# Properties

 Label 464.3.l.c Level 464 Weight 3 Character orbit 464.l Analytic conductor 12.643 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$464 = 2^{4} \cdot 29$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 464.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.6430842663$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 18 x^{6} + 91 x^{4} + 126 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 29) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} + \beta_{5} ) q^{3} + ( \beta_{3} + \beta_{4} ) q^{5} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} + \beta_{5} ) q^{3} + ( \beta_{3} + \beta_{4} ) q^{5} + ( -\beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + ( -\beta_{1} - \beta_{2} - 3 \beta_{3} - 6 \beta_{4} - \beta_{5} - \beta_{6} ) q^{9} + ( 1 - 4 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{2} - 5 \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( -2 \beta_{2} - 5 \beta_{6} ) q^{15} + ( -\beta_{1} + 6 \beta_{5} ) q^{17} + ( 1 - \beta_{1} - 4 \beta_{3} - \beta_{4} + 4 \beta_{5} + 4 \beta_{7} ) q^{19} + ( -5 - 3 \beta_{1} - \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{7} ) q^{23} + ( 14 + \beta_{1} - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{25} + ( 15 + 2 \beta_{2} + 3 \beta_{3} + 15 \beta_{4} + 11 \beta_{6} + 3 \beta_{7} ) q^{27} + ( 15 - 2 \beta_{1} + 5 \beta_{2} - 6 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} ) q^{29} + ( -1 - 7 \beta_{1} + 4 \beta_{3} + \beta_{4} + 9 \beta_{5} - 4 \beta_{7} ) q^{31} + ( 3 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} + 45 \beta_{4} + 12 \beta_{5} + 12 \beta_{6} ) q^{33} + ( -\beta_{1} - \beta_{2} + 5 \beta_{3} - 10 \beta_{4} + \beta_{5} + \beta_{6} ) q^{35} + ( -10 + 6 \beta_{2} + 8 \beta_{3} - 10 \beta_{4} + 2 \beta_{6} + 8 \beta_{7} ) q^{37} + ( 15 + 3 \beta_{2} + 15 \beta_{4} + 15 \beta_{6} ) q^{39} + ( 1 - 7 \beta_{2} - 5 \beta_{3} + \beta_{4} - 6 \beta_{6} - 5 \beta_{7} ) q^{41} + ( 25 + 6 \beta_{1} + \beta_{3} - 25 \beta_{4} - 5 \beta_{5} - \beta_{7} ) q^{43} + ( 36 - 5 \beta_{1} + 5 \beta_{2} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{45} + ( -10 + 6 \beta_{2} + 2 \beta_{3} - 10 \beta_{4} - 11 \beta_{6} + 2 \beta_{7} ) q^{47} + ( -9 + 3 \beta_{1} - 3 \beta_{2} - 12 \beta_{7} ) q^{49} + ( -6 \beta_{1} - 6 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 8 \beta_{5} + 8 \beta_{6} ) q^{51} + ( 35 - 6 \beta_{1} + 6 \beta_{2} - 7 \beta_{5} + 7 \beta_{6} - 10 \beta_{7} ) q^{53} + ( 11 + 7 \beta_{2} + 3 \beta_{3} + 11 \beta_{4} + 7 \beta_{6} + 3 \beta_{7} ) q^{55} + ( \beta_{1} + \beta_{2} - 7 \beta_{3} - 10 \beta_{4} + 23 \beta_{5} + 23 \beta_{6} ) q^{57} + ( 10 - 4 \beta_{1} + 4 \beta_{2} - 10 \beta_{5} + 10 \beta_{6} - 17 \beta_{7} ) q^{59} + ( -24 - 9 \beta_{1} - 2 \beta_{3} + 24 \beta_{4} - 8 \beta_{5} + 2 \beta_{7} ) q^{61} + ( 3 \beta_{1} + 3 \beta_{2} - 10 \beta_{3} + 20 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{63} + ( 5 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} + 11 \beta_{7} ) q^{65} + ( -15 \beta_{1} - 15 \beta_{2} + 6 \beta_{3} + 40 \beta_{4} ) q^{67} + ( 20 - 3 \beta_{1} - 2 \beta_{3} - 20 \beta_{4} - 20 \beta_{5} + 2 \beta_{7} ) q^{69} + ( 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + 62 