# Properties

 Label 261.3.f.a Level 261 Weight 3 Character orbit 261.f Analytic conductor 7.112 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$261 = 3^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 261.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.11173489980$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{5} q^{2} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{4} + ( -\beta_{3} - \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( 5 - \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{8} +O(q^{10})$$ $$q -\beta_{5} q^{2} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{4} + ( -\beta_{3} - \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{7} + ( 5 - \beta_{2} + 2 \beta_{3} + 5 \beta_{4} - \beta_{6} + 2 \beta_{7} ) q^{8} + ( -\beta_{2} + \beta_{3} - 3 \beta_{6} + \beta_{7} ) q^{10} + ( 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} + ( 5 + \beta_{1} + \beta_{3} - 5 \beta_{4} - \beta_{7} ) q^{14} + ( -1 - \beta_{1} + \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{16} + ( \beta_{1} - 6 \beta_{5} ) q^{17} + ( -1 + \beta_{1} + 4 \beta_{3} + \beta_{4} - 4 \beta_{5} - 4 \beta_{7} ) q^{19} + ( -11 - 5 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{20} + ( -5 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{22} + ( 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{7} ) q^{23} + ( 14 + \beta_{1} - \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{25} + ( 5 - 4 \beta_{2} - \beta_{3} + 5 \beta_{4} - 7 \beta_{6} - \beta_{7} ) q^{26} + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + \beta_{5} + \beta_{6} ) q^{28} + ( -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} + ( 10 - 4 \beta_{1} - \beta_{3} - 10 \beta_{4} + 13 \beta_{5} + \beta_{7} ) q^{32} + ( \beta_{1} + \beta_{2} - 5 \beta_{3} - 30 \beta_{4} + 6 \beta_{5} + 6 \beta_{6} ) q^{34} + ( -\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} + ( 5 \beta_{1} + 5 \beta_{2} - 11 \beta_{3} - 20 \beta_{4} + 13 \beta_{5} + 13 \beta_{6} ) q^{38} + ( 25 - 2 \beta_{1} - 3 \beta_{3} - 25 \beta_{4} + 13 \beta_{5} + 3 \beta_{7} ) q^{40} + ( -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} + ( -6 + 3 \beta_{2} - 6 \beta_{4} - 3 \beta_{6} ) q^{44} + ( 7 \beta_{1} - \beta_{3} + 6 \beta_{5} + \beta_{7} ) q^{46} + ( -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} + ( 10 + 2 \beta_{1} - \beta_{3} - 10 \beta_{4} - 10 \beta_{5} + \beta_{7} ) q^{50} + ( -15 - 3 \beta_{1} + 3 \beta_{2} - 6 \beta_{5} + 6 \beta_{6} - 9 \beta_{7} ) q^{52} + ( -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} + ( 25 - 5 \beta_{2} - 8 \beta_{3} + 25 \beta_{4} + 4 \beta_{6} - 8 \beta_{7} ) q^{56} + ( 35 + 7 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 15 \beta_{4} + 19 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{58} + ( 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} + 6 \beta_{3} - 45 \beta_{4} ) q^{62} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} + 61 \beta_{4} - \beta_{5} - \beta_{6} ) q^{64} + ( -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} + ( 30 - 7 \beta_{2} + 12 \beta_{3} + 30 \beta_{4} - 28 \beta_{6} + 12 \beta_{7} ) q^{68} + ( 5 + 3 \beta_{2} - 5 \beta_{3} + 5 \beta_{4} - 2 \beta_{6} - 5 \beta_{7} ) q^{70} + ( 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} + ( 10 - 14 \beta_{1} + 14 \beta_{2} - 4 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{74} + ( 61 - 5 \beta_{2} + 13 \beta_{3} + 61 \beta_{4} - 52 \beta_{6} + 13 \beta_{7} ) q^{76} + ( -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} + ( -5 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} + 21 \beta_{4} - 24 \beta_{5} - 24 \beta_{6} ) q^{80} + ( 30 - 12 \beta_{1} + 12 \beta_{2} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{82} + ( -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} + ( -7 \beta_{1} - 7 \beta_{2} + \beta_{3} + 25 \beta_{4} + 18 \beta_{5} + 18 \beta_{6} ) q^{86} + ( 5 - 23 \beta_{1} + 23 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} + 12 \beta_{7} ) q^{88} + ( 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} + ( -2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 30 \beta_{4} - 8 \beta_{5} - 8 \beta_{6} ) q^{92} + ( -55 - 8 \beta_{1} + 8 \beta_{2} - 5 \beta_{5} + 5 \beta_{6} - 9 \beta_{7} ) q^{94} + ( 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} + ( 18 \beta_{1} - 9 \beta_{3} + 33 \beta_{5} + 9 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{2} - 4q^{7} + 42q^{8} + O(q^{10})$$ $$8q - 2q^{2} - 4q^{7} + 42q^{8} + 6q^{10} + 6q^{11} + 40q^{14} - 32q^{16} - 12q^{17} - 16q^{19} - 108q^{20} + 104q^{25} + 54q^{26} - 128q^{29} - 10q^{31} + 106q^{32} - 84q^{37} + 226q^{40} - 20q^{41} - 190q^{43} - 42q^{44} + 12q^{46} - 58q^{47} - 72q^{49} + 60q^{50} - 144q^{52} - 252q^{53} - 74q^{55} + 192q^{56} + 326q^{58} + 40q^{59} - 208q^{61} - 36q^{65} + 296q^{68} + 44q^{70} - 188q^{73} + 64q^{74} + 592q^{76} - 180q^{77} - 382q^{79} + 228q^{82} - 280q^{83} + 32q^{85} + 20q^{88} + 64q^{89} - 460q^{94} + 380q^{95} - 44q^{97} + 66q^{98} + O(q^{100})$$

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 $$\left(\mathbb{Z}/261\mathbb{Z}\right)^\times$$.

