# Properties

 Label 1568.3.c.h Level 1568 Weight 3 Character orbit 1568.c Analytic conductor 42.725 Analytic rank 0 Dimension 16 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1568.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$42.7249054517$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{28}$$ Twist minimal: no (minimal twist has level 224) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} -\beta_{12} q^{5} + ( -1 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} -\beta_{12} q^{5} + ( -1 + \beta_{3} ) q^{9} + ( -\beta_{10} + \beta_{14} ) q^{11} + ( -\beta_{2} + \beta_{15} ) q^{13} + \beta_{5} q^{15} + ( -2 \beta_{2} + 2 \beta_{12} + \beta_{13} ) q^{17} + ( 3 \beta_{1} + \beta_{6} - 3 \beta_{7} - \beta_{9} ) q^{19} + ( 3 \beta_{4} - \beta_{5} - 2 \beta_{10} + 3 \beta_{14} ) q^{23} + ( -2 - 2 \beta_{3} - 2 \beta_{8} ) q^{25} + ( -\beta_{1} - \beta_{6} - \beta_{7} + \beta_{9} ) q^{27} + ( 7 + \beta_{3} + 2 \beta_{8} - \beta_{11} ) q^{29} + ( -3 \beta_{1} - 2 \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{31} + 5 \beta_{2} q^{33} + ( -1 + 3 \beta_{3} - \beta_{8} - \beta_{11} ) q^{37} + ( \beta_{4} - 4 \beta_{5} - 6 \beta_{10} - \beta_{14} ) q^{39} + ( 7 \beta_{2} + 2 \beta_{12} + 3 \beta_{13} + 2 \beta_{15} ) q^{41} + ( 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{14} ) q^{43} + ( 3 \beta_{2} - 6 \beta_{12} + \beta_{15} ) q^{45} + ( -13 \beta_{1} + 2 \beta_{6} + 3 \beta_{9} ) q^{47} + ( -12 \beta_{4} - 2 \beta_{5} - 3 \beta_{10} - 3 \beta_{14} ) q^{51} + ( 3 + \beta_{3} + 5 \beta_{8} + \beta_{11} ) q^{53} + ( \beta_{1} - 4 \beta_{6} - \beta_{7} - 4 \beta_{9} ) q^{55} + ( -33 + 4 \beta_{3} - 4 \beta_{8} - 2 \beta_{11} ) q^{57} + ( \beta_{1} - 6 \beta_{6} + 4 \beta_{7} + 2 \beta_{9} ) q^{59} + ( -15 \beta_{2} + 3 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{61} + ( 1 - 3 \beta_{3} + 4 \beta_{8} - 2 \beta_{11} ) q^{65} + ( -5 \beta_{4} + 4 \beta_{5} - 12 \beta_{10} + 2 \beta_{14} ) q^{67} + ( -18 \beta_{2} - 3 \beta_{12} + 4 \beta_{13} ) q^{69} + ( -8 \beta_{4} + 2 \beta_{5} - 4 \beta_{10} ) q^{71} + ( -3 \beta_{2} + 6 \beta_{12} - 2 \beta_{13} + 2 \beta_{15} ) q^{73} + ( 18 \beta_{1} + 2 \beta_{6} + 6 \beta_{7} ) q^{75} + ( 11 \beta_{4} + \beta_{5} - 7 \beta_{14} ) q^{79} + ( -4 + 3 \beta_{3} + 2 \beta_{11} ) q^{81} + ( 12 \beta_{1} - 8 \beta_{6} + 4 \beta_{7} + 2 \beta_{9} ) q^{83} + ( 45 + 5 \beta_{3} - \beta_{8} + \beta_{11} ) q^{85} + ( -2 \beta_{1} - 8 \beta_{6} + \beta_{7} + 5 \beta_{9} ) q^{87} + ( 17 \beta_{2} + 10 \beta_{12} + 2 \beta_{15} ) q^{89} + ( 29 - 7 \beta_{3} - 3 \beta_{8} + \beta_{11} ) q^{93} + ( 11 \beta_{4} + 11 \beta_{5} - 14 \beta_{10} - 3 \beta_{14} ) q^{95} + ( 31 \beta_{2} - 6 \beta_{12} - 3 \beta_{13} ) q^{97} + ( 5 \beta_{4} - 9 \beta_{10} + 9 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{9} + O(q^{10})$$ $$16q - 16q^{9} - 32q^{25} + 112q^{29} - 16q^{37} + 48q^{53} - 528q^{57} + 16q^{65} - 64q^{81} + 720q^{85} + 464q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 36 x^{14} + 522 x^{12} + 3644 x^{10} + 12219 x^{8} + 15156 x^{6} + 15478 x^{4} - 10992 x^{2} + 11025$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$113 \nu^{14} + 3818 \nu^{12} + 51075 \nu^{10} + 313505 \nu^{8} + 843923 \nu^{6} + 473319 \nu^{4} + 2292564 \nu^{2} - 557487$$$$)/2711772$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{15} - 158 \nu^{13} - 2599 \nu^{11} - 22087 \nu^{9} - 102335 \nu^{7} - 245347 \nu^{5} - 300255 \nu^{3} - 71343 \nu$$$$)/128520$$ $$\beta_{3}$$ $$=$$ $$($$$$-113 \nu^{14} - 3818 \nu^{12} - 51075 \nu^{10} - 313505 \nu^{8} - 843923 \nu^{6} - 473319 \nu^{4} + 419208 \nu^{2} + 12760461$$$$)/1355886$$ $$\beta_{4}$$ $$=$$ $$($$$$-1349 \nu^{15} - 51424 \nu^{13} - 794798 \nu^{11} - 6024806 \nu^{9} - 22337746 \nu^{7} - 30903344 \nu^{5} - 1517787 \nu^{3} + 17661258 \nu$$$$)/27117720$$ $$\beta_{5}$$ $$=$$ $$($$$$3289 \nu^{15} + 108644 \nu^{13} + 1373628 \nu^{11} + 7277106 \nu^{9} + 11587916 \nu^{7} - 12082056 \nu^{5} + 90706517 \nu^{3} - 113485638 \nu$$$$)/27117720$$ $$\beta_{6}$$ $$=$$ $$($$$$43 \nu^{14} + 1662 \nu^{12} + 26337 \nu^{10} + 209513 \nu^{8} + 861381 \nu^{6} + 1621011 \nu^{4} + 1271992 \nu^{2} - 51135$$$$)/132930$$ $$\beta_{7}$$ $$=$$ $$($$$$-5167 \nu^{14} - 179273 \nu^{12} - 2460748 \nu^{10} - 15568297 \nu^{8} - 42862214 \nu^{6} - 30838549 \nu^{4} - 108675393 \nu^{2} + 24667650$$$$)/13558860$$ $$\beta_{8}$$ $$=$$ $$($$$$477 \nu^{14} + 17097 \nu^{12} + 244996 \nu^{10} + 1653025 \nu^{8} + 4923894 \nu^{6} + 2488801 \nu^{4} - 2284841 \nu^{2} - 5789742$$$$)/903924$$ $$\beta_{9}$$ $$=$$ $$($$$$1307 \nu^{14} + 47303 \nu^{12} + 694458 \nu^{10} + 4988477 \nu^{8} + 17919584 \nu^{6} + 27429309 \nu^{4} + 32265783 \nu^{2} - 3887460$$$$)/1936980$$ $$\beta_{10}$$ $$=$$ $$($$$$-977 \nu^{15} - 34607 \nu^{13} - 490904 \nu^{11} - 3304813 \nu^{9} - 10370438 \nu^{7} - 10587797 \nu^{5} - 12755411 \nu^{3} + 22202004 \nu$$$$)/3389715$$ $$\beta_{11}$$ $$=$$ $$($$$$-969 \nu^{14} - 33812 \nu^{12} - 467441 \nu^{10} - 3001917 \nu^{8} - 8492381 \nu^{6} - 4528475 \nu^{4} + 4080486 \nu^{2} + 