Properties

Label 25.34.a.d
Level $25$
Weight $34$
Character orbit 25.a
Self dual yes
Analytic conductor $172.457$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.457072203\)
Analytic rank: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + 1789020862523351140800 x^{7} + 15057968169589785919302432 x^{6} - 18240376108577281349602470726592 x^{5} - 118793082389247521352101537474727680 x^{4} + 70339384981314044531499864093672157167360 x^{3} - 93184212295829066424280494839341830566058240 x^{2} - 66043164692761651440492767327962940420438352419328 x + 13944892725752680722269718925910686764312949634715648\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{47}\cdot 3^{12}\cdot 5^{30}\cdot 7^{2}\cdot 11^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -854 + \beta_{1} ) q^{2} + ( -7302793 - 124 \beta_{1} + \beta_{2} ) q^{3} + ( 4485290926 + 6477 \beta_{1} - 8 \beta_{2} + \beta_{3} ) q^{4} + ( -1616724083345 - 12748832 \beta_{1} + 5750 \beta_{2} - 201 \beta_{3} + \beta_{4} ) q^{6} + ( -1940372025135 + 67292417 \beta_{1} + 60307 \beta_{2} + 1485 \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{7} + ( 88199534213789 + 4953517386 \beta_{1} - 1291165 \beta_{2} + 15623 \beta_{3} - 21 \beta_{4} + \beta_{5} + 3 \beta_{6} ) q^{8} + ( 2209057425541162 + 6284760631 \beta_{1} - 4515132 \beta_{2} + 58788 \beta_{3} - 223 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q +(-854 + \beta_{1}) q^{2} +(-7302793 - 124 \beta_{1} + \beta_{2}) q^{3} +(4485290926 + 6477 \beta_{1} - 8 \beta_{2} + \beta_{3}) q^{4} +(-1616724083345 - 12748832 \beta_{1} + 5750 \beta_{2} - 201 \beta_{3} + \beta_{4}) q^{6} +(-1940372025135 + 67292417 \beta_{1} + 60307 \beta_{2} + 1485 \beta_{3} - 2 \beta_{4} + \beta_{6}) q^{7} +(88199534213789 + 4953517386 \beta_{1} - 1291165 \beta_{2} + 15623 \beta_{3} - 21 \beta_{4} + \beta_{5} + 3 \beta_{6}) q^{8} +(2209057425541162 + 6284760631 \beta_{1} - 4515132 \beta_{2} + 58788 \beta_{3} - 223 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7}) q^{9} +(-17680643272201881 - 457920144857 \beta_{1} + 327506234 \beta_{2} - 1134161 \beta_{3} - 1631 \beta_{4} + 31 \beta_{5} + 355 \beta_{6} + 5 \beta_{7} + \beta_{8}) q^{11} +(-102582641036988953 - 2473436518865 \beta_{1} + 5030642377 \beta_{2} - 25695174 \beta_{3} + 17081 \beta_{4} - 97 \beta_{5} - 2910 \beta_{6} + 21 \beta_{7} + \beta_{8} - \beta_{9}) q^{12} +(-468288657747260361 + 1584317380310 \beta_{1} - 4447516207 \beta_{2} - 106100692 \beta_{3} + 23620 \beta_{4} + 66 \beta_{5} - 1288 \beta_{6} - 4 \beta_{8} - \beta_{10}) q^{13} +(881362967251566661 + 11602355753555 \beta_{1} - 28043726914 \beta_{2} + 271338729 \beta_{3} - 133889 \beta_{4} + 673 \beta_{5} + 23262 \beta_{6} - 191 \beta_{7} + 4 \beta_{8} + 7 \beta_{9} + \beta_{10}) q^{14} +(26163242192604810856 + 214910603865384 \beta_{1} - 269750111040 \beta_{2} + 4544434396 \beta_{3} - 1289466 \beta_{4} - 11094 \beta_{5} + 65269 \beta_{6} - 1245 \beta_{7} - 259 \beta_{8} + 57 \beta_{9} + 26 \beta_{10}) q^{16} +(-19768233535684881455 - 313208661781569 \beta_{1} + 715645171030 \beta_{2} - 1049372552 \beta_{3} + 5276461 \beta_{4} - 61543 \beta_{5} - 568042 \beta_{6} - 11019 \beta_{7} - 260 \beta_{8} - 136 \beta_{9} + 26 \beta_{10}) q^{17} +(80291140856157179383 + 2802860239957555 \beta_{1} - 3096290197044 \beta_{2} + 15817971279 \beta_{3} - 280003 \beta_{4} - 2591 \beta_{5} - 51118 \beta_{6} - 41879 \beta_{7} - 3714 \beta_{8} + 399 \beta_{9} + 15 \beta_{10}) q^{18} +(-\)\(20\!\cdots\!67\)\( - 1517136263352010 \beta_{1} - 2455725349911 \beta_{2} + 4922057406 \beta_{3} + 6911434 \beta_{4} - 40218 \beta_{5} + 1945190 \beta_{6} - 82518 \beta_{7} + 3710 \beta_{8} + 888 \beta_{9} - 16 \beta_{10}) q^{19} +(\)\(35\!\cdots\!83\)\( - 18008147243938488 \beta_{1} + 3508978413007 \beta_{2} - 231349833696 \beta_{3} + 6225394 \beta_{4} - 4297196 \beta_{5} - 29589504 \beta_{6} + 655374 \beta_{7} - 11140 \beta_{8} - 1976 \beta_{9} - 1443 \beta_{10}) q^{21} +(-\)\(59\!\cdots\!70\)\( - 32704642116276427 \beta_{1} - 14784930287380 \beta_{2} - 196819929234 \beta_{3} + 475538410 \beta_{4} - 9311817 \beta_{5} + 21764346 \beta_{6} - 2509209 \beta_{7} - 10880 \beta_{8} + 6129 \beta_{9} - 1469 \beta_{10}) q^{22} +(-\)\(36\!\cdots\!76\)\( + 11514810947779070 \beta_{1} + 86340287972176 \beta_{2} - 241660585354 \beta_{3} - 245428364 \beta_{4} + 16886072 \beta_{5} - 67899306 \beta_{6} - 1185920 \beta_{7} + 79336 \beta_{8} - 17480 \beta_{9} + 964 \beta_{10}) q^{23} +(-\)\(18\!\cdots\!87\)\( - 264597737460879582 \beta_{1} + 249740896416063 \beta_{2} - 2769259915653 \beta_{3} + 4666547739 \beta_{4} - 39869391 \beta_{5} - 660387447 \beta_{6} - 4845090 \beta_{7} - 83046 \beta_{8} - 50886 \beta_{9} - 948 \beta_{10}) q^{24} +(\)\(21\!\cdots\!51\)\( - 1388646180402755863 \beta_{1} + 406155631593920 \beta_{2} + 1021800022579 \beta_{3} - 4283113359 \beta_{4} - 231180023 \beta_{5} - 1218213766 \beta_{6} + 10220993 \beta_{7} + 611514 \beta_{8} + 23959 \beta_{9} + 37387 \beta_{10}) q^{26} +(-\)\(19\!\cdots\!67\)\( - 1484874311362629041 \beta_{1} + 1239816208419126 \beta_{2} + 2082463135251 \beta_{3} - 9230789917 \beta_{4} - 264502967 \beta_{5} + 1654321847 \beta_{6} + 766891 \beta_{7} + 622395 \beta_{8} - 99096 \beta_{9} + 38856 \beta_{10}) q^{27} +(\)\(16\!\cdots\!58\)\( + 2950491222485981286 \beta_{1} - 2830324353525106 \beta_{2} + 13896045977312 \beta_{3} - 67807158258 \beta_{4} + 445774050 \beta_{5} - 2610498156 \beta_{6} + 9639246 \beta_{7} + 205782 \beta_{8} + 335274 \beta_{9} - 44736 \beta_{10}) q^{28} +(\)\(15\!