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{71} + ( -20 + 6 \beta_{2} - 4 \beta_{3} - 20 \beta_{4} + 14 \beta_{6} - 4 \beta_{7} ) q^{73} + ( 18 \beta_{1} + 3 \beta_{3} + 6 \beta_{5} - 3 \beta_{7} ) q^{75} + ( 25 + 17 \beta_{1} + 2 \beta_{3} - 25 \beta_{4} - 10 \beta_{5} - 2 \beta_{7} ) q^{77} + ( 46 - 14 \beta_{1} - 6 \beta_{3} - 46 \beta_{4} + 7 \beta_{5} + 6 \beta_{7} ) q^{79} + ( -21 + 20 \beta_{1} - 20 \beta_{2} + 11 \beta_{5} - 11 \beta_{6} + 3 \beta_{7} ) q^{81} + ( -30 - 7 \beta_{1} + 7 \beta_{2} - 10 \beta_{5} + 10 \beta_{6} + 21 \beta_{7} ) q^{83} + ( -5 \beta_{2} + 7 \beta_{3} - 16 \beta_{6} + 7 \beta_{7} ) q^{85} + ( -15 + 11 \beta_{1} + 16 \beta_{2} + 8 \beta_{3} + 35 \beta_{4} + 33 \beta_{5} - 10 \beta_{6} + 9 \beta_{7} ) q^{87} + ( -11 - 31 \beta_{1} + 8 \beta_{3} + 11 \beta_{4} + 12 \beta_{5} - 8 \beta_{7} ) q^{89} + ( 4 \beta_{1} + 4 \beta_{2} + 15 \beta_{3} - 50 \beta_{4} + \beta_{5} + \beta_{6} ) q^{91} + ( -14 \beta_{1} - 14 \beta_{2} - 11 \beta_{3} + 25 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} ) q^{93} + ( 41 + 5 \beta_{2} + 2 \beta_{3} + 41 \beta_{4} - 26 \beta_{6} + 2 \beta_{7} ) q^{95} + ( -5 + 7 \beta_{2} + 16 \beta_{3} - 5 \beta_{4} + 2 \beta_{6} + 16 \beta_{7} ) q^{97} + ( -81 - 39 \beta_{2} - 18 \beta_{3} - 81 \beta_{4} - 48 \beta_{6} - 18 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{3} + 4q^{7} + O(q^{10})$$ $$8q + 2q^{3} + 4q^{7} + 6q^{11} + 10q^{15} + 12q^{17} + 16q^{19} - 36q^{21} + 104q^{25} + 98q^{27} + 128q^{29} + 10q^{31} - 84q^{37} + 90q^{39} + 20q^{41} + 190q^{43} + 292q^{45} - 58q^{47} - 72q^{49} + 252q^{53} + 74q^{55} + 40q^{59} - 208q^{61} + 36q^{65} + 120q^{69} - 188q^{73} + 12q^{75} + 180q^{77} + 382q^{79} - 124q^{81} - 280q^{83} + 32q^{85} - 34q^{87} - 64q^{89} + 380q^{95} - 44q^{97} - 552q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 18 x^{6} + 91 x^{4} + 126 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{4} + 9 \nu^{2} + 6 \nu + 5$$$$)/6$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{4} - 9 \nu^{2} + 6 \nu - 5$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{5} - 15 \nu^{3} - 47 \nu$$$$)/6$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 18 \nu^{5} - 86 \nu^{3} - 81 \nu$$$$)/30$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + 17 \nu^{5} + \nu^{4} + 77 \nu^{3} + 15 \nu^{2} + 82 \nu + 35$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 17 \nu^{5} - \nu^{4} + 77 \nu^{3} - 15 \nu^{2} + 82 \nu - 35$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 16 \nu^{4} + 62 \nu^{2} + 35$$$$)/6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{6} + 2 \beta_{5} + \beta_{2} - \beta_{1} - 10$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + 5 \beta_{4} - \beta_{3} - 4 \beta_{2} - 4 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$($$$$18 \beta_{6} - 18 \beta_{5} - 15 \beta_{2} + 15 \beta_{1} + 80$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-30 \beta_{6} - 30 \beta_{5} - 150 \beta_{4} + 18 \beta_{3} + 73 \beta_{2} + 73 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$6 \beta_{7} - 82 \beta_{6} + 82 \beta_{5} + 89 \beta_{2} - 89 \beta_{1} - 365$$ $$\nu^{7}$$ $$=$$ $$($$$$368 \beta_{6} + 368 \beta_{5} + 1780 \beta_{4} - 152 \beta_{3} - 707 \beta_{2} - 707 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$$.

