Properties

Label 10.19.c.a
Level 10
Weight 19
Character orbit 10.c
Analytic conductor 20.539
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(20.5386137710\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{4}\cdot 5^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 256 - 256 \beta_{1} ) q^{2} + ( 4628 + 4628 \beta_{1} - \beta_{2} ) q^{3} -131072 \beta_{1} q^{4} + ( -391556 + 214952 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{5} + ( 2369536 - 256 \beta_{2} - 256 \beta_{3} ) q^{6} + ( -5447327 + 5447324 \beta_{1} + 3 \beta_{2} + 594 \beta_{3} + 2 \beta_{4} - 23 \beta_{5} + 8 \beta_{6} + 11 \beta_{7} ) q^{7} + ( -33554432 - 33554432 \beta_{1} ) q^{8} + ( -47 + 107075815 \beta_{1} - 13066 \beta_{2} + 13160 \beta_{3} - 190 \beta_{4} - 92 \beta_{5} - 235 \beta_{6} + 141 \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 256 - 256 \beta_{1} ) q^{2} + ( 4628 + 4628 \beta_{1} - \beta_{2} ) q^{3} -131072 \beta_{1} q^{4} + ( -391556 + 214952 \beta_{1} - 7 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{5} + ( 2369536 - 256 \beta_{2} - 256 \beta_{3} ) q^{6} + ( -5447327 + 5447324 \beta_{1} + 3 \beta_{2} + 594 \beta_{3} + 2 \beta_{4} - 23 \beta_{5} + 8 \beta_{6} + 11 \beta_{7} ) q^{7} + ( -33554432 - 33554432 \beta_{1} ) q^{8} + ( -47 + 107075815 \beta_{1} - 13066 \beta_{2} + 13160 \beta_{3} - 190 \beta_{4} - 92 \beta_{5} - 235 \beta_{6} + 141 \beta_{7} ) q^{9} + ( -45210624 + 155266048 \beta_{1} - 3328 \beta_{2} - 256 \beta_{3} - 768 \beta_{4} + 256 \beta_{7} ) q^{10} + ( -334817119 - 91 \beta_{1} - 4433 \beta_{2} - 4615 \beta_{3} - 273 \beta_{4} - 455 \beta_{5} - 561 \beta_{6} - 1107 \beta_{7} ) q^{11} + ( 606601216 - 606601216 \beta_{1} - 131072 \beta_{3} ) q^{12} + ( -1702309428 - 1702310609 \beta_{1} + 235809 \beta_{2} - 1181 \beta_{3} + 7288 \beta_{4} - 8469 \beta_{5} - 2564 \beta_{6} - 10831 \beta_{7} ) q^{13} + ( -768 + 2789030656 \beta_{1} - 151296 \beta_{2} + 152832 \beta_{3} + 3328 \beta_{4} - 7936 \beta_{5} - 3840 \beta_{6} + 2304 \beta_{7} ) q^{14} + ( -5612243511 + 2635455932 \beta_{1} + 1378608 \beta_{2} + 279821 \beta_{3} + 4058 \beta_{4} - 34375 \beta_{5} - 28001 \beta_{7} ) q^{15} -17179869184 q^{16} + ( 16422700468 - 16422704174 \beta_{1} + 3706 \beta_{2} - 1996446 \beta_{3} - 51323 \beta_{4} + 25381 \beta_{5} - 43911 \beta_{6} - 40205 \beta_{7} ) q^{17} + ( 27411396608 + 27411420672 \beta_{1} - 6713856 \beta_{2} + 24064 \beta_{3} - 12544 \beta_{4} + 36608 \beta_{5} - 83712 \beta_{6} + 84736 \beta_{7} ) q^{18} + ( -23095 + 45434292615 \beta_{1} - 3479942 \beta_{2} + 3526132 \beta_{3} - 238123 \beta_{4} + 99553 \beta_{5} - 115475 \beta_{6} + 69285 \beta_{7} ) q^{19} + ( 28174188544 + 51322028032 \beta_{1} - 786432 \beta_{2} - 917504 \beta_{3} - 131072 \beta_{4} + 262144 \beta_{7} ) q^{20} + ( -314920186407 - 31504 \beta_{1} + 20086466 \beta_{2} + 20023458 \beta_{3} - 94512 \beta_{4} - 157520 \beta_{5} - 220001 \beta_{6} - 409025 \beta_{7} ) q^{21} + ( -85713205760 + 85713159168 \beta_{1} + 46592 \beta_{2} - 2316288 \beta_{3} - 353280 \beta_{4} + 27136 \beta_{5} - 260096 \beta_{6} - 213504 \beta_{7} ) q^{22} + ( 374094347716 + 374094463613 \beta_{1} - 13816110 \beta_{2} + 115897 \beta_{3} + 792893 \beta_{4} - 676996 \beta_{5} - 1256481 \beta_{6} - 445202 \beta_{7} ) q^{23} + ( -310579822592 \beta_{1} + 33554432 \beta_{2} - 33554432 \beta_{3} ) q^{24} + ( 628165567700 - 535014240150 \beta_{1} + 38606900 \beta_{2} - 123019200 \beta_{3} - 585850 \beta_{4} - 250000 \beta_{5} - 2921875 \beta_{6} - 1465925 \beta_{7} ) q^{25} + ( -871582729472 - 302336 \beta_{1} + 60669440 \beta_{2} + 60064768 \beta_{3} - 907008 \beta_{4} - 1511680 \beta_{5} - 2824448 \beta_{6} - 4638464 \beta_{7} ) q^{26} + ( -4603421271864 + 4603419157656 \beta_{1} + 2114208 \beta_{2} + 163984236 \beta_{3} - 10291284 \beta_{4} - 4508172 \beta_{5} - 6062868 \beta_{6} - 3948660 \beta_{7} ) q^{27} + ( 713991651328 + 713992044544 \beta_{1} - 77856768 \beta_{2} + 393216 \beta_{3} + 1441792 \beta_{4} - 1048576 \beta_{5} - 3014656 \beta_{6} - 262144 \beta_{7} ) q^{28} + ( -2653849 - 1623221232455 \beta_{1} - 137353364 \beta_{2} + 142661062 \beta_{3} - 4575417 \beta_{4} - 11347677 \beta_{5} - 13269245 \beta_{6} + 7961547 \beta_{7} ) q^{29} + ( -762057620224 + 2111411057408 \beta_{1} + 281289472 \beta_{2} + 424557824 \beta_{3} - 6129408 \beta_{4} - 8800000 \beta_{5} - 8800000 \beta_{6} - 8207104 \beta_{7} ) q^{30} + ( -5161641191241 - 718117 \beta_{1} - 639844787 \beta_{2} - 641281021 \beta_{3} - 2154351 \beta_{4} - 3590585 \beta_{5} - 9779131 \beta_{6} - 14087833 \beta_{7} ) q^{31} + ( -4398046511104 + 4398046511104 \beta_{1} ) q^{32} + ( 465381710111 + 465378984211 \beta_{1} + 541075372 \beta_{2} - 2725900 \beta_{3} + 3991723 \beta_{4} - 6717623 \beta_{5} + 6911877 \beta_{6} - 12169423 \beta_{7} ) q^{33} + ( -948736 - 8408423588352 \beta_{1} + 512038912 \beta_{2} - 510141440 \beta_{3} - 23431168 \beta_{4} + 17738752 \beta_{5} - 4743680 \beta_{6} + 2846208 \beta_{7} ) q^{34} + ( 32378352797063 - 4383768705751 \beta_{1} + 1487540586 \beta_{2} - 439826453 \beta_{3} + 9793091 \beta_{4} + 31859375 \beta_{5} + 28721875 \beta_{6} + 10897403 \beta_{7} ) q^{35} + ( 14034641223680 + 6160384 \beta_{1} - 1724907520 \beta_{2} - 1712586752 \beta_{3} + 18481152 \beta_{4} + 30801920 \beta_{5} - 12058624 \beta_{6} + 24903680 \beta_{7} ) q^{36} + ( -23964174398016 + 23964192233473 \beta_{1} - 17835457 \beta_{2} + 6140358195 \beta_{3} + 90442948 \beta_{4} + 34405251 \beta_{5} + 54772034 \beta_{6} + 36936577 \beta_{7} ) q^{37} + ( 11631172997120 + 11631184821760 \beta_{1} - 1793554944 \beta_{2} + 11824640 \beta_{3} - 43222528 \beta_{4} + 55047168 \beta_{5} - 4076032 \beta_{6} + 78696448 \beta_{7} ) q^{38} + ( 35196935 - 122988455378815 \beta_{1} + 5132889319 \beta_{2} - 5203283189 \beta_{3} + 160720135 \beta_{4} + 50461475 \beta_{5} + 175984675 \beta_{6} - 105590805 \beta_{7} ) q^{39} + ( 20351031443456 + 5925846908928 \beta_{1} + 33554432 \beta_{2} - 436207616 \beta_{3} + 33554432 \beta_{4} + 100663296 \beta_{7} ) q^{40} + ( 67319177599498 - 9303643 \beta_{1} - 4550444840 \beta_{2} - 4569052126 \beta_{3} - 27910929 \beta_{4} - 46518215 \beta_{5} + 374634818 \beta_{6} + 318812960 \beta_{7} ) q^{41} + ( -80619575785216 + 80619559655168 \beta_{1} + 16130048 \beta_{2} + 10268140544 \beta_{3} - 128905472 \beta_{4} + 15995136 \beta_{5} - 96645376 \beta_{6} - 80515328 \beta_{7} ) q^{42} + ( 34764406389936 + 34764367853486 \beta_{1} - 30632547111 \beta_{2} - 38536450 \beta_{3} + 1195706 \beta_{4} - 39732156 \beta_{5} + 152950094 \beta_{6} - 116805056 \beta_{7} ) q^{43} + ( -11927552 + 43885149421568 \beta_{1} + 604897280 \beta_{2} - 581042176 \beta_{3} - 145096704 \beta_{4} + 73531392 \beta_{5} - 59637760 \beta_{6} + 35782656 \beta_{7} ) q^{44} + ( -81183954675028 - 483349229574809 \beta_{1} + 34644085934 \beta_{2} - 8585475227 \beta_{3} + 188381814 \beta_{4} - 135840625 \beta_{5} + 500543750 \beta_{6} - 36096178 \beta_{7} ) q^{45} + ( 191536335700224 + 29669632 \beta_{1} - 3566593792 \beta_{2} - 3507254528 \beta_{3} + 89008896 \beta_{4} + 148348160 \beta_{5} - 494970112 \beta_{6} - 316952320 \beta_{7} ) q^{46} + ( -79285730547501 + 79285891003280 \beta_{1} - 160455779 \beta_{2} + 14512801512 \beta_{3} + 1119228014 \beta_{4} + 3962439 \beta_{5} + 798316456 \beta_{6} + 637860677 \beta_{7} ) q^{47} + ( -79508434583552 - 79508434583552 \beta_{1} + 17179869184 \beta_{2} ) q^{48} + ( 62788601 + 692686690732659 \beta_{1} + 1388837470 \beta_{2} - 1514414672 \beta_{3} + 780207140 \beta_{4} - 403475534 \beta_{5} + 313943005 \beta_{6} - 188365803 \beta_{7} ) q^{49} + ( 23846739852800 - 297774030809600 \beta_{1} + 41376281600 \beta_{2} - 21609548800 \beta_{3} - 525254400 \beta_{4} + 684000000 \beta_{5} - 812000000 \beta_{6} - 225299200 \beta_{7} ) q^{50} + ( 1048134911772924 - 15709768 \beta_{1} - 32183768977 \beta_{2} - 32215188513 \beta_{3} - 47129304 \beta_{4} - 78548840 \beta_{5} + 110840884 \beta_{6} + 16582276 \beta_{7} ) q^{51} + ( -223125256142848 + 223125101346816 \beta_{1} + 154796032 \beta_{2} + 30907957248 \beta_{3} - 1419640832 \beta_{4} + 336068608 \beta_{5} - 1110048768 \beta_{6} - 955252736 \beta_{7} ) q^{52} + ( -1623258349980271 - 1623258517272404 \beta_{1} - 34387845627 \beta_{2} - 167292133 \beta_{3} - 339974045 \beta_{4} + 172681912 \beta_{5} + 1009142577 \beta_{6} - 161902354 \beta_{7} ) q^{53} + ( -541237248 + 2356951149957120 \beta_{1} - 41438727168 \beta_{2} + 42521201664 \beta_{3} - 3645425664 \beta_{4} + 398002176 \beta_{5} - 2706186240 \beta_{6} + 1623711744 \beta_{7} ) q^{54} + ( 882683344686868 - 954217240108206 \beta_{1} - 18709312829 \beta_{2} - 17506453193 \beta_{3} + 462717206 \beta_{4} - 1147025000 \beta_{5} - 663618750 \beta_{6} - 488346472 \beta_{7} ) q^{55} + ( 365563826143232 + 100663296 \beta_{1} - 20031995904 \beta_{2} - 19830669312 \beta_{3} + 301989888 \beta_{4} + 503316480 \beta_{5} - 1040187392 \beta_{6} - 436207616 \beta_{7} ) q^{56} + ( -1795570922231848 + 1795569876888518 \beta_{1} + 1045343330 \beta_{2} + 185194514462 \beta_{3} - 4503761078 \beta_{4} - 2813642232 \beta_{5} - 2413074418 \beta_{6} - 1367731088 \beta_{7} ) q^{57} + ( -415545314893824 - 415543956123136 \beta_{1} - 71683693056 \beta_{2} + 1358770688 \beta_{3} + 866849280 \beta_{4} + 491921408 \beta_{5} - 6301932032 \beta_{6} + 3209462784 \beta_{7} ) q^{58} + ( 311356011 + 2709998205405493 \beta_{1} + 226694562330 \beta_{2} - 227317274352 \beta_{3} + 4012536855 \beta_{4} - 2144400789 \beta_{5} + 1556780055 \beta_{6} - 934068033 \beta_{7} ) q^{59} + ( 345434479919104 + 735607981473792 \beta_{1} - 36676698112 \beta_{2} + 180696907776 \beta_{3} - 3670147072 \beta_{4} - 4505600000 \beta_{6} - 531890176 \beta_{7} ) q^{60} + ( 754106774846182 - 908290947 \beta_{1} + 93010238268 \beta_{2} + 91193656374 \beta_{3} - 2724872841 \beta_{4} - 4541454735 \beta_{5} - 2955041762 \beta_{6} - 8404787444 \beta_{7} ) q^{61} + ( -1321380328795648 + 1321379961119744 \beta_{1} + 367675904 \beta_{2} - 327968206848 \beta_{3} - 4157999104 \beta_{4} + 1584267776 \beta_{5} - 3422647296 \beta_{6} - 3054971392 \beta_{7} ) q^{62} + ( -8417348828528600 - 8417350692738843 \beta_{1} + 518002084686 \beta_{2} - 1864210243 \beta_{3} + 6626667253 \beta_{4} - 8490877496 \beta_{5} + 830173719 \beta_{6} - 12219297982 \beta_{7} ) q^{63} + 2251799813685248 \beta_{1} q^{64} + ( 14578565252603709 - 4303746736474758 \beta_{1} - 732717532052 \beta_{2} + 51199142876 \beta_{3} + 4662803723 \beta_{4} - 4128615625 \beta_{5} + 6156281250 \beta_{6} + 2585071944 \beta_{7} ) q^{65} + ( 238274737746432 - 697830400 \beta_{1} + 139213125632 \beta_{2} + 137817464832 \beta_{3} - 2093491200 \beta_{4} - 3489152000 \beta_{5} + 49729024 \beta_{6} - 4137253376 \beta_{7} ) q^{66} + ( -5389276310888638 + 5389279743988932 \beta_{1} - 3433100294 \beta_{2} - 889396267431 \beta_{3} + 22250327324 \beta_{4} + 1781374734 \beta_{5} + 15384126736 \beta_{6} + 11951026442 \beta_{7} ) q^{67} + ( -2152556681494528 - 2152556195741696 \beta_{1} + 261678170112 \beta_{2} + 485752832 \beta_{3} - 5269749760 \beta_{4} + 5755502592 \beta_{5} + 3326738432 \beta_{6} + 6727008256 \beta_{7} ) q^{68} + ( -68943168 + 9486325774406141 \beta_{1} - 933756875714 \beta_{2} + 933894762050 \beta_{3} - 14087946267 \beta_{4} + 13674287259 \beta_{5} - 344715840 \beta_{6} + 206829504 \beta_{7} ) q^{69} + ( 7166613527375872 - 9411103104720384 \beta_{1} + 493405961984 \beta_{2} + 268214818048 \beta_{3} + 5296766464 \beta_{4} + 803200000 \beta_{5} + 15508800000 \beta_{6} + 282703872 \beta_{7} ) q^{70} + ( -2585089106576369 + 5667877403 \beta_{1} + 656940938677 \beta_{2} + 668276693483 \beta_{3} + 17003632209 \beta_{4} + 28339387015 \beta_{5} + 27259206909 \beta_{6} + 61266471327 \beta_{7} ) q^{71} + ( 3592869730320384 - 3592866576203776 \beta_{1} - 3154116608 \beta_{2} - 879998533632 \beta_{3} + 11106516992 \beta_{4} + 10972299264 \beta_{5} + 4798283776 \beta_{6} + 1644167168 \beta_{7} ) q^{72} + ( -10738169723341325 - 10738167458531107 \beta_{1} - 743131237230 \beta_{2} + 2264810218 \beta_{3} - 36827869604 \beta_{4} + 39092679822 \beta_{5} + 27768628732 \beta_{6} + 43622300258 \beta_{7} ) q^{73} + ( 4565876992 + 12269661857661184 \beta_{1} - 1576497574912 \beta_{2} + 1567365820928 \beta_{3} + 32609158400 \beta_{4} - 5213896448 \beta_{5} + 22829384960 \beta_{6} - 13697630976 \beta_{7} ) q^{74} + ( 60905208476518450 - 17224637495895400 \beta_{1} - 1854263160975 \beta_{2} - 1114846722450 \beta_{3} - 3846703100 \beta_{4} + 35880843750 \beta_{5} + 4810437500 \beta_{6} + 2836457950 \beta_{7} ) q^{75} + ( 5955163601633280 + 3027107840 \beta_{1} - 462177173504 \beta_{2} - 456122957824 \beta_{3} + 9081323520 \beta_{4} + 15135539200 \beta_{5} + 13048610816 \beta_{6} + 31211257856 \beta_{7} ) q^{76} + ( -3880417925685831 + 3880420776167989 \beta_{1} - 2850482158 \beta_{2} - 754044268158 \beta_{3} - 1730947455 \beta_{4} + 21684322561 \beta_{5} - 7431911771 \beta_{6} - 10282393929 \beta_{7} ) q^{77} + ( -31485035566561280 - 31485053587392000 \beta_{1} + 2646060162048 \beta_{2} - 18020830720 \beta_{3} + 14113108480 \beta_{4} - 32133939200 \beta_{5} + 57970214400 \beta_{6} - 68175600640 \beta_{7} ) q^{78} + ( 3714094548 + 26756841683550368 \beta_{1} + 3732136236996 \beta_{2} - 3739564426092 \beta_{3} + 49468564792 \beta_{4} - 27183997504 \beta_{5} + 18570472740 \beta_{6} - 11142283644 \beta_{7} ) q^{79} + ( 6726880858210304 - 3692847240839168 \beta_{1} + 120259084288 \beta_{2} - 103079215104 \beta_{3} + 34359738368 \beta_{4} + 17179869184 \beta_{7} ) q^{80} + ( -74755438837939545 - 14888956629 \beta_{1} + 5354250535020 \beta_{2} + 5324472621762 \beta_{3} - 44666869887 \beta_{4} - 74444783145 \beta_{5} - 76252865748 \beta_{6} - 165586605522 \beta_{7} ) q^{81} + ( 17233707083738880 - 17233711847204096 \beta_{1} + 4763465216 \beta_{2} - 2334591223296 \beta_{3} + 74470919936 \beta_{4} - 107815176448 \beta_{5} + 83997850368 \beta_{6} + 88761315584 \beta_{7} ) q^{82} + ( 30888982467726056 + 30889006763834672 \beta_{1} - 2850453941517 \beta_{2} + 24296108616 \beta_{3} + 82694360224 \beta_{4} - 58398251608 \beta_{5} - 179878794688 \beta_{6} - 9806034376 \beta_{7} ) q^{83} + ( -4129292288 + 41277218672738304 \beta_{1} - 2624514686976 \beta_{2} + 2632773271552 \beta_{3} - 53611724800 \beta_{4} + 28835971072 \beta_{5} - 20646461440 \beta_{6} + 12387876864 \beta_{7} ) q^{84} + ( -33213514587404607 - 44439751706289361 \beta_{1} - 1986875509779 \beta_{2} + 633885495717 \beta_{3} - 95769405174 \beta_{4} - 47217234375 \beta_{5} - 130151628125 \beta_{6} - 30679064042 \beta_{7} ) q^{85} + ( 17799366206316032 - 9865331200 \beta_{1} - 7832066729216 \beta_{2} - 7851797391616 \beta_{3} - 29595993600 \beta_{4} - 49326656000 \beta_{5} + 28983792128 \beta_{6} - 30208195072 \beta_{7} ) q^{86} + ( -53907867938596286 + 53907817022061004 \beta_{1} + 50916535282 \beta_{2} + 14495519107738 \beta_{3} - 364502382304 \beta_{4} + 8086635330 \beta_{5} - 262669311740 \beta_{6} - 211752776458 \beta_{7} ) q^{87} + ( 11234595198468096 + 11234601305374720 \beta_{1} + 303600500736 \beta_{2} + 6106906624 \beta_{3} - 27984396288 \beta_{4} + 34091302912 \beta_{5} + 3556769792 \beta_{6} + 46305116160 \beta_{7} ) q^{88} + ( -55546670654 + 127997543338785582 \beta_{1} + 6060348941912 \beta_{2} - 5949255600604 \beta_{3} - 355104677134 \beta_{4} + 21824653210 \beta_{5} - 277733353270 \beta_{6} + 166640011962 \beta_{7} ) q^{89} + ( -144520495167958272 - 102954310374343936 \beta_{1} + 11066767657216 \beta_{2} + 6671004340992 \beta_{3} + 38985122816 \beta_{4} - 162914400000 \beta_{5} + 93364000000 \beta_{6} - 57466365952 \beta_{7} ) q^{90} + ( -43379324219279126 + 22053684822 \beta_{1} - 11536693593009 \beta_{2} - 11492586223365 \beta_{3} + 66161054466 \beta_{4} + 110268424110 \beta_{5} - 113634214538 \beta_{6} + 18687894394 \beta_{7} ) q^{91} + ( 49033309534683136 - 49033294343831552 \beta_{1} - 15190851584 \beta_{2} - 1810905169920 \beta_{3} - 58353516544 \beta_{4} + 164689477632 \beta_{5} - 88735219712 \beta_{6} - 103926071296 \beta_{7} ) q^{92} + ( 264693570328903357 + 264693595216876265 \beta_{1} - 4535399919952 \beta_{2} + 24887972908 \beta_{3} - 13980075823 \beta_{4} + 38868048731 \beta_{5} - 85571815809 \beta_{6} + 88643994547 \beta_{7} ) q^{93} + ( 41076679424 + 40594335116999936 \beta_{1} - 3756353866496 \beta_{2} + 3674200507648 \beta_{3} + 449814704896 \beta_{4} - 203354628352 \beta_{5} + 205383397120 \beta_{6} - 123230038272 \beta_{7} ) q^{94} + ( -207343687131131485 - 212618745145030255 \beta_{1} + 11799465348330 \beta_{2} - 8247856966140 \beta_{3} - 40122906245 \beta_{4} + 14424284375 \beta_{5} - 68188878125 \beta_{6} - 46217370185 \beta_{7} ) q^{95} + ( -40708318506778624 + 4398046511104 \beta_{2} + 4398046511104 \beta_{3} ) q^{96} + ( 238131280543519907 - 238131201262298587 \beta_{1} - 79281221320 \beta_{2} + 17500775869068 \beta_{3} + 533613821406 \beta_{4} + 21354727834 \beta_{5} + 375051378766 \beta_{6} + 295770157446 \beta_{7} ) q^{97} + ( 177327808901442560 + 177327776753678848 \beta_{1} + 743232548352 \beta_{2} - 32147763712 \beta_{3} + 151511382272 \beta_{4} - 183659145984 \beta_{5} - 22920327424 \beta_{6} - 247954673408 \beta_{7} ) q^{98} + ( 20763628727 - 109588005375521167 \beta_{1} + 11780325173125 \beta_{2} - 11821852430579 \beta_{3} - 57775997477 \beta_{4} + 182357769839 \beta_{5} + 103818143635 \beta_{6} - 62290886181 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2048q^{2} + 