Properties

Label 1344.4.c.g
Level $1344$
Weight $4$
Character orbit 1344.c
Analytic conductor $79.299$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + 1646096 x^{3} + 1391904 x^{2} + 174720 x + 43264\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + 3 \beta_{2} q^{3} + ( \beta_{2} + \beta_{11} ) q^{5} + 7 q^{7} -9 q^{9} +O(q^{10})\) \( q + 3 \beta_{2} q^{3} + ( \beta_{2} + \beta_{11} ) q^{5} + 7 q^{7} -9 q^{9} + ( 4 \beta_{2} - \beta_{7} - 2 \beta_{11} ) q^{11} + ( -14 \beta_{2} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{13} + ( -3 - 3 \beta_{1} ) q^{15} + ( 32 + 2 \beta_{1} - \beta_{3} + 2 \beta_{8} ) q^{17} + ( 12 \beta_{2} - 3 \beta_{4} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{19} + 21 \beta_{2} q^{21} + ( 27 - 2 \beta_{1} - 3 \beta_{3} - 4 \beta_{5} + \beta_{6} - \beta_{8} ) q^{23} + ( -14 - \beta_{1} + 2 \beta_{3} + 5 \beta_{5} - 4 \beta_{6} - \beta_{8} ) q^{25} -27 \beta_{2} q^{27} + ( -14 \beta_{2} - 2 \beta_{4} - 7 \beta_{10} - 7 \beta_{11} ) q^{29} + ( -17 - \beta_{1} + 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - 5 \beta_{8} ) q^{31} + ( -12 + 6 \beta_{1} - 3 \beta_{8} ) q^{33} + ( 7 \beta_{2} + 7 \beta_{11} ) q^{35} + ( -40 \beta_{2} - 6 \beta_{4} + 8 \beta_{7} - 4 \beta_{9} - \beta_{10} + \beta_{11} ) q^{37} + ( 42 + 3 \beta_{1} + 3 \beta_{3} + 3 \beta_{6} + 3 \beta_{8} ) q^{39} + ( 40 + 6 \beta_{1} + 3 \beta_{3} + 4 \beta_{5} + 8 \beta_{6} ) q^{41} + ( -26 \beta_{2} - 8 \beta_{4} + 7 \beta_{7} + 7 \beta_{9} - 2 \beta_{10} - 22 \beta_{11} ) q^{43} + ( -9 \beta_{2} - 9 \beta_{11} ) q^{45} + ( 35 - 10 \beta_{1} + 4 \beta_{3} + 6 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} ) q^{47} + 49 q^{49} + ( 96 \beta_{2} - 6 \beta_{7} + 3 \beta_{9} + 6 \beta_{11} ) q^{51} + ( 21 \beta_{2} - 20 \beta_{4} + 3 \beta_{7} - 4 \beta_{9} - 4 \beta_{10} - 17 \beta_{11} ) q^{53} + ( 293 - 9 \beta_{1} + 7 \beta_{5} + 12 \beta_{6} - \beta_{8} ) q^{55} + ( -36 + 3 \beta_{1} - 6 \beta_{3} - 9 \beta_{5} + 3 \beta_{6} ) q^{57} + ( -95 \beta_{2} - 30 \beta_{4} + 3 \beta_{7} - 2 \beta_{9} + 3 \beta_{10} - 8 \beta_{11} ) q^{59} + ( 26 \beta_{2} + 14 \beta_{4} - 11 \beta_{7} + 3 \beta_{9} + 3 \beta_{10} + 9 \beta_{11} ) q^{61} -63 q^{63} + ( 125 + 24 \beta_{1} - 6 \beta_{3} - 10 \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{65} + ( 106 \beta_{2} - 28 \beta_{4} - \beta_{7} + \beta_{9} - 14 \beta_{10} - 10 \beta_{11} ) q^{67} + ( 81 \beta_{2} + 12 \beta_{4} + 3 \beta_{7} + 9 \beta_{9} - 3 \beta_{10} - 6 \beta_{11} ) q^{69} + ( 431 + 12 \beta_{1} - 7 \beta_{3} + 8 \beta_{5} - 5 \beta_{6} + \beta_{8} ) q^{71} + ( -150 - 16 \beta_{1} - 4 \beta_{3} - 16 \beta_{5} - 4 \beta_{6} + 14 \beta_{8} ) q^{73} + ( -42 \beta_{2} - 15 \beta_{4} + 3 \beta_{7} - 6 \beta_{9} + 12 \beta_{10} - 3 \beta_{11} ) q^{75} + ( 28 \beta_{2} - 7 \beta_{7} - 14 \beta_{11} ) q^{77} + ( 128 - 12 \beta_{1} - 12 \beta_{3} - 10 \beta_{5} + 12 \beta_{6} - 14 \beta_{8} ) q^{79} + 81 q^{81} + ( -153 \beta_{2} - 58 \beta_{4} - 5 \beta_{7} - 18 \beta_{9} - 11 \beta_{10} + 20 \beta_{11} ) q^{83} + ( 314 \beta_{2} + 45 \beta_{4} - 12 \beta_{7} + 10 \beta_{9} + \beta_{10} + 53 \beta_{11} ) q^{85} + ( 42 + 21 \beta_{1} - 6 \beta_{5} - 21 \beta_{6} ) q^{87} + ( -308 - 14 \beta_{1} + 5 \beta_{3} - 6 \beta_{5} - 12 \beta_{6} - 12 \beta_{8} ) q^{89} + ( -98 \beta_{2} + 7 \beta_{7} + 7 \beta_{9} + 7 \beta_{10} - 7 \beta_{11} ) q^{91} + ( -51 \beta_{2} - 3 \beta_{4} + 15 \beta_{7} - 6 \beta_{9} - 6 \beta_{10} - 3 \beta_{11} ) q^{93} + ( 228 + 50 \beta_{1} + 4 \beta_{3} - 2 \beta_{5} - 10 \beta_{6} + 18 \beta_{8} ) q^{95} + ( -184 - 46 \beta_{1} + 20 \beta_{3} - 4 \beta_{5} + 20 \beta_{6} ) q^{97} + ( -36 \beta_{2} + 9 \beta_{7} + 18 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 84q^{7} - 108q^{9} + O(q^{10}) \) \( 12q + 84q^{7} - 108q^{9} - 24q^{15} + 376q^{17} + 336q^{23} - 180q^{25} - 192q^{31} - 168q^{33} + 504q^{39} + 488q^{41} + 448q^{47} + 588q^{49} + 3600q^{55} - 432q^{57} - 756q^{63} + 1408q^{65} + 5104q^{71} - 1752q^{73} + 1632q^{79} + 972q^{81} + 336q^{87} - 3688q^{89} + 2496q^{95} - 1944q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 2 x^{11} - x^{10} - 861 x^{8} - 2158 x^{7} + 8654 x^{6} + 118244 x^{5} + 707300 x^{4} + 1646096 x^{3} + 1391904 x^{2} + 174720 x + 43264\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-242416298558977673281 \nu^{11} + 839115149959588070842 \nu^{10} - 1539556398078558853031 \nu^{9} + 7136846219853984935064 \nu^{8} + 185009801648384899478357 \nu^{7} + 278353886789331137012086 \nu^{6} - 2110326912328212739916550 \nu^{5} - 27654763888675064600136548 \nu^{4} - 137547159879372725424231508 \nu^{3} - 226091344416967225763940176 \nu^{2} - 25523145031253870639921152 \nu + 829991262660689314421990720\)\()/ \)\(51\!\cdots\!96\)\( \)
\(\beta_{2}\)\(=\)\((\)\(-1535383228018091 \nu^{11} + 4128406633081010 \nu^{10} - 2425172738800997 \nu^{9} + 6087037497968596 \nu^{8} + 1308864793086229591 \nu^{7} + 2441062851437144142 \nu^{6} - 14073158648524646410 \nu^{5} - 171130245849669440372 \nu^{4} - 978623926910622268236 \nu^{3} - 1963024938268049005216 \nu^{2} - 1384075068435231860992 \nu - 123503150359511081536\)\()/ \)\(21\!\cdots\!88\)\( \)
\(\beta_{3}\)\(=\)\((\)\(21994822978049024610007 \nu^{11} - 37990064679585792986982 \nu^{10} - 127550868237055264825471 \nu^{9} + 424108331359185317500200 \nu^{8} - 20279376285376559069035011 \nu^{7} - 46558707831596320112618122 \nu^{6} + 233318166037966913838261514 \nu^{5} + 2686146385420864307629096636 \nu^{4} + 15898167597817989510683104716 \nu^{3} + 29216044712118758895409424688 \nu^{2} + 3254556345456219099005292544 \nu - 25636381999917589180410176768\)\()/ \)\(15\!\cdots\!84\)\( \)
