416160 \beta_{5} - 37344848547200 \beta_{6} + 22551803601280 \beta_{7}) q^{62} +(-\)\(79\!\cdots\!70\)\( + 6938928786176073576 \beta_{1} - 36921264796188576 \beta_{2} + 17393453827362 \beta_{3} + 3099936254427 \beta_{4} - 160985560722939 \beta_{5} - 26937878414292 \beta_{6} - 4104415310706 \beta_{7}) q^{63} -\)\(37\!\cdots\!68\)\( q^{64} +(-3792730177602328150 \beta_{1} - 78563945512261300 \beta_{2} - 3507840329000 \beta_{3} + 3507840329000 \beta_{4} - 137261786274650 \beta_{5} - 3507840329000 \beta_{6} + 174673707014850 \beta_{7}) q^{65} +(-\)\(18\!\cdots\!68\)\( - 3357850290749481042 \beta_{1} + 75725438027291850 \beta_{2} + 28983958472610 \beta_{3} + 84126744968940 \beta_{4} + 82498702465656 \beta_{5} - 88081740261396 \beta_{6} + 164533918999572 \beta_{7}) q^{66} +(\)\(52\!\cdots\!10\)\( - 8373976647773418 \beta_{1} - 18628530235645944 \beta_{2} + 48572223095307 \beta_{3} + 112827823715517 \beta_{4} + 5227792508301 \beta_{5} + 63970730392203 \beta_{6} + 11955559023024 \beta_{7}) q^{67} +(-12304990343551516672 \beta_{1} + 90319290858733568 \beta_{2} - 43587341385728 \beta_{3} + 43587341385728 \beta_{4} - 114600666202112 \beta_{5} - 43587341385728 \beta_{6} - 59065195560960 \beta_{7}) q^{68} +(-\)\(32\!\cdots\!08\)\( + 47141920729385663148 \beta_{1} + 58893731401208394 \beta_{2} + 395511219216432 \beta_{3} - 1803680732832 \beta_{4} - 172315234471368 \beta_{5} + 177743761992786 \beta_{6} - 223050517000158 \beta_{7}) q^{69} +(\)\(23\!\cdots\!80\)\( - 119846419972399524 \beta_{1} - 266919500499143980 \beta_{2} - 250107765351548 \beta_{3} + 245245647655256 \beta_{4} + 248487059452784 \beta_{5} - 13827901243560 \beta_{6} - 360156866392 \beta_{7}) q^{70} +(-\)\(15\!\cdots\!64\)\( \beta_{1} - 450478646413783444 \beta_{2} + 92261686255702 \beta_{3} - 92261686255702 \beta_{4} - 590676678726698 \beta_{5} + 92261686255702 \beta_{6} + 184243127514744 \beta_{7}) q^{71} +(\)\(78\!\cdots\!40\)\( + 20345090953023848448 \beta_{1} - 31054003710197760 \beta_{2} - 234097125359616 \beta_{3} + 33845818687488 \beta_{4} + 16174578401280 \beta_{5} + 36397029261312 \beta_{6} + 275005279567872 \beta_{7}) q^{72} +(\)\(20\!\cdots\!50\)\( + 328523749599799048 \beta_{1} + 731737446127308120 \beta_{2} + 643165290488120 \beta_{3} - 143084361920576 \beta_{4} - 476471647632272 \beta_{5} + 685944917919536 \beta_{6} + 37043031745744 \beta_{7}) q^{73} +(-\)\(11\!\cdots\!66\)\( \beta_{1} + 179277039321587200 \beta_{2} - 92129769665024 \beta_{3} + 92129769665024 \beta_{4} - 673416757066368 \beta_{5} - 92129769665024 \beta_{6} - 548095119384064 \beta_{7}) q^{74} +(\)\(32\!\cdots\!25\)\( + \)\(34\!\cdots\!15\)\( \beta_{1} - 613005092271150815 \beta_{2} - 651172577149530 \beta_{3} + 397341307509660 \beta_{4} - 931032312741630 \beta_{5} + 53557558510050 \beta_{6} + 462007056384360 \beta_{7}) q^{75} +(-\)\(55\!\cdots\!16\)\( - 48577026688286720 \beta_{1} - 107977272312463360 \beta_{2} + 1020396551798784 \beta_{3} + 392019214073856 \beta_{4} - 549591296507904 \beta_{5} - 458920510357504 \beta_{6} + 104623390064640 \beta_{7}) q^{76} +(-\)\(38\!