Properties

Label 252.9.d.d
Level $252$
Weight $9$
Character orbit 252.d
Analytic conductor $102.659$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} - 17126006720262 x^{3} + 1218271024906002 x^{2} - 21704094932515380 x + 391152460437174276\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{15}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{1} + \beta_{6} ) q^{5} + ( -234 - 2 \beta_{1} + \beta_{7} ) q^{7} +O(q^{10})\) \( q + ( \beta_{1} + \beta_{6} ) q^{5} + ( -234 - 2 \beta_{1} + \beta_{7} ) q^{7} + ( 3760 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( -14 \beta_{1} - \beta_{5} - 26 \beta_{6} - \beta_{7} - 5 \beta_{8} ) q^{13} + ( -104 \beta_{1} + 12 \beta_{5} - 73 \beta_{6} + 12 \beta_{7} + 2 \beta_{9} ) q^{17} + ( 98 \beta_{1} - 21 \beta_{5} + 7 \beta_{6} - 21 \beta_{7} + 19 \beta_{8} + 3 \beta_{9} ) q^{19} + ( 2234 - 3 \beta_{2} + 38 \beta_{3} + 6 \beta_{5} - 6 \beta_{7} ) q^{23} + ( -30402 + \beta_{2} + 18 \beta_{3} + 51 \beta_{4} - 21 \beta_{5} + 21 \beta_{7} ) q^{25} + ( 30898 - 3 \beta_{2} - 53 \beta_{3} + 87 \beta_{4} - 51 \beta_{5} + 51 \beta_{7} ) q^{29} + ( 1403 \beta_{1} + 42 \beta_{5} + 117 \beta_{6} + 42 \beta_{7} - 33 \beta_{8} + 33 \beta_{9} ) q^{31} + ( 148006 + 2273 \beta_{1} - 3 \beta_{2} - 38 \beta_{3} - 156 \beta_{4} - 129 \beta_{5} - 701 \beta_{6} + 21 \beta_{7} + 120 \beta_{8} + 19 \beta_{9} ) q^{35} + ( -547019 + 5 \beta_{2} + 240 \beta_{3} + 87 \beta_{4} + 311 \beta_{5} - 311 \beta_{7} ) q^{37} + ( 1736 \beta_{1} - 318 \beta_{5} - 433 \beta_{6} - 318 \beta_{7} + 546 \beta_{8} - 66 \beta_{9} ) q^{41} + ( 117764 + 46 \beta_{2} - 78 \beta_{3} + 174 \beta_{4} - \beta_{5} + \beta_{7} ) q^{43} + ( -50 \beta_{1} + 696 \beta_{5} - 3716 \beta_{6} + 696 \beta_{7} - 510 \beta_{8} - 192 \beta_{9} ) q^{47} + ( -606102 - 5747 \beta_{1} + 49 \beta_{2} - 294 \beta_{3} + 147 \beta_{4} + 735 \beta_{5} - 1127 \beta_{6} - 483 \beta_{7} - 245 \beta_{8} + 147 \beta_{9} ) q^{49} + ( 13692 - 87 \beta_{2} - 959 \beta_{3} + 311 \beta_{4} + 1225 \beta_{5} - 1225 \beta_{7} ) q^{53} + ( 8403 \beta_{1} - 1409 \beta_{5} + 12301 \beta_{6} - 1409 \beta_{7} - 125 \beta_{8} + 33 \beta_{9} ) q^{55} + ( -6385 \beta_{1} + 1704 \beta_{5} - 234 \beta_{6} + 1704 \beta_{7} + 1962 \beta_{8} - 214 \beta_{9} ) q^{59} + ( -18042 \beta_{1} + 817 \beta_{5} + 4358 \beta_{6} + 817 \beta_{7} - 2881 \beta_{8} + 6 \beta_{9} ) q^{61} + ( 10678694 - 150 \beta_{2} + 652 \beta_{3} - 1838 \beta_{4} - 2422 \beta_{5} + 2422 \beta_{7} ) q^{65} + ( -574824 + 106 \beta_{2} + 3408 \beta_{3} - 2232 \beta_{4} - 3541 \beta_{5} + 3541 \beta_{7} ) q^{67} + ( -2697022 - 87 \beta_{2} - 1660 \beta_{3} - 906 \beta_{4} + 3366 \beta_{5} - 3366 \beta_{7} ) q^{71} + ( -47688 \beta_{1} + 3798 \beta_{5} + 5902 \beta_{6} + 3798 \beta_{7} + 250 \beta_{8} + 384 \beta_{9} ) q^{73} + ( -4879030 - 46626 \beta_{1} - 63 \beta_{2} + 2387 \beta_{3} - 777 \beta_{4} - 2856 \beta_{5} - 22708 \beta_{6} + 4182 \beta_{7} - 1890 \beta_{8} + 7 \beta_{9} ) q^{77} + ( -18115732 + 34 \beta_{2} - 4296 \beta_{3} + 2664 \beta_{4} + 6613 \beta_{5} - 6613 \beta_{7} ) q^{79} + ( 45581 \beta_{1} + 2820 \beta_{5} + 31580 \beta_{6} + 2820 \beta_{7} + 1650 \beta_{8} + 288 \beta_{9} ) q^{83} + ( 33776345 - 111 \beta_{2} - 882 \beta_{3} - 7173 \beta_{4} + 4091 \beta_{5} - 4091 \beta_{7} ) q^{85} + ( -80328 \beta_{1} + 5160 \beta_{5} - 35501 \beta_{6} + 5160 \beta_{7} - 7470 \beta_{8} + 224 \beta_{9} ) q^{89} + ( 6767312 + 34964 \beta_{1} - 348 \beta_{2} + 198 \beta_{3} + 8658 \beta_{4} + 6694 \beta_{5} - 19527 \beta_{6} - 1302 \beta_{7} - 633 \beta_{8} - 99 \beta_{9} ) q^{91} + ( -10319612 + 678 \beta_{2} - 372 \beta_{3} + 10624 \beta_{4} + 21674 \beta_{5} - 21674 \beta_{7} ) q^{95} + ( 2590 \beta_{1} + 6160 \beta_{5} + 15300 \beta_{6} + 6160 \beta_{7} + 4590 \beta_{8} - 1122 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2338 q^{7} + O(q^{10}) \) \( 10 q - 2338 q^{7} + 37596 q^{11} + 22380 q^{23} - 303998 q^{25} + 308892 q^{29} + 1480584 q^{35} - 5471108 q^{37} + 1177324 q^{43} - 6064142 q^{49} + 129132 q^{53} + 106801008 q^{65} - 5722372 q^{67} - 26985540 q^{71} - 48770148 q^{77} - 181197556 q^{79} + 337759224 q^{85} + 67638816 q^{91} - 103302096 q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 14312 x^{8} - 30343 x^{7} + 170123918 x^{6} - 875537263 x^{5} + 496509566533 x^{4} - 17126006720262 x^{3} + 1218271024906002 x^{2} - 21704094932515380 x + 391152460437174276\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-22715830057195306325850114092 \nu^{9} + 105344193606505886858204728016 \nu^{8} - 326362347151646061203891002393060 \nu^{7} + 679478600273813972761191946164980 \nu^{6} - 3876324503307566817030379016370440932 \nu^{5} + 19740580908706479331011900792415421108 \nu^{4} - 11491741615025619440195784281354899030520 \nu^{3} + 227318250075341369289072573739585923721500 \nu^{2} - 28289993221030065480266085484690730477600592 \nu + 255987398899649976872571592339342236014319732\)\()/ \)\(22\!\cdots\!97\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-140695515108095072925245 \nu^{9} - 882084660291498210175777915 \nu^{8} - 6888290006688132327041905840 \nu^{7} - 12422330426067657380918271286189 \nu^{6} - 104138811596915822595890454134920 \nu^{5} - 149981716788947860606165876051182685 \nu^{4} - 747314023253154010210479038120625311 \nu^{3} - 433485327004852791133312478012142614970 \nu^{2} + 7899880671544322217206409884931561034320 \nu - 905013424346862845062522854602966323762392\)\()/ \)\(58\!