\cdots\!57\)\( - 424535781405965496 \beta_{1} - 5811360530257717 \beta_{2} - 6105847705548 \beta_{3} + 74977176430 \beta_{4} - 55438232 \beta_{5} - 11257698300 \beta_{6} + 53614450 \beta_{7} - 122732 \beta_{8} + 1175920 \beta_{9} + 45685 \beta_{10}) q^{29} +(-\)\(17\!\cdots\!10\)\( + 9162636481899116838 \beta_{1} - 7739147217183256 \beta_{2} - 4512702580922 \beta_{3} - 191727939558 \beta_{4} + 101144094 \beta_{5} - 16505968570 \beta_{6} - 358986462 \beta_{7} - 10991654 \beta_{8} - 4008 \beta_{9} - 594644 \beta_{10}) q^{31} +(\)\(20\!\cdots\!86\)\( + 27193628638376350668 \beta_{1} - 14449942805538474 \beta_{2} + 87196478929042 \beta_{3} - 600393836344 \beta_{4} + 2613388176 \beta_{5} + 36270013721 \beta_{6} + 725600077 \beta_{7} - 11614309 \beta_{8} + 576663 \beta_{9} - 633474 \beta_{10}) q^{32} +(\)\(33\!\cdots\!73\)\( - 894212962009436206 \beta_{1} + 5684491865603482 \beta_{2} + 197552104781352 \beta_{3} - 1353719460526 \beta_{4} - 6109648050 \beta_{5} - 70331489172 \beta_{6} + 259461546 \beta_{7} - 25607568 \beta_{8} - 3445776 \beta_{9} + 951414 \beta_{10}) q^{33} +(-\)\(40\!\cdots\!93\)\( - 34720342909304513790 \beta_{1} + 76664078768690948 \beta_{2} - 962125967327555 \beta_{3} + 1696396826151 \beta_{4} + 5536717467 \beta_{5} - 68215557946 \beta_{6} + 818701107 \beta_{7} + 25734010 \beta_{8} - 15804555 \beta_{9} - 997115 \beta_{10}) q^{34} +(\)\(17\!\cdots\!96\)\( + \)\(20\!\cdots\!78\)\( \beta_{1} - 20964684441585720 \beta_{2} + 2700496759466142 \beta_{3} - 5000525332904 \beta_{4} + 41299144520 \beta_{5} + 194830812616 \beta_{6} + 2277115184 \beta_{7} + 99382416 \beta_{8} - 3353136 \beta_{9} + 6387552 \beta_{10}) q^{36} +(-\)\(16\!\cdots\!29\)\( + 57914855695289783822 \beta_{1} + 8122075350780591 \beta_{2} - 1229425977824792 \beta_{3} - 1699001640044 \beta_{4} + 22855905222 \beta_{5} - 121206680364 \beta_{6} - 7056068064 \beta_{7} + 110985844 \beta_{8} + 4761384 \beta_{9} + 7019557 \beta_{10}) q^{37} +(-\)\(19\!\cdots\!77\)\( - \)\(16\!\cdots\!42\)\( \beta_{1} + 87186366828365686 \beta_{2} - 1208400501503133 \beta_{3} - 9103977026187 \beta_{4} - 10774354630 \beta_{5} + 616370685644 \beta_{6} - 2995103494 \beta_{7} + 401159976 \beta_{8} + 15120214 \beta_{9} - 13064390 \beta_{10}) q^{38} +(-\)\(32\!\cdots\!91\)\( + \)\(39\!\cdots\!07\)\( \beta_{1} - 554452307734799585 \beta_{2} + 972172774670235 \beta_{3} + 3357160528958 \beta_{4} - 11009781916 \beta_{5} + 1679763996159 \beta_{6} - 16735812540 \beta_{7} - 426977036 \beta_{8} + 135678392 \beta_{9} + 14060556 \beta_{10}) q^{39} +(\)\(49\!\cdots\!03\)\( - \)\(65\!\cdots\!58\)\( \beta_{1} - 556382777601436318 \beta_{2} + 8947852388961944 \beta_{3} + 2954312123010 \beta_{4} - 217646069330 \beta_{5} + 489343612044 \beta_{6} + 9545339178 \beta_{7} - 376627128 \beta_{8} + 43697424 \beta_{9} - 47415618 \beta_{10}) q^{41} +(-\)\(23\!\cdots\!33\)\( - \)\(18\!\cdots\!25\)\( \beta_{1} + 492488688107159040 \beta_{2} - 57233886262004697 \beta_{3} + 22532489755053 \beta_{4} - 294745835667 \beta_{5} - 3600627492798 \beta_{6} + 2588836677 \beta_{7} - 486990318 \beta_{8} - 123367437 \beta_{9} - 54396345 \beta_{10}) q^{42} +(-\)\(45\!\cdots\!50\)\( + \)\(31\!\cdots\!61\)\( \beta_{1} - 460107813209897259 \beta_{2} - 5246192388199639 \beta_{3} + 40892845408027 \beta_{4} + 341246692701 \beta_{5} + 2580602389285 \beta_{6} - 5249862801 \beta_{7} - 3275689269 \beta_{8} + 89485056 \beta_{9} + 128500728 \beta_{10}) q^{43} +(-\)\(27\!\cdots\!15\)\( - \)\(39\!\cdots\!31\)\( \beta_{1} + 3907158767147173571 \beta_{2} - 88680833343393074 \beta_{3} - 68203469251053 \beta_{4} - 253425868139 \beta_{5} - 150470180962 \beta_{6} + 100399516367 \beta_{7} + 3702539859 \beta_{8} - 723074963 \beta_{9} - 142515584 \beta_{10}) q^{44} +(\)\(15\!\cdots\!76\)\( - \)\(60\!\cdots\!78\)\( \beta_{1} - 2625154768840144080 \beta_{2} + 116656668942543944 \beta_{3} + 241433299103144 \beta_{4} - 1220558744310 \beta_{5} - 8098938796644 \beta_{6} - 181917351126 \beta_{7} - 1660681504 \beta_{8} - 238228506 \beta_{9} + 229238642 \beta_{10}) q^{46} +(-\)\(99\!\cdots\!23\)\( + \)\(15\!\cdots\!45\)\( \beta_{1} - 7164491177379423731 \beta_{2} - 323970841810293199 \beta_{3} + 28596179955024 \beta_{4} + 249947182754 \beta_{5} + 16443072409597 \beta_{6} + 300512811262 \beta_{7} - 1185294354 \beta_{8} + 1164181928 \beta_{9} + 283002956 \beta_{10}) q^{47} +(-\)\(25\!\cdots\!80\)\( - \)\(25\!\cdots\!20\)\( \beta_{1} + 24826957250668894900 \beta_{2} - 533024729621783016 \beta_{3} + 566051811255734 \beta_{4} - 281805585910 \beta_{5} - 26018011256349 \beta_{6} + 256755912297 \beta_{7} + 14968345543 \beta_{8} - 2074851157 \beta_{9} - 945998466 \beta_{10}) q^{48} +(-\)\(22\!\cdots\!43\)\( + \)\(17\!\cdots\!36\)\( \beta_{1} - 23322148030220200376 \beta_{2} - 319823093755936252 \beta_{3} - 311391016402236 \beta_{4} + 928656204420 \beta_{5} - 52494295963796 \beta_{6} - 70229997228 \beta_{7} - 18645072052 \beta_{8} + 1629949800 \beta_{9} + 1087517900 \beta_{10}) q^{49} +(\)\(60\!\cdots\!35\)\( + \)\(50\!\cdots\!53\)\( \beta_{1} - 79131722332555683136 \beta_{2} + 1016423784324652413 \beta_{3} - 77372139448925 \beta_{4} + 12523660073517 \beta_{5} + 10638341496009 \beta_{6} + 741157296735 \beta_{7} + 31723813947 \beta_{8} - 269561088 \beta_{9} - 474784584 \beta_{10}) q^{51} +(-\)\(14\!\cdots\!12\)\( + \)\(46\!\cdots\!92\)\( \beta_{1} - 8021777697288725300 \beta_{2} - 2343524590610330080 \beta_{3} + 170823167145644 \beta_{4} + 7653029201748 \beta_{5} + 69724826597360 \beta_{6} - 1521950849292 \beta_{7} + 33019482372 \beta_{8} - 6008459748 \beta_{9} - 750805344 \beta_{10}) q^{52} +(-\)\(30\!\cdots\!76\)\( - \)\(21\!\cdots\!02\)\( \beta_{1} + 38570212006690032342 \beta_{2} - 160792271832330420 \beta_{3} - 606652849108714 \beta_{4} - 10631017668630 \beta_{5} - 19292254113824 \beta_{6} - 1742443704570 \beta_{7} - 24709696768 \beta_{8} + 18294932920 \beta_{9} + 5285771038 \beta_{10}) q^{53} +(-\)\(19\!