# Properties

 Label 1792.3.d.j Level 1792 Weight 3 Character orbit 1792.d Analytic conductor 48.828 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1792 = 2^{8} \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1792.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$48.8284633734$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{38}$$ Twist minimal: no (minimal twist has level 56) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{4} q^{5} -\beta_{8} q^{7} + ( -6 - \beta_{13} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{4} q^{5} -\beta_{8} q^{7} + ( -6 - \beta_{13} ) q^{9} + ( \beta_{3} - \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{9} ) q^{13} + ( \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{15} + ( -10 + \beta_{15} ) q^{17} + ( \beta_{2} + 2 \beta_{3} ) q^{19} -\beta_{9} q^{21} + ( 5 \beta_{8} - \beta_{10} + \beta_{11} ) q^{23} + ( 2 + \beta_{13} ) q^{25} + ( -4 \beta_{2} + 2 \beta_{7} ) q^{27} + ( -2 \beta_{1} + 3 \beta_{4} - \beta_{5} - \beta_{9} ) q^{29} + ( 6 \beta_{8} + 3 \beta_{10} ) q^{31} + ( 4 - 2 \beta_{13} + \beta_{14} ) q^{33} + ( \beta_{2} + 2 \beta_{3} + \beta_{7} ) q^{35} + ( -5 \beta_{1} - \beta_{4} - 2 \beta_{9} ) q^{37} + ( -\beta_{6} - 5 \beta_{8} + 5 \beta_{10} + \beta_{11} ) q^{39} + ( -16 + \beta_{14} - \beta_{15} ) q^{41} + ( -2 \beta_{2} + \beta_{7} - \beta_{12} ) q^{43} + ( -\beta_{1} - 9 \beta_{4} + 3 \beta_{5} + \beta_{9} ) q^{45} + ( -2 \beta_{6} + 4 \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{47} -7 q^{49} + ( -12 \beta_{2} - 15 \beta_{3} - 2 \beta_{7} + \beta_{12} ) q^{51} + ( \beta_{1} - 5 \beta_{4} + 2 \beta_{9} ) q^{53} + ( -2 \beta_{6} + 14 \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{55} + ( -7 - \beta_{13} + 2 \beta_{15} ) q^{57} + ( 9 \beta_{2} - 2 \beta_{3} - 2 \beta_{7} + 2 \beta_{12} ) q^{59} + ( -6 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{9} ) q^{61} + ( 6 \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{63} + ( -9 + 5 \beta_{13} - 4 \beta_{15} ) q^{65} + ( 8 \beta_{2} + 12 \beta_{3} + \beta_{7} - \beta_{12} ) q^{67} + ( \beta_{1} - 10 \beta_{4} + \beta_{5} + 11 \beta_{9} ) q^{69} + ( 4 \beta_{6} + 8 \beta_{10} ) q^{71} + ( 14 + 2 \beta_{13} - \beta_{14} ) q^{73} + ( 9 \beta_{2} - 2 \beta_{7} ) q^{75} + ( -3 \beta_{1} + 4 \beta_{4} - \beta_{9} ) q^{77} + ( -3 \beta_{6} - 8 \beta_{8} + 6 \beta_{10} + 2 \beta_{11} ) q^{79} + ( 6 - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{81} + ( 3 \beta_{2} + 4 \beta_{3} - 2 \beta_{7} - 2 \beta_{12} ) q^{83} + ( -7 \beta_{1} - 6 \beta_{4} + \beta_{5} - 9 \beta_{9} ) q^{85} + ( 6 \beta_{6} + 6 \beta_{8} - \beta_{10} + 4 \beta_{11} ) q^{87} + ( 64 - 4 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{89} + ( 3 \beta_{2} + 3 \beta_{3} - \beta_{7} - \beta_{12} ) q^{91} + ( -9 \beta_{1} + 3 \beta_{5} - 3 \beta_{9} ) q^{93} + ( -\beta_{6} - 9 \beta_{8} + 3 \beta_{10} + \beta_{11} ) q^{95} + ( 8 - 6 \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{97} + ( -12 \beta_{2} + 5 \beta_{3} + 7 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 96q^{9} + O(q^{10})$$ $$16q - 96q^{9} - 160q^{17} + 32q^{25} + 64q^{33} - 256q^{41} - 112q^{49} - 112q^{57} - 144q^{65} + 224q^{73} + 96q^{81} + 1024q^{89} + 128q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 5 x^{14} + 48 x^{12} + 180 x^{10} + 1056 x^{8} + 2880 x^{6} + 12288 x^{4} + 20480 x^{2} + 65536$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{13} - 5 \nu^{11} - 48 \nu^{9} - 180 \nu^{7} - 1056 \nu^{5} - 1856 \nu^{3} - 7168 \nu$$$$)/1024$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{15} - 81 \nu^{13} - 456 \nu^{11} - 3012 \nu^{9} - 9984 \nu^{7} - 40512 \nu^{5} - 96768 \nu^{3} - 323584 \nu$$$$)/73728$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{15} - 27 \nu^{13} + 24 \nu^{11} - 300 \nu^{9} + 192 \nu^{7} - 3264 \nu^{5} + 4608 \nu^{3} - 32768 \nu$$$$)/73728$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{15} - 13 \nu^{13} - 56 \nu^{11} - 276 \nu^{9} - 832 \nu^{7} - 4032 \nu^{5} - 5120 \nu^{3} - 24576 \nu$$$$)/8192$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + 5 \nu^{13} - 6 \nu^{11} + 36 \nu^{9} + 136 \nu^{7} + 480 \nu^{5} + 3072 \nu^{3} - 6144 \nu$$$$)/4096$$ $$\beta_{6}$$ $$=$$ $$($$$$5 \nu^{14} - 27 \nu^{12} + 60 \nu^{10} - 2028 \nu^{8} - 3120 \nu^{6} - 72960 \nu^{4} - 138240 \nu^{2} - 745472$$$$)/36864$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{15} - 21 \nu^{13} - 156 \nu^{11} - 516 \nu^{9} - 2640 \nu^{7} - 3264 \nu^{5} - 5376 \nu^{3} + 38912 \nu$$$$)/12288$$ $$\beta_{8}$$ $$=$$ $$($$$$-23 \nu^{14} - 135 \nu^{12} - 996 \nu^{10} - 4380 \nu^{8} - 16176 \nu^{6} - 53760 \nu^{4} - 138240 \nu^{2} - 286720$$$$)/73728$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{15} + 4 \nu^{13} + 39 \nu^{11} + 128 \nu^{9} + 700 \nu^{7} + 1808 \nu^{5} + 7296 \nu^{3} + 3072 \nu$$$$)/2048$$ $$\beta_{10}$$ $$=$$ $$($$$$17 \nu^{14} + 9 \nu^{12} + 420 \nu^{10} + 420 \nu^{8} + 8688 \nu^{6} - 2688 \nu^{4} + 129024 \nu^{2} - 57344$$$$)/36864$$ $$\beta_{11}$$ $$=$$ $$($$$$19 \nu^{14} - 93 \nu^{12} - 300 \nu^{10} - 2292 \nu^{8} - 7824 \nu^{6} - 38400 \nu^{4} - 125952 \nu^{2} - 237568$$$$)/24576$$ $$\beta_{12}$$ $$=$$ $$($$$$95 \nu^{15} + 99 \nu^{13} + 2616 \nu^{11} + 780 \nu^{9} + 48000 \nu^{7} + 192 \nu^{5} + 373248 \nu^{3} + 212992 \nu$$$$)/73728$$ $$\beta_{13}$$ $$=$$ $$($$$$-\nu^{14} - \nu^{12} - 28 \nu^{10} + 12 \nu^{8} - 336 \nu^{6} + 320 \nu^{4} - 1792 \nu^{2} + 9216$$$$)/1024$$ $$\beta_{14}$$ $$=$$ $$($$$$\nu^{14} + 5 \nu^{12} - 16 \nu^{10} - 140 \nu^{8} - 992 \nu^{6} - 3520 \nu^{4} - 10240 \nu^{2} - 30720$$$$)/1024$$ $$\beta_{15}$$ $$=$$ $$($$$$-3 \nu^{14} - 23 \nu^{12} - 120 \nu^{10} - 604 \nu^{8} - 2560 \nu^{6} - 8640 \nu^{4} - 18944 \nu^{2} - 49152$$$$)/2048$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{12} - 3 \beta_{9} + 2 \beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3} - 6 \beta_{2} + \beta_{1}$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{15} + \beta_{14} + 2 \beta_{13} - 2 \beta_{11} + 10 \beta_{10} - 4 \beta_{8} - 3 \beta_{6} - 20$$$$)/32$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{12} + 15 \beta_{9} + 14 \beta_{7} + 5 \beta_{5} + 10 \beta_{4} - 15 \beta_{3} - 10 \beta_{2} + 11 \beta_{1}$$$$)/32$$ $$\nu^{4}$$ $$=$$ $$($$$$-2 \beta_{15} - \beta_{14} - 18 \beta_{13} + 2 \beta_{11} - 26 \beta_{10} + 20 \beta_{8} - 21 \beta_{6} - 284$$$$)/32$$ $$\nu^{5}$$ $$=$$ $$($$$$-7 \beta_{12} + 25 \beta_{9} + 2 \beta_{7} - 13 \beta_{5} - 58 \beta_{4} + 23 \beta_{3} + 154 \beta_{2} - 67 \beta_{1}$$$$)/32$$ $$\nu^{6}$$ $$=$$ $$($$$$-78 \beta_{15} - 23 \beta_{14} + 2 \beta_{13} + 14 \beta_{11} + 10 \beta_{10} + 364 \beta_{8} + 61 \beta_{6} + 220$$$$)/32$$ $$\nu^{7}$$ $$=$$ $$($$$$63 \beta_{12} - 225 \beta_{9} - 210 \beta_{7} + 117 \beta_{5} + 266 \beta_{4} - 15 \beta_{3} - 42 \beta_{2} - 229 \beta_{1}$$$$)/32$$ $$\nu^{8}$$ $$=$$ $$($$$$126 \beta_{15} - 17 \beta_{14} + 494 \beta_{13} + 130 \beta_{11} + 230 \beta_{10} - 1420 \beta_{8} + 251 \beta_{6} - 764$$$$)/32$$ $$\nu^{9}$$ $$=$$ $$($$$$-215 \beta_{12} + 73 \beta_{9} - 350 \beta_{7} - 253 \beta_{5} + 1382 \beta_{4} - 473 \beta_{3} - 2374 \beta_{2} + 749 \beta_{1}$$$$)/32$$ $$\nu^{10}$$ $$=$$ $$($$$$1234 \beta_{15} - 7 \beta_{14} - 1566 \beta_{13} - 722 \beta_{11} - 1750 \beta_{10} - 5396 \beta_{8} - 243 \beta_{6} + 7900$$$$)/32$$ $$\nu^{11}$$ $$=$$ $$($$$$-209 \beta_{12} + 1551 \beta_{9} + 1678 \beta_{7} - 2107 \beta_{5} - 7382 \beta_{4} + 11265 \beta_{3} + 2230 \beta_{2} + 2507 \beta_{1}$$$$)/32$$ $$\nu^{12}$$ $$=$$ $$($$$$-3874 \beta_{15} + 2143 \beta_{14} - 946 \beta_{13} - 3550 \beta_{11} + 710 \beta_{10} + 20020 \beta_{8} - 213 \beta_{6} + 20164$$$$)/32$$ $$\nu^{13}$$ $$=$$ $$($$$$2105 \beta_{12} - 3495 \beta_{9} + 3778 \beta_{7} + 13235 \beta_{5} - 20282 \beta_{4} - 20201 \beta_{3} + 9306 \beta_{2} + 3133 \beta_{1}$$$$)/32$$ $$\nu^{14}$$ $$=$$ $$($$$$-7182 \beta_{15} + 3465 \beta_{14} + 7938 \beta_{13} + 24846 \beta_{11} + 21450 \beta_{10} + 5292 \beta_{8} - 11811 \beta_{6} - 84580$$$$)/32$$ $$\nu^{15}$$ $$=$$ $$($$$$31 \beta_{12} + 21759 \beta_{9} - 658 \beta_{7} - 30187 \beta_{5} + 43978 \beta_{4} - 216559 \beta_{3} + 22038 \beta_{2} - 8069 \beta_{1}$$$$)/32$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$$.