37026q^{3} - 3132450q^{5} + 18957312q^{6} - 43579766q^{7} - 268435456q^{8} + O(q^{10}) \) \( 8q + 2048q^{2} + 37026q^{3} - 3132450q^{5} + 18957312q^{6} - 43579766q^{7} - 268435456q^{8} - 361676800q^{10} - 2678523284q^{11} + 4853071872q^{12} - 13618988004q^{13} - 44901376950q^{15} - 137438953472q^{16} + 131385428404q^{17} + 219304891392q^{18} + 225397964800q^{20} - 2519443347204q^{21} - 685701960704q^{22} + 2992780401346q^{23} + 5025487502500q^{25} - 6972921858048q^{26} - 36827718166440q^{27} + 5712087089152q^{28} - 6097905484800q^{30} - 41290623629644q^{31} - 35184372088832q^{32} + 3721928304252q^{33} + 259024770537850q^{35} + 112284104392704q^{36} - 191725492483296q^{37} + 93053262223360q^{38} + 162809459507200q^{40} + 538572935041756q^{41} - 644977496884224q^{42} + 278176126066386q^{43} - 649523899006350q^{45} + 1532303565489152q^{46} - 634311997628766q^{47} - 636101836406784q^{48} + 190733484160000q^{50} + 8385208158427476q^{51} - 1785067995660288q^{52} - 12985998337176064q^{53} + 7061537235641100q^{55} + 2924588589645824q^{56} - 14364945328494720q^{57} - 3324209031454720q^{58} + 2763185671372800q^{60} + 6032452171830396q^{61} - 10570399649188864q^{62} - 67339871781169614q^{63} + 116629895397895800q^{65} + 1905627291777024q^{66} - 43112377024267886q^{67} - 17220950871769088q^{68} + 57331386108262400q^{70} - 20683118221989964q^{71} + 28744730724532224q^{72} - 85903701564675544q^{73} + 487247617377746250q^{75} + 47643270258360320q^{76} - 31041870745561732q^{77} - 251885813313555456q^{78} + 53815081225420800q^{80} - 598065530496252012q^{81} + 137874671370689536q^{82} + 247117472833336746q^{83} - 265705533435464900q^{85} + 142426176545989632q^{86} - 431292883391162160q^{87} + 89876327393394688q^{88} - 1156199666753126400q^{90} - 346988460443022684q^{91} + 392269712765222912q^{92} + 2117557938231099132q^{93} - 1658756785135297000q^{95} - 325684140240273408q^{96} + 1905016584439493544q^{97} + 1418620057223277568q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 1012730 x^{6} + 266454825649 x^{4} + 6118030786524900 x^{2} + 24596382356676000000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -4010593 \nu^{7} - 4062887717390 \nu^{5} - 1049379255348757357 \nu^{3} - 14471383943831866889700 \nu \)\()/ \)\(65\!\cdots\!00\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-11320303741070839 \nu^{7} + 141003731475769500 \nu^{6} - 11253809127845515042220 \nu^{5} + 195830658168495372360000 \nu^{4} - 2858337794848621259820440761 \nu^{3} + 70585302822734554593917080500 \nu^{2} - 43887891498044213734761305063100 \nu + 2473206518249494196636747622420000\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{3}\)\(=\)\((\)\(11320303741070839 \nu^{7} + 141003731475769500 \nu^{6} + 11253809127845515042220 \nu^{5} + 195830658168495372360000 \nu^{4} + 2858337794848621259820440761 \nu^{3} + 70585302822734554593917080500 \nu^{2} + 43887891498044213734761305063100 \nu + 2473206518249494196636747622420000\)\()/ \)\(11\!\cdots\!00\)\( \)
\(\beta_{4}\)\(=\)\((\)\(43323667269564131 \nu^{7} + 25537710858199137000 \nu^{6} + 41867228617668634658380 \nu^{5} + 24086814529103061200010000 \nu^{4} + 9929521155639075141908592269 \nu^{3} + 5757207770150476952166268713000 \nu^{2} - 94639685532711067409659771880100 \nu + 70750517362734599162146862358975000\)\()/ \)\(59\!\cdots\!00\)\( \)
\(\beta_{5}\)\(=\)\((\)\(7508108770501671 \nu^{7} + 2270018742951034400 \nu^{6} + 7783438710236985435330 \nu^{5} + 2141050180364716551112000 \nu^{4} + 2164458795517133666896421979 \nu^{3} + 511751801791153506859223885600 \nu^{2} + 80108740726882941592962114651900 \nu + 6288934876687519925524165543020000\)\()/ \)\(52\!\cdots\!00\)\( \)
\(\beta_{6}\)\(=\)\((\)\(8349956464770049 \nu^{7} - 3333394027431639600 \nu^{6} + 8513682299055486710270 \nu^{5} - 3263831544095036987358000 \nu^{4} + 2287372054162242661636374301 \nu^{3} - 809042645397965339619720290400 \nu^{2} + 63201247299482428846836960176100 \nu - 9049897976500639666056355286580000\)\()/ \)\(52\!\cdots\!00\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-8349956464770049 \nu^{7} - 1019797114824020000 \nu^{6} - 8513682299055486710270 \nu^{5} - 1163421269300620998850000 \nu^{4} - 2287372054162242661636374301 \nu^{3} - 326992221794234761339424330000 \nu^{2} - 63201247299482428846836960176100 \nu - 2578571970752998489350209382384000\)\()/ \)\(52\!\cdots\!00\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(6 \beta_{7} - 10 \beta_{6} + 5 \beta_{5} - 17 \beta_{4} + 163 \beta_{3} - 159 \beta_{2} + 14911 \beta_{1} - 2\)\()/30000\)