\cdots\!38\)\( \beta_{1} + 549241987717581116 \beta_{2} - 183675113228096 \beta_{3} + 183675113228096 \beta_{4} + 1839686929632586 \beta_{5} - 183675113228096 \beta_{6} + 1638366021709050 \beta_{7}) q^{77} +(-\)\(39\!\cdots\!20\)\( + \)\(19\!\cdots\!40\)\( \beta_{1} + 94236815984303350 \beta_{2} + 479268760961310 \beta_{3} - 82555241267180 \beta_{4} + 233437670162600 \beta_{5} - 142552663813100 \beta_{6} - 1070528260858900 \beta_{7}) q^{78} +(-\)\(11\!\cdots\!78\)\( - 605634192381791240 \beta_{1} - 1348821292149011860 \beta_{2} - 342358921080048 \beta_{3} + 301818663882813 \beta_{4} + 328845502014303 \beta_{5} - 1817792591174902 \beta_{6} - 3002982014610 \beta_{7}) q^{79} +(-78406543612613492736 \beta_{1} + 24769797950537728 \beta_{2} + 1125899906842624 \beta_{5} + 1125899906842624 \beta_{7}) q^{80} +(-\)\(93\!\cdots\!43\)\( + \)\(49\!\cdots\!70\)\( \beta_{1} + 1696309399765820952 \beta_{2} - 1088805840774732 \beta_{3} - 1448564360406948 \beta_{4} + 4300084210163616 \beta_{5} - 333724858818696 \beta_{6} + 588554442954810 \beta_{7}) q^{81} +(-\)\(87\!\cdots\!40\)\( + 603766060534435572 \beta_{1} + 1344013112944009756 \beta_{2} - 2141512440830548 \beta_{3} - 2443160277649208 \beta_{4} + 613288201337296 \beta_{5} + 1070119145702088 \beta_{6} - 339605386554056 \beta_{7}) q^{82} +(-\)\(47\!\cdots\!88\)\( \beta_{1} + 1485445922286531114 \beta_{2} + 274667644912061 \beta_{3} - 274667644912061 \beta_{4} - 1778506945563791 \beta_{5} + 274667644912061 \beta_{6} - 9941843347791100 \beta_{7}) q^{83} +(\)\(15\!\cdots\!36\)\( + \)\(36\!\cdots\!12\)\( \beta_{1} - 813819379257966592 \beta_{2} - 888560550936576 \beta_{3} - 267801008799744 \beta_{4} - 1202666575233024 \beta_{5} + 680346978877440 \beta_{6} - 535403610243072 \beta_{7}) q^{84} +(-\)\(51\!\cdots\!20\)\( - 570890460440653304 \beta_{1} - 1272752529968144280 \beta_{2} - 6699224732730208 \beta_{3} - 2237821891021424 \beta_{4} + 3720209191479664 \beta_{5} - 139853694280360 \beta_{6} - 662003453611232 \beta_{7}) q^{85} +(-\)\(64\!\cdots\!78\)\( \beta_{1} - 4543558456335542400 \beta_{2} + 2280904382636928 \beta_{3} - 2280904382636928 \beta_{4} + 6764835802348896 \beta_{5} + 2280904382636928 \beta_{6} + 3244196548065408 \beta_{7}) q^{86} +(\)\(69\!\cdots\!40\)\( + \)\(16\!\cdots\!74\)\( \beta_{1} + 804411634582482 \beta_{2} + 5321250173542878 \beta_{3} + 28412822498055 \beta_{4} + 2809987892887323 \beta_{5} - 2997825273447420 \beta_{6} - 4861528155545262 \beta_{7}) q^{87} +(-\)\(17\!\cdots\!00\)\( + 1261036026961330176 \beta_{1} + 2808390830660780032 \beta_{2} + 1051948287524864 \beta_{3} - 1843144278671360 \beta_{4} - 1315680284573696 \beta_{5} + 2126135647272960 \beta_{6} - 58607110455296 \beta_{7}) q^{88} +(-\)\(29\!\cdots\!82\)\( \beta_{1} + 2107241403349714820 \beta_{2} - 759665137329428 \beta_{3} + 759665137329428 \beta_{4} + 6812410886010964 \beta_{5} - 759665137329428 \beta_{6} + 6523681944883302 \beta_{7}) q^{89} +(\)\(40\!