 $$n$$ $$117$$ $$175$$ $$321$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 2.35663i 1.35225i − 3.22189i − 0.486981i − 2.35663i − 1.35225i 3.22189i 0.486981i
0 −3.81178 + 3.81178i 0 3.14526i 0 −0.342313 0 20.0593i 0
17.2 0 −0.442660 + 0.442660i 0 4.16447i 0 9.68815 0 8.60810i 0
17.3 0 2.14254 2.14254i 0 0.488689i 0 −8.09117 0 0.180982i 0
17.4 0 3.11190 3.11190i 0 4.53053i 0 0.745339 0 10.3678i 0
273.1 0 −3.81178 3.81178i 0 3.14526i 0 −0.342313 0 20.0593i 0
273.2 0 −0.442660 0.442660i 0 4.16447i 0 9.68815 0 8.60810i 0
273.3 0 2.14254 + 2.14254i 0 0.488689i 0 −8.09117 0 0.180982i 0
273.4 0 3.11190 + 3.11190i 0 4.53053i 0 0.745339 0 10.3678i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 273.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 464.3.l.c 8
4.b odd 2 1 29.3.c.a 8
12.b even 2 1 261.3.f.a 8
29.c odd 4 1 inner 464.3.l.c 8
116.e even 4 1 29.3.c.a 8
348.k odd 4 1 261.3.f.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.c.a 8 4.b odd 2 1
29.3.c.a 8 116.e even 4 1
261.3.f.a 8 12.b even 2 1
261.3.f.a 8 348.k odd 4 1
464.3.l.c 8 1.a even 1 1 trivial
464.3.l.c 8 29.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(464, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 2 T + 2 T^{2} - 40 T^{3} + 10 T^{4} + 30 T^{5} + 720 T^{6} - 1242 T^{7} + 3483 T^{8} - 11178 T^{9} + 58320 T^{10} + 21870 T^{11} + 65610 T^{12} - 2361960 T^{13} + 1062882 T^{14} - 9565938 T^{15} + 43046721 T^{16}$$
$5$ $$1 - 152 T^{2} + 11042 T^{4} - 495504 T^{6} + 14942291 T^{8} - 309690000 T^{10} + 4313281250 T^{12} - 37109375000 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 - 2 T + 118 T^{2} - 262 T^{3} + 6782 T^{4} - 12838 T^{5} + 283318 T^{6} - 235298 T^{7} + 5764801 T^{8} )^{2}$$
$11$ $$1 - 6 T + 18 T^{2} + 984 T^{3} + 5458 T^{4} - 25326 T^{5} + 537840 T^{6} + 21947202 T^{7} - 268164717 T^{8} + 2655611442 T^{9} + 7874515440 T^{10} - 44866553886 T^{11} + 1169970772498 T^{12} + 25522425807384 T^{13} + 56491710780978 T^{14} - 2278499001499446 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 - 900 T^{2} + 416090 T^{4} - 121830768 T^{6} + 24605646387 T^{8} - 3479608564848 T^{10} + 339417395700890 T^{12} - 20968276610232900 T^{14} + 665416609183179841 T^{16}$$
$17$ $$1 - 12 T + 72 T^{2} - 156 T^{3} - 11400 T^{4} - 788916 T^{5} + 10299960 T^{6} - 276428772 T^{7} + 7918453918 T^{8} - 79887915108 T^{9} + 860262959160 T^{10} - 19042514385204 T^{11} - 79523634827400 T^{12} - 314495048470044 T^{13} + 41948801080542792 T^{14} - 2020533918712811148 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 - 16 T + 128 T^{2} + 9968 T^{3} - 125288 T^{4} - 124848 T^{5} + 67714944 T^{6} + 