Properties

Label 8004.2.a.j
Level 8004
Weight 2
Character orbit 8004.a
Self dual yes
Analytic conductor 63.912
Analytic rank 0
Dimension 16
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{7} q^{7} + q^{9} -\beta_{6} q^{11} + \beta_{4} q^{13} + \beta_{1} q^{15} + \beta_{9} q^{17} + ( 1 + \beta_{8} ) q^{19} -\beta_{7} q^{21} + q^{23} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{25} + q^{27} - q^{29} + ( 1 - \beta_{7} + \beta_{10} ) q^{31} -\beta_{6} q^{33} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} + \beta_{14} ) q^{35} + ( \beta_{1} + \beta_{14} ) q^{37} + \beta_{4} q^{39} + ( -\beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} ) q^{41} + ( 1 + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{14} ) q^{43} + \beta_{1} q^{45} + ( \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{15} ) q^{47} + ( 2 - \beta_{1} - \beta_{2} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{49} + \beta_{9} q^{51} + ( 1 - \beta_{8} - \beta_{11} - \beta_{12} ) q^{53} + ( 2 + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{55} + ( 1 + \beta_{8} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} + \beta_{12} - 2 \beta_{14} ) q^{59} + ( 2 + \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{13} ) q^{61} -\beta_{7} q^{63} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{7} - \beta_{12} + 2 \beta_{14} ) q^{65} + ( 3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - \beta_{15} ) q^{67} + q^{69} + ( \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{12} - \beta_{15} ) q^{71} + ( 2 - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} ) q^{73} + ( 2 + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{14} + \beta_{15} ) q^{75} + ( 3 + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{77} + ( 1 - \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{79} + q^{81} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{83} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{15} ) q^{85} - q^{87} + ( \beta_{7} - \beta_{11} + \beta_{12} ) q^{89} + ( 1 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - 2 \beta_{12} + \beta_{13} ) q^{91} + ( 1 - \beta_{7} + \beta_{10} ) q^{93} + ( -1 + \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{95} + ( 2 + \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{97} -\beta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{3} + 3q^{5} + 4q^{7} + 16q^{9} + 5q^{11} + 6q^{13} + 3q^{15} + 3q^{17} + 11q^{19} + 4q^{21} + 16q^{23} + 27q^{25} + 16q^{27} - 16q^{29} + 14q^{31} + 5q^{33} + 11q^{35} + 4q^{37} + 6q^{39} + 11q^{41} + 23q^{43} + 3q^{45} - 2q^{47} + 34q^{49} + 3q^{51} + 19q^{53} + 31q^{55} + 11q^{57} + 32q^{59} + 19q^{61} + 4q^{63} + 6q^{65} + 33q^{67} + 16q^{69} - 5q^{71} + 23q^{73} + 27q^{75} + 42q^{77} + 24q^{79} + 16q^{81} + 7q^{83} - 16q^{87} - 2q^{89} + 25q^{91} + 14q^{93} + 7q^{95} + 33q^{97} + 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 3 x^{15} - 49 x^{14} + 130 x^{13} + 932 x^{12} - 2028 x^{11} - 8965 x^{10} + 14400 x^{9} + 46229 x^{8} - 47547 x^{7} - 122604 x^{6} + 65278 x^{5} + 151028 x^{4} - 17988 x^{3} - 67608 x^{2} - 8424 x + 3888\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(128089526929863711 \nu^{15} - 1780989647760093687 \nu^{14} - 2343590043301536281 \nu^{13} + 82428009307722432792 \nu^{12} - 39868293740417462800 \nu^{11} - 1434850612960902643364 \nu^{10} + 1061787358987892121509 \nu^{9} + 12059212146105833185194 \nu^{8} - 6849714181952244815581 \nu^{7} - 49998222735113343315999 \nu^{6} + 12943701821822810175386 \nu^{5} + 87862981358437561842546 \nu^{4} - 985493942779777495944 \nu^{3} - 44010301598073058173068 \nu^{2} - 5734576008998611033216 \nu + 1436096932911927060552\)\()/ \)\(60\!