\(\beta_{4}\)\(=\)\((\)\(-28327139191123 \nu^{11} - 10132937675420 \nu^{10} + 291510454010223 \nu^{9} - 430000792847678 \nu^{8} + 25139450880899087 \nu^{7} + 117084990846699668 \nu^{6} - 206776636077293830 \nu^{5} - 4037784457135523304 \nu^{4} - 26565967578500903204 \nu^{3} - 80872707669910450328 \nu^{2} - 82632463460068205856 \nu - 6407135351419474464\)\()/ 974886278971051668 \)
\(\beta_{5}\)\(=\)\((\)\(263835326032171 \nu^{11} - 754400875389294 \nu^{10} + 205725021722285 \nu^{9} + 502956094678440 \nu^{8} - 228290975057758311 \nu^{7} - 376487922295383970 \nu^{6} + 2773967457302775490 \nu^{5} + 28900428316490463532 \nu^{4} + 159682741535028306300 \nu^{3} + 284813740714640449008 \nu^{2} + 31833290894086908928 \nu - 105438462637402840256\)\()/ 8300266616659676232 \)
\(\beta_{6}\)\(=\)\((\)\(48854060318060129084521 \nu^{11} - 114266327957238710636474 \nu^{10} - 147133461122743768848065 \nu^{9} + 911485319321135100062776 \nu^{8} - 45206261103190335339918973 \nu^{7} - 83098187598358634332828182 \nu^{6} + 556134026805223924156264822 \nu^{5} + 5413222368892719003171507908 \nu^{4} + 31684583555521587158565154996 \nu^{3} + 58561156141781767255747043536 \nu^{2} + 6518536595301540330957283328 \nu + 5179666677626351418057920576\)\()/ \)\(15\!\cdots\!84\)\( \)
\(\beta_{7}\)\(=\)\((\)\(2645810524249166584143 \nu^{11} + 4342062803798924016666 \nu^{10} - 39292523460625556724807 \nu^{9} + 54814978131457982428408 \nu^{8} - 2383209682431926937596907 \nu^{7} - 13785188771489336007845034 \nu^{6} + 16548422104289967852734378 \nu^{5} + 408316883511095928756657276 \nu^{4} + 2843523117115802724526019212 \nu^{3} + 9416262833001292880824525808 \nu^{2} + 10066599404613623928151979456 \nu + 768999923365292220872433280\)\()/ \)\(65\!\cdots\!08\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-4011478261564111962789 \nu^{11} + 10894956004838629606834 \nu^{10} - 499766375003808499555 \nu^{9} - 12288185478164513092664 \nu^{8} + 3481428646907190595549481 \nu^{7} + 6087523274266699465988254 \nu^{6} - 41688335303978700572998110 \nu^{5} - 446513855969458696831699892 \nu^{4} - 2484962848949829065873362148 \nu^{3} - 4440781306380811003756841360 \nu^{2} - 496240961338664391948161024 \nu + 1505910299078276743700658688\)\()/ \)\(65\!\cdots\!08\)\( \)
\(\beta_{9}\)\(=\)\((\)\(-1626562027227495790722837 \nu^{11} + 3351726848499640824586674 \nu^{10} + 854276431231669592877277 \nu^{9} + 3339506463943226973472232 \nu^{8} + 1389093037850359647220237449 \nu^{7} + 3475695832672601612963388734 \nu^{6} - 14171493469604500338150717326 \nu^{5} - 190414554233109686049159176180 \nu^{4} - 1150296187072935608422335762308 \nu^{3} - 2654821715756577309964242361488 \nu^{2} - 2185634213097668506188398289472 \nu - 183213530628970400064057006208\)\()/ \)\(19\!\cdots\!92\)\( \)