28721385$$$$)/1355886$$ $$\beta_{12}$$ $$=$$ $$($$$$-9238 \nu^{15} - 334664 \nu^{13} - 4891055 \nu^{11} - 34544651 \nu^{9} - 118066943 \nu^{7} - 156430715 \nu^{5} - 191515861 \nu^{3} - 61542453 \nu$$$$)/27117720$$ $$\beta_{13}$$ $$=$$ $$($$$$-3602 \nu^{15} - 137612 \nu^{13} - 2171849 \nu^{11} - 17484433 \nu^{9} - 75870593 \nu^{7} - 168229337 \nu^{5} - 205269971 \nu^{3} - 50730891 \nu$$$$)/9039240$$ $$\beta_{14}$$ $$=$$ $$($$$$16689 \nu^{15} + 609984 \nu^{13} + 9012548 \nu^{11} + 64598666 \nu^{9} + 223617876 \nu^{7} + 276178304 \nu^{5} + 121585997 \nu^{3} - 300491778 \nu$$$$)/27117720$$ $$\beta_{15}$$ $$=$$ $$($$$$18622 \nu^{15} + 689692 \nu^{13} + 10442779 \nu^{11} + 78891803 \nu^{9} + 310575283 \nu^{7} + 589390387 \nu^{5} + 717853201 \nu^{3} + 191518401 \nu$$$$)/27117720$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + \beta_{13} + \beta_{10}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{1} - 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{15} - 6 \beta_{14} - 9 \beta_{13} - 6 \beta_{12} - 13 \beta_{10} - 6 \beta_{5} - 12 \beta_{4} + 24 \beta_{2}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{11} + 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{6} - 9 \beta_{3} - 36 \beta_{1} + 63$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$20 \beta_{15} + 45 \beta_{14} + 33 \beta_{13} + 15 \beta_{12} + 92 \beta_{10} + 45 \beta_{5} + 130 \beta_{4} - 140 \beta_{2}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-11 \beta_{11} - 49 \beta_{9} - 6 \beta_{8} + 13 \beta_{7} + 52 \beta_{6} + 53 \beta_{3} + 502 \beta_{1} - 303$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-49 \beta_{15} - 1176 \beta_{14} - 201 \beta_{13} - 2239 \beta_{10} - 1008 \beta_{5} - 3976 \beta_{4} + 1400 \beta_{2}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$21 \beta_{11} + 780 \beta_{9} + 36 \beta_{8} + 12 \beta_{7} - 816 \beta_{6} + 125 \beta_{3} - 6032 \beta_{1} - 1401$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-3575 \beta_{15} + 13362 \beta_{14} - 5061 \beta_{13} - 2730 \beta_{12} + 24133 \beta_{10} + 10050 \beta_{5} + 49284 \beta_{4} + 16920 \beta_{2}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$1577 \beta_{11} - 9437 \beta_{9} + 630 \beta_{8} - 1291 \beta_{7} + 9806 \beta_{6} - 10554 \beta_{3} + 62844 \beta_{1} + 70056$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$40984 \beta_{15} - 64251 \beta_{14} + 71067 \beta_{13} + 27165 \beta_{12} - 112052 \beta_{10} - 44055 \beta_{5} - 249634 \beta_{4} - 308440 \beta_{2}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$-19787 \beta_{11} + 43874 \beta_{9} - 10410 \beta_{8} + 8812 \beta_{7} - 45437 \beta_{6} + 108773 \beta_{3} - 267122 \beta_{1} - 671586$$ $$\nu^{13}$$ $$=$$ $$($$$$-1304123 \beta_{15} + 925548 \beta_{14} - 2431239 \beta_{13} - 810468 \beta_{12} + 1569697 \beta_{10} + 587340 \beta_{5} + 3737656 \beta_{4} + 11304592 \beta_{2}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$644022 \beta_{11} - 505800 \beta_{9} + 363636 \beta_{8} - 133956 \beta_{7} + 522108 \beta_{6} - 3305255 \beta_{3} + 2790062 \beta_{1} + 19796271$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$17302151 \beta_{15} - 1830030 \beta_{14} + 33379797 \beta_{13} + 10397226 \beta_{12} - 2693803 \beta_{10} - 725550 \beta_{5} - 8697756 \beta_{4} - 159843144 \beta_{2}$$$$)/8$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 0.707107 − 3.42121i −0.707107 + 3.42121i −0.707107 + 2.60548i 0.707107 − 2.60548i 0.707107 − 1.17406i −0.707107 + 1.17406i 0.707107 − 0.358323i −0.707107 + 0.358323i −0.707107 − 0.358323i 0.707107 + 0.358323i −0.707107 − 1.17406i 0.707107 + 1.17406i 0.707107 + 2.60548i −0.707107 − 2.60548i −0.707107 − 3.42121i 0.707107 + 3.42121i
0 4.83832i 0 0.0515539i 0 0 0 −14.4094 0
97.2 0 4.83832i 0 0.0515539i 0 0 0 −14.4094 0
97.3 0 3.68470i 0 3.04770i 0 0 0 −4.57700 0
97.4 0 3.68470i 0 3.04770i 0 0 0 −4.57700 0
97.5 0 1.66037i 0 8.40186i 0 0 0 6.24317 0
97.6 0 1.66037i 0 8.40186i 0 0 0 6.24317 0
97.7 0 0.506745i 0 5.30261i 0 0 0 8.74321 0
97.8 0 0.506745i 0 5.30261i 0 0 0 8.74321 0
97.9 0 0.506745i 0 5.30261i 0 0 0 8.74321 0
97.10 0 0.506745i 0 5.30261i 0 0 0 8.74321 0
97.11 0 1.66037i 0 8.40186i 0 0 0 6.24317 0
97.12 0 1.66037i 0 8.40186i 0 0 0 6.24317 0
97.13 0 3.68470i 0 3.04770i 0 0 0 −4.57700 0
97.14 0 3.68470i 0 3.04770i 0 0 0 −4.57700 0
97.15 0 4.83832i 0 0.0515539i 0 0 0 −14.4094 0
97.16 0 4.83832i 0 0.0515539i 0 0 0 −14.4094 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.16 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
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.c.h 16
4.b odd 2 1 inner 1568.3.c.h 16
7.b odd 2 1 inner 1568.3.c.h 16
7.c even 3 1 224.3.s.a 16
7.d odd 6 1 224.3.s.a 16
28.d even 2 1 inner 1568.3.c.h 16
28.f even 6 1 224.3.s.a 16
28.g odd 6 1 224.3.s.a 16
56.j odd 6 1 448.3.s.g 16
56.k odd 6 1 448.3.s.g 16
56.m even 6 1 448.3.s.g 16
56.p even 6 1 448.3.s.g 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.s.a 16 7.c even 3 1
224.3.s.a 16 7.d odd 6 1
224.3.s.a 16 28.f even 6 1
224.3.s.a 16 28.g odd 6 1
448.3.s.g 16 56.j odd 6 1
