Properties

Label 2736.3.m.e
Level $2736$
Weight $3$
Character orbit 2736.m
Analytic conductor $74.551$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.m (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta_{1} ) q^{5} -\beta_{10} q^{7} +O(q^{10})\) \( q + ( 1 + \beta_{1} ) q^{5} -\beta_{10} q^{7} + ( \beta_{7} + \beta_{8} - \beta_{9} ) q^{11} + ( 3 - \beta_{1} - \beta_{4} ) q^{13} + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} ) q^{17} + \beta_{8} q^{19} + ( \beta_{5} + 4 \beta_{8} + 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{23} + ( 1 + 2 \beta_{1} + \beta_{4} ) q^{25} + ( 8 - 2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} - \beta_{6} ) q^{29} + ( 4 \beta_{5} - \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + 3 \beta_{10} + \beta_{11} ) q^{31} + ( -3 \beta_{5} + 4 \beta_{8} - \beta_{11} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} ) q^{37} + ( 1 + 4 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{41} + ( -2 \beta_{5} + \beta_{7} - 4 \beta_{8} + 2 \beta_{9} - \beta_{10} ) q^{43} + ( -3 \beta_{7} + 10 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{47} + ( -9 + 6 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{6} ) q^{49} + ( -19 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 6 \beta_{4} + \beta_{6} ) q^{53} + ( -4 \beta_{5} - 6 \beta_{7} - 8 \beta_{8} - 4 \beta_{9} - 3 \beta_{10} - 2 \beta_{11} ) q^{55} + ( \beta_{5} - 6 \beta_{7} + 4 \beta_{8} - 5 \beta_{10} - 3 \beta_{11} ) q^{59} + ( 30 - 3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 5 \beta_{6} ) q^{61} + ( -13 - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{65} + ( -7 \beta_{5} + 6 \beta_{7} - 4 \beta_{8} - 4 \beta_{9} - \beta_{10} - 5 \beta_{11} ) q^{67} + ( -3 \beta_{5} - 2 \beta_{7} + 12 \beta_{8} - 4 \beta_{9} - 5 \beta_{10} + \beta_{11} ) q^{71} + ( -14 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{73} + ( 25 - 4 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} ) q^{77} + ( 3 \beta_{5} + 3 \beta_{7} - 5 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} ) q^{79} + ( -\beta_{5} + \beta_{7} + 5 \beta_{8} + 3 \beta_{9} + 6 \beta_{10} - \beta_{11} ) q^{83} + ( 10 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} + 7 \beta_{6} ) q^{85} + ( 35 + 8 \beta_{1} - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + 3 \beta_{6} ) q^{89} + ( -\beta_{5} - 16 \beta_{7} - 6 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} + 3 \beta_{11} ) q^{91} + ( -2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{95} + ( 21 + 12 \beta_{1} + 3 \beta_{2} + \beta_{3} - 2 \beta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 10q^{5} + O(q^{10}) \) \( 12q + 10q^{5} + 36q^{13} - 14q^{17} + 10q^{25} + 108q^{29} - 16q^{37} - 16q^{41} - 118q^{49} - 220q^{53} + 366q^{61} - 140q^{65} - 158q^{73} + 286q^{77} + 98q^{85} + 396q^{89} + 224q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} - 195494 x^{4} + 302996 x^{3} + 602332 x^{2} - 2263536 x + 2052928\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(51997441325289764620921 \nu^{11} - 73459412627167568758689 \nu^{10} - 376368559884857089738052 \nu^{9} - 2074383187958425838207906 \nu^{8} - 22310969002623736017588684 \nu^{7} - 34589113615694853943215088 \nu^{6} + 981992577991514469460050707 \nu^{5} + 957764877347901693193909925 \nu^{4} - 11890092294807088563695513768 \nu^{3} - 639691616520158881950125586 \nu^{2} + 53541427663219666647291675720 \nu - 73658615237434719847743352844\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-64153428683245785818187 \nu^{11} + 310947154005637673724987 \nu^{10} + 488084674761945636311292 \nu^{9} + 3561821912524932089148298 \nu^{8} + 25212171466145498186618560 \nu^{7} - 57722877927354117656914608 \nu^{6} - 