\(\beta_{10}\)\(=\)\((\)\(1487397447087817624836695 \nu^{11} - 5588453184969243631144109 \nu^{10} + 8451452062024794219008303 \nu^{9} - 13882333437240929457687865 \nu^{8} - 1266344259435313766814534523 \nu^{7} - 944820305958250134354211959 \nu^{6} + 14296631054117750312982416356 \nu^{5} + 149994796736514682082397200114 \nu^{4} + 791710816560312194244825652824 \nu^{3} + 1083285750931718041448504318980 \nu^{2} + 297058838052462168645782634352 \nu + 44110923105459712386821070160\)\()/ \)\(14\!\cdots\!44\)\( \)
\(\beta_{11}\)\(=\)\((\)\(-109898937302191301113345 \nu^{11} + 286345981730649769999672 \nu^{10} - 139809497068690760668771 \nu^{9} + 365111949595064076795962 \nu^{8} + 94066211854843982636885957 \nu^{7} + 180246153337949313933102096 \nu^{6} - 994179278338770157526785274 \nu^{5} - 12338291889363358549571752960 \nu^{4} - 71035038363780179769095898732 \nu^{3} - 145619560968828110988467400920 \nu^{2} - 105552899073812235631556322080 \nu - 9310043508420624290072095136\)\()/ \)\(10\!\cdots\!72\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-8 \beta_{11} - 4 \beta_{10} + 2 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 12 \beta_{2} - 4 \beta_{1} + 6\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(-14 \beta_{11} - \beta_{10} - \beta_{9} + 9 \beta_{8} - 2 \beta_{7} - 6 \beta_{6} + 27 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 219 \beta_{2} + 12 \beta_{1} + 18\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(74 \beta_{11} + 31 \beta_{10} - 56 \beta_{9} + 37 \beta_{8} - 37 \beta_{7} - 31 \beta_{6} + 82 \beta_{5} - 82 \beta_{4} + 56 \beta_{3} + 63 \beta_{2} + 74 \beta_{1} + 63\)\()/24\)
\(\nu^{4}\)\(=\)\((\)\(158 \beta_{11} + 79 \beta_{10} - 75 \beta_{9} + 43 \beta_{8} - 112 \beta_{7} + 32 \beta_{6} + 22 \beta_{5} - 292 \beta_{4} + 11 \beta_{3} + 159 \beta_{2} - 180 \beta_{1} + 2248\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-2698 \beta_{11} - 749 \beta_{10} - 1073 \beta_{9} + 599 \beta_{8} - 2122 \beta_{7} + 1924 \beta_{6} - 1276 \beta_{5} - 2368 \beta_{4} - 725 \beta_{3} + 40059 \beta_{2} - 5348 \beta_{1} + 36540\)\()/24\)
\(\nu^{6}\)\(=\)\((\)\(-20518 \beta_{11} - 4715 \beta_{10} - 1235 \beta_{9} - 5746 \beta_{7} - 2770 \beta_{4} + 235917 \beta_{2}\)\()/12\)
\(\nu^{7}\)\(=\)\((\)\(-105796 \beta_{11} - 23048 \beta_{10} - 39929 \beta_{9} - 19370 \beta_{8} - 80425 \beta_{7} - 67789 \beta_{6} + 50998 \beta_{5} - 96646 \beta_{4} + 23615 \beta_{3} + 1669272 \beta_{2} + 202634 \beta_{1} - 1510923\)\()/24\)
\(\nu^{8}\)\(=\)\((\)\(234362 \beta_{11} + 112369 \beta_{10} - 144041 \beta_{9} - 78557 \beta_{8} - 161212 \beta_{7} - 65484 \beta_{6} - 39098 \beta_{5} - 394614 \beta_{4} - 16281 \beta_{3} + 396249 \beta_{2} + 260508 \beta_{1} - 2906740\)\()/8\)