 $$n$$ $$118$$ $$146$$ $$\chi(n)$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
46.1
 − 0.486981i 1.35225i − 3.22189i 2.35663i 0.486981i − 1.35225i 3.22189i − 2.35663i
−2.62492 + 2.62492i 0 9.78036i 4.53053i 0 −0.745339 15.1730 + 15.1730i 0 11.8923 + 11.8923i
46.2 −0.909588 + 0.909588i 0 2.34530i 4.16447i 0 −9.68815 −5.77161 5.77161i 0 −3.78796 3.78796i
46.3 1.07935 1.07935i 0 1.67001i 0.488689i 0 8.09117 6.11992 + 6.11992i 0 −0.527467 0.527467i
46.4 1.45515 1.45515i 0 0.234947i 3.14526i 0 0.342313 5.47873 + 5.47873i 0 −4.57683 4.57683i
244.1 −2.62492 2.62492i 0 9.78036i 4.53053i 0 −0.745339 15.1730 15.1730i 0 11.8923 11.8923i
244.2 −0.909588 0.909588i 0 2.34530i 4.16447i 0 −9.68815 −5.77161 + 5.77161i 0 −3.78796 + 3.78796i
244.3 1.07935 + 1.07935i 0 1.67001i 0.488689i 0 8.09117 6.11992 6.11992i 0 −0.527467 + 0.527467i
244.4 1.45515 + 1.45515i 0 0.234947i 3.14526i 0 0.342313 5.47873 5.47873i 0 −4.57683 + 4.57683i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 244.4 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 261.3.f.a 8
3.b odd 2 1 29.3.c.a 8
12.b even 2 1 464.3.l.c 8
29.c odd 4 1 inner 261.3.f.a 8
87.f even 4 1 29.3.c.a 8
348.k odd 4 1 464.3.l.c 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.3.c.a 8 3.b odd 2 1
29.3.c.a 8 87.f even 4 1
261.3.f.a 8 1.a even 1 1 trivial
261.3.f.a 8 29.c odd 4 1 inner
464.3.l.c 8 12.b even 2 1
464.3.l.c 8 348.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(261, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2} - 10 T^{3} - 26 T^{4} - 18 T^{5} + 66 T^{6} + 90 T^{7} + 321 T^{8} + 360 T^{9} + 1056 T^{10} - 1152 T^{11} - 6656 T^{12} - 10240 T^{13} + 8192 T^{14} + 32768 T^{15} + 65536 T^{16}$$
$3$ 1
$5$ $$1 - 152 T^{2} + 11042 T^{4} - 495504 T^{6} + 14942291 T^{8} - 309690000 T^{10} + 4313281250 T^{12} - 37109375000 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 + 2 T + 118 T^{2} + 262 T^{3} + 6782 T^{4} + 12838 T^{5} + 283318 T^{6} + 235298 T^{7} + 5764801 T^{8} )^{2}$$
$11$ $$1 - 6 T + 18 T^{2} + 984 T^{3} + 5458 T^{4} - 25326 T^{5} + 537840 T^{6} + 21947202 T^{7} - 268164717 T^{8} + 2655611442 T^{9} + 7874515440 T^{10} - 44866553886 T^{11} + 1169970772498 T^{12} + 25522425807384 T^{13} + 56491710780978 T^{14} - 2278499001499446 T^{15} + 45949729863572161 T^{16}$$
$13$ $$1 - 900 T^{2} + 416090 T^{4} - 121830768 T^{6} + 24605646387 T^{8} - 3479608564848 T^{10} + 339417395700890 T^{12} - 20968276610232900 T^{14} + 665416609183179841 T^{16}$$
$17$ $$1 + 12 T + 72 T^{2} + 156 T^{3} - 11400 T^{4} + 788916 T^{5} + 10299960 T^{6} + 276428772 T^{7} + 7918453918 T^{8} + 79887915108 T^{9} + 860262959160 T^{10} + 19042514385204 T^{11} - 79523634827400 T^{12} + 314495048470044 T^{13} + 41948801080542792 T^{14} + 2020533918712811148 T^{15} + 48661191875666868481 T^{16}$$
