Properties

Label 36.33.d.b
Level $36$
Weight $33$
Character orbit 36.d
Analytic conductor $233.520$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 33 \)
Character orbit: \([\chi]\) \(=\) 36.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(233.519958512\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 2 x^{13} + \)\(99\!\cdots\!07\)\( x^{12} - \)\(80\!\cdots\!76\)\( x^{11} + \)\(37\!\cdots\!31\)\( x^{10} - \)\(23\!\cdots\!66\)\( x^{9} + \)\(66\!\cdots\!33\)\( x^{8} + \)\(26\!\cdots\!04\)\( x^{7} + \)\(53\!\cdots\!48\)\( x^{6} + \)\(13\!\cdots\!32\)\( x^{5} + \)\(14\!\cdots\!84\)\( x^{4} + \)\(95\!\cdots\!64\)\( x^{3} + \)\(10\!\cdots\!40\)\( x^{2} - \)\(74\!\cdots\!00\)\( x + \)\(14\!\cdots\!00\)\(\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{182}\cdot 3^{29}\cdot 5^{6} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1699 - \beta_{1} ) q^{2} + ( -208774120 - 1782 \beta_{1} - \beta_{2} ) q^{4} + ( -9865592530 - 532715 \beta_{1} + 8 \beta_{2} - \beta_{3} + \beta_{4} ) q^{5} + ( 55143903 - 128669069 \beta_{1} + 1268 \beta_{2} - 794 \beta_{3} - 23 \beta_{4} + 7 \beta_{5} - \beta_{6} ) q^{7} + ( 13668947443586 + 214324147 \beta_{1} - 2967 \beta_{2} + 8744 \beta_{3} + 201 \beta_{4} + 43 \beta_{5} - \beta_{11} ) q^{8} +O(q^{10})\) \( q +(1699 - \beta_{1}) q^{2} +(-208774120 - 1782 \beta_{1} - \beta_{2}) q^{4} +(-9865592530 - 532715 \beta_{1} + 8 \beta_{2} - \beta_{3} + \beta_{4}) q^{5} +(55143903 - 128669069 \beta_{1} + 1268 \beta_{2} - 794 \beta_{3} - 23 \beta_{4} + 7 \beta_{5} - \beta_{6}) q^{7} +(13668947443586 + 214324147 \beta_{1} - 2967 \beta_{2} + 8744 \beta_{3} + 201 \beta_{4} + 43 \beta_{5} - \beta_{11}) q^{8} +(2269681504813112 + 10770848472 \beta_{1} - 566649 \beta_{2} + 39955 \beta_{3} + 9605 \beta_{4} + 2764 \beta_{5} - 28 \beta_{6} + 7 \beta_{7} - 9 \beta_{8} + 2 \beta_{9} + 14 \beta_{10} + 8 \beta_{11} + \beta_{13}) q^{10} +(-38893808054 + 90752802759 \beta_{1} + 1317730 \beta_{2} - 293024 \beta_{3} + 653 \beta_{4} - 5901 \beta_{5} + 107 \beta_{6} + 19 \beta_{7} + 22 \beta_{8} + \beta_{9} + 9 \beta_{10} + 16 \beta_{11} - \beta_{12} - 8 \beta_{13}) q^{11} +(-117172011244620065 - 1718686380454 \beta_{1} + 40343193 \beta_{2} - 3766990 \beta_{3} - 890249 \beta_{4} - 159268 \beta_{5} + 468 \beta_{6} - 734 \beta_{7} + 283 \beta_{8} - 50 \beta_{9} - 981 \beta_{10} + 404 \beta_{11} - 36 \beta_{12}) q^{13} +(-552256883982633915 - 218508500754 \beta_{1} - 119755421 \beta_{2} - 23902584 \beta_{3} - 1126288 \beta_{4} - 348823 \beta_{5} + 21156 \beta_{6} - 3134 \beta_{7} - 220 \beta_{8} - 80 \beta_{9} - 8 \beta_{10} + 576 \beta_{11} + 336 \beta_{12} + 128 \beta_{13}) q^{14} +(-2183625418780215860 - 14542552181852 \beta_{1} + 164196912 \beta_{2} + 733282020 \beta_{3} - 24452788 \beta_{4} - 4378652 \beta_{5} + 330756 \beta_{6} + 15076 \beta_{7} - 33624 \beta_{8} - 1072 \beta_{9} + 3940 \beta_{10} - 23780 \beta_{11} - 816 \beta_{12} + 1104 \beta_{13}) q^{16} +(-2255047583153039491 - 50289465079993 \beta_{1} + 1436162963 \beta_{2} - 149537773 \beta_{3} + 6087725 \beta_{4} - 3566417 \beta_{5} + 3806 \beta_{6} + 202804 \beta_{7} + 19113 \beta_{8} - 26851 \beta_{9} - 44420 \beta_{10} - 68682 \beta_{11} + 1070 \beta_{12}) q^{17} +(-168858879578682 + 393988063340663 \beta_{1} - 1559252798 \beta_{2} - 14978092548 \beta_{3} + 55605701 \beta_{4} - 23777797 \beta_{5} - 210317 \beta_{6} + 2057843 \beta_{7} + 534870 \beta_{8} + 86817 \beta_{9} - 217687 \beta_{10} + 340496 \beta_{11} + 11487 \beta_{12} + 1784 \beta_{13}) q^{19} +(-\)\(14\!\cdots\!60\)\( - 2328377423502020 \beta_{1} + 14058080962 \beta_{2} - 16622311064 \beta_{3} + 968117504 \beta_{4} - 26958040 \beta_{5} - 5660920 \beta_{6} - 159480 \beta_{7} + 163120 \beta_{8} + 322720 \beta_{9} - 548760 \beta_{10} - 388960 \beta_{11} + 21920 \beta_{12} + 28960 \beta_{13}) q^{20} +(\)\(38\!\cdots\!61\)\( + 154113852425537 \beta_{1} + 79351260307 \beta_{2} + 29330345212 \beta_{3} + 13039863241 \beta_{4} + 118325831 \beta_{5} + 17862952 \beta_{6} + 316509 \beta_{7} + 281128 \beta_{8} - 1258016 \beta_{9} + 2755120 \beta_{10} + 1901184 \beta_{11} - 54240 \beta_{12} - 131840 \beta_{13}) q^{22} +(2086976797617861 - 4869578537808575 \beta_{1} - 257137300600 \beta_{2} + 204435315280 \beta_{3} + 1244556063 \beta_{4} + 426390321 \beta_{5} + 8568793 \beta_{6} - 24933690 \beta_{7} - 10188756 \beta_{8} - 869534 \beta_{9} + 1140146 \beta_{10} - 8145376 \beta_{11} - 228194 \beta_{12} - 76560 \beta_{13}) q^{23} +(\)\(73\!\cdots\!75\)\( - 8206794596401100 \beta_{1} - 1901884773760 \beta_{2} + 233189216520 \beta_{3} - 19775386570 \beta_{4} - 870416350 \beta_{5} - 27211700 \beta_{6} + 54036500 \beta_{7} - 15198800 \beta_{8} + 4748850 \beta_{9} + 43734850 \beta_{10} + 19086300 \beta_{11} - 684900 \beta_{12}) q^{25} +(\)\(71\!\cdots\!12\)\( + 120092654595858496 \beta_{1} - 1790697012123 \beta_{2} + 160932813849 \beta_{3} - 107662296049 \beta_{4} + 7396013940 \beta_{5} - 283177876 \beta_{6} - 2789195 \beta_{7} - 31911803 \beta_{8} - 10284218 \beta_{9} - 11684886 \beta_{10} + 26074392 \beta_{11} + 535808 \beta_{12} - 2056157 \beta_{13}) q^{26} +(\)\(20\!\cdots\!28\)\( + 560751938199406676 \beta_{1} - 1068488337324 \beta_{2} + 6233233125474 \beta_{3} - 310086797384 \beta_{4} - 1889905458 \beta_{5} + 557024506 \beta_{6} + 14054426 \beta_{7} + 395299788 \beta_{8} + 43640840 \beta_{9} + 40204034 \beta_{10} + 61971320 \beta_{11} - 8600376 \beta_{12} - 5645720 \beta_{13}) q^{28} +(\)\(27\!\cdots\!54\)\( + 652035968218220733 \beta_{1} + 22151592902016 \beta_{2} - 3002921344665 \beta_{3} + 405207818745 \beta_{4} + 54624770040 \beta_{5} + 478908784 \beta_{6} - 1431392672 \beta_{7} - 172978968 \beta_{8} + 107356840 \beta_{9} - 173046016 \beta_{10} + 143238192 \beta_{11} + 5105392 \beta_{12}) q^{29} +(153878975281976482 - 358976948828387494 \beta_{1} + 108788694434700 \beta_{2} + 2011273200646 \beta_{3} - 800464459518 \beta_{4} - 30110770146 \beta_{5} - 2469956434 \beta_{6} - 1958313958 \beta_{7} + 1059772820 \beta_{8} - 167719362 \beta_{9} + 757025518 \beta_{10} + 1525998560 \beta_{11} + 32108994 \beta_{12} + 35806736 \beta_{13}) q^{31} +(\)\(21\!\cdots\!96\)\( + 2271310234094858448 \beta_{1} - 3768553914368 \beta_{2} - 115152732221168 \beta_{3} - 1574295214608 \beta_{4} + 154244827856 \beta_{5} - 4616074352 \beta_{6} + 869602064 \beta_{7} - 802950752 \beta_{8} + 118211904 \beta_{9} - 1653317360 \beta_{10} - 230163792 \beta_{11} - 107345600 \beta_{12} - 50198720 \beta_{13}) q^{32} +(\)\(21\!