\cdots\!26\)\( - \)\(17\!\cdots\!23\)\( \beta_{1} - \)\(10\!\cdots\!00\)\( \beta_{2} - 3079536585859397586 \beta_{3} + 1484539797202730 \beta_{4} + 6524293720711 \beta_{5} + 60731233042010 \beta_{6} - 2051031740489 \beta_{7} + 42928001280 \beta_{8} + 8501473569 \beta_{9} - 6359273133 \beta_{10}) q^{54} +(\)\(30\!\cdots\!82\)\( + \)\(22\!\cdots\!44\)\( \beta_{1} - \)\(71\!\cdots\!14\)\( \beta_{2} + 4428577167389075830 \beta_{3} - 3992091205483946 \beta_{4} - 11776488370302 \beta_{5} + 67233159580530 \beta_{6} + 561879240316 \beta_{7} - 213454537388 \beta_{8} + 15502158004 \beta_{9} - 2648617928 \beta_{10}) q^{56} +(-\)\(14\!\cdots\!17\)\( + \)\(80\!\cdots\!27\)\( \beta_{1} - \)\(57\!\cdots\!32\)\( \beta_{2} - 5664127739028129528 \beta_{3} - 1885083553508087 \beta_{4} - 28275193861583 \beta_{5} - 450609879090714 \beta_{6} - 1386053922567 \beta_{7} - 246950029564 \beta_{8} + 11355002152 \beta_{9} - 1951615728 \beta_{10}) q^{57} +(-\)\(56\!\cdots\!17\)\( + \)\(12\!\cdots\!41\)\( \beta_{1} + \)\(97\!\cdots\!52\)\( \beta_{2} - 3270625210609512345 \beta_{3} - 11313991559340355 \beta_{4} + 31734416288781 \beta_{5} + 254425062937682 \beta_{6} + 4703852669061 \beta_{7} - 123083292102 \beta_{8} - 102060153741 \beta_{9} - 22108819617 \beta_{10}) q^{58} +(-\)\(61\!\cdots\!68\)\( - \)\(41\!\cdots\!67\)\( \beta_{1} + \)\(14\!\cdots\!93\)\( \beta_{2} - 2968058611176381827 \beta_{3} + 6408780881711789 \beta_{4} - 45101262296841 \beta_{5} + 682076453249033 \beta_{6} + 9667067420797 \beta_{7} + 83832143777 \beta_{8} - 101051924688 \beta_{9} + 28326527616 \beta_{10}) q^{59} +(-\)\(31\!\cdots\!15\)\( - \)\(11\!\cdots\!32\)\( \beta_{1} - \)\(17\!\cdots\!47\)\( \beta_{2} - 2245229193445441688 \beta_{3} - 809020131084178 \beta_{4} - 186893071030548 \beta_{5} - 216958443799624 \beta_{6} - 12876743893566 \beta_{7} + 857149265748 \beta_{8} - 139622854776 \beta_{9} + 30725468907 \beta_{10}) q^{61} +(\)\(11\!\cdots\!68\)\( - \)\(10\!\cdots\!98\)\( \beta_{1} - \)\(26\!\cdots\!08\)\( \beta_{2} + 10098419831309069312 \beta_{3} - 11180394485869568 \beta_{4} - 26675907181134 \beta_{5} - 243843164111972 \beta_{6} + 36379224326642 \beta_{7} + 1102815230056 \beta_{8} + 78878554398 \beta_{9} + 32953737426 \beta_{10}) q^{62} +(\)\(63\!\cdots\!78\)\( + \)\(27\!\cdots\!44\)\( \beta_{1} + \)\(20\!\cdots\!32\)\( \beta_{2} + 23119965867740638356 \beta_{3} - 11583005731469046 \beta_{4} + 152145092809218 \beta_{5} + 2098198547123988 \beta_{6} - 1464830385114 \beta_{7} + 779051222742 \beta_{8} + 382664171184 \beta_{9} + 64816556544 \beta_{10}) q^{63} +(\)\(12\!\cdots\!22\)\( + \)\(12\!\cdots\!40\)\( \beta_{1} - \)\(60\!\cdots\!86\)\( \beta_{2} + 20714655586209849134 \beta_{3} - 15028373818419268 \beta_{4} + 19076330397420 \beta_{5} + 481294618686113 \beta_{6} - 6934134637263 \beta_{7} - 881127484377 \beta_{8} + 485782804227 \beta_{9} - 92068585914 \beta_{10}) q^{64} +(-\)\(14\!\cdots\!42\)\( + \)\(50\!\cdots\!59\)\( \beta_{1} - \)\(18\!\cdots\!68\)\( \beta_{2} - 41608598239024396236 \beta_{3} + 20561488800809740 \beta_{4} + 518038387920196 \beta_{5} + 1330575535200792 \beta_{6} + 21978230458500 \beta_{7} - 2192661009184 \beta_{8} + 747222724636 \beta_{9} - 151907615052 \beta_{10}) q^{66} +(\)\(15\!\cdots\!21\)\( + \)\(40\!\cdots\!69\)\( \beta_{1} - \)\(88\!\cdots\!52\)\( \beta_{2} + 63725837426795343533 \beta_{3} + 69282064575713205 \beta_{4} + 145179798800703 \beta_{5} + 1958952276766457 \beta_{6} - 103505046046563 \beta_{7} - 3262091121395 \beta_{8} - 789338482872 \beta_{9} - 185565336848 \beta_{10}) q^{67} +(-\)\(28\!\cdots\!54\)\( - \)\(10\!\cdots\!13\)\( \beta_{1} + \)\(18\!\cdots\!20\)\( \beta_{2} - 31188211678869317493 \beta_{3} + 194044069050422528 \beta_{4} - 887416372018144 \beta_{5} - 12197391671631864 \beta_{6} + 9548267998952 \beta_{7} - 387755505496 \beta_{8} - 883279896712 \beta_{9} - 100372963552 \beta_{10}) q^{68} +(\)\(65\!\cdots\!04\)\( - \)\(78\!\cdots\!14\)\( \beta_{1} - \)\(65\!\cdots\!94\)\( \beta_{2} + \)\(11\!\cdots\!72\)\( \beta_{3} - 65382820017126066 \beta_{4} + 514552356461154 \beta_{5} - 14942844032636304 \beta_{6} - 54552453747954 \beta_{7} + 1308560909448 \beta_{8} - 1224507106728 \beta_{9} + 186209825646 \beta_{10}) q^{69} +(-\)\(50\!\cdots\!23\)\( + \)\(12\!\cdots\!33\)\( \beta_{1} - \)\(26\!\cdots\!75\)\( \beta_{2} - \)\(20\!\cdots\!23\)\( \beta_{3} + 138023487110284152 \beta_{4} + 1316538444426666 \beta_{5} - 8973546930389891 \beta_{6} + 102372038208838 \beta_{7} + 3945460692558 \beta_{8} - 2630579667976 \beta_{9} + 411840231932 \beta_{10}) q^{71} +(\)\(19\!\cdots\!18\)\( + \)\(19\!\cdots\!12\)\( \beta_{1} - \)\(44\!\cdots\!90\)\( \beta_{2} + \)\(47\!\cdots\!90\)\( \beta_{3} - 32650015672819470 \beta_{4} + 1363843544400534 \beta_{5} + 23404891535422890 \beta_{6} - 106792076136696 \beta_{7} + 6962361848568 \beta_{8} + 3435553921176 \beta_{9} + 595231101936 \beta_{10}) q^{72} +(-\)\(11\!\cdots\!95\)\( - \)\(52\!\cdots\!81\)\( \beta_{1} + \)\(27\!\cdots\!84\)\( \beta_{2} + \)\(12\!\cdots\!20\)\( \beta_{3} + 184273615010629465 \beta_{4} + 32302060229625 \beta_{5} - 1287754102297326 \beta_{6} - 311900587081335 \beta_{7} - 12852897489848 \beta_{8} + 574725785760 \beta_{9} - 134033196272 \beta_{10}) q^{73} +(\)\(77\!\cdots\!49\)\( - \)\(27\!\cdots\!07\)\( \beta_{1} - \)\(21\!\cdots\!04\)\( \beta_{2} + \)\(21\!\cdots\!53\)\( \beta_{3} - 7064348236786729 \beta_{4} - 1383312001166273 \beta_{5} + 11841440781233158 \beta_{6} - 7025400385113 \beta_{7} + 11642735906030 \beta_{8} + 394922102017 \beta_{9} - 24783035819 \beta_{10}) q^{74} +(-\)\(34\!