# Properties

 Label 336.7.bh.a Level $336$ Weight $7$ Character orbit 336.bh Analytic conductor $77.298$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 336.bh (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$77.2981720963$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + 1200774512 x + 5192355364$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -9 - 9 \beta_{1} ) q^{3} + ( -49 + 25 \beta_{1} - \beta_{2} ) q^{5} + ( -37 + 15 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + 243 \beta_{1} q^{9} +O(q^{10})$$ $$q + ( -9 - 9 \beta_{1} ) q^{3} + ( -49 + 25 \beta_{1} - \beta_{2} ) q^{5} + ( -37 + 15 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{7} ) q^{7} + 243 \beta_{1} q^{9} + ( -90 + 96 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + ( 234 - 466 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 4 \beta_{6} + 7 \beta_{7} ) q^{13} + ( 666 - 9 \beta_{1} + 18 \beta_{2} - 9 \beta_{5} ) q^{15} + ( 85 + 64 \beta_{1} + 13 \beta_{2} + 7 \beta_{3} - 11 \beta_{4} - 13 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{17} + ( -615 + 315 \beta_{1} + \beta_{2} + 7 \beta_{3} - 16 \beta_{4} - 16 \beta_{6} - 7 \beta_{7} ) q^{19} + ( 468 + 63 \beta_{1} - 18 \beta_{2} - 18 \beta_{4} + 9 \beta_{5} - 9 \beta_{7} ) q^{21} + ( 10 - 3898 \beta_{1} + 13 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 26 \beta_{5} - 3 \beta_{6} + 7 \beta_{7} ) q^{23} + ( 815 - 861 \beta_{1} + 46 \beta_{2} + 17 \beta_{3} + 30 \beta_{4} + 46 \beta_{5} + 13 \beta_{6} + 13 \beta_{7} ) q^{25} + ( 2187 - 4374 \beta_{1} ) q^{27} + ( -8611 + 91 \beta_{1} - 184 \beta_{2} + 20 \beta_{3} - 20 \beta_{4} + 92 \beta_{5} + \beta_{6} - 21 \beta_{7} ) q^{29} + ( -2012 - 1829 \beta_{1} - 69 \beta_{2} - 3 \beta_{3} + 60 \beta_{4} + 69 \beta_{5} - 57 \beta_{6} + 57 \beta_{7} ) q^{31} + ( 1701 - 936 \beta_{1} + 162 \beta_{2} + 36 \beta_{3} + 9 \beta_{4} + 9 \beta_{6} - 36 \beta_{7} ) q^{33} + ( -16064 + 8808 \beta_{1} + 85 \beta_{2} - 62 \beta_{4} - 199 \beta_{5} - 98 \beta_{6} - 65 \beta_{7} ) q^{35} + ( -61 + 4103 \beta_{1} - 104 \beta_{2} + 94 \beta_{3} - 43 \beta_{4} + 208 \beta_{5} + 43 \beta_{6} + 94 \beta_{7} ) q^{37} + ( -6327 + 6345 \beta_{1} - 18 \beta_{2} + 9 \beta_{3} - 90 \beta_{4} - 18 \beta_{5} - 99 \beta_{6} - 99 \beta_{7} ) q^{39} + ( -13920 + 28164 \beta_{1} + 80 \beta_{3} + 80 \beta_{4} - 262 \beta_{5} + 62 \beta_{6} + 142 \beta_{7} ) q^{41} + ( 21287 - 98 \beta_{1} + 430 \beta_{2} - 35 \beta_{3} + 35 \beta_{4} - 215 \beta_{5} - 117 \beta_{6} + 152 \beta_{7} ) q^{43} + ( -6075 - 5832 \beta_{1} - 243 \beta_{2} + 243 \beta_{5} ) q^{45} + ( -17030 + 8810 \beta_{1} - 671 \beta_{2} - 