\cdots\!82\)\( + 2340545454605621454 \beta_{1} - 18159835421594 \beta_{2} - 278981751991306 \beta_{3} - 12332151558774 \beta_{4} - 103310716912 \beta_{5} + 13251666536 \beta_{6} - 1020611458 \beta_{7} + 1685717086 \beta_{8} - 1215207196 \beta_{9} + 1338378172 \beta_{10} + 512220048 \beta_{11} - 106604160 \beta_{12} + 183886130 \beta_{13}) q^{34} +(3672045346598923786 - 8568353753855089354 \beta_{1} - 453704818725572 \beta_{2} + 52778333276730 \beta_{3} + 2246441552190 \beta_{4} + 722822737282 \beta_{5} + 849075986 \beta_{6} - 31156922294 \beta_{7} - 6691486412 \beta_{8} - 1814410034 \beta_{9} - 1107574338 \beta_{10} + 5355392224 \beta_{11} + 99889970 \beta_{12} + 358545808 \beta_{13}) q^{35} +(-\)\(76\!\cdots\!17\)\( - 16682702186877181122 \beta_{1} + 336918808878261 \beta_{2} - 30701819655666 \beta_{3} - 6577416201801 \beta_{4} - 1443900341044 \beta_{5} + 2040011844 \beta_{6} - 43746010470 \beta_{7} + 9316950671 \beta_{8} + 1556783414 \beta_{9} - 12688153281 \beta_{10} + 11199894532 \beta_{11} - 577594836 \beta_{12}) q^{37} +(\)\(16\!\cdots\!33\)\( + 668700523605424645 \beta_{1} + 35017742491407 \beta_{2} + 3043684693405164 \beta_{3} + 6460275645069 \beta_{4} + 1495649704627 \beta_{5} - 141058634104 \beta_{6} - 24623966255 \beta_{7} + 35975082888 \beta_{8} - 3962458528 \beta_{9} - 17447228048 \beta_{10} + 21532317824 \beta_{11} - 46799968 \beta_{12} + 1099680000 \beta_{13}) q^{38} +(\)\(33\!\cdots\!88\)\( + \)\(14\!\cdots\!58\)\( \beta_{1} - 3188585557251666 \beta_{2} + 7429900713292080 \beta_{3} - 38508541699730 \beta_{4} + 7201518178986 \beta_{5} + 298790954368 \beta_{6} - 17246344832 \beta_{7} - 13674127616 \beta_{8} + 8431205888 \beta_{9} - 21749649024 \beta_{10} + 26441518082 \beta_{11} + 1828462080 \beta_{12} + 2004749824 \beta_{13}) q^{40} +(-\)\(25\!\cdots\!06\)\( + 74827715186252965496 \beta_{1} - 5272729774298780 \beta_{2} + 610335691246764 \beta_{3} - 73568005566422 \beta_{4} + 3621465220670 \beta_{5} - 121806299612 \beta_{6} - 325640228204 \beta_{7} + 130857892188 \beta_{8} + 12317402038 \beta_{9} - 12100454278 \beta_{10} + 67660342548 \beta_{11} - 3073252748 \beta_{12}) q^{41} +(-\)\(16\!\cdots\!99\)\( + \)\(39\!\cdots\!04\)\( \beta_{1} - 14725873599454636 \beta_{2} - 517246645356533 \beta_{3} + 145323896056786 \beta_{4} - 15223572973394 \beta_{5} + 207611649886 \beta_{6} - 592417193570 \beta_{7} - 141852177508 \beta_{8} - 34666395222 \beta_{9} - 59364781062 \beta_{10} + 149638912672 \beta_{11} - 603703722 \beta_{12} - 13509561680 \beta_{13}) q^{43} +(\)\(24\!\cdots\!66\)\( - \)\(38\!\cdots\!54\)\( \beta_{1} + 2731366227117430 \beta_{2} - 26610421604050785 \beta_{3} + 382748560200164 \beta_{4} - 858350241079 \beta_{5} - 1515314321901 \beta_{6} + 130249142147 \beta_{7} - 61595760390 \beta_{8} + 1422687036 \beta_{9} + 353308330575 \beta_{10} + 41259295492 \beta_{11} - 6244226340 \beta_{12} + 9499504908 \beta_{13}) q^{44} +(-\)\(20\!\cdots\!31\)\( - 8265291117368078618 \beta_{1} - 545342989994901 \beta_{2} - 36423814442168664 \beta_{3} - 55779482872472 \beta_{4} + 57785896524689 \beta_{5} + 3000336898532 \beta_{6} + 360083972186 \beta_{7} - 639999363484 \beta_{8} + 39722800816 \beta_{9} + 165562666360 \beta_{10} - 606121888192 \beta_{11} - 1989215664 \beta_{12} - 12086122368 \beta_{13}) q^{46} +(\)\(93\!\cdots\!40\)\( - \)\(21\!\cdots\!04\)\( \beta_{1} + 21125266395012868 \beta_{2} - 756564436654390 \beta_{3} - 382007752264640 \beta_{4} + 115214108427648 \beta_{5} - 1262544820192 \beta_{6} - 1912287039190 \beta_{7} + 91544076212 \beta_{8} - 121563704082 \beta_{9} + 231081064478 \beta_{10} + 715570892512 \beta_{11} + 1621011218 \beta_{12} - 49734621040 \beta_{13}) q^{47} +(-\)\(34\!\cdots\!31\)\( - \)\(77\!\cdots\!72\)\( \beta_{1} - 48986209068214628 \beta_{2} + 9481347226483804 \beta_{3} - 1352681908895604 \beta_{4} - 565835983672092 \beta_{5} + 572995853864 \beta_{6} - 5522223025536 \beta_{7} - 74437831468 \beta_{8} + 379689728124 \beta_{9} + 183236550728 \beta_{10} - 86514109752 \beta_{11} + 60420969000 \beta_{12}) q^{49} +(\)\(36\!\cdots\!85\)\( - \)\(71\!\cdots\!15\)\( \beta_{1} - 4297463877235020 \beta_{2} - 57525086242330700 \beta_{3} + 6586293913317900 \beta_{4} - 272796655482880 \beta_{5} - 8122463724240 \beta_{6} - 758966920540 \beta_{7} + 730814564580 \beta_{8} + 236159917560 \beta_{9} - 1593175777080 \beta_{10} - 1392622000160 \beta_{11} + 28434067200 \beta_{12} - 870521220 \beta_{13}) q^{50} +(-\)\(69\!\cdots\!16\)\( - \)\(74\!\cdots\!16\)\( \beta_{1} + 154642328671998270 \beta_{2} - 231746394248971976 \beta_{3} - 544197861611776 \beta_{4} + 422352261479032 \beta_{5} + 9787540213272 \beta_{6} + 274437973016 \beta_{7} + 3056280223120 \beta_{8} - 554406602272 \beta_{9} + 250398588472 \beta_{10} - 1145363954208 \beta_{11} + 114155189472 \beta_{12} - 123064043680 \beta_{13}) q^{52} +(-\)\(13\!\cdots\!53\)\( + \)\(16\!\cdots\!80\)\( \beta_{1} + 469024893234028417 \beta_{2} - 60067085770753224 \beta_{3} - 7571711393161711 \beta_{4} + 706888790103156 \beta_{5} + 12496618107780 \beta_{6} - 3617241598078 \beta_{7} - 8060244297693 \beta_{8} + 688833090774 \beta_{9} - 2835906728853 \beta_{10} - 1174908991740 \beta_{11} + 182326798092 \beta_{12}) q^{53} +(-\)\(28\!\cdots\!05\)\( + \)\(67\!\cdots\!95\)\( \beta_{1} + 445618355674796500 \beta_{2} + 205052777509200630 \beta_{3} + 4106847695723225 \beta_{4} - 4046336945770345 \beta_{5} + 54915852690575 \beta_{6} - 30764546059120 \beta_{7} + 1254859367200 \beta_{8} - 1822247716560 \beta_{9} + 5957872063920 \beta_{10} + 6857955971840 \beta_{11} + 61266233040 \beta_{12} - 49401727360 \beta_{13}) q^{55} +(-\)\(45\!\cdots\!08\)\( - \)\(20\!\cdots\!84\)\( \beta_{1} + 410652241260923840 \beta_{2} + 1337675978882458400 \beta_{3} + 20292903815274272 \beta_{4} - 5403091968846368 \beta_{5} - 56693707210720 \beta_{6} + 17288450059040 \beta_{7} - 10582434727104 \beta_{8} - 2275394916736 \beta_{9} - 3142422164192 \beta_{10} - 5924359063776 \beta_{11} - 245843138944 \beta_{12} - 295137678720 \beta_{13}) q^{56} +(-\)\(27\!\cdots\!68\)\( - \)\(28\!\cdots\!92\)\( \beta_{1} + 495979207001539495 \beta_{2} + 1705867979166503539 \beta_{3} + 7336699505880229 \beta_{4} + 6393099874717708 \beta_{5} + 38980925168484 \beta_{6} + 12049926032359 \beta_{7} - 10560970104361 \beta_{8} + 5702017585218 \beta_{9} + 7783845256654 \beta_{10} + 27039910461704 \beta_{11} + 244864871424 \beta_{12} - 711721189855 \beta_{13}) q^{58} +(\)\(69\!\cdots\!01\)\( - \)\(16\!\cdots\!78\)\( \beta_{1} + 771257115594493744 \beta_{2} - 339275208467202573 \beta_{3} - 22697854382887456 \beta_{4} + 8789406352728928 \beta_{5} + 278364456062240 \beta_{6} + 24468492808664 \beta_{7} + 26007165908912 \beta_{8} - 70463888184 \beta_{9} + 1880125012616 \beta_{10} + 32046866386048 \beta_{11} + 458515618616 \beta_{12} - 1351777519168 \beta_{13}) q^{59} +(\)\(76\!\cdots\!51\)\( - \)\(29\!