106114512 T^{7} + 873564318 T^{8} + 38307338832 T^{9} + 8824679217024 T^{10} - 5873584151088 T^{11} - 2127836646280808 T^{12} + 61114468457760368 T^{13} + 283304309640468608 T^{14} - 12784106972526145936 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$( 1 + 1798 T^{2} + 204 T^{3} + 1362542 T^{4} + 107916 T^{5} + 503154118 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$1 - 128 T + 7400 T^{2} - 269120 T^{3} + 8004638 T^{4} - 226329920 T^{5} + 5233879400 T^{6} - 76137385088 T^{7} + 500246412961 T^{8}$$
$31$ $$1 - 10 T + 50 T^{2} + 18752 T^{3} - 59126 T^{4} - 5169114 T^{5} + 230466192 T^{6} + 14955517326 T^{7} - 982612297341 T^{8} + 14372252150286 T^{9} + 212840368102032 T^{10} - 4587607702508634 T^{11} - 50428035479736566 T^{12} + 15369669637463980352 T^{13} + 39383139189427488050 T^{14} -$$$$75\!\cdots\!10$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 84 T + 3528 T^{2} + 71652 T^{3} - 2089980 T^{4} - 123205908 T^{5} - 408842280 T^{6} + 218820380604 T^{7} + 13702947378118 T^{8} + 299565101046876 T^{9} - 766236256327080 T^{10} - 316112651900424372 T^{11} - 7341011809105811580 T^{12} +$$$$34\!\cdots\!48$$$$T^{13} +$$$$23\!\cdots\!68$$$$T^{14} +$$$$75\!\cdots\!76$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 - 20 T + 200 T^{2} - 65948 T^{3} + 2077432 T^{4} + 115380012 T^{5} - 548517288 T^{6} + 112249030692 T^{7} - 9153258882402 T^{8} + 188690620593252 T^{9} - 1549978760256168 T^{10} + 548067084327830892 T^{11} + 16588139188583297272 T^{12} -$$$$88\!\cdots\!48$$$$T^{13} +$$$$45\!\cdots\!00$$$$T^{14} -$$$$75\!\cdots\!20$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$1 - 190 T + 18050 T^{2} - 1310896 T^{3} + 89172690 T^{4} - 5513705486 T^{5} + 297261149248 T^{6} - 14457675945470 T^{7} + 648656884794643 T^{8} - 26732242823174030 T^{9} + 1016276714310211648 T^{10} - 34854134122268986814 T^{11} +$$$$10\!\cdots\!90$$$$T^{12} -$$$$28\!\cdots\!04$$$$T^{13} +$$$$72\!\cdots\!50$$$$T^{14} -$$$$14\!\cdots\!10$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 58 T + 1682 T^{2} + 75904 T^{3} + 8447626 T^{4} + 469112154 T^{5} + 15880306608 T^{6} + 750498418338 T^{7} + 30625787988195 T^{8} + 1657851006108642 T^{9} + 77490830429232048 T^{10} + 5056660921417008666 T^{11} +$$$$20\!\cdots\!86$$$$T^{12} +$$$$39\!\cdots\!96$$$$T^{13} +$$$$19\!\cdots\!62$$$$T^{14} +$$$$14\!\cdots\!02$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 - 126 T + 12120 T^{2} - 805920 T^{3} + 48326113 T^{4} - 2263829280 T^{5} + 95632629720 T^{6} - 2792709502254 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$( 1 - 20 T + 4038 T^{2} + 142432 T^{3} + 879134 T^{4} + 495805792 T^{5} + 48929903718 T^{6} - 843610672820 T^{7} + 146830437604321 T^{8} )^{2}$$