 $$n$$ $$1023$$ $$1025$$ $$1541$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1023.1
 −1.45617 + 1.37098i 1.45617 + 1.37098i 1.09337 + 1.67467i −1.09337 + 1.67467i 1.69931 − 1.05468i −1.69931 − 1.05468i 0.739226 − 1.85837i −0.739226 − 1.85837i 0.739226 + 1.85837i −0.739226 + 1.85837i 1.69931 + 1.05468i −1.69931 + 1.05468i 1.09337 − 1.67467i −1.09337 − 1.67467i −1.45617 − 1.37098i 1.45617 − 1.37098i
0 5.22363i 0 −6.26788 0 2.64575i 0 −18.2863 0
1023.2 0 5.22363i 0 6.26788 0 2.64575i 0 −18.2863 0
1023.3 0 4.56747i 0 −5.73252 0 2.64575i 0 −11.8618 0
1023.4 0 4.56747i 0 5.73252 0 2.64575i 0 −11.8618 0
1023.5 0 3.44128i 0 −4.88287 0 2.64575i 0 −2.84239 0
1023.6 0 3.44128i 0 4.88287 0 2.64575i 0 −2.84239 0
1023.7 0 0.0974366i 0 −3.46547 0 2.64575i 0 8.99051 0
1023.8 0 0.0974366i 0 3.46547 0 2.64575i 0 8.99051 0
1023.9 0 0.0974366i 0 −3.46547 0 2.64575i 0 8.99051 0
1023.10 0 0.0974366i 0 3.46547 0 2.64575i 0 8.99051 0
1023.11 0 3.44128i 0 −4.88287 0 2.64575i 0 −2.84239 0
1023.12 0 3.44128i 0 4.88287 0 2.64575i 0 −2.84239 0
1023.13 0 4.56747i 0 −5.73252 0 2.64575i 0 −11.8618 0
1023.14 0 4.56747i 0 5.73252 0 2.64575i 0 −11.8618 0
1023.15 0 5.22363i 0 −6.26788 0 2.64575i 0 −18.2863 0
1023.16 0 5.22363i 0 6.26788 0 2.64575i 0 −18.2863 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1023.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1792.3.d.j 16
4.b odd 2 1 inner 1792.3.d.j 16
8.b even 2 1 inner 1792.3.d.j 16
8.d odd 2 1 inner 1792.3.d.j 16
16.e even 4 1 56.3.g.b 8
16.e even 4 1 224.3.g.b 8
16.f odd 4 1 56.3.g.b 8
16.f odd 4 1 224.3.g.b 8
48.i odd 4 1 504.3.g.b 8
48.i odd 4 1 2016.3.g.b 8
48.k even 4 1 504.3.g.b 8
48.k even 4 1 2016.3.g.b 8
112.j even 4 1 392.3.g.m 8
112.j even 4 1 1568.3.g.m 8
112.l odd 4 1 392.3.g.m 8
112.l odd 4 1 1568.3.g.m 8
112.u odd 12 2 392.3.k.o 16
112.v even 12 2 392.3.k.n 16
112.w even 12 2 392.3.k.o 16
112.x odd 12 2 392.3.k.n 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.b 8 16.e even 4 1
56.3.g.b 8 16.f odd 4 1
224.3.g.b 8 16.e even 4 1
224.3.g.b 8 16.f odd 4 1
392.3.g.m 8 112.j even 4 1
392.3.g.m 8 112.l odd 4 1
392.3.k.n 16 112.v even 12 2
392.3.k.n 16 112.x odd 12 2
392.3.k.o 16 112.u odd 12 2
392.3.k.o 16 112.w even 12 2
504.3.g.b 8 48.i odd 4 1
504.3.g.b 8 48.k even 4 1
1568.3.g.m 8 112.j even 4 1
1568.3.g.m 8 112.l odd 4 1
1792.3.d.j 16 1.a even 1 1 trivial
1792.3.d.j 16 4.b odd 2 1 inner