\cdots\!42\)\( \beta_{1} + 2528540612019837481 \beta_{2} - 193939487838749678 \beta_{3} + 70237411596114975 \beta_{4} - 17574831089436124 \beta_{5} + 22442579387204 \beta_{6} - 75879185574510 \beta_{7} + 10153939535371 \beta_{8} + 3367662859766 \beta_{9} - 41581480678429 \beta_{10} + 41322674643268 \beta_{11} - 2470524923124 \beta_{12}) q^{61} +(-\)\(15\!\cdots\!12\)\( - \)\(60\!\cdots\!68\)\( \beta_{1} - 7688885966973056 \beta_{2} - 2994886231214963296 \beta_{3} - 41230762333527224 \beta_{4} - 31769517679826960 \beta_{5} - 495433145694688 \beta_{6} - 5982544309672 \beta_{7} + 60759208002080 \beta_{8} + 9448863222144 \beta_{9} + 33567955023296 \beta_{10} + 120440178811392 \beta_{11} + 1302741646976 \beta_{12} - 1414508354560 \beta_{13}) q^{62} +(-\)\(14\!\cdots\!16\)\( - \)\(22\!\cdots\!20\)\( \beta_{1} + 2276659987511254016 \beta_{2} + 1695690219472769856 \beta_{3} - 27209282905127744 \beta_{4} + 39491209003700800 \beta_{5} - 18580897893056 \beta_{6} - 333360828212416 \beta_{7} - 107028543378304 \beta_{8} - 23875466247936 \beta_{9} - 58565937843392 \beta_{10} - 42221464980544 \beta_{11} + 2791025119488 \beta_{12} - 1593986864896 \beta_{13}) q^{64} +(-\)\(80\!\cdots\!80\)\( + \)\(67\!\cdots\!40\)\( \beta_{1} - 1979331065275896780 \beta_{2} + 27598030900035980 \beta_{3} - 293664402274640250 \beta_{4} + 35099328264633810 \beta_{5} - 74905335874980 \beta_{6} + 422579104994300 \beta_{7} - 10384340152020 \beta_{8} - 18334223336310 \beta_{9} + 40288072779390 \beta_{10} + 38000217079020 \beta_{11} - 5333475982260 \beta_{12}) q^{65} +(-\)\(37\!\cdots\!94\)\( + \)\(86\!\cdots\!51\)\( \beta_{1} - 5245819158333101414 \beta_{2} - 5761961673563387420 \beta_{3} + 129377391239569001 \beta_{4} - 46400072982472297 \beta_{5} - 3304914454079441 \beta_{6} - 14567777747385 \beta_{7} + 55861842399678 \beta_{8} - 12883401144963 \beta_{9} - 82216325110683 \beta_{10} + 257197851826128 \beta_{11} + 6539842648707 \beta_{12} + 8675214038040 \beta_{13}) q^{67} +(-\)\(25\!\cdots\!88\)\( - \)\(22\!\cdots\!48\)\( \beta_{1} + 4966191440176032898 \beta_{2} - 19777948605071288752 \beta_{3} - 514145224506828288 \beta_{4} - 89436186798932528 \beta_{5} - 1751826755539056 \beta_{6} - 1112318036561008 \beta_{7} - 9174954241184 \beta_{8} - 20223538383040 \beta_{9} - 117669430029232 \beta_{10} - 157631546949824 \beta_{11} - 4610737620672 \beta_{12} - 10133329864640 \beta_{13}) q^{68} +(-\)\(36\!\cdots\!40\)\( - \)\(14\!\cdots\!40\)\( \beta_{1} - 3393811320946890744 \beta_{2} - 40189408044608983312 \beta_{3} - 872606441637025508 \beta_{4} + 87766443351976160 \beta_{5} - 1947805032290480 \beta_{6} + 192413603938500 \beta_{7} + 9996541241680 \beta_{8} + 118586556667840 \beta_{9} + 302237181766240 \beta_{10} - 507826945370880 \beta_{11} + 8913062091840 \beta_{12} - 15929242150400 \beta_{13}) q^{70} +(\)\(88\!\cdots\!67\)\( - \)\(20\!\cdots\!05\)\( \beta_{1} - 25271048667299489880 \beta_{2} - 26061681695802460072 \beta_{3} - 43515590279888195 \beta_{4} + 129374227701336627 \beta_{5} - 7752285971410773 \beta_{6} + 1526175562875194 \beta_{7} + 308949718200788 \beta_{8} + 50325122045086 \beta_{9} - 434497843558322 \beta_{10} + 592774873565664 \beta_{11} + 13606580710242 \beta_{12} - 8440300623088 \beta_{13}) q^{71} +(\)\(10\!\cdots\!85\)\( - \)\(12\!\cdots\!45\)\( \beta_{1} - 27865351365971420333 \beta_{2} + 3584634055678762235 \beta_{3} + 789724747918806875 \beta_{4} - 47660045188399687 \beta_{5} - 1253120599203430 \beta_{6} + 2727527240677592 \beta_{7} + 326215665268697 \beta_{8} - 87295199204361 \beta_{9} + 525703286814182 \beta_{10} + 704347655756610 \beta_{11} - 62781289583238 \beta_{12}) q^{73} +(\)\(70\!\cdots\!48\)\( + \)\(79\!\cdots\!32\)\( \beta_{1} - 22983135501948937611 \beta_{2} + 51147702019723373353 \beta_{3} - 785486177327043073 \beta_{4} + 73894801146616948 \beta_{5} - 4975581653822420 \beta_{6} + 276083100322437 \beta_{7} - 915237966678187 \beta_{8} - 36272989521754 \beta_{9} + 77183802586506 \beta_{10} + 57926203683480 \beta_{11} + 16145459915008 \beta_{12} - 42149816127277 \beta_{13}) q^{74} +(-\)\(55\!\cdots\!54\)\( - \)\(17\!\cdots\!82\)\( \beta_{1} + 1654949747261477838 \beta_{2} + 7004571982567250523 \beta_{3} - 4018250540930087820 \beta_{4} - 328602738691144531 \beta_{5} - 11832219727604577 \beta_{6} + 9744456104564687 \beta_{7} - 299784227694494 \beta_{8} - 48130459011796 \beta_{9} + 829757358003915 \beta_{10} - 396944443928044 \beta_{11} + 53888972233548 \beta_{12} + 32320185802300 \beta_{13}) q^{76} +(\)\(22\!\cdots\!45\)\( - \)\(62\!\cdots\!21\)\( \beta_{1} + \)\(13\!\cdots\!49\)\( \beta_{2} - 14074214293646006795 \beta_{3} + 4513160649115268464 \beta_{4} - 283241266346986684 \beta_{5} + 96570278604212 \beta_{6} + 3822116544819066 \beta_{7} + 578432604535479 \beta_{8} - 397504974815266 \beta_{9} - 1732161526210817 \beta_{10} + 1052020321485876 \beta_{11} - 131930733651172 \beta_{12}) q^{77} +(\)\(39\!\cdots\!22\)\( - \)\(92\!\cdots\!38\)\( \beta_{1} - \)\(11\!\cdots\!32\)\( \beta_{2} - 88270081888408252252 \beta_{3} - 144846367787671606 \beta_{4} + 568846564501277462 \beta_{5} + 21383008905348422 \beta_{6} + 6563968682338152 \beta_{7} + 2667922297683280 \beta_{8} + 99426486127608 \beta_{9} - 1736930952168008 \beta_{10} + 4698705121820544 \beta_{11} + 140819544601608 \beta_{12} + 197517619699776 \beta_{13}) q^{79} +(-\)\(44\!\cdots\!20\)\( - \)\(34\!\cdots\!80\)\( \beta_{1} + \)\(13\!\cdots\!28\)\( \beta_{2} + \)\(12\!\cdots\!24\)\( \beta_{3} - 2258156152024914584 \beta_{4} + 504342265736771640 \beta_{5} - 10033752757717000 \beta_{6} + 24125715747101240 \beta_{7} + 2755934583105200 \beta_{8} + 516103348045920 \beta_{9} - 1611982916964040 \beta_{10} - 1271246720372280 \beta_{11} - 67276785957280 \beta_{12} - 90002708613280 \beta_{13}) q^{80} +(-\)\(32\!\cdots\!38\)\( + \)\(24\!\cdots\!34\)\( \beta_{1} + 22437483277772870700 \beta_{2} + \)\(49\!\cdots\!32\)\( \beta_{3} + 2148031753085689172 \beta_{4} - 1308883922241223872 \beta_{5} - 27776887345686704 \beta_{6} + 2659376267685372 \beta_{7} - 6491756714942020 \beta_{8} + 67414663341256 \beta_{9} + 1896367574142584 \beta_{10} - 8228199706123488 \beta_{11} + 98815515634944 \beta_{12} - 219797194446620 \beta_{13}) q^{82} +(-\)\(16\!\cdots\!33\)\( + \)\(39\!\cdots\!38\)\( \beta_{1} + \)\(37\!\cdots\!84\)\( \beta_{2} + 68593373471422922513 \beta_{3} + 1616960705045963900 \beta_{4} - 2445878199114463036 \beta_{5} + 8392941791257636 \beta_{6} - 5284914284671140 \beta_{7} + 2748123366026232 \beta_{8} - 349611727054412 \beta_{9} + 3212271047118548 \beta_{10} + 1781399365458752 \beta_{11} + 40258283791948 \beta_{12} + 96213142884960 \beta_{13}) q^{83} +(\)\(48\!\cdots\!55\)\( + \)\(64\!\cdots\!35\)\( \beta_{1} + \)\(51\!\cdots\!55\)\( \beta_{2} - 79911694396264611055 \beta_{3} - 27798316421746471050 \beta_{4} + 3193250327074609940 \beta_{5} + 10409985561371980 \beta_{6} + 20297052670093950 \beta_{7} - 9943040882471355 \beta_{8} - 118871286780190 \beta_{9} - 1863025941399515 \beta_{10} + 3338445358451980 \beta_{11} - 255441995143740 \beta_{12}) q^{85} +(\)\(16\!