\cdots\!23\)\( - \)\(19\!\cdots\!47\)\( \beta_{1} - \)\(96\!\cdots\!45\)\( \beta_{2} - 92714414436440217010 \beta_{3} - 224835024297841693 \beta_{4} - 6542908194391899 \beta_{5} + 19387088886799302 \beta_{6} - 305935664726457 \beta_{7} - 10200959633557 \beta_{8} + 5522536367253 \beta_{9} - 210443629696 \beta_{10}) q^{76} +(\)\(22\!\cdots\!37\)\( + \)\(33\!\cdots\!94\)\( \beta_{1} - \)\(22\!\cdots\!79\)\( \beta_{2} - \)\(13\!\cdots\!92\)\( \beta_{3} - 40101716867082716 \beta_{4} - 7329166458366910 \beta_{5} - 101051490677189116 \beta_{6} + 932860866981864 \beta_{7} - 16092606682004 \beta_{8} - 7682020728184 \beta_{9} - 772293623797 \beta_{10}) q^{77} +(\)\(51\!\cdots\!77\)\( - \)\(18\!\cdots\!63\)\( \beta_{1} + \)\(38\!\cdots\!74\)\( \beta_{2} + \)\(64\!\cdots\!65\)\( \beta_{3} - 1927004463161656453 \beta_{4} + 7678655222969427 \beta_{5} + 102527832545974602 \beta_{6} + 1741194998927187 \beta_{7} + 62871134516628 \beta_{8} + 3924825677253 \beta_{9} + 1338391236939 \beta_{10}) q^{78} +(\)\(25\!\cdots\!65\)\( - \)\(24\!\cdots\!77\)\( \beta_{1} - \)\(90\!\cdots\!83\)\( \beta_{2} + 24137444168895569939 \beta_{3} + 400915655263750168 \beta_{4} + 1518556639507206 \beta_{5} + 117972741864101583 \beta_{6} + 1157140812639954 \beta_{7} - 75416431835198 \beta_{8} + 8720692309680 \beta_{9} - 1400519515760 \beta_{10}) q^{79} +(-\)\(41\!\cdots\!95\)\( - \)\(82\!\cdots\!13\)\( \beta_{1} - \)\(64\!\cdots\!68\)\( \beta_{2} + \)\(50\!\cdots\!36\)\( \beta_{3} - 1064139762175185319 \beta_{4} - 2926862951338799 \beta_{5} + 11064798187507790 \beta_{6} - 1234288905389975 \beta_{7} + 46499820634476 \beta_{8} - 1806617439240 \beta_{9} - 3275847514320 \beta_{10}) q^{81} +(-\)\(85\!\cdots\!02\)\( + \)\(12\!\cdots\!85\)\( \beta_{1} + \)\(30\!\cdots\!44\)\( \beta_{2} - \)\(20\!\cdots\!04\)\( \beta_{3} - 1266666390876908484 \beta_{4} + 9618077746564500 \beta_{5} + 15590776847243512 \beta_{6} + 704726560358868 \beta_{7} + 59542129439456 \beta_{8} - 506001174708 \beta_{9} - 2686241408188 \beta_{10}) q^{82} +(-\)\(11\!\cdots\!05\)\( - \)\(11\!\cdots\!43\)\( \beta_{1} - \)\(25\!\cdots\!56\)\( \beta_{2} - \)\(14\!\cdots\!47\)\( \beta_{3} - 806697047778749157 \beta_{4} - 5110894587873939 \beta_{5} - 34338676334009163 \beta_{6} - 4540844772398745 \beta_{7} - 110527211623257 \beta_{8} - 15112349894280 \beta_{9} - 3965255773728 \beta_{10}) q^{83} +(-\)\(27\!\cdots\!72\)\( - \)\(61\!\cdots\!14\)\( \beta_{1} + \)\(35\!\cdots\!00\)\( \beta_{2} - \)\(33\!\cdots\!90\)\( \beta_{3} + 2453592786598123252 \beta_{4} - 15704430708001460 \beta_{5} - 115496346032090016 \beta_{6} - 3819335896206084 \beta_{7} + 187114152936044 \beta_{8} - 29542013634764 \beta_{9} + 5530822249248 \beta_{10}) q^{84} +(\)\(41\!\cdots\!31\)\( - \)\(26\!\cdots\!89\)\( \beta_{1} + \)\(52\!\cdots\!14\)\( \beta_{2} + \)\(56\!\cdots\!79\)\( \beta_{3} - 1148737336296424979 \beta_{4} + 33788078998845297 \beta_{5} - 26401322755263402 \beta_{6} + 8000248177359361 \beta_{7} - 136460611007504 \beta_{8} - 31519205145497 \beta_{9} + 14515485817429 \beta_{10}) q^{86} +(-\)\(44\!\cdots\!73\)\( + \)\(54\!\cdots\!23\)\( \beta_{1} + \)\(23\!\cdots\!07\)\( \beta_{2} - \)\(40\!\cdots\!69\)\( \beta_{3} + 2457193128611382016 \beta_{4} - 356828860361154 \beta_{5} + 362318102804149875 \beta_{6} - 13174868898603246 \beta_{7} - 189865793058798 \beta_{8} + 64213847699976 \beta_{9} + 17765758668732 \beta_{10}) q^{87} +(-\)\(47\!\cdots\!85\)\( - \)\(83\!\cdots\!42\)\( \beta_{1} - \)\(62\!\cdots\!59\)\( \beta_{2} - \)\(45\!\cdots\!59\)\( \beta_{3} + 6333482969796552521 \beta_{4} - 68533618654749381 \beta_{5} - 746076797986494741 \beta_{6} + 6472021926097362 \beta_{7} - 132494619489130 \beta_{8} + 21669963813078 \beta_{9} + 5286829994612 \beta_{10}) q^{88} +(-\)\(21\!\cdots\!11\)\( - \)\(69\!\cdots\!19\)\( \beta_{1} + \)\(13\!\cdots\!22\)\( \beta_{2} - \)\(48\!\cdots\!08\)\( \beta_{3} - 6231323721207584845 \beta_{4} + 99523771630768287 \beta_{5} - 584728380373593086 \beta_{6} + 4436777115278955 \beta_{7} - 42173062225228 \beta_{8} + 23092855983832 \beta_{9} - 10782903542774 \beta_{10}) q^{89} +(-\)\(31\!\cdots\!78\)\( - \)\(64\!\cdots\!02\)\( \beta_{1} + \)\(13\!\cdots\!94\)\( \beta_{2} + \)\(28\!\cdots\!26\)\( \beta_{3} + 10253805079447516596 \beta_{4} + 67707362534516928 \beta_{5} - 270191078078256426 \beta_{6} - 15353554908983064 \beta_{7} + 52946426496612 \beta_{8} + 115475355781656 \beta_{9} - 24710358570192 \beta_{10}) q^{91} +(-\)\(48\!\cdots\!46\)\( + \)\(10\!\cdots\!18\)\( \beta_{1} + \)\(23\!\cdots\!62\)\( \beta_{2} - \)\(15\!\cdots\!16\)\( \beta_{3} + 833567910192515390 \beta_{4} + 76093367536145362 \beta_{5} + 485201658478150796 \beta_{6} + 26462389006450294 \beta_{7} + 228698719951646 \beta_{8} - 204471397210014 \beta_{9} - 43099158254592 \beta_{10}) q^{92} +(-\)\(71\!\cdots\!90\)\( - \)\(15\!\cdots\!20\)\( \beta_{1} - \)\(13\!\cdots\!08\)\( \beta_{2} - \)\(55\!\cdots\!80\)\( \beta_{3} + 19891349381727902972 \beta_{4} + 163155367024254812 \beta_{5} + 1036305824177983236 \beta_{6} - 11743770057019308 \beta_{7} + 1021318171853080 \beta_{8} - 2052284336152 \beta_{9} + 2890991172732 \beta_{10}) q^{93} +(\)\(21\!\cdots\!23\)\( - \)\(38\!\cdots\!89\)\( \beta_{1} + \)\(72\!\cdots\!30\)\( \beta_{2} + \)\(33\!\cdots\!15\)\( \beta_{3} - 11267654702127698767 \beta_{4} - 274077969615612159 \beta_{5} - 248837302053496098 \beta_{6} - 7344920231779935 \beta_{7} - 1054825812595804 \beta_{8} + 109583696022951 \beta_{9} + 6413115644993 \beta_{10}) q^{94} +(-\)\(16\!\cdots\!38\)\( - \)\(53\!\cdots\!00\)\( \beta_{1} + \)\(62\!\cdots\!30\)\( \beta_{2} - \)\(20\!