\(\nu^{9}\)\(=\)\((\)\(3585422 \beta_{11} + 1697149 \beta_{10} - 2359082 \beta_{9} - 2409673 \beta_{8} - 2409673 \beta_{7} + 1697149 \beta_{6} - 5807464 \beta_{5} - 5807464 \beta_{4} - 2359082 \beta_{3} + 7015461 \beta_{2} - 3585422 \beta_{1} - 7015461\)\()/24\)
\(\nu^{10}\)\(=\)\((\)\(-29818874 \beta_{11} - 7736629 \beta_{10} - 1980361 \beta_{9} - 18708813 \beta_{8} - 9394298 \beta_{7} + 12767730 \beta_{6} - 45148446 \beta_{5} - 4917302 \beta_{4} - 17130927 \beta_{3} + 330219363 \beta_{2} - 26503536 \beta_{1} - 54979650\)\()/24\)
\(\nu^{11}\)\(=\)\((\)\(-292246246 \beta_{11} - 93014579 \beta_{10} + 29706397 \beta_{9} - 120706703 \beta_{8} - 26260186 \beta_{7} - 28723636 \beta_{6} - 152111630 \beta_{5} + 73775630 \beta_{4} - 59861971 \beta_{3} + 2305464885 \beta_{2} + 158737460 \beta_{1} - 2613730860\)\()/24\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
673.1
−0.0488747 + 0.182403i
−1.74348 + 0.467165i
−1.04508 + 3.90029i
−3.79295 + 1.01632i
6.03643 1.61746i
1.59395 5.94871i
1.59395 + 5.94871i
6.03643 + 1.61746i
−3.79295 1.01632i
−1.04508 3.90029i
−1.74348 0.467165i
−0.0488747 0.182403i
0 3.00000i 0 17.1345i 0 7.00000 0 −9.00000 0
673.2 0 3.00000i 0 13.9871i 0 7.00000 0 −9.00000 0
673.3 0 3.00000i 0 1.12662i 0 7.00000 0 −9.00000 0
673.4 0 3.00000i 0 2.27324i 0 7.00000 0 −9.00000 0
673.5 0 3.00000i 0 6.24978i 0 7.00000 0 −9.00000 0
673.6 0 3.00000i 0 17.4720i 0 7.00000 0 −9.00000 0
673.7 0 3.00000i 0 17.4720i 0 7.00000 0 −9.00000 0
673.8 0 3.00000i 0 6.24978i 0 7.00000 0 −9.00000 0
673.9 0 3.00000i 0 2.27324i 0 7.00000 0 −9.00000 0
673.10 0 3.00000i 0 1.12662i 0 7.00000 0 −9.00000 0
673.11 0 3.00000i 0 13.9871i 0 7.00000 0 −9.00000 0
673.12 0 3.00000i 0 17.1345i 0 7.00000 0 −9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1344.4.c.g yes 12
4.b odd 2 1 1344.4.c.f 12
8.b even 2 1 inner 1344.4.c.g yes 12
8.d odd 2 1 1344.4.c.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.c.f 12 4.b odd 2 1
1344.4.c.f 12 8.d odd 2 1
1344.4.c.g yes 12 1.a even 1 1 trivial
1344.4.c.g yes 12 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1344, [\chi])\):

\( T_{5}^{12} + 840 T_{5}^{10} + 243192 T_{5}^{8} + 27147584 T_{5}^{6} + 851300112 T_{5}^{4} + 4576494720 T_{5}^{2} + 4492216576 \)
\( T_{23}^{6} - 168 T_{23}^{5} - 23688 T_{23}^{4} + 4379760 T_{23}^{3} - 39519900 T_{23}^{2} - 11108216448 T_{23} + 157604189184 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 9 + T^{2} )^{6} \)
$5$ \( 4492216576 + 4576494720 T^{2} + 851300112 T^{4} + 27147584 T^{6} + 243192 T^{8} + 840 T^{10} + T^{12} \)
$7$ \( ( -7 + T )^{12} \)
$11$ \( 63360835842304 + 45564107938944 T^{2} + 4510696139664 T^{4} + 20937585152 T^{6} + 22440984 T^{8} + 8376 T^{10} + T^{12} \)
$13$ \( 187244715207294976 + 11765216842874880 T^{2} + 96324728328192 T^{4} + 158422014464 T^{6} + 88428048 T^{8} + 16680 T^{10} + T^{12} \)