\cdots\!45\)\( + \)\(66\!\cdots\!75\)\( \beta_{1} + \)\(44\!\cdots\!07\)\( \beta_{2} - \)\(70\!\cdots\!84\)\( \beta_{3} + 13395009837199087355 \beta_{4} + 4405906576588722123 \beta_{5} + 11694437743080784 \beta_{6} + 4167042491108771 \beta_{7} + 769213860521808 \beta_{8} - 859371305402432 \beta_{9} + 13383298652618336 \beta_{10} - 13340332786371328 \beta_{11} + 111472455201856 \beta_{12} - 644317393036800 \beta_{13}) q^{86} +(\)\(79\!\cdots\!12\)\( - \)\(24\!\cdots\!24\)\( \beta_{1} - \)\(31\!\cdots\!32\)\( \beta_{2} - \)\(72\!\cdots\!56\)\( \beta_{3} + 27407772252636450160 \beta_{4} - 3347765880299413488 \beta_{5} - 61469098297053968 \beta_{6} - 92247512626655376 \beta_{7} - 7933677940214688 \beta_{8} + 3814434511121600 \beta_{9} - 1467692377493904 \beta_{10} - 1538951696666256 \beta_{11} + 574367559020736 \beta_{12} + 852007213895360 \beta_{13}) q^{88} +(\)\(13\!\cdots\!75\)\( - \)\(43\!\cdots\!39\)\( \beta_{1} - \)\(59\!\cdots\!71\)\( \beta_{2} + \)\(10\!\cdots\!57\)\( \beta_{3} - 42097287286894835043 \beta_{4} - 4351776848935144337 \beta_{5} - 14052005791961642 \beta_{6} - 75456899925604488 \beta_{7} + 35780509523417607 \beta_{8} - 1460990706760655 \beta_{9} - 25087213369955230 \beta_{10} + 2391631595346798 \beta_{11} - 379543786347722 \beta_{12}) q^{89} +(\)\(36\!\cdots\!82\)\( - \)\(85\!\cdots\!34\)\( \beta_{1} + 28469332992633543644 \beta_{2} + \)\(24\!\cdots\!18\)\( \beta_{3} - 9350788426476760898 \beta_{4} + 4851678987015678082 \beta_{5} + 208206846153933202 \beta_{6} - 50552225849777382 \beta_{7} + 4247229067579604 \beta_{8} - 3908141308891458 \beta_{9} + 2785331403718190 \beta_{10} + 27566787587664864 \beta_{11} + 771839069278530 \beta_{12} + 1969261527116304 \beta_{13}) q^{91} +(-\)\(43\!\cdots\!84\)\( + \)\(19\!\cdots\!48\)\( \beta_{1} + 76834023330323669156 \beta_{2} - \)\(39\!\cdots\!74\)\( \beta_{3} + 37403520043623604376 \beta_{4} + 6385102989637094534 \beta_{5} + 221539384539950690 \beta_{6} - 118087816406371838 \beta_{7} - 8083303377530884 \beta_{8} + 645594201167400 \beta_{9} - 11677508681136502 \beta_{10} - 1858208490226856 \beta_{11} - 1250232911415832 \beta_{12} - 180682699949560 \beta_{13}) q^{92} +(-\)\(93\!\cdots\!62\)\( - \)\(36\!\cdots\!24\)\( \beta_{1} - \)\(18\!\cdots\!98\)\( \beta_{2} - \)\(26\!\cdots\!72\)\( \beta_{3} + 68966539910549485320 \beta_{4} - 3043016055924157890 \beta_{5} + 12190842250756312 \beta_{6} + 7365269186080628 \beta_{7} + 13255524438975448 \beta_{8} - 3797766150731232 \beta_{9} + 48989360689804752 \beta_{10} + 21331200863968640 \beta_{11} + 797494906551264 \beta_{12} - 2171009657380096 \beta_{13}) q^{94} +(-\)\(13\!\cdots\!85\)\( + \)\(32\!\cdots\!15\)\( \beta_{1} - \)\(46\!\cdots\!00\)\( \beta_{2} + \)\(17\!\cdots\!10\)\( \beta_{3} + 36092020875920386325 \beta_{4} + 365904769123492635 \beta_{5} + 686862132323015475 \beta_{6} - 138001682092733240 \beta_{7} - 71851987471286000 \beta_{8} - 5213550269964520 \beta_{9} - 11564944359563560 \beta_{10} - 41225226169749120 \beta_{11} - 775696051658520 \beta_{12} + 1996288541673280 \beta_{13}) q^{95} +(-\)\(14\!\cdots\!89\)\( - \)\(31\!\cdots\!95\)\( \beta_{1} - \)\(30\!\cdots\!71\)\( \beta_{2} + \)\(42\!\cdots\!73\)\( \beta_{3} + \)\(21\!\cdots\!87\)\( \beta_{4} - 12149912464989393075 \beta_{5} - 37610156466640870 \beta_{6} - 272694479880185316 \beta_{7} + 48433945080813179 \beta_{8} + 11572015887406743 \beta_{9} + 20681156584894004 \beta_{10} - 32668170297833310 \beta_{11} + 3986585862331914 \beta_{12}) q^{97} +(\)\(32\!\cdots\!39\)\( + \)\(35\!\cdots\!67\)\( \beta_{1} - \)\(90\!\cdots\!16\)\( \beta_{2} + \)\(77\!\cdots\!40\)\( \beta_{3} + 34943682777228352568 \beta_{4} - 9433050946262309184 \beta_{5} + 652307785801407456 \beta_{6} + 23091840672061224 \beta_{7} + 23338361189956008 \beta_{8} + 16885012171815984 \beta_{9} - 8831611839944880 \beta_{10} - 21050307441749824 \beta_{11} + 47190641164800 \beta_{12} - 5955445916868840 \beta_{13}) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q + 23780q^{2} - 2922848368q^{4} - 138121491740q^{5} + 191366550113600q^{8} + O(q^{10}) \) \( 14q + 23780q^{2} - 2922848368q^{4} - 138121491740q^{5} + 191366550113600q^{8} + 31775605694457400q^{10} - 1640418469677858020q^{13} - 7731597686180285568q^{14} - 30570843123186593536q^{16} - 31570967905797256220q^{17} - \)\(20\!\cdots\!60\)\(q^{20} + \)\(54\!\cdots\!80\)\(q^{22} + \)\(10\!\cdots\!50\)\(q^{25} + \)\(10\!\cdots\!24\)\(q^{26} + \)\(28\!\cdots\!20\)\(q^{28} + \)\(38\!\cdots\!16\)\(q^{29} + \)\(30\!\cdots\!00\)\(q^{32} + \)\(29\!\cdots\!16\)\(q^{34} - \)\(10\!\cdots\!40\)\(q^{37} + \)\(23\!\cdots\!60\)\(q^{38} + \)\(46\!\cdots\!00\)\(q^{40} - \)\(36\!\cdots\!48\)\(q^{41} + \)\(34\!\cdots\!40\)\(q^{44} - \)\(29\!\cdots\!48\)\(q^{46} - \)\(48\!\cdots\!90\)\(q^{49} + \)\(51\!\cdots\!00\)\(q^{50} - \)\(96\!\cdots\!20\)\(q^{52} - \)\(18\!\cdots\!80\)\(q^{53} - \)\(63\!\cdots\!52\)\(q^{56} - \)\(38\!\cdots\!60\)\(q^{58} + \)\(10\!\cdots\!08\)\(q^{61} - \)\(21\!\cdots\!00\)\(q^{62} - \)\(19\!\cdots\!88\)\(q^{64} - \)\(11\!\cdots\!00\)\(q^{65} - \)\(35\!\cdots\!00\)\(q^{68} - \)\(51\!\cdots\!80\)\(q^{70} + \)\(14\!\cdots\!60\)\(q^{73} + \)\(98\!\cdots\!84\)\(q^{74} - \)\(78\!\cdots\!00\)\(q^{76} + \)\(30\!\cdots\!40\)\(q^{77} - \)\(62\!\cdots\!40\)\(q^{80} - \)\(45\!\cdots\!80\)\(q^{82} + \)\(67\!\cdots\!00\)\(q^{85} + \)\(23\!\cdots\!48\)\(q^{86} + \)\(11\!\cdots\!80\)\(q^{88} + \)\(18\!\cdots\!76\)\(q^{89} - \)\(60\!\cdots\!00\)\(q^{92} - \)\(13\!\cdots\!12\)\(q^{94} - \)\(20\!\cdots\!20\)\(q^{97} + \)\(46\!\cdots\!00\)\(q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14} - 2 x^{13} + \)\(99\!\cdots\!07\)\( x^{12} - \)\(80\!\cdots\!76\)\( x^{11} + \)\(37\!\cdots\!31\)\( x^{10} - \)\(23\!\cdots\!66\)\( x^{9} + \)\(66\!\cdots\!33\)\( x^{8} + \)\(26\!\cdots\!04\)\( x^{7} + \)\(53\!\cdots\!48\)\( x^{6} + \)\(13\!\cdots\!32\)\( x^{5} + \)\(14\!\cdots\!84\)\( x^{4} + \)\(95\!\cdots\!64\)\( x^{3} + \)\(10\!\cdots\!40\)\( x^{2} - \)\(74\!\cdots\!00\)\( x + \)\(14\!\cdots\!00\)\(\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(\)\(13\!\cdots\!09\)\( \nu^{13} + \)\(58\!\cdots\!74\)\( \nu^{12} + \)\(13\!\cdots\!75\)\( \nu^{11} + \)\(46\!\cdots\!16\)\( \nu^{10} + \)\(51\!\cdots\!87\)\( \nu^{9} + \)\(18\!\cdots\!62\)\( \nu^{8} + \)\(90\!\cdots\!53\)\( \nu^{7} + \)\(41\!\cdots\!00\)\( \nu^{6} + \)\(72\!\cdots\!32\)\( \nu^{5} + \)\(48\!\cdots\!04\)\( \nu^{4} + \)\(19\!\cdots\!08\)\( \nu^{3} + \)\(21\!\cdots\!80\)\( \nu^{2} + \)\(14\!\cdots\!00\)\( \nu - \)\(54\!\cdots\!00\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(\)\(77\!\cdots\!71\)\( \nu^{13} + \)\(38\!