448.3.s.g 16 56.k odd 6 1
448.3.s.g 16 56.m even 6 1
448.3.s.g 16 56.p even 6 1
1568.3.c.h 16 1.a even 1 1 trivial
1568.3.c.h 16 4.b odd 2 1 inner
1568.3.c.h 16 7.b odd 2 1 inner
1568.3.c.h 16 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{8} + 40 T_{3}^{6} + 430 T_{3}^{4} + 984 T_{3}^{2} + 225$$ $$T_{5}^{8} + 108 T_{5}^{6} + 2902 T_{5}^{4} + 18444 T_{5}^{2} + 49$$ $$T_{11}^{8} - 328 T_{11}^{6} + 10750 T_{11}^{4} - 75000 T_{11}^{2} + 140625$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 32 T^{2} + 538 T^{4} - 6720 T^{6} + 67563 T^{8} - 544320 T^{10} + 3529818 T^{12} - 17006112 T^{14} + 43046721 T^{16} )^{2}$$
$5$ $$( 1 - 92 T^{2} + 4202 T^{4} - 134256 T^{6} + 3554099 T^{8} - 83910000 T^{10} + 1641406250 T^{12} - 22460937500 T^{14} + 152587890625 T^{16} )^{2}$$
$7$ 1
$11$ $$( 1 + 640 T^{2} + 182570 T^{4} + 32301696 T^{6} + 4310016635 T^{8} + 472929131136 T^{10} + 39135500904170 T^{12} + 2008594161101440 T^{14} + 45949729863572161 T^{16} )^{2}$$
$13$ $$( 1 - 192 T^{2} - 16036 T^{4} + 1118400 T^{6} + 1003382406 T^{8} + 31942622400 T^{10} - 13081057841956 T^{12} - 4473232343516352 T^{14} + 665416609183179841 T^{16} )^{2}$$
$17$ $$( 1 - 956 T^{2} + 488618 T^{4} - 201962928 T^{6} + 67623242099 T^{8} - 16868145709488 T^{10} + 3408480649306538 T^{12} - 556986858791651516 T^{14} + 48661191875666868481 T^{16} )^{2}$$
$19$ $$( 1 - 528 T^{2} + 254522 T^{4} - 61599072 T^{6} + 28528481739 T^{8} - 8027652662112 T^{10} + 4322690432321402 T^{12} - 1168630277266933008 T^{14} +$$$$28\!\cdots\!81$$$$T^{16} )^{2}$$
$23$ $$( 1 + 928 T^{2} + 971546 T^{4} + 488810880 T^{6} + 328448702315 T^{8} + 136789325470080 T^{10} + 76082724505814426 T^{12} + 20336771472914857888 T^{14} +$$$$61\!\cdots\!61$$$$T^{16} )^{2}$$
$29$ $$( 1 - 28 T + 1376 T^{2} - 41412 T^{3} + 2055086 T^{4} - 34827492 T^{5} + 973218656 T^{6} - 16655052988 T^{7} + 500246412961 T^{8} )^{4}$$
$31$ $$( 1 - 4480 T^{2} + 10575626 T^{4} - 16536352320 T^{6} + 18543554268251 T^{8} - 15271668630918720 T^{10} + 9019856630728013066 T^{12} -$$$$35\!\cdots\!80$$$$T^{14} +$$$$72\!\cdots\!81$$$$T^{16} )^{2}$$
$37$ $$( 1 + 4 T + 2786 T^{2} + 66528 T^{3} + 3645371 T^{4} + 91076832 T^{5} + 5221412546 T^{6} + 10262905636 T^{7} + 3512479453921 T^{8} )^{4}$$
$41$ $$( 1 - 4384 T^{2} + 14370524 T^{4} - 32104130784 T^{6} + 62557592438726 T^{8} - 90718600708326624 T^{10} +$$$$11\!\cdots\!04$$$$T^{12} -$$$$98\!\cdots\!04$$$$T^{14} +$$$$63\!\cdots\!41$$$$T^{16} )^{2}$$
