# Properties

 Label 7.7.d.b Level 7 Weight 7 Character orbit 7.d Analytic conductor 1.610 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 7.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.61037858534$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{2} + ( 6 + 3 \beta_{1} + 13 \beta_{3} ) q^{3} + ( -30 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{4} + ( -25 + 25 \beta_{1} + 50 \beta_{2} - 50 \beta_{3} ) q^{5} + ( 66 + 132 \beta_{1} - 43 \beta_{2} ) q^{6} + ( 7 - 126 \beta_{1} - 7 \beta_{2} - 126 \beta_{3} ) q^{7} + ( -232 - 62 \beta_{2} + 124 \beta_{3} ) q^{8} + ( 312 + 312 \beta_{1} + 78 \beta_{2} + 78 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -4 - 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{2} + ( 6 + 3 \beta_{1} + 13 \beta_{3} ) q^{3} + ( -30 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{4} + ( -25 + 25 \beta_{1} + 50 \beta_{2} - 50 \beta_{3} ) q^{5} + ( 66 + 132 \beta_{1} - 43 \beta_{2} ) q^{6} + ( 7 - 126 \beta_{1} - 7 \beta_{2} - 126 \beta_{3} ) q^{7} + ( -232 - 62 \beta_{2} + 124 \beta_{3} ) q^{8} + ( 312 + 312 \beta_{1} + 78 \beta_{2} + 78 \beta_{3} ) q^{9} + ( -400 - 200 \beta_{1} + 125 \beta_{3} ) q^{10} + ( -941 \beta_{1} + 362 \beta_{2} - 181 \beta_{3} ) q^{11} + ( 714 - 714 \beta_{1} - 462 \beta_{2} + 462 \beta_{3} ) q^{12} + ( 1470 + 2940 \beta_{1} - 238 \beta_{2} ) q^{13} + ( -1246 - 1582 \beta_{1} + 413 \beta_{2} + 231 \beta_{3} ) q^{14} + ( -4125 + 475 \beta_{2} - 950 \beta_{3} ) q^{15} + ( 124 + 124 \beta_{1} - 992 \beta_{2} - 992 \beta_{3} ) q^{16} + ( 4486 + 2243 \beta_{1} - 78 \beta_{3} ) q^{17} + 156 \beta_{1} q^{18} + ( 3013 - 3013 \beta_{1} + 267 \beta_{2} - 267 \beta_{3} ) q^{19} + ( 3150 + 6300 \beta_{1} + 2100 \beta_{2} ) q^{20} + ( -9408 - 10731 \beta_{1} - 2058 \beta_{2} + 1372 \beta_{3} ) q^{21} + ( -7022 - 1665 \beta_{2} + 3330 \beta_{3} ) q^{22} + ( 1235 + 1235 \beta_{1} + 3431 \beta_{2} + 3431 \beta_{3} ) q^{23} + ( 8280 + 4140 \beta_{1} - 2458 \beta_{3} ) q^{24} + ( -1250 \beta_{1} - 5000 \beta_{2} + 2500 \beta_{3} ) q^{25} + ( 7308 - 7308 \beta_{1} + 5362 \beta_{2} - 5362 \beta_{3} ) q^{26} + ( 4833 + 9666 \beta_{1} - 4719 \beta_{2} ) q^{27} + ( -10500 + 1722 \beta_{1} + 2660 \beta_{2} - 4942 \beta_{3} ) q^{28} + ( -8636 + 3570 \beta_{2} - 7140 \beta_{3} ) q^{29} + ( 7950 + 7950 \beta_{1} - 2225 \beta_{2} - 2225 \beta_{3} ) q^{30} + ( -5734 - 2867 \beta_{1} + 2869 \beta_{3} ) q^{31} + ( -3504 \beta_{1} + 248 \beta_{2} - 124 \beta_{3} ) q^{32} + ( -11295 + 11295 \beta_{1} - 10604 \beta_{2} + 10604 \beta_{3} ) q^{33} + ( -9440 - 18880 \beta_{1} + 7041 \beta_{2} ) q^{34} + ( 42875 + 8575 \beta_{1} + 3675 \beta_{2} + 6125 \beta_{3} ) q^{35} + ( 20592 - 4836 \beta_{2} + 9672 \beta_{3} ) q^{36} + ( -21935 - 21935 \beta_{1} - 6264 \beta_{2} - 6264 \beta_{3} ) q^{37} + ( -27308 - 13654 \beta_{1} + 10107 \beta_{3} ) q^{38} + ( -5334 \beta_{1} + 36792 \beta_{2} - 18396 \beta_{3} ) q^{39} + ( -12800 + 12800 \beta_{1} - 6950 \beta_{2} + 6950 \beta_{3} ) q^{40} + ( -2058 - 4116 \beta_{1} - 9338 \beta_{2} ) q^{41} + ( 15288 + 41748 \beta_{1} - 17395 \beta_{2} + 3822 \beta_{3} ) q^{42} + ( 80054 + 2730 \beta_{2} - 5460 \beta_{3} ) q^{43} + ( -2166 - 2166 \beta_{1} - 2098 \beta_{2} - 2098 \beta_{3} ) q^{44} + ( -62400 - 31200 \beta_{1} - 21450 \beta_{3} ) q^{45} + ( 56818 \beta_{1} - 24978 \beta_{2} + 12489 \beta_{3} ) q^{46} + ( -59119 + 59119 \beta_{1} + 15499 \beta_{2} - 15499 \beta_{3} ) q^{47} + ( -77004 - 154008 \beta_{1} - 7316 \beta_{2} ) q^{48} + ( 79135 + 88200 \beta_{1} + 31654 \beta_{2} - 35280 \beta_{3} ) q^{49} + ( 40000 + 8750 \beta_{2} - 17500 \beta_{3} ) q^{50} + ( 14103 + 14103 \beta_{1} + 28925 \beta_{2} + 28925 \beta_{3} ) q^{51} + ( 65352 + 32676 \beta_{1} + 28140 \beta_{3} ) q^{52} + ( -52265 \beta_{1} - 95160 \beta_{2} + 47580 \beta_{3} ) q^{53} + ( 47646 - 47646 \beta_{1} + 33375 \beta_{2} - 33375 \beta_{3} ) q^{54} + ( -30775 - 61550 \beta_{1} + 33475 \beta_{2} ) q^{55} + ( -97972 - 22848 \beta_{1} - 14434 \beta_{2} + 37912 \beta_{3} ) q^{56} + ( 6291 - 38368 \beta_{2} + 76736 \beta_{3} ) q^{57} + ( -29716 - 29716 \beta_{1} + 5644 \beta_{2} + 5644 \beta_{3} ) q^{58} + ( -20882 - 10441 \beta_{1} - 32299 \beta_{3} ) q^{59} + ( 192150 \beta_{1} + 94500 \beta_{2} - 47250 \beta_{3} ) q^{60} + ( -23447 + 23447 \beta_{1} - 23680 \beta_{2} + 23680 \beta_{3} ) q^{61} + ( 28682 + 57364 \beta_{1} - 20077 \beta_{2} ) q^{62} + ( -14196 - 119028 \beta_{1} - 50778 \beta_{2} + 22386 \beta_{3} ) q^{63} + ( -8312 + 59488 \beta_{2} - 118976 \beta_{3} ) q^{64} + ( -38850 - 38850 \beta_{1} - 67550 \beta_{2} - 67550 \beta_{3} ) q^{65} + ( 