\cdots\!70\)\( \beta_{3} + 17936133125889765680 \beta_{4} - 390924709191950760 \beta_{5} + 803483549986384791 \beta_{6} + 9803101784575563 \beta_{7} + 1072972204016877 \beta_{8} - 197510160970575 \beta_{9} - 24114719494350 \beta_{10}) q^{96} +(\)\(61\!\cdots\!48\)\( + \)\(11\!\cdots\!88\)\( \beta_{1} + \)\(25\!\cdots\!62\)\( \beta_{2} + \)\(13\!\cdots\!20\)\( \beta_{3} + 16289020118885292240 \beta_{4} - 361591427175736476 \beta_{5} - 3208437136575612376 \beta_{6} + 17515198580753064 \beta_{7} + 900727679316520 \beta_{8} + 197332090625616 \beta_{9} + 15917130042034 \beta_{10}) q^{97} +(\)\(23\!\cdots\!70\)\( - \)\(28\!\cdots\!43\)\( \beta_{1} - \)\(40\!\cdots\!96\)\( \beta_{2} + \)\(17\!\cdots\!88\)\( \beta_{3} - 26319880848191044696 \beta_{4} - 20778985258363416 \beta_{5} + 224245744762505696 \beta_{6} + 48726385072124488 \beta_{7} - 1511407241806600 \beta_{8} + 5247394326072 \beta_{9} - 24883237759632 \beta_{10}) q^{98} +(\)\(11\!\cdots\!38\)\( - \)\(85\!\cdots\!05\)\( \beta_{1} + \)\(38\!\cdots\!81\)\( \beta_{2} + \)\(27\!\cdots\!19\)\( \beta_{3} + 5545675460906210791 \beta_{4} + 451194287679277013 \beta_{5} + 4601614219625210683 \beta_{6} + 52994325190118567 \beta_{7} + 2739711982730811 \beta_{8} - 299492629099248 \beta_{9} + 23802223156536 \beta_{10}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 9393q^{2} - 80330849q^{3} + 49338206677q^{4} - 17783977676723q^{6} - 21344025107658q^{7} + 970199832420105q^{8} + 24299637974625658q^{9} + O(q^{10}) \) \( 11q - 9393q^{2} - 80330849q^{3} + 49338206677q^{4} - 17783977676723q^{6} - 21344025107658q^{7} + 970199832420105q^{8} + 24299637974625658q^{9} - 194487534567113523q^{11} - 1128411534853252093q^{12} - 5151173641795199804q^{13} + 9695004297667451034q^{14} + \)\(28\!\cdots\!01\)\(q^{16} - \)\(21\!\cdots\!63\)\(q^{17} + \)\(88\!\cdots\!46\)\(q^{18} - \)\(23\!\cdots\!15\)\(q^{19} + \)\(39\!\cdots\!62\)\(q^{21} - \)\(65\!\cdots\!01\)\(q^{22} - \)\(39\!\cdots\!34\)\(q^{23} - \)\(20\!\cdots\!45\)\(q^{24} + \)\(23\!\cdots\!92\)\(q^{26} - \)\(21\!\cdots\!95\)\(q^{27} + \)\(18\!\cdots\!94\)\(q^{28} + \)\(16\!\cdots\!40\)\(q^{29} - \)\(18\!\cdots\!38\)\(q^{31} + \)\(22\!\cdots\!77\)\(q^{32} + \)\(36\!\cdots\!57\)\(q^{33} - \)\(44\!\cdots\!51\)\(q^{34} + \)\(19\!\cdots\!06\)\(q^{36} - \)\(18\!\cdots\!98\)\(q^{37} - \)\(21\!\cdots\!05\)\(q^{38} - \)\(35\!\cdots\!44\)\(q^{39} + \)\(54\!\cdots\!67\)\(q^{41} - \)\(25\!\cdots\!06\)\(q^{42} - \)\(50\!\cdots\!44\)\(q^{43} - \)\(29\!\cdots\!11\)\(q^{44} + \)\(16\!\cdots\!82\)\(q^{46} - \)\(10\!\cdots\!28\)\(q^{47} - \)\(28\!\cdots\!09\)\(q^{48} - \)\(25\!\cdots\!73\)\(q^{49} + \)\(66\!\cdots\!57\)\(q^{51} - \)\(15\!\cdots\!28\)\(q^{52} - \)\(33\!\cdots\!74\)\(q^{53} - \)\(21\!\cdots\!65\)\(q^{54} + \)\(33\!\cdots\!10\)\(q^{56} - \)\(15\!\cdots\!15\)\(q^{57} - \)\(62\!\cdots\!20\)\(q^{58} - \)\(67\!\cdots\!20\)\(q^{59} - \)\(34\!\cdots\!98\)\(q^{61} + \)\(13\!\cdots\!94\)\(q^{62} + \)\(69\!\cdots\!76\)\(q^{63} + \)\(14\!\cdots\!37\)\(q^{64} - \)\(16\!\cdots\!11\)\(q^{66} + \)\(16\!\cdots\!37\)\(q^{67} - \)\(30\!\cdots\!41\)\(q^{68} + \)\(72\!\cdots\!26\)\(q^{69} - \)\(55\!\cdots\!68\)\(q^{71} + \)\(21\!\cdots\!90\)\(q^{72} - \)\(12\!\cdots\!09\)\(q^{73} + \)\(84\!\cdots\!54\)\(q^{74} - \)\(37\!\cdots\!55\)\(q^{76} + \)\(25\!\cdots\!94\)\(q^{77} + \)\(56\!\cdots\!72\)\(q^{78} + \)\(28\!\cdots\!90\)\(q^{79} - \)\(45\!\cdots\!29\)\(q^{81} - \)\(94\!\cdots\!21\)\(q^{82} - \)\(12\!\cdots\!39\)\(q^{83} - \)\(29\!\cdots\!66\)\(q^{84} + \)\(45\!\cdots\!12\)\(q^{86} - \)\(48\!\cdots\!60\)\(q^{87} - \)\(52\!\cdots\!15\)\(q^{88} - \)\(24\!\cdots\!55\)\(q^{89} - \)\(34\!\cdots\!48\)\(q^{91} - \)\(53\!\cdots\!38\)\(q^{92} - \)\(78\!\cdots\!58\)\(q^{93} + \)\(23\!\cdots\!44\)\(q^{94} - \)\(18\!\cdots\!53\)\(q^{96} + \)\(67\!\cdots\!22\)\(q^{97} + \)\(25\!\cdots\!99\)\(q^{98} + \)\(13\!\cdots\!06\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - x^{10} - 71909733210 x^{9} - 392361971317440 x^{8} + 1789020862523351140800 x^{7} + 15057968169589785919302432 x^{6} - 18240376108577281349602470726592 x^{5} - 118793082389247521352101537474727680 x^{4} + 70339384981314044531499864093672157167360 x^{3} - 93184212295829066424280494839341830566058240 x^{2} - 66043164692761651440492767327962940420438352419328 x + 13944892725752680722269718925910686764312949634715648\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(\)\(29\!\cdots\!81\)\( \nu^{10} - \)\(99\!\cdots\!75\)\( \nu^{9} - \)\(19\!\cdots\!40\)\( \nu^{8} + \)\(46\!\cdots\!80\)\( \nu^{7} + \)\(43\!\cdots\!80\)\( \nu^{6} - \)\(59\!\cdots\!88\)\( \nu^{5} - \)\(33\!\cdots\!40\)\( \nu^{4} + \)\(16\!\cdots\!80\)\( \nu^{3} + \)\(70\!\cdots\!60\)\( \nu^{2} + \)\(18\!\cdots\!80\)\( \nu - \)\(15\!\cdots\!48\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(29\!\cdots\!81\)\( \nu^{10} - \)\(99\!\cdots\!75\)\( \nu^{9} - \)\(19\!\cdots\!40\)\( \nu^{8} + \)\(46\!\cdots\!80\)\( \nu^{7} + \)\(43\!\cdots\!80\)\( \nu^{6} - \)\(59\!\cdots\!88\)\( \nu^{5} - \)\(33\!\cdots\!40\)\( \nu^{4} + \)\(16\!\cdots\!80\)\( \nu^{3} + \)\(10\!\cdots\!40\)\( \nu^{2} - \)\(12\!\cdots\!20\)\( \nu - \)\(51\!\cdots\!08\)\(\)\()/ \)\(38\!\cdots\!80\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-\)\(11\!\cdots\!87\)\( \nu^{10} + \)\(17\!\cdots\!77\)\( \nu^{9} + \)\(68\!\cdots\!36\)\( \nu^{8} - \)\(10\!\cdots\!60\)\( \nu^{7} - \)\(12\!\cdots\!56\)\( \nu^{6} + \)\(20\!\cdots\!16\)\( \nu^{5} + \)\(68\!\cdots\!20\)\( \nu^{4} - \)\(14\!\cdots\!88\)\( \nu^{3} + \)\(13\!\cdots\!64\)\( \nu^{2} + \)\(18\!\cdots\!24\)\( \nu - \)\(27\!\cdots\!28\)\(\)\()/ \)\(30\!\cdots\!24\)\( \)