171 \beta_{3} + 81 \beta_{4} + 81 \beta_{6} + 171 \beta_{7} ) q^{47} + ( 20868 - 3363 \beta_{1} - 86 \beta_{2} + 147 \beta_{3} + 61 \beta_{4} + 920 \beta_{5} - 98 \beta_{6} - 97 \beta_{7} ) q^{49} + ( -252 - 1818 \beta_{1} - 117 \beta_{2} - 27 \beta_{3} + 135 \beta_{4} + 234 \beta_{5} - 135 \beta_{6} - 27 \beta_{7} ) q^{51} + ( 49193 - 49036 \beta_{1} - 157 \beta_{2} - 24 \beta_{3} - 264 \beta_{4} - 157 \beta_{5} - 240 \beta_{6} - 240 \beta_{7} ) q^{53} + ( 71901 - 144837 \beta_{1} + 168 \beta_{3} + 168 \beta_{4} + 1014 \beta_{5} - 21 \beta_{6} + 147 \beta_{7} ) q^{55} + ( 8307 - 216 \beta_{1} - 18 \beta_{2} - 207 \beta_{3} + 207 \beta_{4} + 9 \beta_{5} + 225 \beta_{6} - 18 \beta_{7} ) q^{57} + ( 55813 + 54598 \beta_{1} + 361 \beta_{2} - 14 \beta_{3} - 413 \beta_{4} - 361 \beta_{5} + 427 \beta_{6} - 427 \beta_{7} ) q^{59} + ( -3988 + 2208 \beta_{1} - 556 \beta_{2} - 480 \beta_{3} + 128 \beta_{4} + 128 \beta_{6} + 480 \beta_{7} ) q^{61} + ( -3645 - 5346 \beta_{1} + 243 \beta_{2} + 243 \beta_{4} - 243 \beta_{5} ) q^{63} + ( 1012 + 2652 \beta_{1} + 1285 \beta_{2} - 565 \beta_{3} + 273 \beta_{4} - 2570 \beta_{5} - 273 \beta_{6} - 565 \beta_{7} ) q^{65} + ( -193502 + 194073 \beta_{1} - 571 \beta_{2} + 156 \beta_{3} - 37 \beta_{4} - 571 \beta_{5} - 193 \beta_{6} - 193 \beta_{7} ) q^{67} + ( -35235 + 70110 \beta_{1} - 90 \beta_{3} - 90 \beta_{4} + 351 \beta_{5} - 9 \beta_{6} - 99 \beta_{7} ) q^{69} + ( 90889 - 1824 \beta_{1} + 898 \beta_{2} - 520 \beta_{3} + 520 \beta_{4} - 449 \beta_{5} + 1375 \beta_{6} - 855 \beta_{7} ) q^{71} + ( -99161 - 104097 \beta_{1} + 4054 \beta_{2} - 83 \beta_{3} - 358 \beta_{4} - 4054 \beta_{5} + 441 \beta_{6} - 441 \beta_{7} ) q^{73} + ( -15237 + 8433 \beta_{1} - 1242 \beta_{2} - 36 \beta_{3} - 387 \beta_{4} - 387 \beta_{6} + 36 \beta_{7} ) q^{75} + ( 189243 + 18205 \beta_{1} - 600 \beta_{2} - 833 \beta_{3} - 61 \beta_{4} - 1038 \beta_{5} - 931 \beta_{6} - 21 \beta_{7} ) q^{77} + ( 2882 + 121730 \beta_{1} + 2041 \beta_{2} - 296 \beta_{3} - 841 \beta_{4} - 4082 \beta_{5} + 841 \beta_{6} - 296 \beta_{7} ) q^{79} + ( -59049 + 59049 \beta_{1} ) q^{81} + ( 76056 - 152524 \beta_{1} - 369 \beta_{3} - 369 \beta_{4} + 1861 \beta_{5} + 1449 \beta_{6} + 1080 \beta_{7} ) q^{83} + ( -166404 - 632 \beta_{1} + 124 \beta_{2} - 986 \beta_{3} + 986 \beta_{4} - 62 \beta_{5} + 570 \beta_{6} + 416 \beta_{7} ) q^{85} + ( 78138 + 76050 \beta_{1} + 2484 \beta_{2} - 171 \beta_{3} + 369 \beta_{4} - 2484 \beta_{5} - 198 \beta_{6} + 198 \beta_{7} ) q^{87} + ( 100710 - 48426 \beta_{1} - 2160 \beta_{2} - 378 \beta_{3} - 1698 \beta_{4} - 1698 \beta_{6} + 378 \beta_{7} ) q^{89} + ( -290737 - 