217608 + 108804 \beta_{1} - 76301 \beta_{3} ) q^{66} + ( -216247 \beta_{1} + 45050 \beta_{2} - 22525 \beta_{3} ) q^{67} + ( 63546 - 63546 \beta_{1} - 51492 \beta_{2} + 51492 \beta_{3} ) q^{68} + ( 271323 + 542646 \beta_{1} + 46934 \beta_{2} ) q^{69} + ( -122500 - 75950 \beta_{1} + 12250 \beta_{2} + 40425 \beta_{3} ) q^{70} + ( -370382 - 7266 \beta_{2} + 14532 \beta_{3} ) q^{71} + ( 14664 + 14664 \beta_{1} + 1248 \beta_{2} + 1248 \beta_{3} ) q^{72} + ( -323926 - 161963 \beta_{1} + 159984 \beta_{3} ) q^{73} + ( -25012 \beta_{1} + 6242 \beta_{2} - 3121 \beta_{3} ) q^{74} + ( 198750 - 198750 \beta_{1} - 38750 \beta_{2} + 38750 \beta_{3} ) q^{75} + ( -77574 - 155148 \beta_{1} - 64302 \beta_{2} ) q^{76} + ( 33474 - 254387 \beta_{1} + 143906 \beta_{2} - 103614 \beta_{3} ) q^{77} + ( -352464 - 78918 \beta_{2} + 157836 \beta_{3} ) q^{78} + ( 467995 + 467995 \beta_{1} + 34847 \beta_{2} + 34847 \beta_{3} ) q^{79} + ( 589000 + 294500 \beta_{1} + 68200 \beta_{3} ) q^{80} + ( -552033 \beta_{1} - 16380 \beta_{2} + 8190 \beta_{3} ) q^{81} + ( 47796 - 47796 \beta_{1} + 31178 \beta_{2} - 31178 \beta_{3} ) q^{82} + ( 351036 + 702072 \beta_{1} - 91588 \beta_{2} ) q^{83} + ( -453642 - 204330 \beta_{1} + 23520 \beta_{2} - 181692 \beta_{3} ) q^{84} + ( -144825 + 110200 \beta_{2} - 220400 \beta_{3} ) q^{85} + ( -369356 - 369356 \beta_{1} + 90974 \beta_{2} + 90974 \beta_{3} ) q^{86} + ( -608736 - 304368 \beta_{1} - 144398 \beta_{3} ) q^{87} + ( 420308 \beta_{1} - 200668 \beta_{2} + 100334 \beta_{3} ) q^{88} + ( -113215 + 113215 \beta_{1} + 234984 \beta_{2} - 234984 \beta_{3} ) q^{89} + ( -3900 - 7800 \beta_{1} - 7800 \beta_{2} ) q^{90} + ( 370734 + 385728 \beta_{1} - 382396 \beta_{2} + 175812 \beta_{3} ) q^{91} + ( 531114 - 112810 \beta_{2} + 225620 \beta_{3} ) q^{92} + ( 197979 + 197979 \beta_{1} - 28664 \beta_{2} - 28664 \beta_{3} ) q^{93} + ( 286964 + 143482 \beta_{1} - 115361 \beta_{3} ) q^{94} + ( 145875 \beta_{1} + 287950 \beta_{2} - 143975 \beta_{3} ) q^{95} + ( 840 - 840 \beta_{1} - 44436 \beta_{2} + 44436 \beta_{3} ) q^{96} + ( -388962 - 777924 \beta_{1} + 155722 \beta_{2} ) q^{97} + ( -365344 - 549976 \beta_{1} + 181839 \beta_{2} + 29351 \beta_{3} ) q^{98} + ( 39468 - 16926 \beta_{2} + 33852 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{2} + 18q^{3} + 60q^{4} - 150q^{5} + 