\cdots\!08\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-958087900246403592 \nu^{15} + 2000567955267874333 \nu^{14} + 49357290400200766149 \nu^{13} - 88974714684163101079 \nu^{12} - 963378769466844341132 \nu^{11} + 1415048553601951450082 \nu^{10} + 8893034704374805484880 \nu^{9} - 9960991537155374353069 \nu^{8} - 38488851452268663931116 \nu^{7} + 31123533131595448591055 \nu^{6} + 63124294156284455139465 \nu^{5} - 39843524562808100614170 \nu^{4} - 17036408551707978644276 \nu^{3} + 2778605399574854232452 \nu^{2} + 4140098642180225184060 \nu + 10187796734063718568848\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{4}\)\(=\)\((\)\(4486141389040236929 \nu^{15} - 16191524997021342828 \nu^{14} - 211861740633534736190 \nu^{13} + 721401882595358991197 \nu^{12} + 3820388029937201339968 \nu^{11} - 11804652191031635223666 \nu^{10} - 34218304176590055106445 \nu^{9} + 91025874133171328629977 \nu^{8} + 160820231928040018855981 \nu^{7} - 348051881480837120172498 \nu^{6} - 374464283660322591966429 \nu^{5} + 631086737220708958112600 \nu^{4} + 373450758905607661475524 \nu^{3} - 436002491424203250589680 \nu^{2} - 119911680097050427614324 \nu + 62507843402239991683728\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-2170353582856352493 \nu^{15} + 6394679216719266718 \nu^{14} + 105192616124931196248 \nu^{13} - 273011890888993264339 \nu^{12} - 1963933514692243560836 \nu^{11} + 4159658356999503153722 \nu^{10} + 18290876230917575384385 \nu^{9} - 28530527789909789588287 \nu^{8} - 88888441699606672234641 \nu^{7} + 90582649286250348855860 \nu^{6} + 210029308309103838088587 \nu^{5} - 128266188808560877480848 \nu^{4} - 209073282062352857065304 \nu^{3} + 66867424884872498482256 \nu^{2} + 63961903110921104602836 \nu - 6638130556491638529144\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{6}\)\(=\)\((\)\(1181647930064972053 \nu^{15} - 4488671540266881279 \nu^{14} - 53334067423612189198 \nu^{13} + 193579150520849398669 \nu^{12} + 901332302367398689427 \nu^{11} - 3015546321065278419096 \nu^{10} - 7391050976096937823363 \nu^{9} + 21665705786600716578792 \nu^{8} + 30526909733759416311410 \nu^{7} - 74930972648825634056871 \nu^{6} - 56119572043326164111031 \nu^{5} + 117837296847755457978037 \nu^{4} + 33444989655224619931886 \nu^{3} - 61721460041205402182268 \nu^{2} - 7863407220333139897548 \nu + 3287618491616138537532\)\()/ \)\(90\!\cdots\!12\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-8599410738229651769 \nu^{15} + 31111659866916402345 \nu^{14} + 397851089569802580887 \nu^{13} - 1357434095246951691440 \nu^{12} - 6950409487206069267520 \nu^{11} + 21480555688367312916876 \nu^{10} + 59301366750646584632885 \nu^{9} - 157147318825198786079022 \nu^{8} - 257042371188445739715205 \nu^{7} + 552656578883574074980929 \nu^{6} + 511728863085525731674458 \nu^{5} - 875606750046801999312014 \nu^{4} - 366980592422283677211496 \nu^{3} + 458996610472572057501996 \nu^{2} + 63732795840155393435880 \nu - 41744720482603951993704\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{8}\)\(=\)\((\)\(10166089516631326123 \nu^{15} - 39636043673251564446 \nu^{14} - 463149040304201162734 \nu^{13} + 1741577168182876800871 \nu^{12} + 7937358719018169248978 \nu^{11} - 27920257611472853652366 \nu^{10} - 66526615038749754213403 \nu^{9} + 209053689477697038444789 \nu^{8} + 286804021630695488156393 \nu^{7} - 763778398499676259472184 \nu^{6} - 590218313888419955236599 \nu^{5} + 1275883347371748259536286 \nu^{4} + 501014971463436711617372 \nu^{3} - 734875152379202721919584 \nu^{2} - 188592356674960101516348 \nu + 75381322683826038993408\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{9}\)\(=\)\((\)\(3833924860594975845 \nu^{15} - 15405971411407983416 \nu^{14} - 174748567588173763710 \nu^{13} + 683105279786040092525 \nu^{12} + 3000960197217652805056 \nu^{11} - 11102524477032892698670 \nu^{10} - 25318495554936950579361 \nu^{9} + 84691399626259266906305 \nu^{8} + 111080659953374901792933 \nu^{7} - 315727483540994364481618 \nu^{6} - 237361474884041971348545 \nu^{5} + 530394989727383036599272 \nu^{4} + 206406504455500178785888 \nu^{3} - 290455416050807336719528 \nu^{2} - 58482275477158954969404 \nu + 24368330568281801524128\)\()/ \)\(18\!\cdots\!24\)\( \)
\(\beta_{10}\)\(=\)\((\)\(6055690973521494323 \nu^{15} - 21065591888740470630 \nu^{14} - 282138336758875451579 \nu^{13} + 904938036516128835734 \nu^{12} + 5019613746164753586637 \nu^{11} - 13994381788245243532422 \nu^{10} - 44614067074995671927885 \nu^{9} + 99213383165647751861865 \nu^{8} + 210015545205224104585822 \nu^{7} - 335507138819282232469092 \nu^{6} - 494990400061097230473426 \nu^{5} + 513309222385173256168109 \nu^{4} + 516577122221256491326894 \nu^{3} - 270657506289607141947036 \nu^{2} - 186121404779846364836280 \nu + 18853135064703346138200\)\()/ \)\(27\!\cdots\!36\)\( \)