$19$ $$1 + 16 T + 128 T^{2} - 9968 T^{3} - 125288 T^{4} + 124848 T^{5} + 67714944 T^{6} - 106114512 T^{7} + 873564318 T^{8} - 38307338832 T^{9} + 8824679217024 T^{10} + 5873584151088 T^{11} - 2127836646280808 T^{12} - 61114468457760368 T^{13} + 283304309640468608 T^{14} + 12784106972526145936 T^{15} +$$$$28\!\cdots\!81$$$$T^{16}$$
$23$ $$( 1 + 1798 T^{2} + 204 T^{3} + 1362542 T^{4} + 107916 T^{5} + 503154118 T^{6} + 78310985281 T^{8} )^{2}$$
$29$ $$1 + 128 T + 7400 T^{2} + 269120 T^{3} + 8004638 T^{4} + 226329920 T^{5} + 5233879400 T^{6} + 76137385088 T^{7} + 500246412961 T^{8}$$
$31$ $$1 + 10 T + 50 T^{2} - 18752 T^{3} - 59126 T^{4} + 5169114 T^{5} + 230466192 T^{6} - 14955517326 T^{7} - 982612297341 T^{8} - 14372252150286 T^{9} + 212840368102032 T^{10} + 4587607702508634 T^{11} - 50428035479736566 T^{12} - 15369669637463980352 T^{13} + 39383139189427488050 T^{14} +$$$$75\!\cdots\!10$$$$T^{15} +$$$$72\!\cdots\!81$$$$T^{16}$$
$37$ $$1 + 84 T + 3528 T^{2} + 71652 T^{3} - 2089980 T^{4} - 123205908 T^{5} - 408842280 T^{6} + 218820380604 T^{7} + 13702947378118 T^{8} + 299565101046876 T^{9} - 766236256327080 T^{10} - 316112651900424372 T^{11} - 7341011809105811580 T^{12} +$$$$34\!\cdots\!48$$$$T^{13} +$$$$23\!\cdots\!68$$$$T^{14} +$$$$75\!\cdots\!76$$$$T^{15} +$$$$12\!\cdots\!41$$$$T^{16}$$
$41$ $$1 + 20 T + 200 T^{2} + 65948 T^{3} + 2077432 T^{4} - 115380012 T^{5} - 548517288 T^{6} - 112249030692 T^{7} - 9153258882402 T^{8} - 188690620593252 T^{9} - 1549978760256168 T^{10} - 548067084327830892 T^{11} + 16588139188583297272 T^{12} +$$$$88\!\cdots\!48$$$$T^{13} +$$$$45\!\cdots\!00$$$$T^{14} +$$$$75\!\cdots\!20$$$$T^{15} +$$$$63\!\cdots\!41$$$$T^{16}$$
$43$ $$1 + 190 T + 18050 T^{2} + 1310896 T^{3} + 89172690 T^{4} + 5513705486 T^{5} + 297261149248 T^{6} + 14457675945470 T^{7} + 648656884794643 T^{8} + 26732242823174030 T^{9} + 1016276714310211648 T^{10} + 34854134122268986814 T^{11} +$$$$10\!\cdots\!90$$$$T^{12} +$$$$28\!\cdots\!04$$$$T^{13} +$$$$72\!\cdots\!50$$$$T^{14} +$$$$14\!\cdots\!10$$$$T^{15} +$$$$13\!\cdots\!01$$$$T^{16}$$
$47$ $$1 + 58 T + 1682 T^{2} + 75904 T^{3} + 8447626 T^{4} + 469112154 T^{5} + 15880306608 T^{6} + 750498418338 T^{7} + 30625787988195 T^{8} + 1657851006108642 T^{9} + 77490830429232048 T^{10} + 5056660921417008666 T^{11} +$$$$20\!\cdots\!86$$$$T^{12} +$$$$39\!\cdots\!96$$$$T^{13} +$$$$19\!\cdots\!62$$$$T^{14} +$$$$14\!\cdots\!02$$$$T^{15} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$( 1 + 126 T + 12120 T^{2} + 805920 T^{3} + 48326113 T^{4} + 2263829280 T^{5} + 95632629720 T^{6} + 2792709502254 T^{7} + 62259690411361 T^{8} )^{2}$$