\(\nu^{2}\)\(=\)\((\)\(4669 \beta_{7} + 1885 \beta_{6} + 2320 \beta_{5} + 1392 \beta_{4} + 209387 \beta_{3} + 208459 \beta_{2} + 464 \beta_{1} - 7595372873\)\()/30000\)
\(\nu^{3}\)\(=\)\((\)\(-3171954 \beta_{7} + 5286590 \beta_{6} - 1599845 \beta_{5} + 7943753 \beta_{4} - 66591967 \beta_{3} + 64477331 \beta_{2} + 721943419201 \beta_{1} + 1057318\)\()/30000\)
\(\nu^{4}\)\(=\)\((\)\(-2910978331 \beta_{7} - 1116781915 \beta_{6} - 1495163680 \beta_{5} - 897098208 \beta_{4} - 100892286413 \beta_{3} - 100294220941 \beta_{2} - 299032736 \beta_{1} + 3695294170877327\)\()/30000\)
\(\nu^{5}\)\(=\)\((\)\(1620680781126 \beta_{7} - 2701134635210 \beta_{6} + 845460110855 \beta_{5} - 4086821673107 \beta_{4} + 26212436867173 \beta_{3} - 25131983013089 \beta_{2} - 609636423754866619 \beta_{1} - 540226927042\)\()/30000\)
\(\nu^{6}\)\(=\)\((\)\(1705600419772849 \beta_{7} + 607408334005585 \beta_{6} + 915160071472720 \beta_{5} + 549096042883632 \beta_{4} + 47965898880081527 \beta_{3} + 47599834851492439 \beta_{2} + 183032014294544 \beta_{1} - 1856166415775199917933\)\()/30000\)
\(\nu^{7}\)\(=\)\((\)\(-833515047921675234 \beta_{7} + 1389191746536125390 \beta_{6} - 455923167691584245 \beta_{5} + 2122953263534934713 \beta_{4} - 9718461741361829407 \beta_{3} + 9162785042747379251 \beta_{2} + 423709423955023869749521 \beta_{1} + 277838349307225078\)\()/30000\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
71.7997i
140.480i
735.343i
668.663i
71.7997i
140.480i
735.343i
668.663i
256.000 256.000i −13138.9 13138.9i 131072.i −1.63647e6 1.06614e6i −6.72711e6 −1.89897e7 + 1.89897e7i −3.35544e7 3.35544e7i 4.21603e7i −6.91869e8 + 1.46007e8i
3.2 256.000 256.000i −5080.17 5080.17i 131072.i 82450.2 + 1.95138e6i −2.60105e6 2.79429e7 2.79429e7i −3.35544e7 3.35544e7i 3.35804e8i 5.20662e8 + 4.78447e8i
3.3 256.000 256.000i 10752.0 + 10752.0i 131072.i 1.75470e6 857742.i 5.50503e6 −4.02524e6 + 4.02524e6i −3.35544e7 3.35544e7i 1.56209e8i 2.29622e8 6.68785e8i
3.4 256.000 256.000i 25980.0 + 25980.0i 131072.i −1.76690e6 + 832319.i 1.33018e7 −2.67178e7 + 2.67178e7i −3.35544e7 3.35544e7i 9.62503e8i −2.39253e8 + 6.65401e8i
7.1 256.000 + 256.000i −13138.9 + 13138.9i 131072.i −1.63647e6 + 1.06614e6i −6.72711e6 −1.89897e7 1.89897e7i −3.35544e7 + 3.35544e7i 4.21603e7i −6.91869e8 1.46007e8i
7.2 256.000 + 256.000i −5080.17 + 5080.17i 131072.i 82450.2 1.95138e6i −2.60105e6 2.79429e7 + 2.79429e7i −3.35544e7 + 3.35544e7i 3.35804e8i 5.20662e8 4.78447e8i
7.3 256.000 + 256.000i 10752.0 10752.0i 131072.i 1.75470e6 + 857742.i 5.50503e6 −4.02524e6 4.02524e6i −3.35544e7 + 3.35544e7i 1.56209e8i 2.29622e8 + 6.68785e8i
7.4 256.000 + 256.000i 25980.0 25980.0i 131072.i −1.76690e6 832319.i 1.33018e7 −2.67178e7 2.67178e7i −3.35544e7 + 3.35544e7i 9.62503e8i −2.39253e8 6.65401e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.19.c.a 8
3.b odd 2 1 90.19.g.a 8
5.b even 2 1 50.19.c.c 8
5.c odd 4 1 inner 10.19.c.a 8
5.c odd 4 1 50.19.c.c 8
15.e even 4 1 90.19.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.19.c.a 8 1.a even 1 1 trivial
10.19.c.a 8 5.c odd 4 1 inner
50.19.c.c 8 5.b even 2 1
50.19.c.c 8 5.c odd 4 1
90.19.g.a 8 3.b odd 2 1
90.19.g.a 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} - \cdots\) acting on \(S_{19}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 512 T + 131072 T^{2} )^{4} \)
$3$ \( 1 - 37026 T + 685462338 T^{2} - 965613795354 T^{3} - 132631992466885152 T^{4} + \)\(80\!\cdots\!74\)\( T^{5} + \)\(61\!\cdots\!10\)\( T^{6} - \)\(11\!\cdots\!30\)\( T^{7} + \)\(25\!\cdots\!26\)\( T^{8} - \)\(45\!\cdots\!70\)\( T^{9} + \)\(92\!\cdots\!10\)\( T^{10} + \)\(46\!\cdots\!06\)\( T^{11} - \)\(29\!\cdots\!32\)\( T^{12} - \)\(84\!\cdots\!46\)\( T^{13} + \)\(23\!\cdots\!18\)\( T^{14} - \)\(48\!\cdots\!54\)\( T^{15} + \)\(50\!\cdots\!81\)\( T^{16} \)