\cdots\!54\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-5135404741358155394363806 \nu^{9} - 194830021937554840263409099 \nu^{8} - 1521447717582787702129949104 \nu^{7} - 4485769162593752036394598137115 \nu^{6} - 23001609552163702867182255263752 \nu^{5} - 33127139023782985912736921664421261 \nu^{4} + 6497815145946161055828400682895822744 \nu^{3} - 95745861561693925884627343188178656282 \nu^{2} + 1744882315527871994934227459980895149392 \nu - 7217779739844659793283234239909517101444\)\()/ \)\(16\!\cdots\!12\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-5118588525803755880590936 \nu^{9} + 2735505860707389085202993535 \nu^{8} + 21361847146645296076486533360 \nu^{7} + 39374678147602833506595902436243 \nu^{6} + 322953501261801199127113312744680 \nu^{5} + 465120734714444329724807655018623865 \nu^{4} + 9762905784020766439273571640223439746 \nu^{3} + 1344317281473376677975058422477088646130 \nu^{2} - 24498974813548067741096587254686629067280 \nu + 1430490330729750894958779198042407636814024\)\()/ \)\(81\!\cdots\!56\)\( \)
\(\beta_{5}\)\(=\)\((\)\(8254248497279727596255182450630360 \nu^{9} + 71376433178111413850471260662027313 \nu^{8} + 125935298315221474610623591775231627582 \nu^{7} + 886896434211297266626991642504181536953 \nu^{6} + 1535230621675173662751277319572502056589330 \nu^{5} + 6801291729124384703748949527129939951920371 \nu^{4} + 5678067022678665627605871812102227547565934660 \nu^{3} - 107771304046400475288502088005243699871623280010 \nu^{2} + 13641880260387463796648679369996652225914486583944 \nu - 165592763278623251927118836070024565130451416918028\)\()/ \)\(10\!\cdots\!12\)\( \)
\(\beta_{6}\)\(=\)\((\)\(612548716544573251155006223017524 \nu^{9} + 10607735151878610244067060573463763 \nu^{8} + 8480576739553125177350735208629476114 \nu^{7} + 115124189102507255255335967582955849523 \nu^{6} + 100203989615208188099745431862551824252270 \nu^{5} + 993550001262344596259756820713464292096041 \nu^{4} + 248227881583087960239200566494527245918408640 \nu^{3} - 8715124381504477036041794233334003801414288286 \nu^{2} + 413843246119068963565503196851274552988607341544 \nu - 4175488307294017980274945676070337892433177220164\)\()/ \)\(73\!\cdots\!08\)\( \)
\(\beta_{7}\)\(=\)\((\)\(8969149098708362739021114570300808 \nu^{9} + 60532125626337139535330546075731201 \nu^{8} + 125850614000097284542932148350547988030 \nu^{7} + 784642829700919693983802684222436442601 \nu^{6} + 1533950343996770283671389255339394486999954 \nu^{5} + 4957423518713636300038603365584452307888803 \nu^{4} + 4278924345846930032110808666310128643405388852 \nu^{3} - 113100552379172297372603859848242435279503850026 \nu^{2} + 13739001023825559625494372952447993564918238453640 \nu - 89534970068373157896341275262909382366266581643340\)\()/ \)\(10\!\cdots\!12\)\( \)