1517499010423618250065755245 \nu^{5} + 1027241616247046163589843453 \nu^{4} + 16596882951535903126145714784 \nu^{3} - 25465637922911669259130524290 \nu^{2} - 8600983603266963856792147704 \nu + 123375392685516809200436072248\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{3}\)\(=\)\((\)\(-84892865770599251162339 \nu^{11} + 168118329099878366426107 \nu^{10} + 723012291447320513297992 \nu^{9} + 3533037606014996393196298 \nu^{8} + 36195392263413134067880368 \nu^{7} + 35879927549824755904232704 \nu^{6} - 1743035770545872875094948277 \nu^{5} - 1050959033981795356921783923 \nu^{4} + 20628493787297198541294186596 \nu^{3} - 5680893582225625367089468930 \nu^{2} - 75799513659359725579054177144 \nu + 163507193570116392877294745476\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{4}\)\(=\)\((\)\(99544283707287549393653 \nu^{11} - 125181146363758083951437 \nu^{10} - 406718968367570415284520 \nu^{9} - 3277699230573006098902810 \nu^{8} - 42198570170669361261183364 \nu^{7} - 78026685225665428629074512 \nu^{6} + 1710207624740396366280405263 \nu^{5} + 1205326107019945794872851649 \nu^{4} - 19870118216551028486489402484 \nu^{3} + 8685774484586922238019223606 \nu^{2} + 63568101195091550952213604008 \nu - 139555867932238585032328422144\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-168258373562793920591915 \nu^{11} - 124228512679122943003347 \nu^{10} + 150730219315212471580168 \nu^{9} + 6092808024956870779887502 \nu^{8} + 83207860050661196690810404 \nu^{7} + 308650908404778384349773404 \nu^{6} - 2134627392568785921547380849 \nu^{5} - 5899944872959435029165030825 \nu^{4} + 17477848115776488615564996320 \nu^{3} + 17338965366054497389943460614 \nu^{2} - 68081701536668245024399137544 \nu + 69005215129095585460048480896\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-43893138267987812709820 \nu^{11} + 42447692262194396050588 \nu^{10} + 200797068306193098933167 \nu^{9} + 1504109505127442397885783 \nu^{8} + 18745988896273876487590893 \nu^{7} + 38519538601465730565312664 \nu^{6} - 742800474069023568368046031 \nu^{5} - 747091570485838504448531565 \nu^{4} + 8850090084279830046284853719 \nu^{3} - 818461976748029610960015862 \nu^{2} - 36171422778518273841909922184 \nu + 39176940743514085440262810034\)\()/ \)\(16\!\cdots\!27\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-157524610518805 \nu^{11} - 6067399667026 \nu^{10} + 201453765859666 \nu^{9} + 4883740095972916 \nu^{8} + 72437356465362456 \nu^{7} + 233340579213886720 \nu^{6} - 2172506813770474015 \nu^{5} - 3779165499267309010 \nu^{4} + 21733840351284574574 \nu^{3} - 3953241666051162292 \nu^{2} - 96285444021538449388 \nu + 153431969355162127352\)\()/ 4031297913092380756 \)
\(\beta_{8}\)\(=\)\((\)\(-237830692375 \nu^{11} - 6844355458 \nu^{10} + 326852665758 \nu^{9} + 7584570552796 \nu^{8} + 109922313318736 \nu^{7} + 355278937573312 \nu^{6} - 3287098690953053 \nu^{5} - 5754231775852978 \nu^{4} + 32921184637856658 \nu^{3} - 5841144248678836 \nu^{2} - 146057139468543444 \nu + 232304768678122312\)\()/ 4818191046081112 \)
\(\beta_{9}\)\(=\)\((\)\(880963784071120031416741 \nu^{11} + 370305367263462111837984 \nu^{10} - 1119685666340330206241094 \nu^{9} - 30174340328293560163623624 \nu^{8} - 422228084238779138708928296 \nu^{7} - 1485792635294709509696590776 \nu^{6} + 11652026428595855807054000215 \nu^{5} + 26778235012978911074266101356 \nu^{4} - 107522646401875155097270752602 \nu^{3} - 37621682456038514990396343464 \nu^{2} + 457040698985277454669966593540 \nu - 617940713562505870867010134792\)\()/ \)\(12\!\cdots\!16\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-882176042542487277342189 \nu^{11} - 304057356664857798652429 \nu^{10} + 1356706251598515074862448 \nu^{9} + 27976093570240698616039382 \nu^{8} + 418375543338988828669840648 \nu^{7} + 1427649909136390954899715516 \nu^{6} - 11860288114252872608120850355 \nu^{5} - 25496872577654676020423046395 \nu^{4} + 116824854937613937351872881128 \nu^{3} + 14295197238439818764591642738 \nu^{2} - 520469124215665830988970176408 \nu + 761462040151515795303297972096\)\()/ \)\(64\!\cdots\!08\)\( \)