\(\beta_{11}\)\(=\)\((\)\(2103742732645796968 \nu^{15} - 7587807286478686338 \nu^{14} - 98318269336507859293 \nu^{13} + 333589527564189827935 \nu^{12} + 1746467731155074504273 \nu^{11} - 5342141610410401144176 \nu^{10} - 15313513871554454343490 \nu^{9} + 39771223605111128968242 \nu^{8} + 69581428527495728527523 \nu^{7} - 143065840395346199349174 \nu^{6} - 151733582643722727860283 \nu^{5} + 230632231887950394137473 \nu^{4} + 131479917376481033077250 \nu^{3} - 118630530405786786024672 \nu^{2} - 36722337309588432352164 \nu + 7696294671096247442904\)\()/ \)\(90\!\cdots\!12\)\( \)
\(\beta_{12}\)\(=\)\((\)\(15643727938311237652 \nu^{15} - 56547423258025579917 \nu^{14} - 736811796288429578563 \nu^{13} + 2511604996502047287625 \nu^{12} + 13220075965007561551802 \nu^{11} - 40821134148960918199182 \nu^{10} - 117349102552007926729696 \nu^{9} + 310045122176464699783689 \nu^{8} + 543132566223574961590886 \nu^{7} - 1144928931914839786999539 \nu^{6} - 1234507192266499635469335 \nu^{5} + 1908106630392058166642008 \nu^{4} + 1205285884230039076834004 \nu^{3} - 1029539354898820273092228 \nu^{2} - 431698814922534389321508 \nu + 60226142682101943655944\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{13}\)\(=\)\((\)\(16902750278691257515 \nu^{15} - 66100600868765845944 \nu^{14} - 765501985844103172942 \nu^{13} + 2878724443289246873407 \nu^{12} + 13050836157168499238396 \nu^{11} - 45539064879024801708138 \nu^{10} - 109453059483503010262423 \nu^{9} + 334561429000218091255911 \nu^{8} + 478818628546944868872983 \nu^{7} - 1190123853407848150127478 \nu^{6} - 1026586069984405062827043 \nu^{5} + 1916731753201095741369916 \nu^{4} + 932234121437022270337808 \nu^{3} - 1043599212580558264999584 \nu^{2} - 292239039888885554789364 \nu + 83307899572874489830104\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{14}\)\(=\)\((\)\(18310254666719483314 \nu^{15} - 64997796606289655715 \nu^{14} - 854944160891350069327 \nu^{13} + 2842406557507315492831 \nu^{12} + 15152969014719198018848 \nu^{11} - 45130633610269192266030 \nu^{10} - 132199513307530952648122 \nu^{9} + 331680872425345273285917 \nu^{8} + 594566658421572741407942 \nu^{7} - 1173217721169586183526145 \nu^{6} - 1274906221868961120611337 \nu^{5} + 1870018247056491778135546 \nu^{4} + 1105580567571758766871040 \nu^{3} - 982373741782699986430740 \nu^{2} - 330829154134196740304148 \nu + 73093928798098161025416\)\()/ \)\(54\!\cdots\!72\)\( \)
\(\beta_{15}\)\(=\)\((\)\(-28101973564733554969 \nu^{15} + 108682923995574038268 \nu^{14} + 1292569355600345279962 \nu^{13} - 4807355654391981949669 \nu^{12} - 22457115817186195403960 \nu^{11} + 77765457791517503993622 \nu^{10} + 191723502462768280879765 \nu^{9} - 588180818937536644168137 \nu^{8} - 848072989308638681641817 \nu^{7} + 2164766731936135949379246 \nu^{6} + 1814874225627403113533529 \nu^{5} - 3591447951250514794518160 \nu^{4} - 1594204885835619991562096 \nu^{3} + 1969768145562148310520360 \nu^{2} + 517658891675791265035020 \nu - 193169117447690193359208\)\()/ \)\(54\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{15} - \beta_{14} + \beta_{11} + \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 7\)
\(\nu^{3}\)\(=\)\(\beta_{13} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{8} + 2 \beta_{7} + 13 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(17 \beta_{15} - 16 \beta_{14} - 5 \beta_{13} + 16 \beta_{11} + \beta_{10} - \beta_{9} + 21 \beta_{8} - 22 \beta_{7} - 18 \beta_{5} + 16 \beta_{4} - 14 \beta_{3} + 18 \beta_{2} + \beta_{1} + 92\)
\(\nu^{5}\)\(=\)\(6 \beta_{14} + 27 \beta_{13} + 3 \beta_{12} + 42 \beta_{11} - 27 \beta_{10} - 43 \beta_{9} + 18 \beta_{8} + 45 \beta_{7} - 4 \beta_{6} + 5 \beta_{5} - 6 \beta_{4} + 8 \beta_{3} - 6 \beta_{2} + 198 \beta_{1} + 48\)
\(\nu^{6}\)\(=\)\(292 \beta_{15} - 263 \beta_{14} - 140 \beta_{13} + 20 \beta_{12} + 250 \beta_{11} + 27 \beta_{10} - 22 \beta_{9} + 385 \beta_{8} - 435 \beta_{7} + 18 \beta_{6} - 310 \beta_{5} + 250 \beta_{4} - 196 \beta_{3} + 312 \beta_{2} + 2 \beta_{1} + 1422\)
