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

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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:

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