\(\beta_{11}\)\(=\)\((\)\(745648445784192410477188 \nu^{11} + 226751202998417829946529 \nu^{10} - 1298069524273119551289236 \nu^{9} - 23220848667899750043359316 \nu^{8} - 354588534592121464881430098 \nu^{7} - 1188152538959713926738525240 \nu^{6} + 10093443958151896510700250922 \nu^{5} + 21131835060632637536868612345 \nu^{4} - 101435684173973780692532131252 \nu^{3} - 5028287258072877166764497038 \nu^{2} + 458192662062736796254079333436 \nu - 685272875849983540049645764328\)\()/ \)\(32\!\cdots\!54\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{9} - \beta_{8} - \beta_{5} - \beta_{4} + \beta_{1} + 1\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{11} - 3 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 5 \beta_{4} - 3 \beta_{3} - \beta_{2} - 7 \beta_{1}\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(4 \beta_{11} + 14 \beta_{10} + 20 \beta_{9} + 46 \beta_{8} - 61 \beta_{7} + 2 \beta_{6} + 14 \beta_{5} - 12 \beta_{4} - 10 \beta_{3} - 2 \beta_{1} + 32\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(-19 \beta_{11} - 55 \beta_{10} - \beta_{9} + 49 \beta_{8} - 19 \beta_{7} + 89 \beta_{6} + 42 \beta_{5} + 75 \beta_{4} + 76 \beta_{3} - 13 \beta_{2} + 230 \beta_{1} + 718\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-173 \beta_{11} - 461 \beta_{10} - 943 \beta_{9} - 979 \beta_{8} + 756 \beta_{7} + 140 \beta_{6} - 898 \beta_{5} + 43 \beta_{4} - 223 \beta_{3} + 55 \beta_{2} - 23 \beta_{1} + 2464\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-40 \beta_{11} - 488 \beta_{10} - 1704 \beta_{9} + 160 \beta_{8} - 628 \beta_{7} - 2681 \beta_{6} - 1680 \beta_{5} - 3834 \beta_{4} - 1133 \beta_{3} - 632 \beta_{2} - 4549 \beta_{1} - 27120\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(-989 \beta_{11} + 1759 \beta_{10} + 24805 \beta_{9} + 33945 \beta_{8} - 31016 \beta_{7} - 3368 \beta_{6} + 28894 \beta_{5} - 5227 \beta_{4} - 4433 \beta_{3} - 127 \beta_{2} - 9425 \beta_{1} - 20864\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(10239 \beta_{11} + 23999 \beta_{10} + 27437 \beta_{9} + 92599 \beta_{8} - 105587 \beta_{7} + 85287 \beta_{6} + 19282 \beta_{5} + 108035 \beta_{4} + 48638 \beta_{3} + 10767 \beta_{2} + 172752 \beta_{1} + 798374\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-146444 \beta_{11} - 426874 \beta_{10} - 861636 \beta_{9} - 1104378 \beta_{8} + 1043371 \beta_{7} + 143482 \beta_{6} - 784530 \beta_{5} + 185268 \beta_{4} + 181210 \beta_{3} - 5540 \beta_{2} + 424862 \beta_{1} + 874220\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-263331 \beta_{11} - 835819 \beta_{10} - 1880729 \beta_{9} - 1984355 \beta_{8} + 1768251 \beta_{7} - 2614417 \beta_{6} - 1756674 \beta_{5} - 3235969 \beta_{4} - 2145414 \beta_{3} - 132573 \beta_{2} - 6274188 \beta_{1} - 21212802\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(3045003 \beta_{11} + 10000603 \beta_{10} + 25128717 \beta_{9} + 36690833 \beta_{8} - 35739552 \beta_{7} - 6331648 \beta_{6} + 24330242 \beta_{5} - 7683217 \beta_{4} - 1946731 \beta_{3} - 1084305 \beta_{2} - 10843863 \beta_{1} - 66587876\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1711.1
−3.76811 + 2.72136i
−3.76811 2.72136i
−3.01700 + 0.588235i
−3.01700 0.588235i
2.21305 0.177784i
2.21305 + 0.177784i
4.06962 0.767506i
4.06962 + 0.767506i
1.30395 + 1.41343i
1.30395 1.41343i
0.198491 + 5.66831i
0.198491 5.66831i
0 0 0 −4.92825 0 9.95984i 0 0 0
1711.2 0 0 0 −4.92825 0 9.95984i 0 0 0