\(\beta_{5}\)\(=\)\((\)\(\)\(65\!\cdots\!73\)\( \nu^{10} + \)\(35\!\cdots\!09\)\( \nu^{9} - \)\(13\!\cdots\!04\)\( \nu^{8} - \)\(20\!\cdots\!40\)\( \nu^{7} + \)\(65\!\cdots\!48\)\( \nu^{6} + \)\(34\!\cdots\!84\)\( \nu^{5} - \)\(11\!\cdots\!96\)\( \nu^{4} + \)\(14\!\cdots\!08\)\( \nu^{3} + \)\(61\!\cdots\!92\)\( \nu^{2} - \)\(30\!\cdots\!52\)\( \nu - \)\(33\!\cdots\!80\)\(\)\()/ \)\(10\!\cdots\!08\)\( \)
\(\beta_{6}\)\(=\)\((\)\(\)\(10\!\cdots\!23\)\( \nu^{10} + \)\(15\!\cdots\!75\)\( \nu^{9} - \)\(54\!\cdots\!80\)\( \nu^{8} - \)\(12\!\cdots\!60\)\( \nu^{7} + \)\(51\!\cdots\!40\)\( \nu^{6} + \)\(30\!\cdots\!76\)\( \nu^{5} + \)\(91\!\cdots\!20\)\( \nu^{4} - \)\(28\!\cdots\!00\)\( \nu^{3} - \)\(11\!\cdots\!80\)\( \nu^{2} + \)\(74\!\cdots\!00\)\( \nu + \)\(13\!\cdots\!16\)\(\)\()/ \)\(10\!\cdots\!60\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(83\!\cdots\!47\)\( \nu^{10} + \)\(94\!\cdots\!05\)\( \nu^{9} + \)\(49\!\cdots\!20\)\( \nu^{8} - \)\(55\!\cdots\!00\)\( \nu^{7} - \)\(92\!\cdots\!60\)\( \nu^{6} + \)\(10\!\cdots\!16\)\( \nu^{5} + \)\(49\!\cdots\!00\)\( \nu^{4} - \)\(67\!\cdots\!00\)\( \nu^{3} + \)\(19\!\cdots\!80\)\( \nu^{2} + \)\(94\!\cdots\!80\)\( \nu - \)\(61\!\cdots\!84\)\(\)\()/ \)\(15\!\cdots\!20\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(17\!\cdots\!37\)\( \nu^{10} + \)\(22\!\cdots\!35\)\( \nu^{9} + \)\(10\!\cdots\!80\)\( \nu^{8} - \)\(13\!\cdots\!20\)\( \nu^{7} - \)\(20\!\cdots\!20\)\( \nu^{6} + \)\(24\!\cdots\!16\)\( \nu^{5} + \)\(12\!\cdots\!20\)\( \nu^{4} - \)\(14\!\cdots\!80\)\( \nu^{3} - \)\(44\!\cdots\!60\)\( \nu^{2} + \)\(10\!\cdots\!00\)\( \nu - \)\(60\!\cdots\!04\)\(\)\()/ \)\(15\!\cdots\!20\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(26\!\cdots\!99\)\( \nu^{10} + \)\(19\!\cdots\!03\)\( \nu^{9} - \)\(19\!\cdots\!60\)\( \nu^{8} - \)\(13\!\cdots\!08\)\( \nu^{7} + \)\(49\!\cdots\!64\)\( \nu^{6} + \)\(30\!\cdots\!00\)\( \nu^{5} - \)\(49\!\cdots\!08\)\( \nu^{4} - \)\(22\!\cdots\!88\)\( \nu^{3} + \)\(17\!\cdots\!60\)\( \nu^{2} + \)\(24\!\cdots\!52\)\( \nu - \)\(61\!\cdots\!84\)\(\)\()/ \)\(61\!\cdots\!48\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(52\!\cdots\!87\)\( \nu^{10} + \)\(31\!\cdots\!25\)\( \nu^{9} + \)\(33\!\cdots\!20\)\( \nu^{8} - \)\(17\!\cdots\!60\)\( \nu^{7} - \)\(71\!\cdots\!80\)\( \nu^{6} + \)\(27\!\cdots\!56\)\( \nu^{5} + \)\(56\!\cdots\!60\)\( \nu^{4} - \)\(13\!\cdots\!60\)\( \nu^{3} - \)\(14\!\cdots\!40\)\( \nu^{2} + \)\(35\!\cdots\!60\)\( \nu + \)\(14\!\cdots\!56\)\(\)\()/ \)\(30\!\cdots\!40\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 8 \beta_{2} + 8185 \beta_{1} + 13074496202\)
\(\nu^{3}\)\(=\)\(3 \beta_{6} + \beta_{5} - 21 \beta_{4} + 18185 \beta_{3} - 1311661 \beta_{2} + 22152168592 \beta_{1} + 107025408036041\)
\(\nu^{4}\)\(=\)\(26 \beta_{10} + 57 \beta_{9} - 259 \beta_{8} - 1245 \beta_{7} + 75517 \beta_{6} - 7678 \beta_{5} - 1361202 \beta_{4} + 30371982236 \beta_{3} - 480354168056 \beta_{2} + 457460105467504 \beta_{1} + 289630647449891501568\)
\(\nu^{5}\)\(=\)\(-522454 \beta_{10} + 820053 \beta_{9} - 12720239 \beta_{8} + 720283927 \beta_{7} + 139649806935 \beta_{6} + 36933048324 \beta_{5} - 1327607518252 \beta_{4} + 753560637844066 \beta_{3} - 60855630366386674 \beta_{2} + 568121915156930789516 \beta_{1} + 5981793113833608302335762\)
\(\nu^{6}\)\(=\)\(1021661423510 \beta_{10} + 2937492549339 \beta_{9} - 12067437892993 \beta_{8} - 56702122654215 \beta_{7} + 3999353667653553 \beta_{6} - 268065949767644 \beta_{5} - 77198548130077276 \beta_{4} + 883508615002371693974 \beta_{3} - 23210815365795341696790 \beta_{2} + 17668697105502812929772660 \beta_{1} + 7428010330323390958308094406870\)
\(\nu^{7}\)\(=\)\(-18752634392138278 \beta_{10} + 73799300071379357 \beta_{9} - 490145279946084071 \beta_{8} + 33996302684484788655 \beta_{7} + 4949852783902716882135 \beta_{6} + 1159661872458095219420 \beta_{5} - 58103888913323869738468 \beta_{4} + 27416727320958556873627082 \beta_{3} - 2354360423052530567842402746 \beta_{2} + 15606756947536761446932272706172 \beta_{1} + 231046544828472573087791995299075866\)
\(\nu^{8}\)\(=\)\(32486852636732970673782 \beta_{10} + 114221164428490093771787 \beta_{9} - 426061629225746481422993 \beta_{8} - 2283168859839947487991799 \beta_{7} + 196398526624584869964473105 \beta_{6} - 5994040405425135733041324 \beta_{5} - 3445991560717720337053167820 \beta_{4} + 25840221206009896190380469762326 \beta_{3} - 935239252234620289561216728732582 \beta_{2} + 615428713511929943939116438042825892 \beta_{1} + 204054455696223675886623500208097384934022\)
\(\nu^{9}\)\(=\)\(-502192266370756465911712806 \beta_{10} + 4078089406968561715272000861 \beta_{9} - 14743403489376313890311031975 \beta_{8} + 1216172179609502889404449781487 \beta_{7} + 163941605547670362614531500346631 \beta_{6} + 34987283308230823883945436105004 \beta_{5} - 2190441848665868052441895870138740 \beta_{4} + 961010613455547844082932008056979482 \beta_{3} - 86782786641809608283099899371816384202 \beta_{2} + 445229453454057812219849307677184287705980 \beta_{1} + 8047951093270435707192794204751458200492003946\)
\(\nu^{10}\)\(=\)\(978291485485151072137314674538518 \beta_{10} + 4022116325345494622511337504380699 \beta_{9} - 13767853546535790036840557935534657 \beta_{8} - 82228414116020784536674040624589639 \beta_{7} + 8427966488491798414563660437238791905 \beta_{6} - 71174140740058062904014112511205388 \beta_{5} - 139774381124640982435662307296331373676 \beta_{4} + 763467033768120945798546792782367667155158 \beta_{3} - 34049824014337662443661797547079883594488742 \beta_{2} + 20706992379905215668798659381192179104353827108 \beta_{1} + 5821297335996504827543449723726144881180333860744454\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−170539.