\cdots\!20\)\( + \)\(25\!\cdots\!62\)\( \beta_{1} - 961994827273864374 \beta_{2} + 5207669259474978 \beta_{3} - 4639276738468116 \beta_{4} - 8241731834506056 \beta_{5} + 2766205070727660 \beta_{6} - 776758530967020 \beta_{7}) q^{90} +(-\)\(89\!\cdots\!40\)\( - 598010682125827080 \beta_{1} - 1330787891771909520 \beta_{2} + 4987542304592520 \beta_{3} + 2075948198687730 \beta_{4} - 2633045470165770 \beta_{5} - 3493050031273500 \beta_{6} + 523221518761500 \beta_{7}) q^{91} +(-\)\(13\!\cdots\!88\)\( \beta_{1} - 5120836520680357888 \beta_{2} + 1156201639313408 \beta_{3} - 1156201639313408 \beta_{4} - 11211372377407488 \beta_{5} + 1156201639313408 \beta_{6} - 3753063219200000 \beta_{7}) q^{92} +(\)\(23\!\cdots\!90\)\( + \)\(76\!\cdots\!41\)\( \beta_{1} - 842447326817387872 \beta_{2} - 16178475497928120 \beta_{3} + 5448874483477980 \beta_{4} - 5441047000879545 \beta_{5} + 1374066474869970 \beta_{6} + 18582965184062835 \beta_{7}) q^{93} +(\)\(84\!\cdots\!40\)\( + 1565042773446768200 \beta_{1} + 3487733524155736920 \beta_{2} + 13160013611970040 \beta_{3} + 1300570835946320 \beta_{4} - 8339818795997920 \beta_{5} - 1572771565523120 \beta_{6} + 1071154403549360 \beta_{7}) q^{94} +(-\)\(11\!\cdots\!12\)\( \beta_{1} + 16123059448764642476 \beta_{2} - 9374398000119050 \beta_{3} + 9374398000119050 \beta_{4} - 13705732463177942 \beta_{5} - 9374398000119050 \beta_{6} + 11359730830673808 \beta_{7}) q^{95} +(\)\(12\!\cdots\!56\)\( + \)\(30\!\cdots\!88\)\( \beta_{1} - 1125899906842624 \beta_{2} + 3377699720527872 \beta_{3} + 2251799813685248 \beta_{4} + 4503599627370496 \beta_{5} + 2251799813685248 \beta_{6} - 2251799813685248 \beta_{7}) q^{96} +(-\)\(11\!\cdots\!10\)\( - 4120068635364770214 \beta_{1} - 9171107325569749462 \beta_{2} + 10495072828092196 \beta_{3} + 25285895468798156 \beta_{4} + 1431916604204588 \beta_{5} + 5675503732019034 \beta_{6} + 2650442096065952 \beta_{7}) q^{97} +(-\)\(37\!\cdots\!21\)\( \beta_{1} - 11048576973127898624 \beta_{2} + 2182155859147264 \beta_{3} - 2182155859147264 \beta_{4} - 33791372542241664 \beta_{5} + 2182155859147264 \beta_{6} - 15619101499025920 \beta_{7}) q^{98} +(\)\(61\!\cdots\!92\)\( + \)\(10\!\cdots\!96\)\( \beta_{1} + 1220085986025986910 \beta_{2} - 7522842944564199 \beta_{3} + 31205894157979029 \beta_{4} + 5324807664889821 \beta_{5} - 2371044861788283 \beta_{6} - 20068906086959796 \beta_{7}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2194920q^{3} - 268435456q^{4} + 8978890752q^{6} + 183211263760q^{7} - 4853931303096q^{9} + O(q^{10}) \) \( 8q + 2194920q^{3} - 268435456q^{4} + 8978890752q^{6} + 183211263760q^{7} - 4853931303096q^{9} + 18696278507520q^{10} - 73649293885440q^{12} + 512429022548560q^{13} - 1064426665559040q^{15} + 9007199254740992q^{16} - 1881283573186560q^{18} + 13250502825732304q^{19} - 376087240418977584q^{21} + 428662713355468800q^{22} - 301281579173412864q^{24} - 5623831579932691000q^{25} + 11144542948278241320q^{27} - 6147549891468984320q^{28} - 12961438025281044480q^{30} - 31284409294003569776q^{31} - 27135100230528798720q^{33} - 98764413849435635712q^{34} + \)\(16\!