\(\nu^{7}\)\(=\)\(-18 \beta_{15} + 218 \beta_{14} + 566 \beta_{13} + 97 \beta_{12} + 742 \beta_{11} - 578 \beta_{10} - 802 \beta_{9} + 301 \beta_{8} + 923 \beta_{7} - 114 \beta_{6} + 160 \beta_{5} - 222 \beta_{4} + 266 \beta_{3} - 204 \beta_{2} + 3255 \beta_{1} + 626\)
\(\nu^{8}\)\(=\)\(5100 \beta_{15} - 4491 \beta_{14} - 3030 \beta_{13} + 664 \beta_{12} + 3993 \beta_{11} + 555 \beta_{10} - 404 \beta_{9} + 6878 \beta_{8} - 8259 \beta_{7} + 636 \beta_{6} - 5445 \beta_{5} + 4089 \beta_{4} - 2905 \beta_{3} + 5467 \beta_{2} - 413 \beta_{1} + 23542\)
\(\nu^{9}\)\(=\)\(-819 \beta_{15} + 5553 \beta_{14} + 10895 \beta_{13} + 2315 \beta_{12} + 12537 \beta_{11} - 11449 \beta_{10} - 14473 \beta_{9} + 4797 \beta_{8} + 18229 \beta_{7} - 2238 \beta_{6} + 3752 \beta_{5} - 5663 \beta_{4} + 6505 \beta_{3} - 5024 \beta_{2} + 55717 \beta_{1} + 6965\)
\(\nu^{10}\)\(=\)\(90066 \beta_{15} - 78119 \beta_{14} - 60472 \beta_{13} + 15735 \beta_{12} + 65104 \beta_{11} + 10695 \beta_{10} - 6786 \beta_{9} + 122233 \beta_{8} - 153768 \beta_{7} + 16003 \beta_{6} - 97105 \beta_{5} + 69364 \beta_{4} - 45408 \beta_{3} + 96976 \beta_{2} - 15564 \beta_{1} + 403524\)
\(\nu^{11}\)\(=\)\(-24915 \beta_{15} + 124569 \beta_{14} + 202728 \beta_{13} + 48849 \beta_{12} + 209058 \beta_{11} - 218564 \beta_{10} - 259251 \beta_{9} + 72776 \beta_{8} + 352935 \beta_{7} - 38377 \beta_{6} + 80018 \beta_{5} - 125013 \beta_{4} + 141337 \beta_{3} - 109660 \beta_{2} + 975756 \beta_{1} + 55139\)
\(\nu^{12}\)\(=\)\(1602574 \beta_{15} - 1369817 \beta_{14} - 1167210 \beta_{13} + 327646 \beta_{12} + 1078384 \beta_{11} + 203896 \beta_{10} - 105992 \beta_{9} + 2172396 \beta_{8} - 2833616 \beta_{7} + 353182 \beta_{6} - 1745053 \beta_{5} + 1206681 \beta_{4} - 739735 \beta_{3} + 1735350 \beta_{2} - 414872 \beta_{1} + 7050086\)
\(\nu^{13}\)\(=\)\(-638200 \beta_{15} + 2634590 \beta_{14} + 3719888 \beta_{13} + 964071 \beta_{12} + 3477248 \beta_{11} - 4089626 \beta_{10} - 4643295 \beta_{9} + 1043482 \beta_{8} + 6758749 \beta_{7} - 620343 \beta_{6} + 1646430 \beta_{5} - 2570854 \beta_{4} + 2894128 \beta_{3} - 2258798 \beta_{2} + 17322068 \beta_{1} - 143779\)
\(\nu^{14}\)\(=\)\(28665480 \beta_{15} - 24111775 \beta_{14} - 22171021 \beta_{13} + 6402010 \beta_{12} + 18076637 \beta_{11} + 3905179 \beta_{10} - 1539030 \beta_{9} + 38666159 \beta_{8} - 51935720 \beta_{7} + 7302225 \beta_{6} - 31465887 \beta_{5} + 21356178 \beta_{4} - 12425342 \beta_{3} + 31239529 \beta_{2} - 9671295 \beta_{1} + 124565501\)
\(\nu^{15}\)\(=\)\(-14893147 \beta_{15} + 53944933 \beta_{14} + 67887756 \beta_{13} + 18262187 \beta_{12} + 57915120 \beta_{11} - 75643357 \beta_{10} - 83307736 \beta_{9} + 13834904 \beta_{8} + 128598654 \beta_{7} - 9769310 \beta_{6} + 33307089 \beta_{5} - 50852857 \beta_{4} + 57298007 \beta_{3} - 45079355 \beta_{2} + 310116137 \beta_{1} - 21232931\)

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
−4.29726
−3.12066
−2.45101
−2.34509
−1.86734
−0.900032
−0.812118
−0.434018
0.187793
0.897331
1.63007
1.72426
2.96949
3.50323
4.08902
4.22634
0 1.00000 0 −4.29726 0 3.54512 0 1.00000 0
1.2 0 1.00000 0 −3.12066 0 −0.760812 0 1.00000 0
1.3 0 1.00000 0 −2.45101 0 3.23607 0 1.00000 0
1.4 0 1.00000 0 −2.34509 0 −1.25807 0 1.00000 0
1.5 0 1.00000 0 −1.86734 0 −4.84365 0 1.00000 0
1.6 0 1.00000 0 −0.900032 0 0.226627 0 1.00000 0
1.7 0 1.00000 0 −0.812118 0 −4.94593 0 1.00000 0
1.8 0 1.00000 0 −0.434018 0 −2.25969 0 1.00000 0
1.9 0 1.00000 0 0.187793 0 3.11113 0 1.00000 0
1.10 0 1.00000 0 0.897331 0 3.81467 0 1.00000 0
1.11 0 1.00000 0 1.63007 0 −0.577524 0 1.00000 0
1.12 0 1.00000 0 1.72426 0 4.78449 0 1.00000 0
1.13 0 1.00000 0 2.96949 0 −3.61534 0 1.00000 0
1.14 0 1.00000 0 3.50323 0 0.271751 0 1.00000 0
1.15 0 1.00000 0 4.08902 0 2.57555 0 1.00000 0
1.16 0 1.00000 0 4.22634 0 0.695618 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

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 8004.2.a.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.j 16 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{16} - \cdots\)
\(T_{7}^{16} - \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( ( 1 - T )^{16} \)