−139697.
−103767.
−99441.3
−33894.6
211.095
42069.3
62096.8
110093.
154848.
178021.
−171393. −3.66048e7 2.07856e10 0 6.27381e12 −4.77732e13 −2.09024e15 −4.21915e15 0
1.2 −140551. 1.29136e8 1.11647e10 0 −1.81502e13 4.00085e13 −3.61884e14 1.11170e16 0
1.3 −104621. 2.08087e7 2.35554e9 0 −2.17702e12 5.34780e13 6.52247e14 −5.12606e15 0
1.4 −100295. −1.18201e8 1.46921e9 0 1.18550e13 −1.07417e13 7.14175e14 8.41244e15 0
1.5 −34748.6 4.98225e7 −7.38247e9 0 −1.73126e12 −1.47813e14 5.55018e14 −3.07678e15 0
1.6 −642.905 −5.85306e7 −8.58952e9 0 3.76296e10 1.03914e14 1.10448e13 −2.13323e15 0
1.7 41215.3 1.10688e8 −6.89123e9 0 4.56203e12 7.02715e13 −6.38061e14 6.69270e15 0
1.8 61242.8 −1.16397e8 −4.83925e9 0 −7.12847e12 −1.52684e14 −8.22441e14 7.98915e15 0
1.9 109239. −1.96178e7 3.34318e9 0 −2.14302e12 7.24227e12 −5.73150e14 −5.17420e15 0
1.10 153994. 7.94763e7 1.51243e10 0 1.22389e13 −4.85062e13 1.00626e15 7.57418e14 0
1.11 177167. −1.20911e8 2.27982e10 0 −2.14214e13 1.11259e14 2.51723e15 9.06036e15 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.34.a.d 11
5.b even 2 1 25.34.a.e yes 11
5.c odd 4 2 25.34.b.d 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.34.a.d 11 1.a even 1 1 trivial
25.34.a.e yes 11 5.b even 2 1
25.34.b.d 22 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(94\!\cdots\!60\)\( T_{2}^{8} + \)\(17\!\cdots\!60\)\( T_{2}^{7} + \)\(25\!\cdots\!80\)\( T_{2}^{6} - \)\(18\!\cdots\!40\)\( T_{2}^{5} - \)\(19\!\cdots\!20\)\( T_{2}^{4} + \)\(69\!\cdots\!80\)\( T_{2}^{3} + \)\(86\!\cdots\!40\)\( T_{2}^{2} - \)\(66\!\cdots\!24\)\( T_{2} - \)\(42\!\cdots\!32\)\( \)">\(T_{2}^{11} + \cdots\) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -\)\(42\!\cdots\!32\)\( - \)\(66\!\cdots\!24\)\( T + \)\(86\!\cdots\!40\)\( T^{2} + \)\(69\!\cdots\!80\)\( T^{3} - \)\(19\!\cdots\!20\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(25\!\cdots\!80\)\( T^{6} + \)\(17\!\cdots\!60\)\( T^{7} - 944957445671160 T^{8} - 71869629370 T^{9} + 9393 T^{10} + T^{11} \)
$3$ \( -\)\(82\!\cdots\!49\)\( - \)\(19\!\cdots\!01\)\( T + \)\(26\!\cdots\!45\)\( T^{2} + \)\(53\!\cdots\!05\)\( T^{3} - \)\(16\!\cdots\!90\)\( T^{4} - \)\(29\!\cdots\!10\)\( T^{5} + \)\(36\!\cdots\!90\)\( T^{6} + \)\(53\!\cdots\!10\)\( T^{7} - \)\(29\!\cdots\!45\)\( T^{8} - 39498129452837805 T^{9} + 80330849 T^{10} + T^{11} \)
$5$ \( T^{11} \)
$7$ \( \)\(70\!\cdots\!08\)\( - \)\(47\!\cdots\!24\)\( T - \)\(90\!\cdots\!60\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(78\!\cdots\!80\)\( T^{4} - \)\(19\!\cdots\!40\)\( T^{5} - \)\(18\!\cdots\!20\)\( T^{6} + \)\(53\!\cdots\!60\)\( T^{7} + \)\(43\!\cdots\!40\)\( T^{8} - \)\(41\!\cdots\!20\)\( T^{9} + 21344025107658 T^{10} + T^{11} \)
$11$ \( \)\(13\!\cdots\!57\)\( + \)\(63\!\cdots\!39\)\( T + \)\(37\!\cdots\!15\)\( T^{2} - \)\(18\!\cdots\!35\)\( T^{3} - \)\(26\!\cdots\!90\)\( T^{4} - \)\(86\!\cdots\!62\)\( T^{5} + \)\(15\!\cdots\!66\)\( T^{6} + \)\(51\!\cdots\!30\)\( T^{7} - \)\(30\!\cdots\!95\)\( T^{8} - \)\(13\!\cdots\!05\)\( T^{9} + 194487534567113523 T^{10} + T^{11} \)
$13$ \( -\)\(12\!\cdots\!04\)\( + \)\(64\!\cdots\!24\)\( T + \)\(16\!\cdots\!20\)\( T^{2} - \)\(46\!\cdots\!20\)\( T^{3} - \)\(85\!\cdots\!40\)\( T^{4} + \)\(63\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} - \)\(18\!\cdots\!20\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + 5151173641795199804 T^{10} + T^{11} \)
$17$ \( \)\(39\!\cdots\!13\)\( - \)\(22\!\cdots\!49\)\( T - \)\(22\!\cdots\!85\)\( T^{2} + \)\(76\!\cdots\!05\)\( T^{3} + \)\(36\!\cdots\!30\)\( T^{4} - \)\(21\!\cdots\!90\)\( T^{5} + \)\(41\!\cdots\!30\)\( T^{6} + \)\(10\!\cdots\!10\)\( T^{7} - \)\(22\!\cdots\!35\)\( T^{8} - \)\(18\!\cdots\!45\)\( T^{9} + \)\(21\!\cdots\!63\)\( T^{10} + T^{11} \)
$19$ \( -\)\(26\!\cdots\!75\)\( - \)\(18\!\cdots\!25\)\( T + \)\(82\!\cdots\!75\)\( T^{2} + \)\(19\!\cdots\!25\)\( T^{3} - \)\(57\!\cdots\!50\)\( T^{4} - \)\(60\!\cdots\!50\)\( T^{5} + \)\(12\!\cdots\!50\)\( T^{6} + \)\(73\!\cdots\!50\)\( T^{7} - \)\(98\!\cdots\!75\)\( T^{8} - \)\(43\!\cdots\!25\)\( T^{9} + \)\(23\!\cdots\!15\)\( T^{10} + T^{11} \)
$23$ \( -\)\(46\!\cdots\!84\)\( + \)\(64\!\cdots\!24\)\( T - \)\(23\!\cdots\!80\)\( T^{2} + \)\(28\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!60\)\( T^{4} - \)\(20\!\cdots\!60\)\( T^{5} + \)\(36\!\cdots\!40\)\( T^{6} + \)\(43\!\cdots\!60\)\( T^{7} - \)\(83\!\cdots\!20\)\( T^{8} - \)\(33\!\cdots\!80\)\( T^{9} + \)\(39\!\cdots\!34\)\( T^{10} + T^{11} \)
$29$ \( -\)\(16\!\cdots\!00\)\( - \)\(65\!\cdots\!00\)\( T - \)\(42\!\cdots\!00\)\( T^{2} - \)\(18\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!00\)\( T^{4} - \)\(90\!\cdots\!00\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(62\!\cdots\!00\)\( T^{7} + \)\(24\!\cdots\!00\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} - \)\(16\!\cdots\!40\)\( T^{10} + T^{11} \)
$31$ \( -\)\(24\!\cdots\!08\)\( - \)\(46\!\cdots\!36\)\( T + \)\(23\!\cdots\!40\)\( T^{2} + \)\(78\!\cdots\!40\)\( T^{3} - \)\(58\!\cdots\!40\)\( T^{4} - \)\(23\!\cdots\!72\)\( T^{5} + \)\(48\!\cdots\!16\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} - \)\(15\!\cdots\!20\)\( T^{8} - \)\(83\!\cdots\!80\)\( T^{9} + \)\(18\!\cdots\!38\)\( T^{10} + T^{11} \)