$17$ \( ( 3823312 - 7198544 T - 34064716 T^{2} + 911008 T^{3} + 2420 T^{4} - 188 T^{5} + T^{6} )^{2} \)
$19$ \( 108393962674323456 + 35684761490030592 T^{2} + 2853325702621440 T^{4} + 1797987338496 T^{6} + 385968096 T^{8} + 33744 T^{10} + T^{12} \)
$23$ \( ( 157604189184 - 11108216448 T - 39519900 T^{2} + 4379760 T^{3} - 23688 T^{4} - 168 T^{5} + T^{6} )^{2} \)
$29$ \( \)\(16\!\cdots\!56\)\( + \)\(78\!\cdots\!60\)\( T^{2} + 9761019510810687744 T^{4} + 533656017329408 T^{6} + 14591287008 T^{8} + 194448 T^{10} + T^{12} \)
$31$ \( ( -3105857137664 + 76558655232 T + 1112513232 T^{2} - 5687936 T^{3} - 65880 T^{4} + 96 T^{5} + T^{6} )^{2} \)
$37$ \( \)\(52\!\cdots\!04\)\( + \)\(12\!\cdots\!96\)\( T^{2} + 93548598253785682176 T^{4} + 3150487185688832 T^{6} + 50381895264 T^{8} + 370416 T^{10} + T^{12} \)
$41$ \( ( 1874674085200 - 84603237040 T + 139556948 T^{2} + 25033376 T^{3} - 147100 T^{4} - 244 T^{5} + T^{6} )^{2} \)
$43$ \( \)\(68\!\cdots\!44\)\( + \)\(11\!\cdots\!68\)\( T^{2} + \)\(37\!\cdots\!08\)\( T^{4} + 49009360789845248 T^{6} + 305335073520 T^{8} + 897432 T^{10} + T^{12} \)
$47$ \( ( 2057937719296 - 41559719936 T - 1864249600 T^{2} + 32596480 T^{3} - 109456 T^{4} - 224 T^{5} + T^{6} )^{2} \)
$53$ \( \)\(53\!\cdots\!44\)\( + \)\(50\!\cdots\!76\)\( T^{2} + \)\(13\!\cdots\!64\)\( T^{4} + 9223461876858624 T^{6} + 143366228064 T^{8} + 723024 T^{10} + T^{12} \)
$59$ \( \)\(46\!\cdots\!84\)\( + \)\(29\!\cdots\!48\)\( T^{2} + \)\(69\!\cdots\!52\)\( T^{4} + 78330761003565056 T^{6} + 438275278080 T^{8} + 1133856 T^{10} + T^{12} \)
$61$ \( \)\(20\!\cdots\!56\)\( + \)\(46\!\cdots\!76\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{4} + 6941522331726848 T^{6} + 149901067920 T^{8} + 761352 T^{10} + T^{12} \)
$67$ \( \)\(22\!\cdots\!76\)\( + \)\(79\!\cdots\!80\)\( T^{2} + \)\(23\!\cdots\!88\)\( T^{4} + 223704546994721024 T^{6} + 908233980912 T^{8} + 1601400 T^{10} + T^{12} \)
$71$ \( ( 270319500291328 - 12240629811008 T + 180560118596 T^{2} - 985686416 T^{3} + 2368520 T^{4} - 2552 T^{5} + T^{6} )^{2} \)
$73$ \( ( -903469003040448 + 1469172219840 T + 45117003312 T^{2} - 112235616 T^{3} - 417156 T^{4} + 876 T^{5} + T^{6} )^{2} \)
$79$ \( ( -1565834797285376 - 38167396128768 T - 11794123968 T^{2} + 772517120 T^{3} - 887232 T^{4} - 816 T^{5} + T^{6} )^{2} \)
$83$ \( \)\(32\!\cdots\!96\)\( + \)\(29\!\cdots\!60\)\( T^{2} + \)\(10\!\cdots\!36\)\( T^{4} + 16671348624250552320 T^{6} + 14244714847872 T^{8} + 6030336 T^{10} + T^{12} \)
$89$ \( ( -432185216528816 - 61821291269584 T - 374847226924 T^{2} - 491019232 T^{3} + 685220 T^{4} + 1844 T^{5} + T^{6} )^{2} \)
$97$ \( ( -112893910101349568 - 312061906018368 T + 2043088721712 T^{2} - 928001248 T^{3} - 2627172 T^{4} + 972 T^{5} + T^{6} )^{2} \)
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