\cdots\!46\)\( \nu^{12} + \)\(76\!\cdots\!85\)\( \nu^{11} + \)\(34\!\cdots\!84\)\( \nu^{10} + \)\(28\!\cdots\!53\)\( \nu^{9} + \)\(14\!\cdots\!98\)\( \nu^{8} + \)\(51\!\cdots\!87\)\( \nu^{7} + \)\(31\!\cdots\!20\)\( \nu^{6} + \)\(41\!\cdots\!68\)\( \nu^{5} + \)\(35\!\cdots\!76\)\( \nu^{4} + \)\(11\!\cdots\!92\)\( \nu^{3} + \)\(14\!\cdots\!20\)\( \nu^{2} + \)\(81\!\cdots\!00\)\( \nu - \)\(15\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(26\!\cdots\!87\)\( \nu^{13} - \)\(81\!\cdots\!82\)\( \nu^{12} - \)\(26\!\cdots\!25\)\( \nu^{11} - \)\(63\!\cdots\!88\)\( \nu^{10} - \)\(99\!\cdots\!41\)\( \nu^{9} - \)\(26\!\cdots\!66\)\( \nu^{8} - \)\(17\!\cdots\!79\)\( \nu^{7} - \)\(68\!\cdots\!00\)\( \nu^{6} - \)\(14\!\cdots\!76\)\( \nu^{5} - \)\(85\!\cdots\!72\)\( \nu^{4} - \)\(37\!\cdots\!44\)\( \nu^{3} - \)\(39\!\cdots\!40\)\( \nu^{2} - \)\(27\!\cdots\!00\)\( \nu + \)\(98\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(31\!\cdots\!81\)\( \nu^{13} - \)\(11\!\cdots\!34\)\( \nu^{12} + \)\(31\!\cdots\!75\)\( \nu^{11} - \)\(10\!\cdots\!56\)\( \nu^{10} + \)\(11\!\cdots\!83\)\( \nu^{9} - \)\(35\!\cdots\!42\)\( \nu^{8} + \)\(20\!\cdots\!77\)\( \nu^{7} - \)\(53\!\cdots\!00\)\( \nu^{6} + \)\(16\!\cdots\!88\)\( \nu^{5} - \)\(35\!\cdots\!64\)\( \nu^{4} + \)\(44\!\cdots\!72\)\( \nu^{3} - \)\(65\!\cdots\!80\)\( \nu^{2} + \)\(32\!\cdots\!00\)\( \nu - \)\(45\!\cdots\!00\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(11\!\cdots\!79\)\( \nu^{13} - \)\(41\!\cdots\!94\)\( \nu^{12} - \)\(11\!\cdots\!25\)\( \nu^{11} - \)\(32\!\cdots\!96\)\( \nu^{10} - \)\(41\!\cdots\!97\)\( \nu^{9} - \)\(12\!\cdots\!22\)\( \nu^{8} - \)\(74\!\cdots\!43\)\( \nu^{7} - \)\(30\!\cdots\!00\)\( \nu^{6} - \)\(59\!\cdots\!92\)\( \nu^{5} - \)\(36\!\cdots\!24\)\( \nu^{4} - \)\(15\!\cdots\!48\)\( \nu^{3} - \)\(16\!\cdots\!80\)\( \nu^{2} - \)\(11\!\cdots\!00\)\( \nu + \)\(34\!\cdots\!00\)\(\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(14\!\cdots\!13\)\( \nu^{13} - \)\(41\!\cdots\!18\)\( \nu^{12} - \)\(14\!\cdots\!75\)\( \nu^{11} - \)\(28\!\cdots\!12\)\( \nu^{10} - \)\(55\!\cdots\!59\)\( \nu^{9} - \)\(11\!\cdots\!34\)\( \nu^{8} - \)\(98\!\cdots\!21\)\( \nu^{7} - \)\(30\!\cdots\!00\)\( \nu^{6} - \)\(78\!\cdots\!24\)\( \nu^{5} - \)\(40\!\cdots\!28\)\( \nu^{4} - \)\(21\!\cdots\!56\)\( \nu^{3} - \)\(19\!\cdots\!60\)\( \nu^{2} - \)\(15\!\cdots\!00\)\( \nu + \)\(63\!\cdots\!00\)\(\)\()/ \)\(14\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(18\!\cdots\!01\)\( \nu^{13} - \)\(52\!\cdots\!86\)\( \nu^{12} - \)\(18\!\cdots\!75\)\( \nu^{11} - \)\(42\!\cdots\!24\)\( \nu^{10} - \)\(70\!\cdots\!43\)\( \nu^{9} - \)\(19\!\cdots\!18\)\( \nu^{8} - \)\(12\!\cdots\!17\)\( \nu^{7} - \)\(51\!\cdots\!00\)\( \nu^{6} - \)\(10\!\cdots\!48\)\( \nu^{5} - \)\(65\!\cdots\!56\)\( \nu^{4} - \)\(26\!\cdots\!12\)\( \nu^{3} - \)\(29\!\cdots\!20\)\( \nu^{2} - \)\(19\!\cdots\!00\)\( \nu + \)\(33\!\cdots\!00\)\(\)\()/ \)\(58\!\cdots\!00\)\( \)
\(\beta_{8}\)\(=\)\((\)\(\)\(86\!\cdots\!93\)\( \nu^{13} - \)\(52\!\cdots\!82\)\( \nu^{12} + \)\(85\!\cdots\!55\)\( \nu^{11} - \)\(50\!\cdots\!28\)\( \nu^{10} + \)\(32\!\cdots\!99\)\( \nu^{9} - \)\(15\!\cdots\!66\)\( \nu^{8} + \)\(57\!\cdots\!21\)\( \nu^{7} - \)\(16\!\cdots\!40\)\( \nu^{6} + \)\(46\!\cdots\!44\)\( \nu^{5} + \)\(33\!\cdots\!08\)\( \nu^{4} + \)\(12\!\cdots\!36\)\( \nu^{3} + \)\(58\!\cdots\!60\)\( \nu^{2} + \)\(90\!\cdots\!00\)\( \nu - \)\(76\!\cdots\!00\)\(\)\()/ \)\(15\!\cdots\!00\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(13\!\cdots\!71\)\( \nu^{13} + \)\(73\!\cdots\!06\)\( \nu^{12} + \)\(13\!\cdots\!25\)\( \nu^{11} + \)\(56\!\cdots\!04\)\( \nu^{10} + \)\(51\!\cdots\!53\)\( \nu^{9} + \)\(20\!\cdots\!78\)\( \nu^{8} + \)\(91\!\cdots\!07\)\( \nu^{7} + \)\(41\!\cdots\!00\)\( \nu^{6} + \)\(73\!\cdots\!08\)\( \nu^{5} + \)\(46\!\cdots\!76\)\( \nu^{4} + \)\(19\!\cdots\!52\)\( \nu^{3} + \)\(21\!\cdots\!20\)\( \nu^{2} + \)\(18\!\cdots\!00\)\( \nu + \)\(30\!\cdots\!00\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{10}\)\(=\)\((\)\(\)\(75\!\cdots\!19\)\( \nu^{13} + \)\(16\!\cdots\!34\)\( \nu^{12} + \)\(74\!\cdots\!25\)\( \nu^{11} + \)\(61\!\cdots\!56\)\( \nu^{10} + \)\(28\!\cdots\!17\)\( \nu^{9} + \)\(12\!\cdots\!42\)\( \nu^{8} + \)\(50\!\cdots\!23\)\( \nu^{7} + \)\(46\!\cdots\!00\)\( \nu^{6} + \)\(40\!\cdots\!12\)\( \nu^{5} + \)\(99\!\cdots\!64\)\( \nu^{4} + \)\(10\!\cdots\!28\)\( \nu^{3} + \)\(69\!\cdots\!80\)\( \nu^{2} + \)\(78\!\cdots\!00\)\( \nu - \)\(42\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!00\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-\)\(52\!\cdots\!91\)\( \nu^{13} - \)\(10\!\cdots\!06\)\( \nu^{12} - \)\(51\!\cdots\!45\)\( \nu^{11} - \)\(71\!\cdots\!44\)\( \nu^{10} - \)\(19\!\cdots\!13\)\( \nu^{9} - \)\(34\!\cdots\!78\)\( \nu^{8} - \)\(34\!\cdots\!07\)\( \nu^{7} - \)\(10\!\cdots\!40\)\( \nu^{6} - \)\(27\!\cdots\!88\)\( \nu^{5} - \)\(14\!\cdots\!96\)\( \nu^{4} - \)\(74\!\cdots\!72\)\( \nu^{3} - \)\(71\!\cdots\!20\)\( \nu^{2} - \)\(56\!\cdots\!00\)\( \nu + \)\(23\!\cdots\!00\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(74\!\cdots\!57\)\( \nu^{13} - \)\(34\!\cdots\!02\)\( \nu^{12} - \)\(74\!\cdots\!75\)\( \nu^{11} - \)\(26\!\cdots\!68\)\( \nu^{10} - \)\(28\!\cdots\!51\)\( \nu^{9} - \)\(99\!\cdots\!26\)\( \nu^{8} - \)\(49\!\cdots\!69\)\( \nu^{7} - \)\(21\!\cdots\!00\)\( \nu^{6} - \)\(39\!\cdots\!36\)\( \nu^{5} - \)\(24\!\cdots\!92\)\( \nu^{4} - \)\(10\!\cdots\!84\)\( \nu^{3} - \)\(10\!\cdots\!40\)\( \nu^{2} - \)\(80\!\cdots\!00\)\( \nu + \)\(30\!\cdots\!00\)\(\)\()/ \)\(23\!\cdots\!00\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(37\!\cdots\!63\)\( \nu^{13} + \)\(13\!\cdots\!18\)\( \nu^{12} + \)\(36\!\cdots\!25\)\( \nu^{11} + \)\(99\!\cdots\!12\)\( \nu^{10} + \)\(13\!\cdots\!09\)\( \nu^{9} + \)\(39\!\cdots\!34\)\( \nu^{8} + \)\(24\!\cdots\!71\)\( \nu^{7} + \)\(94\!\cdots\!00\)\( \nu^{6} + \)\(19\!\cdots\!24\)\( \nu^{5} + \)\(11\!\cdots\!28\)\( \nu^{4} + \)\(52\!\cdots\!56\)\( \nu^{3} + \)\(53\!\cdots\!60\)\( \nu^{2} + \)\(40\!\cdots\!00\)\( \nu - \)\(14\!\cdots\!00\)\(\)\()/ \)\(38\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - 7 \beta_{5} + 22 \beta_{4} + 795 \beta_{3} - 1276 \beta_{2} + 129201784 \beta_{1} - 55372207\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(358545808 \beta_{13} + 30310032020 \beta_{12} - 37892119502 \beta_{11} + 90532568451 \beta_{10} + 188032828453 \beta_{9} - 43918001546 \beta_{8} - 2792241416812 \beta_{7} + 277467576234 \beta_{6} - 282126543561101 \beta_{5} - 674321448647245 \beta_{4} + 4785725213563862 \beta_{3} - 24935171507835582 \beta_{2} - 3905821127638844894146 \beta_{1} - 724740339183049824280571770\)\()/512\)