$5$ \( 1 - 3 T + 31 T^{2} - 95 T^{3} + 502 T^{4} - 1453 T^{5} + 5480 T^{6} - 14265 T^{7} + 43779 T^{8} - 99922 T^{9} + 265306 T^{10} - 525742 T^{11} + 1264458 T^{12} - 2188838 T^{13} + 5201327 T^{14} - 8479994 T^{15} + 23022008 T^{16} - 42399970 T^{17} + 130033175 T^{18} - 273604750 T^{19} + 790286250 T^{20} - 1642943750 T^{21} + 4145406250 T^{22} - 7806406250 T^{23} + 17101171875 T^{24} - 27861328125 T^{25} + 53515625000 T^{26} - 70947265625 T^{27} + 122558593750 T^{28} - 115966796875 T^{29} + 189208984375 T^{30} - 91552734375 T^{31} + 152587890625 T^{32} \)
$7$ \( 1 - 4 T + 47 T^{2} - 137 T^{3} + 972 T^{4} - 1999 T^{5} + 12134 T^{6} - 16308 T^{7} + 112920 T^{8} - 92052 T^{9} + 974324 T^{10} - 611743 T^{11} + 8628020 T^{12} - 6093673 T^{13} + 73145687 T^{14} - 56106964 T^{15} + 549676366 T^{16} - 392748748 T^{17} + 3584138663 T^{18} - 2090129839 T^{19} + 20715876020 T^{20} - 10281564601 T^{21} + 114628244276 T^{22} - 75808780236 T^{23} + 650961328920 T^{24} - 658086622956 T^{25} + 3427554671366 T^{26} - 3952676159257 T^{27} + 13453731159372 T^{28} - 13273794425759 T^{29} + 31876484423903 T^{30} - 18990246039772 T^{31} + 33232930569601 T^{32} \)
$11$ \( 1 - 5 T + 90 T^{2} - 368 T^{3} + 3750 T^{4} - 12402 T^{5} + 96387 T^{6} - 248512 T^{7} + 1715621 T^{8} - 3107483 T^{9} + 22553162 T^{10} - 20117519 T^{11} + 231881226 T^{12} + 67176915 T^{13} + 2074874657 T^{14} + 3224397132 T^{15} + 20170399972 T^{16} + 35468368452 T^{17} + 251059833497 T^{18} + 89412473865 T^{19} + 3394973029866 T^{20} - 3239946552469 T^{21} + 39954302225882 T^{22} - 60556052600593 T^{23} + 367758597780101 T^{24} - 585978296585792 T^{25} + 2500030545016587 T^{26} - 3538435338917622 T^{27} + 11769106412703750 T^{28} - 12704358068966608 T^{29} + 34177485022491690 T^{30} - 20886240847078255 T^{31} + 45949729863572161 T^{32} \)
$13$ \( 1 - 6 T + 97 T^{2} - 359 T^{3} + 3862 T^{4} - 8739 T^{5} + 97576 T^{6} - 109826 T^{7} + 1948915 T^{8} - 237632 T^{9} + 34316745 T^{10} + 20907884 T^{11} + 569527538 T^{12} + 521147076 T^{13} + 8794275353 T^{14} + 8484394754 T^{15} + 121320902098 T^{16} + 110297131802 T^{17} + 1486232534657 T^{18} + 1144960125972 T^{19} + 16266276012818 T^{20} + 7762950974012 T^{21} + 165640373616705 T^{22} - 14911055591744 T^{23} + 1589789838117715 T^{24} - 1164649748139098 T^{25} + 13451680200658024 T^{26} - 15661689683489343 T^{27} + 89977204743021622 T^{28} - 108732163266618827 T^{29} + 381925509412831033 T^{30} - 307115358084544542 T^{31} + 665416609183179841 T^{32} \)
$17$ \( 1 - 3 T + 83 T^{2} - 125 T^{3} + 2861 T^{4} - 992 T^{5} + 69519 T^{6} - 20198 T^{7} + 1771472 T^{8} - 1756390 T^{9} + 39865506 T^{10} - 38769572 T^{11} + 711027721 T^{12} - 699954287 T^{13} + 13151255436 T^{14} - 19625255617 T^{15} + 245439220562 T^{16} - 333629345489 T^{17} + 3800712821004 T^{18} - 3438875412031 T^{19} + 59385746285641 T^{20} - 55047248191204 T^{21} + 962256401794914 T^{22} - 720714741870470 T^{23} + 12357358985523152 T^{24} - 2395237929486406 T^{25} + 140149879965314031 T^{26} - 33997721137171936 T^{27} + 1666882220714346221 T^{28} - 1238072254113242125 T^{29} + 13975359604430277107 T^{30} - 8587269154529447379 T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 11 T + 210 T^{2} - 1815 T^{3} + 20282 T^{4} - 145097 T^{5} + 1217138 T^{6} - 7441281 T^{7} + 51415617 T^{8} - 274818036 T^{9} + 1645993344 T^{10} - 7853002292 T^{11} + 42366418590 T^{12} - 184882176290 T^{13} + 930230079980 T^{14} - 3820111749718 T^{15} + 18399434920492 T^{16} - 72582123244642 T^{17} + 335813058872780 T^{18} - 1268106847173110 T^{19} + 5521234037067390 T^{20} - 19444811122218908 T^{21} + 77437206988616064 T^{22} - 245652075747884604 T^{23} + 873220372611411297 T^{24} - 2401209834416614899 T^{25} + 7462353722887393538 T^{26} - 16902387095354882243 T^{27} + 44890453188499877402 T^{28} - 76326164983996562085 T^{29} + \)\(16\!\cdots\!10\)\( T^{30} - \)\(16\!\cdots\!89\)\( T^{31} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( ( 1 - T )^{16} \)
$29$ \( ( 1 + T )^{16} \)