1711.3 0 0 0 −4.69015 0 3.50408i 0 0 0
1711.4 0 0 0 −4.69015 0 3.50408i 0 0 0
1711.5 0 0 0 −2.67107 0 11.9371i 0 0 0
1711.6 0 0 0 −2.67107 0 11.9371i 0 0 0
1711.7 0 0 0 4.65777 0 4.49506i 0 0 0
1711.8 0 0 0 4.65777 0 4.49506i 0 0 0
1711.9 0 0 0 6.08630 0 8.85163i 0 0 0
1711.10 0 0 0 6.08630 0 8.85163i 0 0 0
1711.11 0 0 0 6.54540 0 0.687253i 0 0 0
1711.12 0 0 0 6.54540 0 0.687253i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1711.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.m.e 12
3.b odd 2 1 912.3.m.b 12
4.b odd 2 1 inner 2736.3.m.e 12
12.b even 2 1 912.3.m.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.3.m.b 12 3.b odd 2 1
912.3.m.b 12 12.b even 2 1
2736.3.m.e 12 1.a even 1 1 trivial
2736.3.m.e 12 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 5 T_{5}^{5} - 65 T_{5}^{4} + 245 T_{5}^{3} + 1468 T_{5}^{2} - 2964 T_{5} - 11456 \) acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( T^{12} \)
$5$ \( ( -11456 - 2964 T + 1468 T^{2} + 245 T^{3} - 65 T^{4} - 5 T^{5} + T^{6} )^{2} \)
$7$ \( 129777664 + 295635712 T^{2} + 45249504 T^{4} + 2281875 T^{6} + 43883 T^{8} + 353 T^{10} + T^{12} \)
$11$ \( 67566724096 + 32832860800 T^{2} + 1492920004 T^{4} + 25067351 T^{6} + 195415 T^{8} + 717 T^{10} + T^{12} \)
$13$ \( ( -7808 - 2912 T + 4912 T^{2} + 1568 T^{3} - 104 T^{4} - 18 T^{5} + T^{6} )^{2} \)
$17$ \( ( -65017064 + 1348460 T + 567398 T^{2} - 7503 T^{3} - 1443 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$19$ \( ( 19 + T^{2} )^{6} \)
$23$ \( 750170395352498176 + 5469449725411328 T^{2} + 15611170193408 T^{4} + 22316218368 T^{6} + 16941840 T^{8} + 6520 T^{10} + T^{12} \)
$29$ \( ( 351804928 - 25920800 T - 1570640 T^{2} + 161696 T^{3} - 2384 T^{4} - 54 T^{5} + T^{6} )^{2} \)
$31$ \( 2966294913920598016 + 22699404346351616 T^{2} + 61629602849024 T^{4} + 73097515776 T^{6} + 41118048 T^{8} + 10576 T^{10} + T^{12} \)
$37$ \( ( 343318528 - 64365056 T + 3149568 T^{2} + 22512 T^{3} - 3532 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$41$ \( ( -622678016 + 101894656 T + 3814016 T^{2} - 159696 T^{3} - 5820 T^{4} + 8 T^{5} + T^{6} )^{2} \)
$43$ \( 1466000349331456 + 255470220331776 T^{2} + 9314058426400 T^{4} + 21351281651 T^{6} + 18577003 T^{8} + 7105 T^{10} + T^{12} \)
$47$ \( 15488299031782887424 + 178514526888483376 T^{2} + 373594376176408 T^{4} + 299826927179 T^{6} + 107264563 T^{8} + 17145 T^{10} + T^{12} \)
$53$ \( ( -5005174784 + 520004448 T - 2069168 T^{2} - 643472 T^{3} - 4184 T^{4} + 110 T^{5} + T^{6} )^{2} \)
$59$ \( 3460586750869504 + 36298090508713984 T^{2} + 165434032177152 T^{4} + 203759490048 T^{6} + 91889984 T^{8} + 16688 T^{10} + T^{12} \)
$61$ \( ( 21645030952 + 1124956852 T - 99570302 T^{2} + 1713407 T^{3} - 431 T^{4} - 183 T^{5} + T^{6} )^{2} \)
$67$ \( 28052182439418658816 + 1454666049764720640 T^{2} + 2903819598217216 T^{4} + 1825570411520 T^{6} + 388453696 T^{8} + 33328 T^{10} + T^{12} \)
$71$ \( 53197075220103430144 + 996005425323180032 T^{2} + 3021269224521728 T^{4} + 1594819243008 T^{6} + 331635264 T^{8} + 30160 T^{10} + T^{12} \)
$73$ \( ( 30337444648 + 1298274860 T - 13506814 T^{2} - 1133175 T^{3} - 10287 T^{4} + 79 T^{5} + T^{6} )^{2} \)
$79$ \( 310812422670647296 + 22934215835582464 T^{2} + 417164522500096 T^{4} + 1470386856704 T^{6} + 407864704 T^{8} + 35772 T^{10} + T^{12} \)
$83$ \( \)\(14\!\cdots\!24\)\( + 586288346635698176 T^{2} + 835527729348608 T^{4} + 539432847360 T^{6} + 163560192 T^{8} + 21676 T^{10} + T^{12} \)
$89$ \( ( -12103330176 - 3450377952 T - 22493808 T^{2} + 1558224 T^{3} + 48 T^{4} - 198 T^{5} + T^{6} )^{2} \)
$97$ \( ( -12366236864 - 1338468864 T + 68080 T^{2} + 2089024 T^{3} - 18356 T^{4} - 112 T^{5} + T^{6} )^{2} \)
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