\(\nu^{3}\)\(=\)\((\)\(609588255266028085590704 \beta_{13} - 112817361593502064630898 \beta_{12} - 5686750552336802385105808 \beta_{11} - 1001275266687356307132525 \beta_{10} - 2065264669383685183033574 \beta_{9} - 15534286215331814599237045 \beta_{8} - 43130737204891107747699076 \beta_{7} - 1124401822034308334683636966 \beta_{6} + 13163538497227379836894117888 \beta_{5} - 29471464477890969362728068388 \beta_{4} - 2626653520857047477006807556017 \beta_{3} + 885691031065847339890074991441 \beta_{2} - 236921470454333605071642962167656459 \beta_{1} + 28493176082446705007200429233956209351\)\()/16384\)
\(\nu^{4}\)\(=\)\((\)\(-\)\(14\!\cdots\!36\)\( \beta_{13} - \)\(78\!\cdots\!80\)\( \beta_{12} + \)\(42\!\cdots\!36\)\( \beta_{11} - \)\(22\!\cdots\!21\)\( \beta_{10} - \)\(31\!\cdots\!44\)\( \beta_{9} - \)\(20\!\cdots\!11\)\( \beta_{8} + \)\(56\!\cdots\!86\)\( \beta_{7} - \)\(15\!\cdots\!04\)\( \beta_{6} + \)\(40\!\cdots\!06\)\( \beta_{5} + \)\(20\!\cdots\!82\)\( \beta_{4} - \)\(10\!\cdots\!89\)\( \beta_{3} + \)\(71\!\cdots\!95\)\( \beta_{2} + \)\(52\!\cdots\!65\)\( \beta_{1} + \)\(10\!\cdots\!83\)\(\)\()/32768\)
\(\nu^{5}\)\(=\)\((\)\(-\)\(60\!\cdots\!00\)\( \beta_{13} - \)\(87\!\cdots\!30\)\( \beta_{12} - \)\(28\!\cdots\!40\)\( \beta_{11} + \)\(13\!\cdots\!45\)\( \beta_{10} + \)\(16\!\cdots\!10\)\( \beta_{9} + \)\(70\!\cdots\!45\)\( \beta_{8} + \)\(26\!\cdots\!00\)\( \beta_{7} + \)\(43\!\cdots\!18\)\( \beta_{6} - \)\(66\!\cdots\!60\)\( \beta_{5} + \)\(12\!\cdots\!12\)\( \beta_{4} + \)\(16\!\cdots\!41\)\( \beta_{3} - \)\(20\!\cdots\!69\)\( \beta_{2} + \)\(11\!\cdots\!19\)\( \beta_{1} - \)\(52\!\cdots\!23\)\(\)\()/262144\)
\(\nu^{6}\)\(=\)\((\)\(\)\(19\!\cdots\!12\)\( \beta_{13} + \)\(88\!\cdots\!32\)\( \beta_{12} - \)\(57\!\cdots\!08\)\( \beta_{11} + \)\(26\!\cdots\!27\)\( \beta_{10} + \)\(30\!\cdots\!16\)\( \beta_{9} + \)\(35\!\cdots\!81\)\( \beta_{8} - \)\(58\!\cdots\!06\)\( \beta_{7} - \)\(92\!\cdots\!32\)\( \beta_{6} - \)\(27\!\cdots\!74\)\( \beta_{5} - \)\(23\!\cdots\!42\)\( \beta_{4} + \)\(10\!\cdots\!27\)\( \beta_{3} - \)\(81\!\cdots\!73\)\( \beta_{2} - \)\(31\!\cdots\!19\)\( \beta_{1} - \)\(89\!\cdots\!77\)\(\)\()/1048576\)
\(\nu^{7}\)\(=\)\((\)\(\)\(10\!\cdots\!44\)\( \beta_{13} + \)\(26\!\cdots\!64\)\( \beta_{12} + \)\(98\!\cdots\!72\)\( \beta_{11} - \)\(28\!\cdots\!37\)\( \beta_{10} - \)\(24\!\cdots\!20\)\( \beta_{9} - \)\(72\!\cdots\!35\)\( \beta_{8} - \)\(35\!\cdots\!34\)\( \beta_{7} - \)\(45\!\cdots\!76\)\( \beta_{6} + \)\(83\!\cdots\!68\)\( \beta_{5} - \)\(14\!\cdots\!98\)\( \beta_{4} - \)\(22\!\cdots\!77\)\( \beta_{3} + \)\(51\!\cdots\!17\)\( \beta_{2} - \)\(14\!\cdots\!15\)\( \beta_{1} + \)\(11\!\cdots\!41\)\(\)\()/1048576\)
\(\nu^{8}\)\(=\)\((\)\(-\)\(38\!\cdots\!72\)\( \beta_{13} - \)\(14\!\cdots\!76\)\( \beta_{12} + \)\(10\!\cdots\!32\)\( \beta_{11} - \)\(46\!\cdots\!21\)\( \beta_{10} - \)\(46\!\cdots\!60\)\( \beta_{9} - \)\(73\!\cdots\!07\)\( \beta_{8} + \)\(95\!\cdots\!86\)\( \beta_{7} + \)\(47\!\cdots\!48\)\( \beta_{6} + \)\(26\!\cdots\!27\)\( \beta_{5} + \)\(39\!\cdots\!28\)\( \beta_{4} - \)\(14\!\cdots\!21\)\( \beta_{3} + \)\(13\!\cdots\!54\)\( \beta_{2} + \)\(26\!\cdots\!61\)\( \beta_{1} + \)\(12\!\cdots\!40\)\(\)\()/524288\)
\(\nu^{9}\)\(=\)\((\)\(-\)\(19\!\cdots\!16\)\( \beta_{13} - \)\(58\!\cdots\!68\)\( \beta_{12} - \)\(22\!\cdots\!12\)\( \beta_{11} + \)\(60\!\cdots\!23\)\( \beta_{10} + \)\(43\!\cdots\!72\)\( \beta_{9} + \)\(97\!\cdots\!41\)\( \beta_{8} + \)\(56\!\cdots\!90\)\( \beta_{7} + \)\(62\!\cdots\!72\)\( \beta_{6} - \)\(12\!\cdots\!90\)\( \beta_{5} + \)\(19\!\cdots\!22\)\( \beta_{4} + \)\(38\!\cdots\!15\)\( \beta_{3} + \)\(15\!\cdots\!71\)\( \beta_{2} + \)\(23\!\cdots\!53\)\( \beta_{1} - \)\(27\!\cdots\!61\)\(\)\()/524288\)
\(\nu^{10}\)\(=\)\((\)\(\)\(28\!\cdots\!60\)\( \beta_{13} + \)\(95\!\cdots\!24\)\( \beta_{12} - \)\(67\!\cdots\!60\)\( \beta_{11} + \)\(30\!\cdots\!81\)\( \beta_{10} + \)\(29\!\cdots\!48\)\( \beta_{9} + \)\(55\!\cdots\!15\)\( \beta_{8} - \)\(62\!\cdots\!46\)\( \beta_{7} - \)\(49\!\cdots\!76\)\( \beta_{6} - \)\(98\!\cdots\!90\)\( \beta_{5} - \)\(25\!\cdots\!74\)\( \beta_{4} + \)\(84\!\cdots\!73\)\( \beta_{3} - \)\(91\!\cdots\!95\)\( \beta_{2} - \)\(69\!\cdots\!25\)\( \beta_{1} - \)\(67\!\cdots\!79\)\(\)\()/1048576\)
\(\nu^{11}\)\(=\)\((\)\(\)\(13\!\cdots\!32\)\( \beta_{13} + \)\(44\!\cdots\!20\)\( \beta_{12} + \)\(17\!\cdots\!96\)\( \beta_{11} - \)\(45\!\cdots\!79\)\( \beta_{10} - \)\(29\!\cdots\!84\)\( \beta_{9} - \)\(54\!\cdots\!25\)\( \beta_{8} - \)\(35\!\cdots\!54\)\( \beta_{7} - \)\(35\!\cdots\!68\)\( \beta_{6} + \)\(81\!\cdots\!84\)\( \beta_{5} - \)\(11\!\cdots\!54\)\( \beta_{4} - \)\(25\!\cdots\!91\)\( \beta_{3} - \)\(21\!\cdots\!85\)\( \beta_{2} - \)\(14\!\cdots\!89\)\( \beta_{1} + \)\(23\!\cdots\!43\)\(\)\()/1048576\)
\(\nu^{12}\)\(=\)\((\)\(-\)\(51\!\cdots\!68\)\( \beta_{13} - \)\(15\!\cdots\!92\)\( \beta_{12} + \)\(10\!\cdots\!28\)\( \beta_{11} - \)\(49\!\cdots\!26\)\( \beta_{10} - \)\(46\!\cdots\!16\)\( \beta_{9} - \)\(99\!\cdots\!38\)\( \beta_{8} + \)\(10\!\cdots\!16\)\( \beta_{7} + \)\(10\!\cdots\!20\)\( \beta_{6} + \)\(80\!\cdots\!83\)\( \beta_{5} + \)\(42\!\cdots\!14\)\( \beta_{4} - \)\(11\!\cdots\!18\)\( \beta_{3} + \)\(15\!\cdots\!23\)\( \beta_{2} + \)\(66\!\cdots\!02\)\( \beta_{1} + \)\(97\!\cdots\!77\)\(\)\()/524288\)
\(\nu^{13}\)\(=\)\((\)\(-\)\(11\!\cdots\!16\)\( \beta_{13} - \)\(38\!\cdots\!18\)\( \beta_{12} - \)\(15\!\cdots\!72\)\( \beta_{11} + \)\(40\!\cdots\!73\)\( \beta_{10} + \)\(24\!\cdots\!22\)\( \beta_{9} + \)\(40\!\cdots\!41\)\( \beta_{8} + \)\(27\!\cdots\!80\)\( \beta_{7} + \)\(26\!\cdots\!54\)\( \beta_{6} - \)\(63\!\cdots\!65\)\( \beta_{5} + \)\(79\!\cdots\!30\)\( \beta_{4} + \)\(20\!\cdots\!49\)\( \beta_{3} + \)\(23\!\cdots\!00\)\( \beta_{2} + \)\(11\!\cdots\!19\)\( \beta_{1} - \)\(23\!\cdots\!88\)\(\)\()/262144\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
7.49079e9 + 1.45317e12i
7.49079e9 1.45317e12i
−1.32519e9 + 1.48274e12i
−1.32519e9 1.48274e12i
−1.70126e9 1.38274e12i
−1.70126e9 + 1.38274e12i
−4.89209e9 + 1.78639e12i
−4.89209e9 1.78639e12i
3.48102e9 + 9.91448e8i
3.48102e9 9.91448e8i
−7.09569e9 8.80649e10i
−7.09569e9 + 8.80649e10i
4.04243e9 6.98294e11i
4.04243e9 + 6.98294e11i
−60712.0 24678.5i 0 3.07691e9 + 2.99655e9i −2.49571e11 0 4.65014e13i −1.12855e14 2.57860e14i 0 1.51520e16 + 6.15903e15i
19.2 −60712.0 + 24678.5i 0 3.07691e9 2.99655e9i −2.49571e11 0 4.65014e13i −1.12855e14 + 2.57860e14i 0 1.51520e16 6.15903e15i
19.3 −46092.7 46587.9i 0 −4.59006e7 + 4.29472e9i 3.25403e10 0 4.74477e13i 2.02198e14 1.95817e14i 0 −1.49987e15 1.51599e15i
19.4 −46092.7 + 46587.9i 0 −4.59006e7 4.29472e9i 3.25403e10 0 4.74477e13i 2.02198e14 + 1.95817e14i 0 −1.49987e15 + 1.51599e15i
19.5 −32011.4 57186.0i 0 −2.24550e9 + 3.66121e9i 4.45746e10 0 4.42477e13i 2.81252e14 + 1.12108e13i 0 −1.42690e15 2.54904e15i