382153 \beta_{1} - 2793 \beta_{2} - 245 \beta_{3} + 196 \beta_{4} + 6362 \beta_{5} - 882 \beta_{6} + 41 \beta_{7} ) q^{91} + ( 1674 + 50490 \beta_{1} + 621 \beta_{2} - 486 \beta_{3} - 1053 \beta_{4} - 1242 \beta_{5} + 1053 \beta_{6} - 486 \beta_{7} ) q^{93} + ( -113245 + 110474 \beta_{1} + 2771 \beta_{2} - 753 \beta_{3} + 985 \beta_{4} + 2771 \beta_{5} + 1738 \beta_{6} + 1738 \beta_{7} ) q^{95} + ( -571 - 3798 \beta_{1} - 613 \beta_{3} - 613 \beta_{4} + 6140 \beta_{5} + 1200 \beta_{6} + 587 \beta_{7} ) q^{97} + ( -24057 + 1944 \beta_{1} - 2916 \beta_{2} - 243 \beta_{3} + 243 \beta_{4} + 1458 \beta_{5} - 486 \beta_{6} + 729 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q - 108 q^{3} - 294 q^{5} - 232 q^{7} + 972 q^{9} + O(q^{10})$$ $$8 q - 108 q^{3} - 294 q^{5} - 232 q^{7} + 972 q^{9} - 378 q^{11} + 5292 q^{15} + 852 q^{17} - 3690 q^{19} + 3942 q^{21} - 15600 q^{23} + 3386 q^{25} - 68604 q^{29} - 23028 q^{31} + 10206 q^{33} - 93828 q^{35} + 15914 q^{37} - 25326 q^{39} + 170044 q^{43} - 71442 q^{45} - 102180 q^{47} + 157340 q^{49} - 7668 q^{51} + 196410 q^{53} + 66420 q^{57} + 662550 q^{59} - 23928 q^{61} - 50058 q^{63} + 14892 q^{65} - 774838 q^{67} + 721896 q^{71} - 1219050 q^{73} - 91422 q^{75} + 1584738 q^{77} + 493868 q^{79} - 236196 q^{81} - 1329816 q^{85} + 926154 q^{87} + 604260 q^{89} - 3831690 q^{91} + 207252 q^{93} - 448944 q^{95} - 183708 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} + 1061 x^{6} + 35442 x^{5} + 1155979 x^{4} + 17325616 x^{3} + 201523590 x^{2} + 1200774512 x + 5192355364$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-896877863432 \nu^{7} + 6041228263913 \nu^{6} - 1021792250749710 \nu^{5} - 26857091404894553 \nu^{4} - 957859907329660610 \nu^{3} - 11775455453028959461 \nu^{2} - 174297213726189823636 \nu - 314616662632701367834$$$$)/$$$$73\!\cdots\!34$$ $$\beta_{2}$$ $$=$$ $$($$$$-183863909546285 \nu^{7} + 18229925255082312 \nu^{6} - 391000809124369202 \nu^{5} + 12386413249503110037 \nu^{4} + 279788965160726470064 \nu^{3} + 11155237836159615910206 \nu^{2} + 101426167143129490957192 \nu + 890254789664279551983072$$$$)/$$$$14\!\cdots\!68$$ $$\beta_{3}$$ $$=$$ $$($$$$39954530257266 \nu^{7} + 880422014972009 \nu^{6} + 28776968386772265 \nu^{5} + 2391093571624293239 \nu^{4} + 70457180636704215266 \nu^{3} + 1297671150940055659922 \nu^{2} + 10908788665654789245724 \nu + 51400375695712227877126$$$$)/ 43047994130847708002$$ $$\beta_{4}$$ $$=$$ $$($$$$718139017476094 \nu^{7} + 11747984640696868 \nu^{6} + 432526205355998853 \nu^{5} + 43129073166676770636 \nu^{4} + 965806004381997551388 \nu^{3} + 19555608599214840237151 \nu^{2} + 140929163194706907910800 \nu + 1016125809744114733509902$$$$)/$$$$73\!