280q^{7} - 928q^{8} + 624q^{9} + O(q^{10})$$ $$4q - 8q^{2} + 18q^{3} + 60q^{4} - 150q^{5} + 280q^{7} - 928q^{8} + 624q^{9} - 1200q^{10} + 1882q^{11} + 4284q^{12} - 1820q^{14} - 16500q^{15} + 248q^{16} + 13458q^{17} - 312q^{18} + 18078q^{19} - 16170q^{21} - 28088q^{22} + 2470q^{23} + 24840q^{24} + 2500q^{25} + 43848q^{26} - 45444q^{28} - 34544q^{29} + 15900q^{30} - 17202q^{31} + 7008q^{32} - 67770q^{33} + 154350q^{35} + 82368q^{36} - 43870q^{37} - 81924q^{38} + 10668q^{39} - 76800q^{40} - 22344q^{42} + 320216q^{43} - 4332q^{44} - 187200q^{45} - 113636q^{46} - 354714q^{47} + 140140q^{49} + 160000q^{50} + 28206q^{51} + 196056q^{52} + 104530q^{53} + 285876q^{54} - 346192q^{56} + 25164q^{57} - 59432q^{58} - 62646q^{59} - 384300q^{60} - 140682q^{61} + 181272q^{63} - 33248q^{64} - 77700q^{65} + 652824q^{66} + 432494q^{67} + 381276q^{68} - 338100q^{70} - 1481528q^{71} + 29328q^{72} - 971778q^{73} + 50024q^{74} + 1192500q^{75} + 642670q^{77} - 1409856q^{78} + 935990q^{79} + 1767000q^{80} + 1104066q^{81} + 286776q^{82} - 1405908q^{84} - 579300q^{85} - 738712q^{86} - 1826208q^{87} - 840616q^{88} - 679290q^{89} + 711480q^{91} + 2124456q^{92} + 395958q^{93} + 860892q^{94} - 291750q^{95} + 5040q^{96} - 361424q^{98} + 157872q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 2 \beta_{2}$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−4.12132 7.13834i −23.0772 13.3236i −1.97056 + 3.41311i 68.5660 39.5866i 219.643i 337.286 + 62.3451i −495.044 −9.46299 16.3904i −565.165 326.298i
3.2 0.121320 + 0.210133i 32.0772 + 18.5198i 31.9706 55.3746i −143.566 + 82.8879i 8.98729i −197.286 280.583i 31.0437 321.463 + 556.790i −34.8350 20.1120i
5.1 −4.12132 + 7.13834i −23.0772 + 13.3236i −1.97056 3.41311i 68.5660 + 39.5866i 219.643i 337.286 62.3451i −495.044 −9.46299 + 16.3904i −565.165 + 326.298i
5.2 0.121320 0.210133i 32.0772 18.5198i 31.9706 + 55.3746i −143.566 82.8879i 8.98729i −197.286 + 280.583i 31.0437 321.463 556.790i −34.8350 + 20.1120i
 $$n$$: e.g. 2-40 or 990-1000 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 7.7.d.b 4
3.b odd 2 1 63.7.m.b 4
4.b odd 2 1 112.7.s.b 4
7.b odd 2 1 49.7.d.c 4
7.c even 3 1 49.7.b.b 4
7.c even 3 1 49.7.d.c 4
7.d odd 6 1 inner 7.7.d.b 4
7.d odd 6 1 49.7.b.b 4
21.g even 6 1 63.7.m.b 4
21.g even 6 1 441.7.d.b 4
21.h odd 6 1 441.7.d.b 4