$37$ \( \)\(23\!\cdots\!48\)\( + \)\(13\!\cdots\!76\)\( T - \)\(24\!\cdots\!60\)\( T^{2} - \)\(11\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!80\)\( T^{4} + \)\(25\!\cdots\!60\)\( T^{5} + \)\(20\!\cdots\!80\)\( T^{6} - \)\(15\!\cdots\!40\)\( T^{7} - \)\(24\!\cdots\!60\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{9} + \)\(18\!\cdots\!98\)\( T^{10} + T^{11} \)
$41$ \( -\)\(20\!\cdots\!53\)\( - \)\(10\!\cdots\!61\)\( T + \)\(13\!\cdots\!65\)\( T^{2} - \)\(29\!\cdots\!35\)\( T^{3} + \)\(14\!\cdots\!10\)\( T^{4} + \)\(12\!\cdots\!98\)\( T^{5} - \)\(11\!\cdots\!34\)\( T^{6} - \)\(22\!\cdots\!70\)\( T^{7} + \)\(19\!\cdots\!55\)\( T^{8} - \)\(30\!\cdots\!05\)\( T^{9} - \)\(54\!\cdots\!67\)\( T^{10} + T^{11} \)
$43$ \( \)\(18\!\cdots\!56\)\( + \)\(17\!\cdots\!24\)\( T - \)\(13\!\cdots\!80\)\( T^{2} + \)\(63\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!60\)\( T^{4} - \)\(95\!\cdots\!60\)\( T^{5} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(50\!\cdots\!60\)\( T^{7} + \)\(97\!\cdots\!80\)\( T^{8} - \)\(43\!\cdots\!80\)\( T^{9} + \)\(50\!\cdots\!44\)\( T^{10} + T^{11} \)
$47$ \( -\)\(24\!\cdots\!72\)\( + \)\(10\!\cdots\!76\)\( T + \)\(20\!\cdots\!40\)\( T^{2} - \)\(63\!\cdots\!20\)\( T^{3} - \)\(17\!\cdots\!20\)\( T^{4} + \)\(54\!\cdots\!60\)\( T^{5} + \)\(21\!\cdots\!80\)\( T^{6} - \)\(61\!\cdots\!40\)\( T^{7} - \)\(88\!\cdots\!60\)\( T^{8} - \)\(49\!\cdots\!20\)\( T^{9} + \)\(10\!\cdots\!28\)\( T^{10} + T^{11} \)
$53$ \( -\)\(26\!\cdots\!24\)\( - \)\(40\!\cdots\!76\)\( T + \)\(16\!\cdots\!20\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} - \)\(17\!\cdots\!40\)\( T^{4} - \)\(10\!\cdots\!60\)\( T^{5} + \)\(61\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!60\)\( T^{7} - \)\(79\!\cdots\!20\)\( T^{8} - \)\(29\!\cdots\!80\)\( T^{9} + \)\(33\!\cdots\!74\)\( T^{10} + T^{11} \)
$59$ \( \)\(23\!\cdots\!00\)\( - \)\(81\!\cdots\!00\)\( T - \)\(16\!\cdots\!00\)\( T^{2} + \)\(67\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!00\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!00\)\( T^{7} - \)\(94\!\cdots\!00\)\( T^{8} - \)\(15\!\cdots\!00\)\( T^{9} + \)\(67\!\cdots\!20\)\( T^{10} + T^{11} \)
$61$ \( \)\(26\!\cdots\!32\)\( - \)\(63\!\cdots\!36\)\( T + \)\(16\!\cdots\!40\)\( T^{2} + \)\(40\!\cdots\!40\)\( T^{3} - \)\(13\!\cdots\!40\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(23\!\cdots\!16\)\( T^{6} + \)\(11\!\cdots\!80\)\( T^{7} - \)\(15\!\cdots\!20\)\( T^{8} - \)\(55\!\cdots\!80\)\( T^{9} + \)\(34\!\cdots\!98\)\( T^{10} + T^{11} \)
$67$ \( \)\(46\!\cdots\!13\)\( - \)\(48\!\cdots\!49\)\( T + \)\(19\!\cdots\!15\)\( T^{2} - \)\(39\!\cdots\!95\)\( T^{3} + \)\(32\!\cdots\!30\)\( T^{4} + \)\(18\!\cdots\!10\)\( T^{5} - \)\(18\!\cdots\!70\)\( T^{6} + \)\(63\!\cdots\!10\)\( T^{7} + \)\(31\!\cdots\!65\)\( T^{8} - \)\(15\!\cdots\!45\)\( T^{9} - \)\(16\!\cdots\!37\)\( T^{10} + T^{11} \)
$71$ \( -\)\(61\!\cdots\!88\)\( + \)\(82\!\cdots\!64\)\( T + \)\(70\!\cdots\!40\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} - \)\(29\!\cdots\!40\)\( T^{4} - \)\(12\!\cdots\!92\)\( T^{5} + \)\(22\!\cdots\!16\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} - \)\(62\!\cdots\!20\)\( T^{8} - \)\(99\!\cdots\!80\)\( T^{9} + \)\(55\!\cdots\!68\)\( T^{10} + T^{11} \)
$73$ \( -\)\(80\!\cdots\!09\)\( + \)\(54\!\cdots\!99\)\( T + \)\(70\!\cdots\!45\)\( T^{2} + \)\(17\!\cdots\!05\)\( T^{3} - \)\(13\!\cdots\!90\)\( T^{4} - \)\(45\!\cdots\!10\)\( T^{5} + \)\(81\!\cdots\!90\)\( T^{6} + \)\(33\!\cdots\!10\)\( T^{7} - \)\(18\!\cdots\!45\)\( T^{8} - \)\(98\!\cdots\!05\)\( T^{9} + \)\(12\!\cdots\!09\)\( T^{10} + T^{11} \)
$79$ \( -\)\(92\!\cdots\!00\)\( + \)\(59\!\cdots\!00\)\( T - \)\(94\!\cdots\!00\)\( T^{2} - \)\(27\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!00\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{5} - \)\(38\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!00\)\( T^{7} + \)\(56\!\cdots\!00\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} - \)\(28\!\cdots\!90\)\( T^{10} + T^{11} \)
$83$ \( \)\(17\!\cdots\!61\)\( - \)\(28\!\cdots\!01\)\( T - \)\(15\!\cdots\!05\)\( T^{2} - \)\(10\!\cdots\!95\)\( T^{3} + \)\(31\!\cdots\!10\)\( T^{4} + \)\(40\!\cdots\!90\)\( T^{5} + \)\(39\!\cdots\!90\)\( T^{6} - \)\(34\!\cdots\!90\)\( T^{7} - \)\(88\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!05\)\( T^{9} + \)\(12\!\cdots\!39\)\( T^{10} + T^{11} \)
$89$ \( -\)\(12\!\cdots\!75\)\( - \)\(28\!\cdots\!25\)\( T + \)\(65\!\cdots\!75\)\( T^{2} + \)\(12\!\cdots\!25\)\( T^{3} - \)\(77\!\cdots\!50\)\( T^{4} - \)\(14\!\cdots\!50\)\( T^{5} + \)\(35\!\cdots\!50\)\( T^{6} + \)\(67\!\cdots\!50\)\( T^{7} - \)\(62\!\cdots\!75\)\( T^{8} - \)\(13\!\cdots\!25\)\( T^{9} + \)\(24\!\cdots\!55\)\( T^{10} + T^{11} \)
$97$ \( -\)\(85\!\cdots\!72\)\( + \)\(13\!\cdots\!76\)\( T - \)\(49\!\cdots\!60\)\( T^{2} - \)\(22\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!80\)\( T^{4} - \)\(46\!\cdots\!40\)\( T^{5} - \)\(82\!\cdots\!20\)\( T^{6} + \)\(73\!\cdots\!60\)\( T^{7} + \)\(14\!\cdots\!40\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} - \)\(67\!\cdots\!22\)\( T^{10} + T^{11} \)
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