\(\beta_{8}\)\(=\)\((\)\(21360868759590956507072048376973800 \nu^{9} + 256730314485598794123730264965728681 \nu^{8} + 298802854199258576719697604127708734054 \nu^{7} + 2966639848338038863882645193106025764353 \nu^{6} + 3507672372369505133019751550028156364612986 \nu^{5} + 23934909199132693133851998534577208461744691 \nu^{4} + 9041441100305369007813914880802257854805902316 \nu^{3} - 286757895538723064797277146091074662991981149570 \nu^{2} + 16666008831284924419093955641118485607271771493032 \nu - 163076766797024481868226036394815640618582207103332\)\()/ \)\(25\!\cdots\!28\)\( \)
\(\beta_{9}\)\(=\)\((\)\(19917400937634063960868470171623765 \nu^{9} + 51673003079893434165107861206923877 \nu^{8} + 276747551386333334995358859296626340848 \nu^{7} + 989796274975492066667822630673365478235 \nu^{6} + 3236895486026941590910561733345701531071556 \nu^{5} + 3354373944062505310959742667227253530978867 \nu^{4} + 8078655652374519967711905183713010513312539195 \nu^{3} - 236899906794230892271960523940830439501857996394 \nu^{2} + 13903815418447126650041704084138969088677779483036 \nu - 132422452148323603735525731322360382650880502344940\)\()/ \)\(12\!\cdots\!14\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{9} + 12 \beta_{8} + 21 \beta_{7} - 75 \beta_{6} - 15 \beta_{5} - 31 \beta_{1} + 144\)\()/1512\)
\(\nu^{2}\)\(=\)\((\)\(-156 \beta_{9} - 258 \beta_{8} - 990 \beta_{7} + 3156 \beta_{6} - 1218 \beta_{5} - 126 \beta_{4} + 252 \beta_{3} - 42 \beta_{2} - 38257 \beta_{1} - 4327902\)\()/1512\)
\(\nu^{3}\)\(=\)\((\)\(-4177 \beta_{7} + 4177 \beta_{5} - 236 \beta_{4} - 424 \beta_{3} - 46 \beta_{2} + 88296\)\()/18\)
\(\nu^{4}\)\(=\)\((\)\(2360298 \beta_{9} + 1884570 \beta_{8} + 11791416 \beta_{7} - 44267694 \beta_{6} + 11683260 \beta_{5} - 1932966 \beta_{4} + 3936492 \beta_{3} - 487914 \beta_{2} + 356776717 \beta_{1} - 40940647038\)\()/1512\)
\(\nu^{5}\)\(=\)\((\)\(288162975 \beta_{9} - 1539687612 \beta_{8} + 2895386511 \beta_{7} + 8697280971 \beta_{6} - 881555157 \beta_{5} + 140970396 \beta_{4} + 256540200 \beta_{3} + 27199032 \beta_{2} + 5781327631 \beta_{1} + 359061858468\)\()/1512\)
\(\nu^{6}\)\(=\)\((\)\(295509413 \beta_{7} - 295509413 \beta_{5} + 552613029 \beta_{4} - 1136452554 \beta_{3} + 131168311 \beta_{2} + 10386708923577\)\()/18\)
\(\nu^{7}\)\(=\)\((\)\(-3244728644013 \beta_{9} + 17169107512296 \beta_{8} + 8134927877979 \beta_{7} - 95657156087169 \beta_{6} - 33625307929305 \beta_{5} + 1732305478008 \beta_{4} + 2935906212624 \beta_{3} + 343082029668 \beta_{2} - 95278369593991 \beta_{1} + 7496152774289568\)\()/1512\)
\(\nu^{8}\)\(=\)\((\)\(-360210090197190 \beta_{9} - 38474555411130 \beta_{8} - 1713581966291688 \beta_{7} + 5024419910498274 \beta_{6} - 1177716692539380 \beta_{5} - 269858762639046 \beta_{4} + 538124206347756 \beta_{3} - 62852268585498 \beta_{2} - 42078487349648437 \beta_{1} - 4819447486926447534\)\()/1512\)