$43$ $$( 1 + 11176 T^{2} + 58293116 T^{4} + 188186249880 T^{6} + 415246366681670 T^{8} + 643371339275993880 T^{10} +$$$$68\!\cdots\!16$$$$T^{12} +$$$$44\!\cdots\!76$$$$T^{14} +$$$$13\!\cdots\!01$$$$T^{16} )^{2}$$
$47$ $$( 1 - 7504 T^{2} + 35127098 T^{4} - 112856452512 T^{6} + 286269593458379 T^{8} - 550703487050208672 T^{10} +$$$$83\!\cdots\!78$$$$T^{12} -$$$$87\!\cdots\!64$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16} )^{2}$$
$53$ $$( 1 - 12 T + 5458 T^{2} - 94464 T^{3} + 15366507 T^{4} - 265349376 T^{5} + 43066245298 T^{6} - 265972333548 T^{7} + 62259690411361 T^{8} )^{4}$$
$59$ $$( 1 - 12496 T^{2} + 86558762 T^{4} - 425221081440 T^{6} + 1629915742998971 T^{8} - 5152557348618879840 T^{10} +$$$$12\!\cdots\!02$$$$T^{12} -$$$$22\!\cdots\!76$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16} )^{2}$$
$61$ $$( 1 - 23052 T^{2} + 242874170 T^{4} - 1566594340176 T^{6} + 6924881882544003 T^{8} - 21690816145576808016 T^{10} +$$$$46\!\cdots\!70$$$$T^{12} -$$$$61\!\cdots\!92$$$$T^{14} +$$$$36\!\cdots\!61$$$$T^{16} )^{2}$$
$67$ $$( 1 + 5792 T^{2} + 57775162 T^{4} + 343117582400 T^{6} + 1556492106048139 T^{8} + 6914203920169870400 T^{10} +$$$$23\!\cdots\!42$$$$T^{12} +$$$$47\!\cdots\!12$$$$T^{14} +$$$$16\!\cdots\!81$$$$T^{16} )^{2}$$
$71$ $$( 1 + 27624 T^{2} + 370534588 T^{4} + 3151580461272 T^{6} + 18747759294515334 T^{8} + 80086957327676918232 T^{10} +$$$$23\!\cdots\!68$$$$T^{12} +$$$$45\!\cdots\!84$$$$T^{14} +$$$$41\!\cdots\!21$$$$T^{16} )^{2}$$
$73$ $$( 1 - 25452 T^{2} + 341755514 T^{4} - 3010449993360 T^{6} + 18880763286886851 T^{8} - 85491484429885679760 T^{10} +$$$$27\!\cdots\!34$$$$T^{12} -$$$$58\!\cdots\!92$$$$T^{14} +$$$$65\!\cdots\!61$$$$T^{16} )^{2}$$
$79$ $$( 1 + 10112 T^{2} + 157634746 T^{4} + 1079520612032 T^{6} + 9316361241248779 T^{8} + 42047415279815974592 T^{10} +$$$$23\!\cdots\!06$$$$T^{12} +$$$$59\!\cdots\!92$$$$T^{14} +$$$$23\!\cdots\!21$$$$T^{16} )^{2}$$
$83$ $$( 1 - 24232 T^{2} + 235912988 T^{4} - 1035732354840 T^{6} + 3398417338650758 T^{8} - 49154118566082623640 T^{10} +$$$$53\!\cdots\!08$$$$T^{12} -$$$$25\!\cdots\!52$$$$T^{14} +$$$$50\!\cdots\!81$$$$T^{16} )^{2}$$
$89$ $$( 1 - 44348 T^{2} + 940582730 T^{4} - 12584560545264 T^{6} + 117584952413543123 T^{8} -$$$$78\!\cdots\!24$$$$T^{10} +$$$$37\!\cdots\!30$$$$T^{12} -$$$$10\!\cdots\!08$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16} )^{2}$$
$97$ $$( 1 - 51968 T^{2} + 1305731356 T^{4} - 20766356196608 T^{6} + 230652504659485894 T^{8} -$$$$18\!\cdots\!48$$$$T^{10} +$$$$10\!\cdots\!16$$$$T^{12} -$$$$36\!\cdots\!88$$$$T^{14} +$$$$61\!\cdots\!21$$$$T^{16} )^{2}$$