\cdots\!72\)\(q^{36} - \)\(88\!\cdots\!00\)\(q^{37} + \)\(15\!\cdots\!40\)\(q^{39} - \)\(62\!\cdots\!40\)\(q^{40} + \)\(29\!\cdots\!60\)\(q^{42} - \)\(51\!\cdots\!40\)\(q^{43} + \)\(20\!\cdots\!20\)\(q^{45} - \)\(10\!\cdots\!48\)\(q^{46} + \)\(24\!\cdots\!80\)\(q^{48} - \)\(29\!\cdots\!40\)\(q^{49} + \)\(55\!\cdots\!84\)\(q^{51} - \)\(17\!\cdots\!20\)\(q^{52} + \)\(63\!\cdots\!00\)\(q^{54} - \)\(76\!\cdots\!20\)\(q^{55} + \)\(66\!\cdots\!80\)\(q^{57} - \)\(14\!\cdots\!40\)\(q^{58} + \)\(35\!\cdots\!80\)\(q^{60} + \)\(23\!\cdots\!32\)\(q^{61} - \)\(63\!\cdots\!60\)\(q^{63} - \)\(30\!\cdots\!44\)\(q^{64} - \)\(15\!\cdots\!44\)\(q^{66} + \)\(42\!\cdots\!80\)\(q^{67} - \)\(26\!\cdots\!64\)\(q^{69} + \)\(18\!\cdots\!40\)\(q^{70} + \)\(63\!\cdots\!20\)\(q^{72} + \)\(16\!\cdots\!00\)\(q^{73} + \)\(25\!\cdots\!00\)\(q^{75} - \)\(44\!\cdots\!28\)\(q^{76} - \)\(31\!\cdots\!60\)\(q^{78} - \)\(93\!\cdots\!24\)\(q^{79} - \)\(74\!\cdots\!44\)\(q^{81} - \)\(69\!\cdots\!20\)\(q^{82} + \)\(12\!\cdots\!88\)\(q^{84} - \)\(41\!\cdots\!60\)\(q^{85} + \)\(55\!\cdots\!20\)\(q^{87} - \)\(14\!\cdots\!00\)\(q^{88} + \)\(32\!\cdots\!60\)\(q^{90} - \)\(71\!\cdots\!20\)\(q^{91} + \)\(18\!\cdots\!20\)\(q^{93} + \)\(67\!\cdots\!20\)\(q^{94} + \)\(10\!\cdots\!48\)\(q^{96} - \)\(92\!\cdots\!80\)\(q^{97} + \)\(49\!\cdots\!36\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 11609466 x^{6} - 3416571600 x^{5} + 38618090622117 x^{4} + 19832313245955600 x^{3} - 25659229618759655524 x^{2} - 8410094250198512820000 x + 6059395276689883547331396\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-1584093760565750944768 \nu^{7} + 682806232215571051827200 \nu^{6} + 16745556275384504378097391616 \nu^{5} - 1209610537070777201048302182400 \nu^{4} - 48893394663140704807990500945402880 \nu^{3} - 11494872742795795060149698535130009600 \nu^{2} + 18503844884559516235665654812193147844608 \nu + 117627066671777958985826633240996492697600\)\()/ \)\(71\!\cdots\!67\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-23771668536858186847709471950 \nu^{7} + 52290159152897806908247250728583 \nu^{6} + 245541329519506923416385015670135226 \nu^{5} - 458967971939290240835944160660190100630 \nu^{4} - 772501875943416134598853790239365150776608 \nu^{3} + 1083151641318939537251344420303618545135867965 \nu^{2} + 600837329867679563926086928357927003430916173904 \nu - 482635364801645747904141675118969882722798584800974\)\()/ \)\(11\!\cdots\!10\)\( \)