$31$ \( 1 - 14 T + 294 T^{2} - 3265 T^{3} + 42055 T^{4} - 395598 T^{5} + 3968314 T^{6} - 32714604 T^{7} + 278210372 T^{8} - 2057052635 T^{9} + 15470768158 T^{10} - 104234601543 T^{11} + 710239861465 T^{12} - 4407340071450 T^{13} + 27603613889578 T^{14} - 158782070053245 T^{15} + 921279848105094 T^{16} - 4922244171650595 T^{17} + 26527072947884458 T^{18} - 131299068068566950 T^{19} + 655921427100018265 T^{20} - 2984148146999379993 T^{21} + 13730363688122589598 T^{22} - 56594895352770732485 T^{23} + \)\(23\!\cdots\!52\)\( T^{24} - \)\(86\!\cdots\!84\)\( T^{25} + \)\(32\!\cdots\!14\)\( T^{26} - \)\(10\!\cdots\!38\)\( T^{27} + \)\(33\!\cdots\!55\)\( T^{28} - \)\(79\!\cdots\!15\)\( T^{29} + \)\(22\!\cdots\!74\)\( T^{30} - \)\(32\!\cdots\!14\)\( T^{31} + \)\(72\!\cdots\!81\)\( T^{32} \)
$37$ \( 1 - 4 T + 348 T^{2} - 1381 T^{3} + 60800 T^{4} - 246520 T^{5} + 7069922 T^{6} - 29586173 T^{7} + 612259255 T^{8} - 2635710207 T^{9} + 41930613664 T^{10} - 183054493959 T^{11} + 2355141441220 T^{12} - 10182672001961 T^{13} + 110988909462834 T^{14} - 460428621018917 T^{15} + 4445235835686068 T^{16} - 17035858977699929 T^{17} + 151943817054619746 T^{18} - 515782884915330533 T^{19} + 4413914238618316420 T^{20} - 12693722957749655763 T^{21} + \)\(10\!\cdots\!76\)\( T^{22} - \)\(25\!\cdots\!31\)\( T^{23} + \)\(21\!\cdots\!55\)\( T^{24} - \)\(38\!\cdots\!21\)\( T^{25} + \)\(33\!\cdots\!78\)\( T^{26} - \)\(43\!\cdots\!60\)\( T^{27} + \)\(40\!\cdots\!00\)\( T^{28} - \)\(33\!\cdots\!57\)\( T^{29} + \)\(31\!\cdots\!72\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( 1 - 11 T + 351 T^{2} - 3509 T^{3} + 64654 T^{4} - 576009 T^{5} + 7980108 T^{6} - 64037115 T^{7} + 735336275 T^{8} - 5364634568 T^{9} + 53746803598 T^{10} - 359761312208 T^{11} + 3241410212330 T^{12} - 20060735863402 T^{13} + 165714106021951 T^{14} - 953277968669194 T^{15} + 7299109474364504 T^{16} - 39084396715436954 T^{17} + 278565412222899631 T^{18} - 1382605976441529242 T^{19} + 9159450563003833130 T^{20} - 41680578899193801808 T^{21} + \)\(25\!\cdots\!18\)\( T^{22} - \)\(10\!\cdots\!08\)\( T^{23} + \)\(58\!\cdots\!75\)\( T^{24} - \)\(20\!\cdots\!15\)\( T^{25} + \)\(10\!\cdots\!08\)\( T^{26} - \)\(31\!\cdots\!69\)\( T^{27} + \)\(14\!\cdots\!74\)\( T^{28} - \)\(32\!\cdots\!89\)\( T^{29} + \)\(13\!\cdots\!11\)\( T^{30} - \)\(17\!\cdots\!11\)\( T^{31} + \)\(63\!\cdots\!41\)\( T^{32} \)
$43$ \( 1 - 23 T + 621 T^{2} - 10629 T^{3} + 176736 T^{4} - 2424625 T^{5} + 31250850 T^{6} - 360299505 T^{7} + 3893683705 T^{8} - 38882456890 T^{9} + 365372984754 T^{10} - 3221700793654 T^{11} + 26840051705792 T^{12} - 211481434110410 T^{13} + 1578864928177479 T^{14} - 11191510143979808 T^{15} + 75271909946487532 T^{16} - 481234936191131744 T^{17} + 2919321252200158671 T^{18} - 16814254381816367870 T^{19} + 91760795611813395392 T^{20} - \)\(47\!\cdots\!22\)\( T^{21} + \)\(23\!\cdots\!46\)\( T^{22} - \)\(10\!\cdots\!30\)\( T^{23} + \)\(45\!\cdots\!05\)\( T^{24} - \)\(18\!\cdots\!15\)\( T^{25} + \)\(67\!\cdots\!50\)\( T^{26} - \)\(22\!\cdots\!75\)\( T^{27} + \)\(70\!\cdots\!36\)\( T^{28} - \)\(18\!\cdots\!47\)\( T^{29} + \)\(45\!\cdots\!29\)\( T^{30} - \)\(73\!\cdots\!61\)\( T^{31} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( 1 + 2 T + 318 T^{2} - 4 T^{3} + 50079 T^{4} - 77447 T^{5} + 5344865 T^{6} - 14968679 T^{7} + 432946692 T^{8} - 1678285661 T^{9} + 28484569213 T^{10} - 136345336793 T^{11} + 1603599668977 T^{12} - 8855144414230 T^{13} + 81258028210116 T^{14} - 483256164976500 T^{15} + 3890208356154710 T^{16} - 22713039753895500 T^{17} + 179498984316146244 T^{18} - 919367658518601290 T^{19} + 7825054836313356337 T^{20} - 31270122221207942551 T^{21} + \)\(30\!\cdots\!77\)\( T^{22} - \)\(85\!\cdots\!43\)\( T^{23} + \)\(10\!\cdots\!12\)\( T^{24} - \)\(16\!\cdots\!93\)\( T^{25} + \)\(28\!\cdots\!85\)\( T^{26} - \)\(19\!\cdots\!41\)\( T^{27} + \)\(58\!\cdots\!39\)\( T^{28} - \)\(21\!\cdots\!08\)\( T^{29} + \)\(81\!\cdots\!42\)\( T^{30} + \)\(24\!\cdots\!86\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 - 19 T + 431 T^{2} - 5819 T^{3} + 85523 T^{4} - 950502 T^{5} + 11164835 T^{6} - 107392850 T^{7} + 1081832710 T^{8} - 9255764980 T^{9} + 83301576533 T^{10} - 649451637176 T^{11} + 5397180784669 T^{12} - 39327106176827 T^{13} + 311239365622937 T^{14} - 2179312691515715 T^{15} + 16832485566076786 T^{16} - 115503572650332895 T^{17} + 874271378034830033 T^{18} - 5854901586287473279 T^{19} + 42586352434995835789 T^{20} - \)\(27\!