\(\nu^{9}\)\(=\)\((\)\(-5553250483737853 \beta_{7} + 5553250483737853 \beta_{5} - 496654370929802 \beta_{4} - 742347711340156 \beta_{3} - 100260922182328 \beta_{2} - 2750036276421755886\)\()/18\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Embeddings

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

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

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
19.5272 + 33.8221i
−51.3922 + 89.0139i
−32.7935 56.7999i
10.9569 18.9779i
54.2016 + 93.8799i
54.2016 93.8799i
10.9569 + 18.9779i
−32.7935 + 56.7999i
−51.3922 89.0139i
19.5272 33.8221i
0 0 0 1053.92i 0 582.142 + 2329.36i 0 0 0
181.2 0 0 0 803.413i 0 −2396.47 + 147.366i 0 0 0
181.3 0 0 0 429.816i 0 −1615.33 1776.38i 0 0 0
181.4 0 0 0 307.159i 0 222.189 2390.70i 0 0 0
181.5 0 0 0 264.232i 0 2038.47 1268.64i 0 0 0
181.6 0 0 0 264.232i 0 2038.47 + 1268.64i 0 0 0
181.7 0 0 0 307.159i 0 222.189 + 2390.70i 0 0 0
181.8 0 0 0 429.816i 0 −1615.33 + 1776.38i 0 0 0
181.9 0 0 0 803.413i 0 −2396.47 147.366i 0 0 0
181.10 0 0 0 1053.92i 0 582.142 2329.36i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.9.d.d 10
3.b odd 2 1 84.9.d.a 10
7.b odd 2 1 inner 252.9.d.d 10
12.b even 2 1 336.9.f.a 10
21.c even 2 1 84.9.d.a 10
84.h odd 2 1 336.9.f.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.d.a 10 3.b odd 2 1
84.9.d.a 10 21.c even 2 1
252.9.d.d 10 1.a even 1 1 trivial
252.9.d.d 10 7.b odd 2 1 inner
336.9.f.a 10 12.b even 2 1
336.9.f.a 10 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 2105124 T_{5}^{8} + \)\(13\!\cdots\!24\)\( T_{5}^{6} + \)\(31\!\cdots\!68\)\( T_{5}^{4} + \)\(28\!\cdots\!00\)\( T_{5}^{2} + \)\(87\!\cdots\!00\)\( \) acting on \(S_{9}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( T^{10} \)
$5$ \( \)\(87\!\cdots\!00\)\( + \)\(28\!\cdots\!00\)\( T^{2} + 316199300854863168 T^{4} + 1366627233924 T^{6} + 2105124 T^{8} + T^{10} \)
$7$ \( \)\(63\!\cdots\!01\)\( + \)\(25\!\cdots\!38\)\( T + \)\(11\!\cdots\!93\)\( T^{2} + \)\(68\!\cdots\!24\)\( T^{3} + \)\(15\!\cdots\!06\)\( T^{4} + 48744189974570076 T^{5} + 26180372439606 T^{6} + 20538691824 T^{7} + 5765193 T^{8} + 2338 T^{9} + T^{10} \)
$11$ \( ( 18689723376907771728 + 12724758338896728 T + 1771518821940 T^{2} - 120395826 T^{3} - 18798 T^{4} + T^{5} )^{2} \)
$13$ \( \)\(16\!\cdots\!88\)\( + \)\(79\!\cdots\!64\)\( T^{2} + \)\(10\!\cdots\!64\)\( T^{4} + 5121446888629021248 T^{6} + 4349063232 T^{8} + T^{10} \)
$17$ \( \)\(52\!\cdots\!68\)\( + \)\(55\!\cdots\!76\)\( T^{2} + \)\(21\!\cdots\!36\)\( T^{4} + \)\(39\!\cdots\!52\)\( T^{6} + 33037951668 T^{8} + T^{10} \)
$19$ \( \)\(17\!\cdots\!52\)\( + \)\(97\!\cdots\!12\)\( T^{2} + \)\(32\!\cdots\!92\)\( T^{4} + \)\(29\!\cdots\!48\)\( T^{6} + 93797905320 T^{8} + T^{10} \)