1792.3.d.j 16 8.b even 2 1 inner
1792.3.d.j 16 8.d odd 2 1 inner
2016.3.g.b 8 48.i odd 4 1
2016.3.g.b 8 48.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1792, [\chi])$$:

 $$T_{3}^{8} + 60 T_{3}^{6} + 1140 T_{3}^{4} + 6752 T_{3}^{2} + 64$$ $$T_{5}^{8} - 108 T_{5}^{6} + 4164 T_{5}^{4} - 66944 T_{5}^{2} + 369664$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 12 T^{2} + 168 T^{4} - 2212 T^{6} + 17038 T^{8} - 179172 T^{10} + 1102248 T^{12} - 6377292 T^{14} + 43046721 T^{16} )^{2}$$
$5$ $$( 1 + 92 T^{2} + 5464 T^{4} + 211956 T^{6} + 6231214 T^{8} + 132472500 T^{10} + 2134375000 T^{12} + 22460937500 T^{14} + 152587890625 T^{16} )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{8}$$
$11$ $$( 1 - 400 T^{2} + 47836 T^{4} + 1686928 T^{6} - 819692794 T^{8} + 24698312848 T^{10} + 10254071431516 T^{12} - 1255371350688400 T^{14} + 45949729863572161 T^{16} )^{2}$$
$13$ $$( 1 + 444 T^{2} + 142936 T^{4} + 35361044 T^{6} + 6575436334 T^{8} + 1009946777684 T^{10} + 116597286336856 T^{12} + 10344349794381564 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 + 40 T + 1308 T^{2} + 31512 T^{3} + 588230 T^{4} + 9106968 T^{5} + 109245468 T^{6} + 965502760 T^{7} + 6975757441 T^{8} )^{4}$$
$19$ $$( 1 - 2636 T^{2} + 3120040 T^{4} - 2166005604 T^{6} + 964702997902 T^{8} - 282276016318884 T^{10} + 52989396030441640 T^{12} - 5834298126658400396 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$( 1 - 1744 T^{2} + 1272156 T^{4} - 412076080 T^{6} + 99307893702 T^{8} - 115315782303280 T^{10} + 99623789791135836 T^{12} - 38219105009443439824 T^{14} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$( 1 + 3384 T^{2} + 6555580 T^{4} + 8754768776 T^{6} + 8490907402822 T^{8} + 6192081614658056 T^{10} + 3279405379878872380 T^{12} +$$$$11\!\cdots\!44$$$$T^{14} +$$$$25\!\cdots\!21$$$$T^{16} )^{2}$$
$31$ $$( 1 - 3944 T^{2} + 8438620 T^{4} - 12447428312 T^{6} + 13694235978694 T^{8} - 11495461442126552 T^{10} + 7197223366370371420 T^{12} -$$$$31\!\cdots\!84$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$( 1 + 3512 T^{2} + 9188668 T^{4} + 18622781448 T^{6} + 27544347275206 T^{8} + 34902090701365128 T^{10} + 32275007558901367228 T^{12} +$$$$23\!\cdots\!72$$$$T^{14} +$$$$12\!\cdots\!41$$$$T^{16} )^{2}$$
$41$ $$( 1 + 64 T + 4956 T^{2} + 221760 T^{3} + 11848326 T^{4} + 372778560 T^{5} + 14004471516 T^{6} + 304006671424 T^{7} + 7984925229121 T^{8} )^{4}$$