\(\beta_{3}\)\(=\)\((\)\(10472828415467624028693327053990 \nu^{7} - 59550776530164021130435106404351699 \nu^{6} - 99323488217725276919656353238540413058 \nu^{5} + 671676878824106020436031924378853190560910 \nu^{4} + 350282011585734395569791572999715782842795424 \nu^{3} - 2078217852733308767503732097830832231449274363905 \nu^{2} - 677576573863881738767598752856306793858005911747472 \nu + 851297339896747881222666111808260323430005462452233702\)\()/ \)\(33\!\cdots\!30\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-14649123168149302896572342723414 \nu^{7} + 51831441118402953028199918573163679 \nu^{6} + 141939672290653764982363773803304450274 \nu^{5} - 502706129743721155715089422213266227434086 \nu^{4} - 455315952886567229795074604868832254666471872 \nu^{3} + 1347664420567985647843275195670772027478473865733 \nu^{2} + 531250469566830810685567625678662052778487711659888 \nu - 559115823971172311224748188926193380573098917776117486\)\()/ \)\(16\!\cdots\!15\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-179037373225328256454683461524 \nu^{7} + 7490999588469613397441446443338 \nu^{6} + 1987342693549973541070212782460277532 \nu^{5} + 1175398665764559296392627184681322684284 \nu^{4} - 5955344762625162598593170296739989607797696 \nu^{3} - 6901820155404200740310024000098268127855148402 \nu^{2} + 768779010765490740721019614392909701883301670880 \nu + 2247716680784567884636519779150001181005096536073452\)\()/ \)\(16\!\cdots\!85\)\( \)
\(\beta_{6}\)\(=\)\((\)\(40004282257370968348627515853942 \nu^{7} - 88268259337088648723552627636079707 \nu^{6} - 412240675147550438395360601825051806322 \nu^{5} + 774302930108036458909481179418547652494238 \nu^{4} + 1291841500770879335876334501583283994457222816 \nu^{3} - 1823543971182979128225220514568148363096040082329 \nu^{2} - 996035323361372647253219490020164677937202072342544 \nu + 806458126559182115795702124641593233597165511281207798\)\()/ \)\(33\!\cdots\!30\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-13213394915339168538561840431828 \nu^{7} - 3657421675370141954504771289037682 \nu^{6} + 158469283862685011311946053693426285708 \nu^{5} + 65730941354286673478556807150308290522468 \nu^{4} - 500971514056089067402763688715650694591055264 \nu^{3} - 275953943049751272645586147517540288983856528054 \nu^{2} + 163978460951324930396203755008784359127471991354176 \nu + 53970789496238104179862167051957174906660695018493108\)\()/ \)\(55\!\cdots\!05\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(10 \beta_{7} - 10 \beta_{6} + 28 \beta_{5} + 118 \beta_{4} + 17 \beta_{3} - 75867 \beta_{2} - 73439 \beta_{1}\)\()/ 161243136 \)
\(\nu^{2}\)\(=\)\((\)\(-578 \beta_{7} + 4042 \beta_{6} + 5453 \beta_{5} + 314 \beta_{4} - 7991 \beta_{3} - 1098075 \beta_{2} - 492511 \beta_{1} + 19499444845056\)\()/6718464\)
\(\nu^{3}\)\(=\)\((\)\(14591978 \beta_{7} - 52975850 \beta_{6} + 140814986 \beta_{5} + 266010278 \beta_{4} - 77504459 \beta_{3} - 233745563847 \beta_{2} - 276345948991 \beta_{1} + 103293509841100800\)\()/80621568\)
\(\nu^{4}\)\(=\)\((\)\(-1334816132 \beta_{7} + 11380999528 \beta_{6} + 16898970563 \beta_{5} + 5105334572 \beta_{4} - 22582217672 \beta_{3} - 5875070021436 \beta_{2} - 6839742490816 \beta_{1} + 48325508075418782976\)\()/3359232\)