\cdots\!68\)\( T^{21} + \)\(18\!\cdots\!57\)\( T^{22} - \)\(10\!\cdots\!60\)\( T^{23} + \)\(67\!\cdots\!10\)\( T^{24} - \)\(35\!\cdots\!50\)\( T^{25} + \)\(19\!\cdots\!15\)\( T^{26} - \)\(88\!\cdots\!94\)\( T^{27} + \)\(42\!\cdots\!43\)\( T^{28} - \)\(15\!\cdots\!87\)\( T^{29} + \)\(59\!\cdots\!39\)\( T^{30} - \)\(13\!\cdots\!83\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 32 T + 822 T^{2} - 16290 T^{3} + 280616 T^{4} - 4266252 T^{5} + 59002420 T^{6} - 750388520 T^{7} + 8889961715 T^{8} - 98683415146 T^{9} + 1034078840112 T^{10} - 10260068417802 T^{11} + 96817508417900 T^{12} - 870380894105704 T^{13} + 7473128002490358 T^{14} - 61327399691512338 T^{15} + 481586259456684752 T^{16} - 3618316581799227942 T^{17} + 26013958576668936198 T^{18} - \)\(17\!\cdots\!16\)\( T^{19} + \)\(11\!\cdots\!00\)\( T^{20} - \)\(73\!\cdots\!98\)\( T^{21} + \)\(43\!\cdots\!92\)\( T^{22} - \)\(24\!\cdots\!74\)\( T^{23} + \)\(13\!\cdots\!15\)\( T^{24} - \)\(65\!\cdots\!80\)\( T^{25} + \)\(30\!\cdots\!20\)\( T^{26} - \)\(12\!\cdots\!68\)\( T^{27} + \)\(49\!\cdots\!96\)\( T^{28} - \)\(17\!\cdots\!10\)\( T^{29} + \)\(50\!\cdots\!42\)\( T^{30} - \)\(11\!\cdots\!68\)\( T^{31} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 - 19 T + 724 T^{2} - 12046 T^{3} + 257102 T^{4} - 3745752 T^{5} + 59115952 T^{6} - 758558610 T^{7} + 9828612801 T^{8} - 112001359983 T^{9} + 1252735431022 T^{10} - 12783502348115 T^{11} + 126737259090406 T^{12} - 1166087086221651 T^{13} + 10393498646127598 T^{14} - 86613947283856460 T^{15} + 698885105370952532 T^{16} - 5283450784315244060 T^{17} + 38674208462240792158 T^{18} - \)\(26\!\cdots\!31\)\( T^{19} + \)\(17\!\cdots\!46\)\( T^{20} - \)\(10\!\cdots\!15\)\( T^{21} + \)\(64\!\cdots\!42\)\( T^{22} - \)\(35\!\cdots\!43\)\( T^{23} + \)\(18\!\cdots\!81\)\( T^{24} - \)\(88\!\cdots\!10\)\( T^{25} + \)\(42\!\cdots\!52\)\( T^{26} - \)\(16\!\cdots\!72\)\( T^{27} + \)\(68\!\cdots\!42\)\( T^{28} - \)\(19\!\cdots\!26\)\( T^{29} + \)\(71\!\cdots\!84\)\( T^{30} - \)\(11\!\cdots\!19\)\( T^{31} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 - 33 T + 949 T^{2} - 18749 T^{3} + 341207 T^{4} - 5197800 T^{5} + 75073803 T^{6} - 969967468 T^{7} + 12034150648 T^{8} - 137435325418 T^{9} + 1515871577664 T^{10} - 15620337250128 T^{11} + 156015993236885 T^{12} - 1468272486068645 T^{13} + 13422503694450576 T^{14} - 116108234925557999 T^{15} + 976415923955764262 T^{16} - 7779251740012385933 T^{17} + 60253619084388635664 T^{18} - \)\(44\!\cdots\!35\)\( T^{19} + \)\(31\!\cdots\!85\)\( T^{20} - \)\(21\!\cdots\!96\)\( T^{21} + \)\(13\!\cdots\!16\)\( T^{22} - \)\(83\!\cdots\!14\)\( T^{23} + \)\(48\!\cdots\!68\)\( T^{24} - \)\(26\!\cdots\!96\)\( T^{25} + \)\(13\!\cdots\!47\)\( T^{26} - \)\(63\!\cdots\!00\)\( T^{27} + \)\(27\!\cdots\!27\)\( T^{28} - \)\(10\!\cdots\!63\)\( T^{29} + \)\(34\!\cdots\!21\)\( T^{30} - \)\(81\!\cdots\!19\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( 1 + 5 T + 524 T^{2} + 2352 T^{3} + 149201 T^{4} + 595217 T^{5} + 29572318 T^{6} + 103995194 T^{7} + 4508719255 T^{8} + 13939173581 T^{9} + 557562404560 T^{10} + 1520509439145 T^{11} + 57725098788352 T^{12} + 140465013275417 T^{13} + 5102567896652094 T^{14} + 11293730464414365 T^{15} + 389271182680022622 T^{16} + 801854862973419915 T^{17} + 25722044767023205854 T^{18} + 50273973366417773887 T^{19} + \)\(14\!\cdots\!12\)\( T^{20} + \)\(27\!\cdots\!95\)\( T^{21} + \)\(71\!\cdots\!60\)\( T^{22} + \)\(12\!\cdots\!71\)\( T^{23} + \)\(29\!\cdots\!55\)\( T^{24} + \)\(47\!\cdots\!14\)\( T^{25} + \)\(96\!\cdots\!18\)\( T^{26} + \)\(13\!\cdots\!07\)\( T^{27} + \)\(24\!\cdots\!41\)\( T^{28} + \)\(27\!\cdots\!72\)\( T^{29} + \)\(43\!\cdots\!44\)\( T^{30} + \)\(29\!\cdots\!55\)\( T^{31} + \)\(41\!\cdots\!21\)\( T^{32} \)