\cdots\!34$$ $$\beta_{5}$$ $$=$$ $$($$$$1615003316699301 \nu^{7} - 1023724879792003 \nu^{6} + 1519321793044335929 \nu^{5} + 61629572271591505654 \nu^{4} + 1747422265337618494262 \nu^{3} + 25339769573425769017226 \nu^{2} + 217432763210238211497844 \nu + 973864425514123824592632$$$$)/$$$$14\!\cdots\!68$$ $$\beta_{6}$$ $$=$$ $$($$$$-85566866050425 \nu^{7} + 1641279106520201 \nu^{6} - 91481808542443869 \nu^{5} - 1725692575549182196 \nu^{4} - 42654633113513188174 \nu^{3} + 66584118021960249722 \nu^{2} + 2818480818201421499716 \nu + 39694930904486128572266$$$$)/ 56293530786493156618$$ $$\beta_{7}$$ $$=$$ $$($$$$1502047347016715 \nu^{7} - 28769556653539830 \nu^{6} + 1845278294272487085 \nu^{5} + 26148375894322597861 \nu^{4} + 990648586051920832168 \nu^{3} + 4842950476900008799095 \nu^{2} + 55591103128615125986216 \nu - 155002313043282618164356$$$$)/$$$$73\!\cdots\!34$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{7} + 2 \beta_{6} + 12 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} + 41 \beta_{1} - 4$$$$)/84$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{6} + 44 \beta_{5} - 15 \beta_{4} - 17 \beta_{3} + 44 \beta_{2} + 11089 \beta_{1} - 11133$$$$)/21$$ $$\nu^{3}$$ $$=$$ $$($$$$1324 \beta_{7} - 1727 \beta_{6} - 3930 \beta_{5} - 403 \beta_{4} + 403 \beta_{3} + 7860 \beta_{2} - 2203 \beta_{1} - 590288$$$$)/42$$ $$\nu^{4}$$ $$=$$ $$($$$$30080 \beta_{7} - 27625 \beta_{6} - 150868 \beta_{5} + 27625 \beta_{4} + 30080 \beta_{3} + 75434 \beta_{2} - 14026917 \beta_{1} + 47809$$$$)/21$$ $$\nu^{5}$$ $$=$$ $$($$$$202582 \beta_{7} + 202582 \beta_{6} - 3069176 \beta_{5} + 1276883 \beta_{4} + 1074301 \beta_{3} - 3069176 \beta_{2} - 524702247 \beta_{1} + 527771423$$$$)/21$$ $$\nu^{6}$$ $$=$$ $$($$$$-43604781 \beta_{7} + 49818560 \beta_{6} + 121787402 \beta_{5} + 6213779 \beta_{4} - 6213779 \beta_{3} - 243574804 \beta_{2} + 71968842 \beta_{1} + 21577820576$$$$)/21$$ $$\nu^{7}$$ $$=$$ $$($$$$-2012777654 \beta_{7} + 1730791115 \beta_{6} + 9772639964 \beta_{5} - 1730791115 \beta_{4} - 2012777654 \beta_{3} - 4886319982 \beta_{2} + 854760798785 \beta_{1} - 3155528867$$$$)/21$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1
 −4.36471 + 7.55990i −8.68580 + 15.0442i −5.94197 + 10.2918i 19.9925 − 34.6280i −4.36471 − 7.55990i −8.68580 − 15.0442i −5.94197 − 10.2918i 19.9925 + 34.6280i
0 −13.5000 + 7.79423i 0 −167.007 96.4218i 0 243.806 + 241.263i 0 121.500 210.444i 0
145.2 0 −13.5000 + 7.79423i 0 −93.2075 53.8134i 0 −280.014 198.094i 0 121.500 210.444i 0
145.3 0 −13.5000 + 7.79423i 0 −0.0783677 0.0452456i 0 218.562 264.348i 0 121.500 210.444i 0