$5$ \( 1 + 3132450 T + 2393377750000 T^{2} - 2709717895019531250 T^{3} - \)\(41\!\cdots\!50\)\( T^{4} - \)\(10\!\cdots\!50\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!50\)\( T^{7} + \)\(21\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 + 43579766 T + 949598002307378 T^{2} + \)\(79\!\cdots\!74\)\( T^{3} + \)\(39\!\cdots\!88\)\( T^{4} + \)\(16\!\cdots\!66\)\( T^{5} + \)\(13\!\cdots\!30\)\( T^{6} - \)\(40\!\cdots\!30\)\( T^{7} - \)\(84\!\cdots\!94\)\( T^{8} - \)\(65\!\cdots\!70\)\( T^{9} + \)\(35\!\cdots\!30\)\( T^{10} + \)\(71\!\cdots\!34\)\( T^{11} + \)\(27\!\cdots\!88\)\( T^{12} + \)\(90\!\cdots\!26\)\( T^{13} + \)\(17\!\cdots\!78\)\( T^{14} + \)\(13\!\cdots\!34\)\( T^{15} + \)\(49\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 + 1339261642 T + 21379988756645042848 T^{2} + \)\(20\!\cdots\!74\)\( T^{3} + \)\(17\!\cdots\!70\)\( T^{4} + \)\(11\!\cdots\!94\)\( T^{5} + \)\(66\!\cdots\!28\)\( T^{6} + \)\(23\!\cdots\!22\)\( T^{7} + \)\(95\!\cdots\!21\)\( T^{8} )^{2} \)
$13$ \( 1 + 13618988004 T + 92738417126547952008 T^{2} + \)\(76\!\cdots\!76\)\( T^{3} - \)\(29\!\cdots\!92\)\( T^{4} - \)\(20\!\cdots\!56\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!40\)\( T^{7} + \)\(64\!\cdots\!06\)\( T^{8} + \)\(27\!\cdots\!60\)\( T^{9} - \)\(14\!\cdots\!00\)\( T^{10} - \)\(29\!\cdots\!84\)\( T^{11} - \)\(47\!\cdots\!52\)\( T^{12} + \)\(13\!\cdots\!24\)\( T^{13} + \)\(18\!\cdots\!68\)\( T^{14} + \)\(30\!\cdots\!36\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$17$ \( 1 - 131385428404 T + \)\(86\!\cdots\!08\)\( T^{2} - \)\(19\!\cdots\!96\)\( T^{3} + \)\(47\!\cdots\!28\)\( T^{4} - \)\(32\!\cdots\!64\)\( T^{5} + \)\(20\!\cdots\!40\)\( T^{6} - \)\(42\!\cdots\!60\)\( T^{7} + \)\(86\!\cdots\!26\)\( T^{8} - \)\(59\!\cdots\!40\)\( T^{9} + \)\(40\!\cdots\!40\)\( T^{10} - \)\(91\!\cdots\!56\)\( T^{11} + \)\(18\!\cdots\!08\)\( T^{12} - \)\(10\!\cdots\!04\)\( T^{13} + \)\(66\!\cdots\!28\)\( T^{14} - \)\(14\!\cdots\!76\)\( T^{15} + \)\(15\!\cdots\!21\)\( T^{16} \)
$19$ \( 1 - \)\(63\!\cdots\!28\)\( T^{2} + \)\(19\!\cdots\!68\)\( T^{4} - \)\(35\!\cdots\!76\)\( T^{6} + \)\(44\!\cdots\!70\)\( T^{8} - \)\(38\!\cdots\!56\)\( T^{10} + \)\(22\!\cdots\!48\)\( T^{12} - \)\(81\!\cdots\!48\)\( T^{14} + \)\(13\!\cdots\!21\)\( T^{16} \)
$23$ \( 1 - 2992780401346 T + \)\(44\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!14\)\( T^{3} + \)\(18\!\cdots\!68\)\( T^{4} + \)\(68\!\cdots\!54\)\( T^{5} - \)\(24\!\cdots\!30\)\( T^{6} + \)\(10\!\cdots\!50\)\( T^{7} - \)\(34\!\cdots\!34\)\( T^{8} + \)\(33\!\cdots\!50\)\( T^{9} - \)\(26\!\cdots\!30\)\( T^{10} + \)\(23\!\cdots\!86\)\( T^{11} + \)\(20\!\cdots\!28\)\( T^{12} - \)\(45\!\cdots\!86\)\( T^{13} + \)\(52\!\cdots\!98\)\( T^{14} - \)\(11\!\cdots\!94\)\( T^{15} + \)\(12\!\cdots\!41\)\( T^{16} \)
$29$ \( 1 + \)\(11\!\cdots\!12\)\( T^{2} + \)\(15\!\cdots\!88\)\( T^{4} + \)\(15\!\cdots\!64\)\( T^{6} + \)\(95\!\cdots\!70\)\( T^{8} + \)\(68\!\cdots\!44\)\( T^{10} + \)\(29\!\cdots\!08\)\( T^{12} + \)\(97\!\cdots\!32\)\( T^{14} + \)\(38\!\cdots\!81\)\( T^{16} \)
$31$ \( ( 1 + 20645311814822 T + \)\(19\!\cdots\!08\)\( T^{2} + \)\(35\!\cdots\!34\)\( T^{3} + \)\(19\!\cdots\!70\)\( T^{4} + \)\(24\!\cdots\!94\)\( T^{5} + \)\(94\!\cdots\!48\)\( T^{6} + \)\(70\!\cdots\!62\)\( T^{7} + \)\(23\!\cdots\!61\)\( T^{8} )^{2} \)
$37$ \( 1 + 191725492483296 T + \)\(18\!\cdots\!08\)\( T^{2} - \)\(19\!\cdots\!76\)\( T^{3} - \)\(62\!\cdots\!92\)\( T^{4} - \)\(76\!\cdots\!44\)\( T^{5} - \)\(33\!\cdots\!00\)\( T^{6} + \)\(58\!\cdots\!60\)\( T^{7} + \)\(19\!\cdots\!06\)\( T^{8} + \)\(99\!\cdots\!40\)\( T^{9} - \)\(94\!\cdots\!00\)\( T^{10} - \)\(36\!\cdots\!16\)\( T^{11} - \)\(50\!\cdots\!52\)\( T^{12} - \)\(26\!\cdots\!24\)\( T^{13} + \)\(42\!\cdots\!68\)\( T^{14} + \)\(75\!\cdots\!64\)\( T^{15} + \)\(66\!\cdots\!61\)\( T^{16} \)
$41$ \( ( 1 - 269286467520878 T + \)\(87\!\cdots\!28\)\( T^{2} - \)\(18\!\cdots\!86\)\( T^{3} - \)\(16\!\cdots\!30\)\( T^{4} - \)\(19\!\cdots\!06\)\( T^{5} + \)\(99\!\cdots\!48\)\( T^{6} - \)\(33\!\cdots\!58\)\( T^{7} + \)\(13\!\cdots\!81\)\( T^{8} )^{2} \)
$43$ \( 1 - 278176126066386 T + \)\(38\!\cdots\!98\)\( T^{2} - \)\(14\!\cdots\!54\)\( T^{3} - \)\(11\!\cdots\!12\)\( T^{4} - \)\(14\!\cdots\!06\)\( T^{5} + \)\(14\!\cdots\!50\)\( T^{6} - \)\(43\!\cdots\!10\)\( T^{7} + \)\(69\!\cdots\!26\)\( T^{8} - \)\(10\!\cdots\!90\)\( T^{9} + \)\(92\!\cdots\!50\)\( T^{10} - \)\(23\!\cdots\!94\)\( T^{11} - \)\(47\!\cdots\!12\)\( T^{12} - \)\(14\!\cdots\!46\)\( T^{13} + \)\(10\!\cdots\!98\)\( T^{14} - \)\(18\!\cdots\!14\)\( T^{15} + \)\(16\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 + 634311997628766 T + \)\(20\!\cdots\!78\)\( T^{2} - \)\(14\!\cdots\!86\)\( T^{3} + \)\(34\!\cdots\!48\)\( T^{4} + \)\(30\!\cdots\!06\)\( T^{5} + \)\(28\!\cdots\!50\)\( T^{6} + \)\(11\!\cdots\!10\)\( T^{7} - \)\(46\!\cdots\!34\)\( T^{8} + \)\(14\!\cdots\!90\)\( T^{9} + \)\(44\!\cdots\!50\)\( T^{10} + \)\(59\!\cdots\!14\)\( T^{11} + \)\(85\!\cdots\!68\)\( T^{12} - \)\(43\!\cdots\!14\)\( T^{13} + \)\(77\!\cdots\!58\)\( T^{14} + \)\(30\!\cdots\!14\)\( T^{15} + \)\(60\!\cdots\!81\)\( T^{16} \)