$59$ $$( 1 - 20 T + 4038 T^{2} + 142432 T^{3} + 879134 T^{4} + 495805792 T^{5} + 48929903718 T^{6} - 843610672820 T^{7} + 146830437604321 T^{8} )^{2}$$
$61$ $$1 + 208 T + 21632 T^{2} + 2017096 T^{3} + 184647288 T^{4} + 13664724728 T^{5} + 882310746016 T^{6} + 58889137003472 T^{7} + 3818097286447198 T^{8} + 219126478789919312 T^{9} + 12216334301928919456 T^{10} +$$$$70\!\cdots\!08$$$$T^{11} +$$$$35\!\cdots\!28$$$$T^{12} +$$$$14\!\cdots\!96$$$$T^{13} +$$$$57\!\cdots\!72$$$$T^{14} +$$$$20\!\cdots\!28$$$$T^{15} +$$$$36\!\cdots\!61$$$$T^{16}$$
$67$ $$1 - 11008 T^{2} + 27285916 T^{4} + 201250654976 T^{6} - 1694675724118778 T^{8} + 4055426299750628096 T^{10} +$$$$11\!\cdots\!56$$$$T^{12} -$$$$90\!\cdots\!88$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16}$$
$71$ $$1 - 21324 T^{2} + 250110380 T^{4} - 1967023408740 T^{6} + 11455271890903830 T^{8} - 49985371382433491940 T^{10} +$$$$16\!\cdots\!80$$$$T^{12} -$$$$34\!\cdots\!84$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16}$$
$73$ $$1 + 188 T + 17672 T^{2} + 1559180 T^{3} + 182719716 T^{4} + 18486242500 T^{5} + 1461911905048 T^{6} + 113727191207188 T^{7} + 8685700663672390 T^{8} + 606052201943104852 T^{9} + 41515726600322220568 T^{10} +$$$$27\!\cdots\!00$$$$T^{11} +$$$$14\!\cdots\!96$$$$T^{12} +$$$$67\!\cdots\!20$$$$T^{13} +$$$$40\!\cdots\!12$$$$T^{14} +$$$$22\!\cdots\!92$$$$T^{15} +$$$$65\!\cdots\!61$$$$T^{16}$$
$79$ $$1 + 382 T + 72962 T^{2} + 10210992 T^{3} + 1201614634 T^{4} + 122408323406 T^{5} + 11219951427216 T^{6} + 967189248125926 T^{7} + 78871519191926211 T^{8} + 6036228097553904166 T^{9} +$$$$43\!\cdots\!96$$$$T^{10} +$$$$29\!\cdots\!26$$$$T^{11} +$$$$18\!\cdots\!74$$$$T^{12} +$$$$96\!\cdots\!92$$$$T^{13} +$$$$43\!\cdots\!42$$$$T^{14} +$$$$14\!\cdots\!42$$$$T^{15} +$$$$23\!\cdots\!21$$$$T^{16}$$
$83$ $$( 1 + 140 T + 22718 T^{2} + 2108544 T^{3} + 207732422 T^{4} + 14525759616 T^{5} + 1078158136478 T^{6} + 45771652271660 T^{7} + 2252292232139041 T^{8} )^{2}$$
$89$ $$1 - 64 T + 2048 T^{2} + 960272 T^{3} - 17872712 T^{4} + 244915632 T^{5} + 481989870720 T^{6} + 41121933793344 T^{7} - 1442843088822882 T^{8} + 325726837577077824 T^{9} + 30241124628273083520 T^{10} +$$$$12\!\cdots\!52$$$$T^{11} -$$$$70\!\cdots\!72$$$$T^{12} +$$$$29\!\cdots\!72$$$$T^{13} +$$$$50\!\cdots\!08$$$$T^{14} -$$$$12\!\cdots\!24$$$$T^{15} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$1 + 44 T + 968 T^{2} - 45508 T^{3} + 12434104 T^{4} + 5439303060 T^{5} + 228328611000 T^{6} + 46830606041796 T^{7} + 9294107220541086 T^{8} + 440629172247258564 T^{9} + 20213767763558691000 T^{10} +$$$$45\!\cdots\!40$$$$T^{11} +$$$$97\!\cdots\!44$$$$T^{12} -$$$$33\!\cdots\!92$$$$T^{13} +$$$$67\!\cdots\!88$$$$T^{14} +$$$$28\!\cdots\!36$$$$T^{15} +$$$$61\!\cdots\!21$$$$T^{16}$$