\(\nu^{5}\)\(=\)\((\)\(31345738661674 \beta_{7} - 106295080362730 \beta_{6} + 448722316174849 \beta_{5} + 702783641569390 \beta_{4} - 317436085215601 \beta_{3} - 642566675944074453 \beta_{2} - 996431497920160445 \beta_{1} + 499658682426986534400000\)\()/40310784\)
\(\nu^{6}\)\(=\)\((\)\(-4031504815391588 \beta_{7} + 38214853713302860 \beta_{6} + 67864218917851133 \beta_{5} + 32118945541071356 \beta_{4} - 84507017522072828 \beta_{3} - 32848431346031639652 \beta_{2} - 55407824651133078100 \beta_{1} + 175912484985411593316678144\)\()/2239488\)
\(\nu^{7}\)\(=\)\((\)\(141485327727071136388 \beta_{7} - 319070598049079124100 \beta_{6} + 2786371089635688186625 \beta_{5} + 3857278019380935416692 \beta_{4} - 2210719859184331803904 \beta_{3} - 3560079215684868603126864 \beta_{2} - 7009969823590916359966052 \beta_{1} + 3763887841144034315317244620800\)\()/40310784\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2468.36 + 1.41421i
−891.970 + 1.41421i
530.624 + 1.41421i
−2107.01 + 1.41421i
2468.36 1.41421i
−891.970 1.41421i
530.624 1.41421i
−2107.01 1.41421i
5792.62i −847324. + 1.35052e6i −3.35544e7 2.16695e9i 7.82305e9 + 4.90823e9i 1.32110e11 1.94368e11i −1.10595e12 2.28866e12i 1.25523e13
5.2 5792.62i −388622. 1.54623e6i −3.35544e7 1.99247e8i −8.95674e9 + 2.25114e9i −4.03334e9 1.94368e11i −2.23981e12 + 1.20180e12i −1.15416e12
5.3 5792.62i 793293. + 1.38295e6i −3.35544e7 1.58663e9i 8.01091e9 4.59524e9i 2.87923e10 1.94368e11i −1.28324e12 + 2.19417e12i −9.19074e12
5.4 5792.62i 1.54011e6 412209.i −3.35544e7 1.23273e9i −2.38777e9 8.92129e9i −6.52628e10 1.94368e11i 2.20203e12 1.26970e12i 7.14075e12
5.5 5792.62i −847324. 1.35052e6i −3.35544e7 2.16695e9i 7.82305e9 4.90823e9i 1.32110e11 1.94368e11i −1.10595e12 + 2.28866e12i 1.25523e13
5.6 5792.62i −388622. + 1.54623e6i −3.35544e7 1.99247e8i −8.95674e9 2.25114e9i −4.03334e9 1.94368e11i −2.23981e12 1.20180e12i −1.15416e12
5.7 5792.62i 793293. 1.38295e6i −3.35544e7 1.58663e9i 8.01091e9 + 4.59524e9i 2.87923e10 1.94368e11i −1.28324e12 2.19417e12i −9.19074e12
5.8 5792.62i 1.54011e6 + 412209.i −3.35544e7 1.23273e9i −2.38777e9 + 8.92129e9i −6.52628e10 1.94368e11i 2.20203e12 + 1.26970e12i 7.14075e12
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.27.b.a 8
3.b odd 2 1 inner 6.27.b.a 8
4.b odd 2 1 48.27.e.b 8
12.b even 2 1 48.27.e.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.27.b.a 8 1.a even 1 1 trivial
6.27.b.a 8 3.b odd 2 1 inner
48.27.e.b 8 4.b odd 2 1
48.27.e.b 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{27}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 33554432 + T^{2} )^{4} \)
$3$ \( \)\(41\!\cdots\!81\)\( - \)\(36\!\cdots\!80\)\( T + \)\(31\!\cdots\!68\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!38\)\( T^{4} - 10804244529179064600 T^{5} + 4835802554748 T^{6} - 2194920 T^{7} + T^{8} \)
$5$ \( \)\(71\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( T^{2} + \)\(23\!\cdots\!00\)\( T^{4} + 8772380267505408000 T^{6} + T^{8} \)
$7$ \( ( \)\(10\!\cdots\!56\)\( + \)\(22\!\cdots\!60\)\( T - \)\(70\!\cdots\!88\)\( T^{2} - 91605631880 T^{3} + T^{4} )^{2} \)