19.6 −32011.4 + 57186.0i 0 −2.24550e9 3.66121e9i 4.45746e10 0 4.42477e13i 2.81252e14 1.12108e13i 0 −1.42690e15 + 2.54904e15i
19.7 15224.0 63743.2i 0 −3.83143e9 1.94085e9i 1.46681e11 0 5.71644e13i −1.82046e14 + 2.14680e14i 0 2.23307e15 9.34993e15i
19.8 15224.0 + 63743.2i 0 −3.83143e9 + 1.94085e9i 1.46681e11 0 5.71644e13i −1.82046e14 2.14680e14i 0 2.23307e15 + 9.34993e15i
19.9 18058.8 62998.8i 0 −3.64273e9 2.27536e9i −1.21258e11 0 3.17263e10i −2.09128e14 + 1.88397e14i 0 −2.18978e15 + 7.63913e15i
19.10 18058.8 + 62998.8i 0 −3.64273e9 + 2.27536e9i −1.21258e11 0 3.17263e10i −2.09128e14 1.88397e14i 0 −2.18978e15 7.63913e15i
19.11 56022.2 34007.1i 0 1.98200e9 3.81031e9i 2.17196e11 0 2.81808e12i −1.85418e13 2.80864e14i 0 1.21678e16 7.38622e15i
19.12 56022.2 + 34007.1i 0 1.98200e9 + 3.81031e9i 2.17196e11 0 2.81808e12i −1.85418e13 + 2.80864e14i 0 1.21678e16 + 7.38622e15i
19.13 61401.1 22910.1i 0 3.24522e9 2.81341e9i −1.39224e11 0 2.23454e13i 1.34805e14 2.47095e14i 0 −8.54848e15 + 3.18962e15i
19.14 61401.1 + 22910.1i 0 3.24522e9 + 2.81341e9i −1.39224e11 0 2.23454e13i 1.34805e14 + 2.47095e14i 0 −8.54848e15 3.18962e15i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.33.d.b 14
3.b odd 2 1 4.33.b.b 14
4.b odd 2 1 inner 36.33.d.b 14
12.b even 2 1 4.33.b.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.33.b.b 14 3.b odd 2 1
4.33.b.b 14 12.b even 2 1
36.33.d.b 14 1.a even 1 1 trivial
36.33.d.b 14 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} + 69060745870 T_{5}^{6} - \)81669251881758433411500

'>\(81\!\cdots\!00\)\( T_{5}^{5} - \)3582839712286042804283048693025000
'>\(35\!\cdots\!00\)\( T_{5}^{4} + \)1691497483111433499262778436348931058418750000'>\(16\!\cdots\!00\)\( T_{5}^{3} + \)37726750193783531578663453906693952860056222539062500000'>\(37\!\cdots\!00\)\( T_{5}^{2} - \)8790602739051590164802042439957880994980502000925276806640625000000'>\(87\!\cdots\!00\)\( T_{5} + \)194695030971705992795169005603188289602422978392661942832712707519531250000000'>\(19\!\cdots\!00\)\( \) acting on \(S_{33}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 23780 T + 1744168384 T^{2} - 100782736158720 T^{3} + 10654023216358490112 T^{4} - \)\(93\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!88\)\( T^{6} + \)\(27\!\cdots\!20\)\( T^{7} + \)\(28\!\cdots\!48\)\( T^{8} - \)\(17\!\cdots\!80\)\( T^{9} + \)\(84\!\cdots\!32\)\( T^{10} - \)\(34\!\cdots\!20\)\( T^{11} + \)\(25\!\cdots\!84\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{13} + \)\(26\!\cdots\!16\)\( T^{14} \)
$3$ 1
$5$ \( ( 1 + 69060745870 T + \)\(81\!\cdots\!75\)\( T^{2} + \)\(60\!\cdots\!00\)\( T^{3} + \)\(35\!\cdots\!25\)\( T^{4} + \)\(26\!\cdots\!50\)\( T^{5} + \)\(10\!\cdots\!75\)\( T^{6} + \)\(77\!\cdots\!00\)\( T^{7} + \)\(25\!\cdots\!75\)\( T^{8} + \)\(14\!\cdots\!50\)\( T^{9} + \)\(45\!\cdots\!25\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{11} + \)\(55\!\cdots\!75\)\( T^{12} + \)\(11\!\cdots\!50\)\( T^{13} + \)\(37\!\cdots\!25\)\( T^{14} )^{2} \)
$7$ \( 1 - \)\(53\!\cdots\!62\)\( T^{2} + \)\(15\!\cdots\!35\)\( T^{4} - \)\(35\!\cdots\!08\)\( T^{6} + \)\(65\!\cdots\!89\)\( T^{8} - \)\(10\!\cdots\!10\)\( T^{10} + \)\(13\!\cdots\!75\)\( T^{12} - \)\(15\!\cdots\!40\)\( T^{14} + \)\(16\!\cdots\!75\)\( T^{16} - \)\(14\!\cdots\!10\)\( T^{18} + \)\(11\!\cdots\!89\)\( T^{20} - \)\(78\!\cdots\!08\)\( T^{22} + \)\(42\!\cdots\!35\)\( T^{24} - \)\(17\!\cdots\!62\)\( T^{26} + \)\(40\!\cdots\!01\)\( T^{28} \)
$11$ \( 1 - \)\(13\!\cdots\!94\)\( T^{2} + \)\(94\!\cdots\!91\)\( T^{4} - \)\(47\!\cdots\!04\)\( T^{6} + \)\(18\!\cdots\!81\)\( T^{8} - \)\(59\!\cdots\!02\)\( T^{10} + \)\(16\!\cdots\!63\)\( T^{12} - \)\(36\!\cdots\!32\)\( T^{14} + \)\(71\!\cdots\!83\)\( T^{16} - \)\(11\!\cdots\!62\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} - \)\(18\!\cdots\!44\)\( T^{22} + \)\(16\!\cdots\!91\)\( T^{24} - \)\(10\!\cdots\!54\)\( T^{26} + \)\(34\!\cdots\!81\)\( T^{28} \)
$13$ \( ( 1 + 820209234838929010 T + \)\(24\!\cdots\!39\)\( T^{2} + \)\(16\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!57\)\( T^{4} + \)\(16\!\cdots\!30\)\( T^{5} + \)\(19\!\cdots\!43\)\( T^{6} + \)\(91\!\cdots\!80\)\( T^{7} + \)\(85\!\cdots\!83\)\( T^{8} + \)\(31\!\cdots\!30\)\( T^{9} + \)\(24\!\cdots\!37\)\( T^{10} + \)\(65\!\cdots\!00\)\( T^{11} + \)\(41\!\cdots\!39\)\( T^{12} + \)\(61\!\cdots\!10\)\( T^{13} + \)\(33\!\cdots\!61\)\( T^{14} )^{2} \)
$17$ \( ( 1 + 15785483952898628110 T + \)\(81\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(40\!\cdots\!57\)\( T^{4} + \)\(52\!\cdots\!50\)\( T^{5} + \)\(13\!\cdots\!03\)\( T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(31\!\cdots\!83\)\( T^{8} + \)\(29\!\cdots\!50\)\( T^{9} + \)\(53\!\cdots\!17\)\( T^{10} + \)\(36\!\cdots\!00\)\( T^{11} + \)\(60\!\cdots\!39\)\( T^{12} + \)\(27\!\cdots\!10\)\( T^{13} + \)\(41\!\cdots\!21\)\( T^{14} )^{2} \)
$19$ \( 1 - \)\(56\!\cdots\!14\)\( T^{2} + \)\(15\!\cdots\!91\)\( T^{4} - \)\(29\!\cdots\!44\)\( T^{6} + \)\(41\!\cdots\!41\)\( T^{8} - \)\(49\!\cdots\!02\)\( T^{10} + \)\(51\!\cdots\!03\)\( T^{12} - \)\(45\!\cdots\!32\)\( T^{14} + \)\(35\!\cdots\!63\)\( T^{16} - \)\(23\!\cdots\!82\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} - \)\(67\!\cdots\!64\)\( T^{22} + \)\(25\!\cdots\!91\)\( T^{24} - \)\(62\!\cdots\!94\)\( T^{26} + \)\(76\!\cdots\!41\)\( T^{28} \)
$23$ \( 1 - \)\(39\!\cdots\!22\)\( T^{2} + \)\(75\!\cdots\!95\)\( T^{4} - \)\(90\!\cdots\!48\)\( T^{6} + \)\(78\!\cdots\!29\)\( T^{8} - \)\(51\!\cdots\!90\)\( T^{10} + \)\(26\!\cdots\!95\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{14} + \)\(37\!\cdots\!95\)\( T^{16} - \)\(10\!\cdots\!90\)\( T^{18} + \)\(22\!\cdots\!09\)\( T^{20} - \)\(36\!\cdots\!28\)\( T^{22} + \)\(42\!\cdots\!95\)\( T^{24} - \)\(31\!\cdots\!02\)\( T^{26} + \)\(11\!\cdots\!81\)\( T^{28} \)
$29$ \( ( 1 - \)\(19\!\cdots\!58\)\( T + \)\(22\!\cdots\!75\)\( T^{2} - \)\(23\!\cdots\!32\)\( T^{3} + \)\(19\!\cdots\!89\)\( T^{4} - \)\(11\!\cdots\!70\)\( T^{5} + \)\(88\!\cdots\!55\)\( T^{6} + \)\(79\!\cdots\!60\)\( T^{7} + \)\(55\!\cdots\!55\)\( T^{8} - \)\(46\!\cdots\!70\)\( T^{9} + \)\(46\!\cdots\!69\)\( T^{10} - \)\(35\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!75\)\( T^{12} - \)\(11\!\cdots\!78\)\( T^{13} + \)\(37\!\cdots\!81\)\( T^{14} )^{2} \)
$31$ \( 1 - \)\(24\!\cdots\!14\)\( T^{2} + \)\(34\!\cdots\!91\)\( T^{4} - \)\(33\!\cdots\!44\)\( T^{6} + \)\(25\!\cdots\!41\)\( T^{8} - \)\(16\!\cdots\!02\)\( T^{10} + \)\(93\!\cdots\!03\)\( T^{12} - \)\(50\!\cdots\!32\)\( T^{14} + \)\(26\!\cdots\!63\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(56\!\cdots\!01\)\( T^{20} - \)\(20\!\cdots\!64\)\( T^{22} + \)\(59\!\cdots\!91\)\( T^{24} - \)\(12\!\cdots\!94\)\( T^{26} + \)\(13\!\cdots\!41\)\( T^{28} \)