145.4 0 −13.5000 + 7.79423i 0 113.293 + 65.4099i 0 −298.353 + 169.217i 0 121.500 210.444i 0
241.1 0 −13.5000 7.79423i 0 −167.007 + 96.4218i 0 243.806 241.263i 0 121.500 + 210.444i 0
241.2 0 −13.5000 7.79423i 0 −93.2075 + 53.8134i 0 −280.014 + 198.094i 0 121.500 + 210.444i 0
241.3 0 −13.5000 7.79423i 0 −0.0783677 + 0.0452456i 0 218.562 + 264.348i 0 121.500 + 210.444i 0
241.4 0 −13.5000 7.79423i 0 113.293 65.4099i 0 −298.353 169.217i 0 121.500 + 210.444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 241.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.7.bh.a 8
4.b odd 2 1 84.7.m.b 8
7.d odd 6 1 inner 336.7.bh.a 8
12.b even 2 1 252.7.z.e 8
28.d even 2 1 588.7.m.b 8
28.f even 6 1 84.7.m.b 8
28.f even 6 1 588.7.d.a 8
28.g odd 6 1 588.7.d.a 8
28.g odd 6 1 588.7.m.b 8
84.j odd 6 1 252.7.z.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.7.m.b 8 4.b odd 2 1
84.7.m.b 8 28.f even 6 1
252.7.z.e 8 12.b even 2 1
252.7.z.e 8 84.j odd 6 1
336.7.bh.a 8 1.a even 1 1 trivial
336.7.bh.a 8 7.d odd 6 1 inner
588.7.d.a 8 28.f even 6 1
588.7.d.a 8 28.g odd 6 1
588.7.m.b 8 28.d even 2 1
588.7.m.b 8 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$73\!\cdots\!00$$$$T_{5}^{2} +$$$$11\!\cdots\!00$$$$T_{5} + 60368490000$$">$$T_{5}^{8} + \cdots$$ acting on $$S_{7}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 243 + 27 T + T^{2} )^{4}$$
$5$ $$60368490000 + 1156200318000 T + 7385884190100 T^{2} + 87230302380 T^{3} - 117296451 T^{4} - 5449878 T^{5} + 10275 T^{6} + 294 T^{7} + T^{8}$$
$7$ $$19\!\cdots\!01$$$$+ 377791954715224168 T - 716397342949358 T^{2} + 2284336985056 T^{3} + 31366517539 T^{4} + 19416544 T^{5} - 51758 T^{6} + 232 T^{7} + T^{8}$$
$11$ $$20\!\cdots\!00$$$$-$$$$50\!\cdots\!00$$$$T + 19108394396175766140 T^{2} + 15673800750043500 T^{3} + 22991548494429 T^{4} + 5156604234 T^{5} + 4951071 T^{6} + 378 T^{7} + T^{8}$$
$13$ $$79\!\cdots\!36$$$$+$$$$63\!\cdots\!80$$$$T^{2} + 270020210289417 T^{4} + 30055590 T^{6} + T^{8}$$
$17$ $$90\!\cdots\!84$$$$+$$$$37\!\cdots\!36$$$$T +$$$$65\!\cdots\!16$$$$T^{2} + 5290803690075775872 T^{3} + 1833916076679600 T^{4} + 35826296688 T^{5} - 41807676 T^{6} - 852 T^{7} + T^{8}$$
$19$ $$35\!\cdots\!96$$$$-$$$$22\!\cdots\!60$$$$T +$$$$34\!\cdots\!16$$$$T^{2} + 29416836842592638940 T^{3} + 4813318365101397 T^{4} - 282940642890 T^{5} - 72138981 T^{6} + 3690 T^{7} + T^{8}$$
$23$ $$13\!\cdots\!84$$$$-$$$$14\!\cdots\!32$$$$T +$$$$36\!\cdots\!16$$$$T^{2} + 3364994032014205440 T^{3} + 2431680639233472 T^{4} + 777056709888 T^{5} + 188417160 T^{6} + 15600 T^{7} + T^{8}$$
$29$ $$( -86809576902337152 - 17865254182464 T - 553061151 T^{2} + 34302 T^{3} + T^{4} )^{2}$$
