# Properties

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

# Related objects

## 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
 1.59395 + 5.94871i 6.03643 + 1.61746i −3.79295 − 1.01632i −1.04508 − 3.90029i −1.74348 − 0.467165i −0.0488747 − 0.182403i −0.0488747 + 0.182403i −1.74348 + 0.467165i −1.04508 + 3.90029i −3.79295 + 1.01632i 6.03643 − 1.61746i 1.59395 − 5.94871i
0 3.00000i 0 17.4720i 0 −7.00000 0 −9.00000 0
673.2 0 3.00000i 0 6.24978i 0 −7.00000 0 −9.00000 0
673.3 0 3.00000i 0 2.27324i 0 −7.00000 0 −9.00000 0
673.4 0 3.00000i 0 1.12662i 0 −7.00000 0 −9.00000 0
673.5 0 3.00000i 0 13.9871i 0 −7.00000 0 −9.00000 0
673.6 0 3.00000i 0 17.1345i 0 −7.00000 0 −9.00000 0
673.7 0 3.00000i 0 17.1345i 0 −7.00000 0 −9.00000 0
673.8 0 3.00000i 0 13.9871i 0 −7.00000 0 −9.00000 0
673.9 0 3.00000i 0 1.12662i 0 −7.00000 0 −9.00000 0
673.10 0 3.00000i 0 2.27324i 0 −7.00000 0 −9.00000 0
673.11 0 3.00000i 0 6.24978i 0 −7.00000 0 −9.00000 0
673.12 0 3.00000i 0 17.4720i 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.f 12
4.b odd 2 1 1344.4.c.g yes 12
8.b even 2 1 inner 1344.4.c.f 12
8.d odd 2 1 1344.4.c.g yes 12

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

## 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

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