$11$ \( \)\(74\!\cdots\!16\)\( + \)\(81\!\cdots\!32\)\( T^{2} + \)\(12\!\cdots\!24\)\( T^{4} + \)\(60\!\cdots\!08\)\( T^{6} + T^{8} \)
$13$ \( ( \)\(36\!\cdots\!00\)\( + \)\(21\!\cdots\!00\)\( T - \)\(50\!\cdots\!00\)\( T^{2} - 256214511274280 T^{3} + T^{4} )^{2} \)
$17$ \( \)\(37\!\cdots\!76\)\( + \)\(47\!\cdots\!48\)\( T^{2} + \)\(58\!\cdots\!24\)\( T^{4} + \)\(22\!\cdots\!52\)\( T^{6} + T^{8} \)
$19$ \( ( \)\(42\!\cdots\!76\)\( + \)\(37\!\cdots\!52\)\( T - \)\(58\!\cdots\!76\)\( T^{2} - 6625251412866152 T^{3} + T^{4} )^{2} \)
$23$ \( \)\(22\!\cdots\!36\)\( + \)\(44\!\cdots\!08\)\( T^{2} + \)\(31\!\cdots\!44\)\( T^{4} + \)\(95\!\cdots\!32\)\( T^{6} + T^{8} \)
$29$ \( \)\(19\!\cdots\!00\)\( + \)\(55\!\cdots\!00\)\( T^{2} + \)\(93\!\cdots\!00\)\( T^{4} + \)\(53\!\cdots\!60\)\( T^{6} + T^{8} \)
$31$ \( ( \)\(58\!\cdots\!56\)\( - \)\(39\!\cdots\!08\)\( T - \)\(24\!\cdots\!96\)\( T^{2} + 15642204647001784888 T^{3} + T^{4} )^{2} \)
$37$ \( ( -\)\(29\!\cdots\!64\)\( - \)\(10\!\cdots\!40\)\( T + \)\(14\!\cdots\!92\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{3} + T^{4} )^{2} \)
$41$ \( \)\(33\!\cdots\!56\)\( + \)\(19\!\cdots\!12\)\( T^{2} + \)\(51\!\cdots\!24\)\( T^{4} + \)\(41\!\cdots\!68\)\( T^{6} + T^{8} \)
$43$ \( ( -\)\(15\!\cdots\!84\)\( - \)\(24\!\cdots\!00\)\( T - \)\(73\!\cdots\!72\)\( T^{2} + \)\(25\!\cdots\!20\)\( T^{3} + T^{4} )^{2} \)
$47$ \( \)\(28\!\cdots\!00\)\( + \)\(70\!\cdots\!00\)\( T^{2} + \)\(44\!\cdots\!00\)\( T^{4} + \)\(56\!\cdots\!00\)\( T^{6} + T^{8} \)
$53$ \( \)\(28\!\cdots\!76\)\( + \)\(29\!\cdots\!48\)\( T^{2} + \)\(11\!\cdots\!24\)\( T^{4} + \)\(17\!\cdots\!52\)\( T^{6} + T^{8} \)
$59$ \( \)\(39\!\cdots\!56\)\( + \)\(78\!\cdots\!72\)\( T^{2} + \)\(13\!\cdots\!84\)\( T^{4} + \)\(68\!\cdots\!08\)\( T^{6} + T^{8} \)
$61$ \( ( -\)\(37\!\cdots\!04\)\( + \)\(79\!\cdots\!24\)\( T - \)\(59\!\cdots\!64\)\( T^{2} - \)\(11\!\cdots\!16\)\( T^{3} + T^{4} )^{2} \)
$67$ \( ( \)\(21\!\cdots\!16\)\( - \)\(37\!\cdots\!00\)\( T + \)\(14\!\cdots\!12\)\( T^{2} - \)\(21\!\cdots\!40\)\( T^{3} + T^{4} )^{2} \)
$71$ \( \)\(51\!\cdots\!00\)\( + \)\(54\!\cdots\!00\)\( T^{2} + \)\(15\!\cdots\!00\)\( T^{4} + \)\(73\!\cdots\!60\)\( T^{6} + T^{8} \)
$73$ \( ( \)\(12\!\cdots\!00\)\( + \)\(55\!\cdots\!00\)\( T - \)\(86\!\cdots\!60\)\( T^{2} - \)\(82\!\cdots\!00\)\( T^{3} + T^{4} )^{2} \)
$79$ \( ( -\)\(10\!\cdots\!04\)\( - \)\(14\!\cdots\!32\)\( T - \)\(35\!\cdots\!36\)\( T^{2} + \)\(46\!\cdots\!12\)\( T^{3} + T^{4} )^{2} \)
$83$ \( \)\(45\!\cdots\!56\)\( + \)\(25\!\cdots\!12\)\( T^{2} + \)\(70\!\cdots\!24\)\( T^{4} + \)\(50\!\cdots\!68\)\( T^{6} + T^{8} \)
$89$ \( \)\(20\!\cdots\!56\)\( + \)\(68\!\cdots\!72\)\( T^{2} + \)\(65\!\cdots\!84\)\( T^{4} + \)\(15\!\cdots\!08\)\( T^{6} + T^{8} \)
$97$ \( ( \)\(80\!\cdots\!56\)\( - \)\(20\!\cdots\!60\)\( T - \)\(63\!\cdots\!52\)\( T^{2} + \)\(46\!\cdots\!40\)\( T^{3} + T^{4} )^{2} \)
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