$43$ $$( 1 - 9360 T^{2} + 42404572 T^{4} - 124584286064 T^{6} + 265774122344710 T^{8} - 425928881779889264 T^{10} +$$$$49\!\cdots\!72$$$$T^{12} -$$$$37\!\cdots\!60$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 - 8392 T^{2} + 39566748 T^{4} - 127207295352 T^{6} + 316693927920198 T^{8} - 620731022190542712 T^{10} +$$$$94\!\cdots\!28$$$$T^{12} -$$$$97\!\cdots\!72$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$( 1 + 18920 T^{2} + 162796828 T^{4} + 840091728600 T^{6} + 2864724835962118 T^{8} + 6628727822775456600 T^{10} +$$$$10\!\cdots\!08$$$$T^{12} +$$$$92\!\cdots\!20$$$$T^{14} +$$$$38\!\cdots\!21$$$$T^{16} )^{2}$$
$59$ $$( 1 - 1804 T^{2} - 12929112 T^{4} + 2548527324 T^{6} + 280743846121998 T^{8} + 30881425603271964 T^{10} -$$$$18\!\cdots\!52$$$$T^{12} -$$$$32\!\cdots\!24$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$61$ $$( 1 + 16316 T^{2} + 140172120 T^{4} + 816942037524 T^{6} + 3499102878259502 T^{8} + 11311249557773337684 T^{10} +$$$$26\!\cdots\!20$$$$T^{12} +$$$$43\!\cdots\!36$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$( 1 - 21344 T^{2} + 191426940 T^{4} - 1006794243744 T^{6} + 4374042881960006 T^{8} - 20288032627788837024 T^{10} +$$$$77\!\cdots\!40$$$$T^{12} -$$$$17\!\cdots\!84$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$( 1 - 9864 T^{2} + 51888284 T^{4} - 294531431096 T^{6} + 1789421441990854 T^{8} - 7484538771485032376 T^{10} +$$$$33\!\cdots\!24$$$$T^{12} -$$$$16\!\cdots\!24$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 56 T + 18460 T^{2} - 736008 T^{3} + 138223494 T^{4} - 3922186632 T^{5} + 524231528860 T^{6} - 8474716672184 T^{7} + 806460091894081 T^{8} )^{4}$$
$79$ $$( 1 - 24968 T^{2} + 330869788 T^{4} - 3106127956152 T^{6} + 22189846569597766 T^{8} -$$$$12\!\cdots\!12$$$$T^{10} +$$$$50\!\cdots\!68$$$$T^{12} -$$$$14\!\cdots\!88$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$( 1 - 31660 T^{2} + 529313064 T^{4} - 5903046467332 T^{6} + 47298953549416590 T^{8} -$$$$28\!\cdots\!72$$$$T^{10} +$$$$11\!\cdots\!24$$$$T^{12} -$$$$33\!\cdots\!60$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 256 T + 48252 T^{2} - 6269952 T^{3} + 638304966 T^{4} - 49664289792 T^{5} + 3027438612732 T^{6} - 127227210486016 T^{7} + 3936588805702081 T^{8} )^{4}$$
$97$ $$( 1 - 32 T + 19484 T^{2} - 1437536 T^{3} + 199130566 T^{4} - 13525776224 T^{5} + 1724904511004 T^{6} - 26655104157728 T^{7} + 7837433594376961 T^{8} )^{4}$$