$73$ \( 1 - 23 T + 1033 T^{2} - 19008 T^{3} + 492054 T^{4} - 7612833 T^{5} + 146285379 T^{6} - 1961092652 T^{7} + 30725787368 T^{8} - 363799575798 T^{9} + 4869148361641 T^{10} - 51538421046465 T^{11} + 604954307644026 T^{12} - 5765014485858262 T^{13} + 60276142525015787 T^{14} - 518759265856954575 T^{15} + 4874762472412736270 T^{16} - 37869426407557683975 T^{17} + \)\(32\!\cdots\!23\)\( T^{18} - \)\(22\!\cdots\!54\)\( T^{19} + \)\(17\!\cdots\!66\)\( T^{20} - \)\(10\!\cdots\!45\)\( T^{21} + \)\(73\!\cdots\!49\)\( T^{22} - \)\(40\!\cdots\!06\)\( T^{23} + \)\(24\!\cdots\!08\)\( T^{24} - \)\(11\!\cdots\!76\)\( T^{25} + \)\(62\!\cdots\!71\)\( T^{26} - \)\(23\!\cdots\!41\)\( T^{27} + \)\(11\!\cdots\!34\)\( T^{28} - \)\(31\!\cdots\!64\)\( T^{29} + \)\(12\!\cdots\!97\)\( T^{30} - \)\(20\!\cdots\!11\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 - 24 T + 976 T^{2} - 17809 T^{3} + 423439 T^{4} - 6352844 T^{5} + 113351534 T^{6} - 1460320242 T^{7} + 21484645028 T^{8} - 244361185721 T^{9} + 3116047394294 T^{10} - 31936971160069 T^{11} + 364590920008457 T^{12} - 3423996579040382 T^{13} + 35802802462447500 T^{14} - 311998492413383683 T^{15} + 3031192079903180294 T^{16} - 24647880900657310957 T^{17} + \)\(22\!\cdots\!00\)\( T^{18} - \)\(16\!\cdots\!98\)\( T^{19} + \)\(14\!\cdots\!17\)\( T^{20} - \)\(98\!\cdots\!31\)\( T^{21} + \)\(75\!\cdots\!74\)\( T^{22} - \)\(46\!\cdots\!39\)\( T^{23} + \)\(32\!\cdots\!08\)\( T^{24} - \)\(17\!\cdots\!98\)\( T^{25} + \)\(10\!\cdots\!34\)\( T^{26} - \)\(47\!\cdots\!76\)\( T^{27} + \)\(25\!\cdots\!99\)\( T^{28} - \)\(83\!\cdots\!51\)\( T^{29} + \)\(35\!\cdots\!56\)\( T^{30} - \)\(69\!\cdots\!76\)\( T^{31} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 7 T + 721 T^{2} - 5329 T^{3} + 268417 T^{4} - 1996258 T^{5} + 67821703 T^{6} - 490771140 T^{7} + 12924366426 T^{8} - 89194290838 T^{9} + 1962515432901 T^{10} - 12771659562060 T^{11} + 245541472047935 T^{12} - 1496562645439495 T^{13} + 25858985819788019 T^{14} - 146790708985858061 T^{15} + 2320478010739533098 T^{16} - 12183628845826219063 T^{17} + \)\(17\!\cdots\!91\)\( T^{18} - \)\(85\!\cdots\!65\)\( T^{19} + \)\(11\!\cdots\!35\)\( T^{20} - \)\(50\!\cdots\!80\)\( T^{21} + \)\(64\!\cdots\!69\)\( T^{22} - \)\(24\!\cdots\!26\)\( T^{23} + \)\(29\!\cdots\!66\)\( T^{24} - \)\(91\!\cdots\!20\)\( T^{25} + \)\(10\!\cdots\!47\)\( T^{26} - \)\(25\!\cdots\!86\)\( T^{27} + \)\(28\!\cdots\!37\)\( T^{28} - \)\(47\!\cdots\!27\)\( T^{29} + \)\(53\!\cdots\!09\)\( T^{30} - \)\(42\!\cdots\!49\)\( T^{31} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 + 2 T + 737 T^{2} + 1433 T^{3} + 278119 T^{4} + 460613 T^{5} + 71322669 T^{6} + 92702047 T^{7} + 13910198950 T^{8} + 13584922067 T^{9} + 2189781117450 T^{10} + 1600640366051 T^{11} + 287863420852619 T^{12} + 164145436153087 T^{13} + 32189395580626840 T^{14} + 15491547806714788 T^{15} + 3088744141027715150 T^{16} + 1378747754797616132 T^{17} + \)\(25\!\cdots\!40\)\( T^{18} + \)\(11\!\cdots\!03\)\( T^{19} + \)\(18\!\cdots\!79\)\( T^{20} + \)\(89\!\cdots\!99\)\( T^{21} + \)\(10\!\cdots\!50\)\( T^{22} + \)\(60\!\cdots\!43\)\( T^{23} + \)\(54\!\cdots\!50\)\( T^{24} + \)\(32\!\cdots\!23\)\( T^{25} + \)\(22\!\cdots\!69\)\( T^{26} + \)\(12\!\cdots\!57\)\( T^{27} + \)\(68\!\cdots\!99\)\( T^{28} + \)\(31\!\cdots\!77\)\( T^{29} + \)\(14\!\cdots\!17\)\( T^{30} + \)\(34\!\cdots\!98\)\( T^{31} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( 1 - 33 T + 1684 T^{2} - 42506 T^{3} + 1258657 T^{4} - 25952915 T^{5} + 570633520 T^{6} - 9975451732 T^{7} + 178170946482 T^{8} - 2701117024970 T^{9} + 40944616452912 T^{10} - 545983773475415 T^{11} + 7196188837938271 T^{12} - 85124759399088460 T^{13} + 989278876034505676 T^{14} - 10425323056413780161 T^{15} + \)\(10\!\cdots\!10\)\( T^{16} - \)\(10\!\cdots\!17\)\( T^{17} + \)\(93\!\cdots\!84\)\( T^{18} - \)\(77\!\cdots\!80\)\( T^{19} + \)\(63\!\cdots\!51\)\( T^{20} - \)\(46\!\cdots\!55\)\( T^{21} + \)\(34\!\cdots\!48\)\( T^{22} - \)\(21\!\cdots\!10\)\( T^{23} + \)\(13\!\cdots\!02\)\( T^{24} - \)\(75\!\cdots\!44\)\( T^{25} + \)\(42\!\cdots\!80\)\( T^{26} - \)\(18\!\cdots\!95\)\( T^{27} + \)\(87\!\cdots\!37\)\( T^{28} - \)\(28\!\cdots\!62\)\( T^{29} + \)\(10\!\cdots\!96\)\( T^{30} - \)\(20\!\cdots\!69\)\( T^{31} + \)\(61\!\cdots\!21\)\( T^{32} \)
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