28.f even 6 1 112.7.s.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.d.b 4 1.a even 1 1 trivial
7.7.d.b 4 7.d odd 6 1 inner
49.7.b.b 4 7.c even 3 1
49.7.b.b 4 7.d odd 6 1
49.7.d.c 4 7.b odd 2 1
49.7.d.c 4 7.c even 3 1
63.7.m.b 4 3.b odd 2 1
63.7.m.b 4 21.g even 6 1
112.7.s.b 4 4.b odd 2 1
112.7.s.b 4 28.f even 6 1
441.7.d.b 4 21.g even 6 1
441.7.d.b 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 8 T_{2}^{3} + 66 T_{2}^{2} - 16 T_{2} + 4$$ acting on $$S_{7}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T - 62 T^{2} - 16 T^{3} + 7684 T^{4} - 1024 T^{5} - 253952 T^{6} + 2097152 T^{7} + 16777216 T^{8}$$
$3$ $$1 - 18 T + 579 T^{2} - 8478 T^{3} - 230868 T^{4} - 6180462 T^{5} + 307704339 T^{6} - 6973568802 T^{7} + 282429536481 T^{8}$$
$5$ $$( 1 + 50 T + 15625 T^{2} )^{2}( 1 + 50 T - 13125 T^{2} + 781250 T^{3} + 244140625 T^{4} )$$
$7$ $$1 - 280 T - 30870 T^{2} - 32941720 T^{3} + 13841287201 T^{4}$$
$11$ $$1 - 1882 T - 296981 T^{2} - 556663606 T^{3} + 5324028798940 T^{4} - 986163534508966 T^{5} - 932053597746979301 T^{6} -$$$$10\!\cdots\!42$$$$T^{7} +$$$$98\!\cdots\!41$$$$T^{8}$$
$13$ $$1 - 5662108 T^{2} + 45798091584678 T^{4} -$$$$13\!\cdots\!48$$$$T^{6} +$$$$54\!\cdots\!61$$$$T^{8}$$
$17$ $$1 - 13458 T + 123704369 T^{2} - 852319108698 T^{3} + 4885539755960772 T^{4} - 20572911296216475162 T^{5} +$$$$72\!\cdots\!09$$$$T^{6} -$$$$18\!\cdots\!22$$$$T^{7} +$$$$33\!\cdots\!21$$$$T^{8}$$
$19$ $$1 - 18078 T + 229836563 T^{2} - 2185603715730 T^{3} + 17528226347742732 T^{4} -$$$$10\!\cdots\!30$$$$T^{5} +$$$$50\!\cdots\!43$$$$T^{6} -$$$$18\!\cdots\!98$$$$T^{7} +$$$$48\!\cdots\!21$$$$T^{8}$$
$23$ $$1 - 2470 T - 79604405 T^{2} + 519605188310 T^{3} - 15472377282077396 T^{4} + 76920215980483257590 T^{5} -$$$$17\!\cdots\!05$$$$T^{6} -$$$$80\!\cdots\!30$$$$T^{7} +$$$$48\!\cdots\!41$$$$T^{8}$$
$29$ $$( 1 + 17272 T + 1034818938 T^{2} + 10273788400312 T^{3} + 353814783205469041 T^{4} )^{2}$$
$31$ $$1 + 17202 T + 1848915731 T^{2} + 30108307322526 T^{3} + 2363355465741121116 T^{4} +$$$$26\!\cdots\!06$$$$T^{5} +$$$$14\!\cdots\!91$$$$T^{6} +$$$$12\!\cdots\!82$$$$T^{7} +$$$$62\!\cdots\!21$$$$T^{8}$$
$37$ $$1 + 43870 T - 2981741615 T^{2} - 9876641872610 T^{3} + 12551071586310417844 T^{4} -$$$$25\!\cdots\!90$$$$T^{5} -$$$$19\!\cdots\!15$$$$T^{6} +$$$$74\!\cdots\!30$$$$T^{7} +$$$$43\!\cdots\!61$$$$T^{8}$$