$61$ $$1 + 208 T + 21632 T^{2} + 2017096 T^{3} + 184647288 T^{4} + 13664724728 T^{5} + 882310746016 T^{6} + 58889137003472 T^{7} + 3818097286447198 T^{8} + 219126478789919312 T^{9} + 12216334301928919456 T^{10} +$$$$70\!\cdots\!08$$$$T^{11} +$$$$35\!\cdots\!28$$$$T^{12} +$$$$14\!\cdots\!96$$$$T^{13} +$$$$57\!\cdots\!72$$$$T^{14} +$$$$20\!\cdots\!28$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$1 - 11008 T^{2} + 27285916 T^{4} + 201250654976 T^{6} - 1694675724118778 T^{8} + 4055426299750628096 T^{10} +$$$$11\!\cdots\!56$$$$T^{12} -$$$$90\!\cdots\!88$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$1 - 21324 T^{2} + 250110380 T^{4} - 1967023408740 T^{6} + 11455271890903830 T^{8} - 49985371382433491940 T^{10} +$$$$16\!\cdots\!80$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$1 + 188 T + 17672 T^{2} + 1559180 T^{3} + 182719716 T^{4} + 18486242500 T^{5} + 1461911905048 T^{6} + 113727191207188 T^{7} + 8685700663672390 T^{8} + 606052201943104852 T^{9} + 41515726600322220568 T^{10} +$$$$27\!\cdots\!00$$$$T^{11} +$$$$14\!\cdots\!96$$$$T^{12} +$$$$67\!\cdots\!20$$$$T^{13} +$$$$40\!\cdots\!12$$$$T^{14} +$$$$22\!\cdots\!92$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 - 382 T + 72962 T^{2} - 10210992 T^{3} + 1201614634 T^{4} - 122408323406 T^{5} + 11219951427216 T^{6} - 967189248125926 T^{7} + 78871519191926211 T^{8} - 6036228097553904166 T^{9} +$$$$43\!\cdots\!96$$$$T^{10} -$$$$29\!\cdots\!26$$$$T^{11} +$$$$18\!\cdots\!74$$$$T^{12} -$$$$96\!\cdots\!92$$$$T^{13} +$$$$43\!\cdots\!42$$$$T^{14} -$$$$14\!\cdots\!42$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$( 1 + 140 T + 22718 T^{2} + 2108544 T^{3} + 207732422 T^{4} + 14525759616 T^{5} + 1078158136478 T^{6} + 45771652271660 T^{7} + 2252292232139041 T^{8} )^{2}$$
$89$ $$1 + 64 T + 2048 T^{2} - 960272 T^{3} - 17872712 T^{4} - 244915632 T^{5} + 481989870720 T^{6} - 41121933793344 T^{7} - 1442843088822882 T^{8} - 325726837577077824 T^{9} + 30241124628273083520 T^{10} -$$$$12\!\cdots\!52$$$$T^{11} -$$$$70\!\cdots\!72$$$$T^{12} -$$$$29\!\cdots\!72$$$$T^{13} +$$$$50\!\cdots\!08$$$$T^{14} +$$$$12\!\cdots\!24$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 + 44 T + 968 T^{2} - 45508 T^{3} + 12434104 T^{4} + 5439303060 T^{5} + 228328611000 T^{6} + 46830606041796 T^{7} + 9294107220541086 T^{8} + 440629172247258564 T^{9} + 20213767763558691000 T^{10} +$$$$45\!\cdots\!40$$$$T^{11} +$$$$97\!\cdots\!44$$$$T^{12} -$$$$33\!\cdots\!92$$$$T^{13} +$$$$67\!\cdots\!88$$$$T^{14} +$$$$28\!\cdots\!36$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$