$23$ \( ( -\)\(99\!\cdots\!48\)\( + \)\(17\!\cdots\!00\)\( T + 9804207125002068 T^{2} - 265054586706 T^{3} - 11190 T^{4} + T^{5} )^{2} \)
$29$ \( ( \)\(76\!\cdots\!32\)\( + \)\(32\!\cdots\!12\)\( T - 88506331549109712 T^{2} - 1362442041576 T^{3} - 154446 T^{4} + T^{5} )^{2} \)
$31$ \( \)\(21\!\cdots\!92\)\( + \)\(28\!\cdots\!04\)\( T^{2} + \)\(12\!\cdots\!08\)\( T^{4} + \)\(14\!\cdots\!72\)\( T^{6} + 3329499569760 T^{8} + T^{10} \)
$37$ \( ( -\)\(88\!\cdots\!00\)\( - \)\(29\!\cdots\!60\)\( T - 30552823185723122248 T^{2} - 7689283819892 T^{3} + 2735554 T^{4} + T^{5} )^{2} \)
$41$ \( \)\(25\!\cdots\!00\)\( + \)\(59\!\cdots\!04\)\( T^{2} + \)\(38\!\cdots\!88\)\( T^{4} + \)\(68\!\cdots\!64\)\( T^{6} + 45642885508500 T^{8} + T^{10} \)
$43$ \( ( -\)\(15\!\cdots\!52\)\( + \)\(19\!\cdots\!32\)\( T + 5028899381236620584 T^{2} - 31717906384028 T^{3} - 588662 T^{4} + T^{5} )^{2} \)
$47$ \( \)\(76\!\cdots\!88\)\( + \)\(18\!\cdots\!52\)\( T^{2} + \)\(27\!\cdots\!04\)\( T^{4} + \)\(12\!\cdots\!88\)\( T^{6} + 197441545003296 T^{8} + T^{10} \)
$53$ \( ( \)\(68\!\cdots\!40\)\( + \)\(94\!\cdots\!08\)\( T - \)\(13\!\cdots\!84\)\( T^{2} - 201420882958104 T^{3} - 64566 T^{4} + T^{5} )^{2} \)
$59$ \( \)\(12\!\cdots\!08\)\( + \)\(68\!\cdots\!80\)\( T^{2} + \)\(19\!\cdots\!84\)\( T^{4} + \)\(18\!\cdots\!96\)\( T^{6} + 748454294068656 T^{8} + T^{10} \)
$61$ \( \)\(62\!\cdots\!32\)\( + \)\(11\!\cdots\!48\)\( T^{2} + \)\(32\!\cdots\!20\)\( T^{4} + \)\(30\!\cdots\!24\)\( T^{6} + 1037467907874288 T^{8} + T^{10} \)
$67$ \( ( -\)\(43\!\cdots\!52\)\( + \)\(72\!\cdots\!68\)\( T + \)\(26\!\cdots\!24\)\( T^{2} - 1700255621008916 T^{3} + 2861186 T^{4} + T^{5} )^{2} \)
$71$ \( ( \)\(22\!\cdots\!52\)\( + \)\(71\!\cdots\!64\)\( T - \)\(58\!\cdots\!32\)\( T^{2} - 622511191807938 T^{3} + 13492770 T^{4} + T^{5} )^{2} \)
$73$ \( \)\(86\!\cdots\!00\)\( + \)\(65\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!92\)\( T^{4} + \)\(78\!\cdots\!96\)\( T^{6} + 1722914019623952 T^{8} + T^{10} \)
$79$ \( ( -\)\(10\!\cdots\!96\)\( - \)\(28\!\cdots\!88\)\( T - \)\(12\!\cdots\!20\)\( T^{2} + 461227617072652 T^{3} + 90598778 T^{4} + T^{5} )^{2} \)
$83$ \( \)\(15\!\cdots\!00\)\( + \)\(35\!\cdots\!00\)\( T^{2} + \)\(18\!\cdots\!32\)\( T^{4} + \)\(37\!\cdots\!00\)\( T^{6} + 3244136671767120 T^{8} + T^{10} \)
$89$ \( \)\(84\!\cdots\!00\)\( + \)\(34\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!52\)\( T^{4} + \)\(21\!\cdots\!24\)\( T^{6} + 11173236604151796 T^{8} + T^{10} \)
$97$ \( \)\(90\!\cdots\!72\)\( + \)\(28\!\cdots\!64\)\( T^{2} + \)\(15\!\cdots\!88\)\( T^{4} + \)\(20\!\cdots\!60\)\( T^{6} + 8238372664007376 T^{8} + T^{10} \)
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