$41$ $$1 - 17928625852 T^{2} +$$$$12\!\cdots\!86$$$$T^{4} -$$$$40\!\cdots\!12$$$$T^{6} +$$$$50\!\cdots\!61$$$$T^{8}$$
$43$ $$( 1 - 160108 T + 18917216814 T^{2} - 1012100795049292 T^{3} + 39959630797262576401 T^{4} )^{2}$$
$47$ $$1 + 354714 T + 72542959067 T^{2} + 10855058969376390 T^{3} +$$$$12\!\cdots\!12$$$$T^{4} +$$$$11\!\cdots\!10$$$$T^{5} +$$$$84\!\cdots\!47$$$$T^{6} +$$$$44\!\cdots\!46$$$$T^{7} +$$$$13\!\cdots\!81$$$$T^{8}$$
$53$ $$1 - 104530 T + 4615583617 T^{2} + 3973999063436750 T^{3} -$$$$69\!\cdots\!52$$$$T^{4} +$$$$88\!\cdots\!50$$$$T^{5} +$$$$22\!\cdots\!97$$$$T^{6} -$$$$11\!\cdots\!70$$$$T^{7} +$$$$24\!\cdots\!81$$$$T^{8}$$
$59$ $$1 + 62646 T + 79736932091 T^{2} + 4913247993652074 T^{3} +$$$$44\!\cdots\!32$$$$T^{4} +$$$$20\!\cdots\!34$$$$T^{5} +$$$$14\!\cdots\!71$$$$T^{6} +$$$$47\!\cdots\!66$$$$T^{7} +$$$$31\!\cdots\!61$$$$T^{8}$$
$61$ $$1 + 140682 T + 107922721457 T^{2} + 14254685210248818 T^{3} +$$$$79\!\cdots\!68$$$$T^{4} +$$$$73\!\cdots\!98$$$$T^{5} +$$$$28\!\cdots\!97$$$$T^{6} +$$$$19\!\cdots\!42$$$$T^{7} +$$$$70\!\cdots\!41$$$$T^{8}$$
$67$ $$1 - 432494 T - 31495708061 T^{2} - 16274750845744946 T^{3} +$$$$22\!\cdots\!64$$$$T^{4} -$$$$14\!\cdots\!74$$$$T^{5} -$$$$25\!\cdots\!21$$$$T^{6} -$$$$32\!\cdots\!46$$$$T^{7} +$$$$66\!\cdots\!21$$$$T^{8}$$
$71$ $$( 1 + 740764 T + 392433088158 T^{2} + 94892078718455644 T^{3} +$$$$16\!\cdots\!41$$$$T^{4} )^{2}$$
$73$ $$1 + 971778 T + 542579371577 T^{2} + 221366374699952922 T^{3} +$$$$76\!\cdots\!72$$$$T^{4} +$$$$33\!\cdots\!58$$$$T^{5} +$$$$12\!\cdots\!17$$$$T^{6} +$$$$33\!\cdots\!82$$$$T^{7} +$$$$52\!\cdots\!41$$$$T^{8}$$
$79$ $$1 - 935990 T + 192740690395 T^{2} - 184541359611781370 T^{3} +$$$$19\!\cdots\!84$$$$T^{4} -$$$$44\!\cdots\!70$$$$T^{5} +$$$$11\!\cdots\!95$$$$T^{6} -$$$$13\!\cdots\!90$$$$T^{7} +$$$$34\!\cdots\!81$$$$T^{8}$$
$83$ $$1 - 467743512772 T^{2} +$$$$19\!\cdots\!90$$$$T^{4} -$$$$49\!\cdots\!92$$$$T^{6} +$$$$11\!\cdots\!21$$$$T^{8}$$
$89$ $$1 + 679290 T + 854922243761 T^{2} + 476257425629046690 T^{3} +$$$$32\!\cdots\!00$$$$T^{4} +$$$$23\!\cdots\!90$$$$T^{5} +$$$$21\!\cdots\!81$$$$T^{6} +$$$$83\!\cdots\!90$$$$T^{7} +$$$$61\!\cdots\!41$$$$T^{8}$$
$97$ $$1 - 2133147299644 T^{2} +$$$$22\!\cdots\!54$$$$T^{4} -$$$$14\!\cdots\!04$$$$T^{6} +$$$$48\!\cdots\!81$$$$T^{8}$$