$31$ $$33\!\cdots\!49$$$$+$$$$10\!\cdots\!96$$$$T +$$$$13\!\cdots\!74$$$$T^{2} +$$$$79\!\cdots\!84$$$$T^{3} + 1620341120821447755 T^{4} - 31590626097816 T^{5} - 1195072494 T^{6} + 23028 T^{7} + T^{8}$$
$37$ $$30\!\cdots\!64$$$$+$$$$12\!\cdots\!16$$$$T +$$$$60\!\cdots\!12$$$$T^{2} -$$$$44\!\cdots\!76$$$$T^{3} + 42433240754448136381 T^{4} - 37702695523610 T^{5} + 6694605295 T^{6} - 15914 T^{7} + T^{8}$$
$41$ $$59\!\cdots\!64$$$$+$$$$86\!\cdots\!68$$$$T^{2} + 93609431443627753824 T^{4} + 20901269040 T^{6} + T^{8}$$
$43$ $$( -27487775463560893820 + 1293449617930540 T - 11938887867 T^{2} - 85022 T^{3} + T^{4} )^{2}$$
$47$ $$49\!\cdots\!00$$$$-$$$$48\!\cdots\!00$$$$T -$$$$11\!\cdots\!80$$$$T^{2} +$$$$12\!\cdots\!00$$$$T^{3} +$$$$26\!\cdots\!56$$$$T^{4} - 1940814698350320 T^{5} - 15513825324 T^{6} + 102180 T^{7} + T^{8}$$
$53$ $$41\!\cdots\!24$$$$+$$$$43\!\cdots\!24$$$$T +$$$$46\!\cdots\!84$$$$T^{2} -$$$$22\!\cdots\!00$$$$T^{3} +$$$$54\!\cdots\!37$$$$T^{4} - 2107412918810586 T^{5} + 49673098395 T^{6} - 196410 T^{7} + T^{8}$$
$59$ $$26\!\cdots\!76$$$$-$$$$54\!\cdots\!52$$$$T +$$$$40\!\cdots\!60$$$$T^{2} -$$$$59\!\cdots\!84$$$$T^{3} -$$$$54\!\cdots\!87$$$$T^{4} + 11741687203809150 T^{5} + 128602203567 T^{6} - 662550 T^{7} + T^{8}$$
$61$ $$19\!\cdots\!44$$$$+$$$$43\!\cdots\!20$$$$T +$$$$36\!\cdots\!00$$$$T^{2} +$$$$79\!\cdots\!00$$$$T^{3} +$$$$63\!\cdots\!52$$$$T^{4} - 1913558463561600 T^{5} - 79780667472 T^{6} + 23928 T^{7} + T^{8}$$
$67$ $$71\!\cdots\!04$$$$-$$$$33\!\cdots\!28$$$$T +$$$$16\!\cdots\!00$$$$T^{2} +$$$$23\!\cdots\!76$$$$T^{3} +$$$$23\!\cdots\!09$$$$T^{4} + 115198012225634902 T^{5} + 418958273371 T^{6} + 774838 T^{7} + T^{8}$$
$71$ $$($$$$59\!\cdots\!24$$$$+ 123986084669035776 T - 517996206468 T^{2} - 360948 T^{3} + T^{4} )^{2}$$
$73$ $$95\!\cdots\!00$$$$-$$$$87\!\cdots\!00$$$$T +$$$$17\!\cdots\!20$$$$T^{2} +$$$$82\!\cdots\!60$$$$T^{3} -$$$$61\!\cdots\!79$$$$T^{4} - 353722300894062450 T^{5} + 205198709271 T^{6} + 1219050 T^{7} + T^{8}$$
$79$ $$13\!\cdots\!89$$$$+$$$$52\!\cdots\!28$$$$T +$$$$35\!\cdots\!70$$$$T^{2} -$$$$23\!\cdots\!40$$$$T^{3} +$$$$20\!\cdots\!19$$$$T^{4} - 83509393737314288 T^{5} + 658387505482 T^{6} - 493868 T^{7} + T^{8}$$
$83$ $$11\!\cdots\!36$$$$+$$$$20\!\cdots\!44$$$$T^{2} +$$$$96\!\cdots\!21$$$$T^{4} + 1710139830126 T^{6} + T^{8}$$
$89$ $$13\!\cdots\!56$$$$+$$$$51\!\cdots\!32$$$$T -$$$$49\!\cdots\!00$$$$T^{2} -$$$$19\!\cdots\!96$$$$T^{3} +$$$$15\!\cdots\!48$$$$T^{4} + 843522787997502480 T^{5} - 1274249948148 T^{6} - 604260 T^{7} + T^{8}$$
$97$ $$15\!\cdots\!84$$$$+$$$$45\!\cdots\!12$$$$T^{2} +$$$$31\!\cdots\!29$$$$T^{4} + 3383453221110 T^{6} + T^{8}$$