$37$ \( ( 1 + \)\(53\!\cdots\!70\)\( T + \)\(85\!\cdots\!99\)\( T^{2} + \)\(40\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!97\)\( T^{4} + \)\(13\!\cdots\!30\)\( T^{5} + \)\(78\!\cdots\!63\)\( T^{6} + \)\(26\!\cdots\!60\)\( T^{7} + \)\(11\!\cdots\!03\)\( T^{8} + \)\(31\!\cdots\!30\)\( T^{9} + \)\(11\!\cdots\!77\)\( T^{10} + \)\(21\!\cdots\!20\)\( T^{11} + \)\(69\!\cdots\!99\)\( T^{12} + \)\(66\!\cdots\!70\)\( T^{13} + \)\(18\!\cdots\!61\)\( T^{14} )^{2} \)
$41$ \( ( 1 + \)\(18\!\cdots\!74\)\( T + \)\(16\!\cdots\!11\)\( T^{2} + \)\(37\!\cdots\!44\)\( T^{3} + \)\(12\!\cdots\!61\)\( T^{4} + \)\(25\!\cdots\!02\)\( T^{5} + \)\(68\!\cdots\!23\)\( T^{6} + \)\(11\!\cdots\!72\)\( T^{7} + \)\(27\!\cdots\!63\)\( T^{8} + \)\(42\!\cdots\!22\)\( T^{9} + \)\(84\!\cdots\!01\)\( T^{10} + \)\(10\!\cdots\!24\)\( T^{11} + \)\(17\!\cdots\!11\)\( T^{12} + \)\(81\!\cdots\!94\)\( T^{13} + \)\(18\!\cdots\!61\)\( T^{14} )^{2} \)
$43$ \( 1 - \)\(10\!\cdots\!42\)\( T^{2} + \)\(66\!\cdots\!55\)\( T^{4} - \)\(29\!\cdots\!68\)\( T^{6} + \)\(10\!\cdots\!29\)\( T^{8} - \)\(29\!\cdots\!50\)\( T^{10} + \)\(71\!\cdots\!15\)\( T^{12} - \)\(14\!\cdots\!80\)\( T^{14} + \)\(24\!\cdots\!15\)\( T^{16} - \)\(36\!\cdots\!50\)\( T^{18} + \)\(43\!\cdots\!29\)\( T^{20} - \)\(43\!\cdots\!68\)\( T^{22} + \)\(33\!\cdots\!55\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{26} + \)\(62\!\cdots\!01\)\( T^{28} \)
$47$ \( 1 - \)\(28\!\cdots\!22\)\( T^{2} + \)\(37\!\cdots\!15\)\( T^{4} - \)\(33\!\cdots\!08\)\( T^{6} + \)\(21\!\cdots\!89\)\( T^{8} - \)\(11\!\cdots\!30\)\( T^{10} + \)\(47\!\cdots\!35\)\( T^{12} - \)\(16\!\cdots\!60\)\( T^{14} + \)\(49\!\cdots\!35\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(24\!\cdots\!49\)\( T^{20} - \)\(38\!\cdots\!68\)\( T^{22} + \)\(44\!\cdots\!15\)\( T^{24} - \)\(34\!\cdots\!82\)\( T^{26} + \)\(12\!\cdots\!61\)\( T^{28} \)
$53$ \( ( 1 + \)\(92\!\cdots\!90\)\( T + \)\(83\!\cdots\!59\)\( T^{2} + \)\(46\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!57\)\( T^{4} + \)\(11\!\cdots\!10\)\( T^{5} + \)\(54\!\cdots\!63\)\( T^{6} + \)\(20\!\cdots\!60\)\( T^{7} + \)\(82\!\cdots\!83\)\( T^{8} + \)\(26\!\cdots\!10\)\( T^{9} + \)\(89\!\cdots\!97\)\( T^{10} + \)\(23\!\cdots\!20\)\( T^{11} + \)\(63\!\cdots\!59\)\( T^{12} + \)\(10\!\cdots\!90\)\( T^{13} + \)\(17\!\cdots\!81\)\( T^{14} )^{2} \)
$59$ \( 1 - \)\(29\!\cdots\!14\)\( T^{2} + \)\(31\!\cdots\!71\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{6} - \)\(51\!\cdots\!79\)\( T^{8} + \)\(42\!\cdots\!98\)\( T^{10} + \)\(82\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!92\)\( T^{14} + \)\(17\!\cdots\!83\)\( T^{16} + \)\(19\!\cdots\!58\)\( T^{18} - \)\(52\!\cdots\!99\)\( T^{20} - \)\(23\!\cdots\!64\)\( T^{22} + \)\(14\!\cdots\!71\)\( T^{24} - \)\(29\!\cdots\!54\)\( T^{26} + \)\(21\!\cdots\!21\)\( T^{28} \)
$61$ \( ( 1 - \)\(53\!\cdots\!54\)\( T + \)\(52\!\cdots\!51\)\( T^{2} - \)\(16\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} - \)\(31\!\cdots\!02\)\( T^{5} + \)\(22\!\cdots\!83\)\( T^{6} - \)\(55\!\cdots\!52\)\( T^{7} + \)\(30\!\cdots\!43\)\( T^{8} - \)\(57\!\cdots\!82\)\( T^{9} + \)\(28\!\cdots\!01\)\( T^{10} - \)\(53\!\cdots\!04\)\( T^{11} + \)\(23\!\cdots\!51\)\( T^{12} - \)\(32\!\cdots\!34\)\( T^{13} + \)\(82\!\cdots\!41\)\( T^{14} )^{2} \)
$67$ \( 1 - \)\(17\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!55\)\( T^{4} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(48\!\cdots\!89\)\( T^{8} - \)\(18\!\cdots\!10\)\( T^{10} + \)\(62\!\cdots\!15\)\( T^{12} - \)\(18\!\cdots\!40\)\( T^{14} + \)\(46\!\cdots\!15\)\( T^{16} - \)\(10\!\cdots\!10\)\( T^{18} + \)\(19\!\cdots\!29\)\( T^{20} - \)\(30\!\cdots\!08\)\( T^{22} + \)\(35\!\cdots\!55\)\( T^{24} - \)\(29\!\cdots\!02\)\( T^{26} + \)\(12\!\cdots\!41\)\( T^{28} \)
$71$ \( 1 - \)\(96\!\cdots\!14\)\( T^{2} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{6} + \)\(55\!\cdots\!61\)\( T^{8} - \)\(13\!\cdots\!02\)\( T^{10} + \)\(27\!\cdots\!03\)\( T^{12} - \)\(49\!\cdots\!72\)\( T^{14} + \)\(81\!\cdots\!43\)\( T^{16} - \)\(12\!\cdots\!22\)\( T^{18} + \)\(15\!\cdots\!01\)\( T^{20} - \)\(16\!\cdots\!64\)\( T^{22} + \)\(13\!\cdots\!11\)\( T^{24} - \)\(73\!\cdots\!34\)\( T^{26} + \)\(23\!\cdots\!61\)\( T^{28} \)
$73$ \( ( 1 - \)\(71\!\cdots\!30\)\( T + \)\(19\!\cdots\!99\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(17\!\cdots\!17\)\( T^{4} - \)\(89\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!03\)\( T^{6} - \)\(46\!\cdots\!80\)\( T^{7} + \)\(44\!\cdots\!63\)\( T^{8} - \)\(15\!\cdots\!30\)\( T^{9} + \)\(13\!\cdots\!37\)\( T^{10} - \)\(35\!\cdots\!20\)\( T^{11} + \)\(26\!\cdots\!99\)\( T^{12} - \)\(41\!\cdots\!30\)\( T^{13} + \)\(24\!\cdots\!41\)\( T^{14} )^{2} \)
$79$ \( 1 - \)\(32\!\cdots\!34\)\( T^{2} + \)\(63\!\cdots\!51\)\( T^{4} - \)\(84\!\cdots\!04\)\( T^{6} + \)\(88\!\cdots\!61\)\( T^{8} - \)\(73\!\cdots\!02\)\( T^{10} + \)\(50\!\cdots\!43\)\( T^{12} - \)\(29\!\cdots\!52\)\( T^{14} + \)\(14\!\cdots\!83\)\( T^{16} - \)\(57\!\cdots\!22\)\( T^{18} + \)\(19\!\cdots\!01\)\( T^{20} - \)\(52\!\cdots\!84\)\( T^{22} + \)\(10\!\cdots\!51\)\( T^{24} - \)\(16\!\cdots\!54\)\( T^{26} + \)\(13\!\cdots\!61\)\( T^{28} \)
$83$ \( 1 - \)\(22\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!95\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{6} + \)\(97\!\cdots\!69\)\( T^{8} - \)\(41\!\cdots\!90\)\( T^{10} + \)\(14\!\cdots\!95\)\( T^{12} - \)\(40\!\cdots\!20\)\( T^{14} + \)\(95\!\cdots\!95\)\( T^{16} - \)\(18\!\cdots\!90\)\( T^{18} + \)\(28\!\cdots\!09\)\( T^{20} - \)\(34\!\cdots\!48\)\( T^{22} + \)\(31\!\cdots\!95\)\( T^{24} - \)\(18\!\cdots\!42\)\( T^{26} + \)\(55\!\cdots\!41\)\( T^{28} \)
$89$ \( ( 1 - \)\(92\!\cdots\!38\)\( T + \)\(73\!\cdots\!55\)\( T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(36\!\cdots\!89\)\( T^{4} - \)\(55\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!15\)\( T^{6} - \)\(16\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!15\)\( T^{8} - \)\(32\!\cdots\!50\)\( T^{9} + \)\(50\!\cdots\!29\)\( T^{10} - \)\(40\!\cdots\!52\)\( T^{11} + \)\(58\!\cdots\!55\)\( T^{12} - \)\(17\!\cdots\!98\)\( T^{13} + \)\(46\!\cdots\!41\)\( T^{14} )^{2} \)
$97$ \( ( 1 + \)\(10\!\cdots\!10\)\( T + \)\(16\!\cdots\!39\)\( T^{2} + \)\(11\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!57\)\( T^{4} + \)\(42\!\cdots\!30\)\( T^{5} + \)\(33\!\cdots\!63\)\( T^{6} + \)\(10\!\cdots\!40\)\( T^{7} + \)\(12\!\cdots\!83\)\( T^{8} + \)\(60\!\cdots\!30\)\( T^{9} + \)\(55\!\cdots\!97\)\( T^{10} + \)\(22\!\cdots\!20\)\( T^{11} + \)\(12\!\cdots\!39\)\( T^{12} + \)\(29\!\cdots\!10\)\( T^{13} + \)\(10\!\cdots\!81\)\( T^{14} )^{2} \)
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