$53$ \( 1 + 12985998337176064 T + \)\(84\!\cdots\!48\)\( T^{2} + \)\(45\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!48\)\( T^{4} + \)\(11\!\cdots\!04\)\( T^{5} + \)\(45\!\cdots\!20\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(59\!\cdots\!86\)\( T^{8} + \)\(18\!\cdots\!00\)\( T^{9} + \)\(53\!\cdots\!20\)\( T^{10} + \)\(14\!\cdots\!76\)\( T^{11} + \)\(34\!\cdots\!68\)\( T^{12} + \)\(69\!\cdots\!44\)\( T^{13} + \)\(14\!\cdots\!28\)\( T^{14} + \)\(23\!\cdots\!56\)\( T^{15} + \)\(19\!\cdots\!81\)\( T^{16} \)
$59$ \( 1 - \)\(34\!\cdots\!68\)\( T^{2} + \)\(62\!\cdots\!48\)\( T^{4} - \)\(74\!\cdots\!16\)\( T^{6} + \)\(64\!\cdots\!70\)\( T^{8} - \)\(41\!\cdots\!56\)\( T^{10} + \)\(19\!\cdots\!88\)\( T^{12} - \)\(61\!\cdots\!28\)\( T^{14} + \)\(10\!\cdots\!61\)\( T^{16} \)
$61$ \( ( 1 - 3016226085915198 T + \)\(44\!\cdots\!88\)\( T^{2} - \)\(11\!\cdots\!26\)\( T^{3} + \)\(84\!\cdots\!70\)\( T^{4} - \)\(15\!\cdots\!06\)\( T^{5} + \)\(82\!\cdots\!68\)\( T^{6} - \)\(77\!\cdots\!18\)\( T^{7} + \)\(34\!\cdots\!21\)\( T^{8} )^{2} \)
$67$ \( 1 + 43112377024267886 T + \)\(92\!\cdots\!98\)\( T^{2} + \)\(39\!\cdots\!14\)\( T^{3} + \)\(45\!\cdots\!28\)\( T^{4} - \)\(18\!\cdots\!34\)\( T^{5} - \)\(47\!\cdots\!70\)\( T^{6} - \)\(21\!\cdots\!30\)\( T^{7} - \)\(95\!\cdots\!34\)\( T^{8} - \)\(15\!\cdots\!70\)\( T^{9} - \)\(25\!\cdots\!70\)\( T^{10} - \)\(76\!\cdots\!86\)\( T^{11} + \)\(13\!\cdots\!08\)\( T^{12} + \)\(87\!\cdots\!86\)\( T^{13} + \)\(15\!\cdots\!18\)\( T^{14} + \)\(52\!\cdots\!34\)\( T^{15} + \)\(90\!\cdots\!21\)\( T^{16} \)
$71$ \( ( 1 + 10341559110994982 T + \)\(30\!\cdots\!28\)\( T^{2} + \)\(18\!\cdots\!54\)\( T^{3} + \)\(33\!\cdots\!70\)\( T^{4} + \)\(38\!\cdots\!94\)\( T^{5} + \)\(13\!\cdots\!88\)\( T^{6} + \)\(96\!\cdots\!42\)\( T^{7} + \)\(19\!\cdots\!41\)\( T^{8} )^{2} \)
$73$ \( 1 + 85903701564675544 T + \)\(36\!\cdots\!68\)\( T^{2} + \)\(16\!\cdots\!96\)\( T^{3} - \)\(50\!\cdots\!32\)\( T^{4} - \)\(25\!\cdots\!16\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!26\)\( T^{8} + \)\(78\!\cdots\!20\)\( T^{9} + \)\(12\!\cdots\!80\)\( T^{10} - \)\(10\!\cdots\!44\)\( T^{11} - \)\(73\!\cdots\!72\)\( T^{12} + \)\(81\!\cdots\!04\)\( T^{13} + \)\(63\!\cdots\!08\)\( T^{14} + \)\(51\!\cdots\!16\)\( T^{15} + \)\(20\!\cdots\!41\)\( T^{16} \)
$79$ \( 1 - \)\(55\!\cdots\!88\)\( T^{2} + \)\(16\!\cdots\!88\)\( T^{4} - \)\(35\!\cdots\!36\)\( T^{6} + \)\(59\!\cdots\!70\)\( T^{8} - \)\(74\!\cdots\!56\)\( T^{10} + \)\(70\!\cdots\!08\)\( T^{12} - \)\(48\!\cdots\!68\)\( T^{14} + \)\(18\!\cdots\!81\)\( T^{16} \)
$83$ \( 1 - 247117472833336746 T + \)\(30\!\cdots\!58\)\( T^{2} - \)\(11\!\cdots\!54\)\( T^{3} + \)\(31\!\cdots\!28\)\( T^{4} + \)\(30\!\cdots\!14\)\( T^{5} - \)\(19\!\cdots\!10\)\( T^{6} + \)\(79\!\cdots\!10\)\( T^{7} - \)\(27\!\cdots\!74\)\( T^{8} + \)\(27\!\cdots\!90\)\( T^{9} - \)\(23\!\cdots\!10\)\( T^{10} + \)\(12\!\cdots\!06\)\( T^{11} + \)\(46\!\cdots\!08\)\( T^{12} - \)\(59\!\cdots\!46\)\( T^{13} + \)\(55\!\cdots\!78\)\( T^{14} - \)\(15\!\cdots\!74\)\( T^{15} + \)\(22\!\cdots\!21\)\( T^{16} \)
$89$ \( 1 - \)\(25\!\cdots\!48\)\( T^{2} + \)\(53\!\cdots\!08\)\( T^{4} - \)\(85\!\cdots\!96\)\( T^{6} + \)\(12\!\cdots\!70\)\( T^{8} - \)\(12\!\cdots\!56\)\( T^{10} + \)\(12\!\cdots\!68\)\( T^{12} - \)\(88\!\cdots\!88\)\( T^{14} + \)\(51\!\cdots\!41\)\( T^{16} \)
$97$ \( 1 - 1905016584439493544 T + \)\(18\!\cdots\!68\)\( T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!48\)\( T^{4} - \)\(92\!\cdots\!64\)\( T^{5} + \)\(60\!\cdots\!40\)\( T^{6} - \)\(44\!\cdots\!60\)\( T^{7} + \)\(33\!\cdots\!06\)\( T^{8} - \)\(25\!\cdots\!40\)\( T^{9} + \)\(20\!\cdots\!40\)\( T^{10} - \)\(17\!\cdots\!16\)\( T^{11} + \)\(15\!\cdots\!68\)\( T^{12} - \)\(10\!\cdots\!24\)\( T^{13} + \)\(67\!\cdots\!48\)\( T^{14} - \)\(41\!\cdots\!76\)\( T^{15} + \)\(12\!\cdots\!81\)\( T^{16} \)
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