Properties

Label 80.6.c.d
Level 80
Weight 6
Character orbit 80.c
Analytic conductor 12.831
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 80.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.8307055850\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{31}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + ( -125 + \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 - \beta_{2} ) q^{5} + ( -\beta_{2} - \beta_{6} ) q^{7} + ( -125 + \beta_{2} + \beta_{3} ) q^{9} + ( 92 + \beta_{2} + \beta_{5} ) q^{11} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{13} + ( 124 + 10 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{15} + ( -40 \beta_{1} - 9 \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{17} + ( -172 - 8 \beta_{1} + 19 \beta_{2} - \beta_{5} + 4 \beta_{7} ) q^{19} + ( 248 - 14 \beta_{1} + 27 \beta_{2} - 4 \beta_{3} - 4 \beta_{5} + 7 \beta_{7} ) q^{21} + ( 22 \beta_{1} - \beta_{2} + 10 \beta_{4} + 9 \beta_{6} + 4 \beta_{7} ) q^{23} + ( -267 + 80 \beta_{1} + 3 \beta_{3} + 9 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{25} + ( -212 \beta_{1} - 42 \beta_{2} - 8 \beta_{4} + 6 \beta_{6} + 8 \beta_{7} ) q^{27} + ( 734 - 12 \beta_{1} + 28 \beta_{2} - 6 \beta_{3} + 4 \beta_{5} + 6 \beta_{7} ) q^{29} + ( -528 + 12 \beta_{2} + 12 \beta_{3} ) q^{31} + ( 188 \beta_{1} - 87 \beta_{2} - 43 \beta_{4} - 24 \beta_{6} + 4 \beta_{7} ) q^{33} + ( -2404 + 215 \beta_{1} - 19 \beta_{2} + 12 \beta_{3} - 34 \beta_{4} + \beta_{5} + 18 \beta_{6} + 8 \beta_{7} ) q^{35} + ( -410 \beta_{1} + 5 \beta_{2} + 26 \beta_{4} - 16 \beta_{6} + \beta_{7} ) q^{37} + ( 376 + 16 \beta_{1} - 46 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} - 8 \beta_{7} ) q^{39} + ( 2950 + 28 \beta_{1} - 39 \beta_{2} + 15 \beta_{3} + 16 \beta_{5} - 14 \beta_{7} ) q^{41} + ( 243 \beta_{1} + 54 \beta_{2} + 108 \beta_{4} - 54 \beta_{6} ) q^{43} + ( -3541 + 430 \beta_{1} + 151 \beta_{2} - 10 \beta_{3} + 75 \beta_{4} - 20 \beta_{5} + 60 \beta_{6} + 5 \beta_{7} ) q^{45} + ( -532 \beta_{1} + 81 \beta_{2} - 50 \beta_{4} - 9 \beta_{6} - 28 \beta_{7} ) q^{47} + ( -5625 + 24 \beta_{1} - 75 \beta_{2} + 9 \beta_{3} - 24 \beta_{5} - 12 \beta_{7} ) q^{49} + ( 15600 - 56 \beta_{1} + 52 \beta_{2} - 72 \beta_{3} - 16 \beta_{5} + 28 \beta_{7} ) q^{51} + ( -134 \beta_{1} - 70 \beta_{2} - 191 \beta_{4} + 36 \beta_{6} - 17 \beta_{7} ) q^{53} + ( -1876 + 560 \beta_{1} - 194 \beta_{2} - 30 \beta_{3} - 125 \beta_{4} + 15 \beta_{5} - 20 \beta_{6} - 70 \beta_{7} ) q^{55} + ( -756 \beta_{1} + 327 \beta_{2} + 83 \beta_{4} + 144 \beta_{6} - 20 \beta_{7} ) q^{57} + ( -11460 + 8 \beta_{1} + 41 \beta_{2} + 48 \beta_{3} + 13 \beta_{5} - 4 \beta_{7} ) q^{59} + ( 15482 + 118 \beta_{1} - 269 \beta_{2} + 18 \beta_{3} + 8 \beta_{5} - 59 \beta_{7} ) q^{61} + ( 330 \beta_{1} + 693 \beta_{2} + 274 \beta_{4} + 39 \beta_{6} - 76 \beta_{7} ) q^{63} + ( -9008 - 340 \beta_{1} + 57 \beta_{2} - 21 \beta_{3} + 182 \beta_{4} + 12 \beta_{5} - 144 \beta_{6} - 34 \beta_{7} ) q^{65} + ( 115 \beta_{1} - 214 \beta_{2} - 124 \beta_{4} - 10 \beta_{6} + 16 \beta_{7} ) q^{67} + ( -9592 + 78 \beta_{1} - 133 \beta_{2} + 26 \beta_{3} + 36 \beta_{5} - 39 \beta_{7} ) q^{69} + ( 15704 + 240 \beta_{1} - 494 \beta_{2} + 108 \beta_{3} - 2 \beta_{5} - 120 \beta_{7} ) q^{71} + ( 1124 \beta_{1} - 103 \beta_{2} - 311 \beta_{4} + 8 \beta_{6} - 40 \beta_{7} ) q^{73} + ( -27848 - 995 \beta_{1} - 180 \beta_{2} + 52 \beta_{3} - 194 \beta_{4} - 14 \beta_{5} - 72 \beta_{6} + 88 \beta_{7} ) q^{75} + ( 2384 \beta_{1} - 217 \beta_{2} + 111 \beta_{4} - 468 \beta_{6} - 28 \beta_{7} ) q^{77} + ( -5408 + 208 \beta_{1} - 608 \beta_{2} - 60 \beta_{3} - 28 \beta_{5} - 104 \beta_{7} ) q^{79} + ( 51209 - 236 \beta_{1} + 381 \beta_{2} - 121 \beta_{3} - 88 \beta_{5} + 118 \beta_{7} ) q^{81} + ( -1669 \beta_{1} - 302 \beta_{2} + 236 \beta_{4} + 342 \beta_{6} + 176 \beta_{7} ) q^{83} + ( -36720 + 320 \beta_{1} + 55 \beta_{2} + 90 \beta_{3} + 125 \beta_{4} + 80 \beta_{5} + 60 \beta_{6} - 40 \beta_{7} ) q^{85} + ( 2366 \beta_{1} + 264 \beta_{2} - 64 \beta_{4} + 48 \beta_{6} - 56 \beta_{7} ) q^{87} + ( -5238 - 156 \beta_{1} + 312 \beta_{2} - 126 \beta_{3} + 48 \beta_{5} + 78 \beta_{7} ) q^{89} + ( 60952 - 200 \beta_{1} + 730 \beta_{2} + 120 \beta_{3} + 110 \beta_{5} + 100 \beta_{7} ) q^{91} + ( -4488 \beta_{1} - 504 \beta_{2} - 96 \beta_{4} + 72 \beta_{6} + 96 \beta_{7} ) q^{93} + ( -55324 - 2160 \beta_{1} + 234 \beta_{2} - 30 \beta_{3} - 55 \beta_{4} - 95 \beta_{5} + 180 \beta_{6} + 30 \beta_{7} ) q^{95} + ( 1544 \beta_{1} + 107 \beta_{2} + 49 \beta_{4} + 488 \beta_{6} + 86 \beta_{7} ) q^{97} + ( -33356 - 808 \beta_{1} + 1803 \beta_{2} - 152 \beta_{3} - 65 \beta_{5} + 404 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{5} - 1000q^{9} + O(q^{10}) \) \( 8q + 8q^{5} - 1000q^{9} + 736q^{11} + 992q^{15} - 1376q^{19} + 1984q^{21} - 2136q^{25} + 5872q^{29} - 4224q^{31} - 19232q^{35} + 3008q^{39} + 23600q^{41} - 28328q^{45} - 45000q^{49} + 124800q^{51} - 15008q^{55} - 91680q^{59} + 123856q^{61} - 72064q^{65} - 76736q^{69} + 125632q^{71} - 222784q^{75} - 43264q^{79} + 409672q^{81} - 293760q^{85} - 41904q^{89} + 487616q^{91} - 442592q^{95} - 266848q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 41 x^{6} + 460 x^{4} + 969 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 182 \nu^{7} + 7420 \nu^{5} + 82106 \nu^{3} + 160458 \nu \)\()/639\)
\(\beta_{2}\)\(=\)\((\)\( -176 \nu^{7} - 66 \nu^{6} - 7372 \nu^{5} - 2232 \nu^{4} - 86042 \nu^{3} - 19752 \nu^{2} - 203676 \nu - 29412 \)\()/639\)
\(\beta_{3}\)\(=\)\((\)\( 176 \nu^{7} + 438 \nu^{6} + 7372 \nu^{5} + 15432 \nu^{4} + 86042 \nu^{3} + 113112 \nu^{2} + 203676 \nu - 106884 \)\()/639\)
\(\beta_{4}\)\(=\)\((\)\( -656 \nu^{7} + 66 \nu^{6} - 26548 \nu^{5} + 2232 \nu^{4} - 295142 \nu^{3} + 19752 \nu^{2} - 646692 \nu + 29412 \)\()/639\)
\(\beta_{5}\)\(=\)\((\)\( 176 \nu^{7} + 282 \nu^{6} + 7372 \nu^{5} + 17592 \nu^{4} + 86042 \nu^{3} + 259752 \nu^{2} + 203676 \nu + 390564 \)\()/639\)
\(\beta_{6}\)\(=\)\((\)\( -974 \nu^{7} + 66 \nu^{6} - 40168 \nu^{5} + 2232 \nu^{4} - 454172 \nu^{3} + 19752 \nu^{2} - 991374 \nu + 29412 \)\()/639\)
\(\beta_{7}\)\(=\)\((\)\( 1244 \nu^{7} - 330 \nu^{6} + 51700 \nu^{5} - 11160 \nu^{4} + 594422 \nu^{3} - 98760 \nu^{2} + 1339296 \nu - 147060 \)\()/639\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-8 \beta_{7} - 24 \beta_{6} - 3 \beta_{4} + 13 \beta_{2} - 72 \beta_{1}\)\()/1280\)
\(\nu^{2}\)\(=\)\((\)\(-7 \beta_{7} + \beta_{5} - 13 \beta_{3} - 47 \beta_{2} + 14 \beta_{1} - 6560\)\()/640\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + 6 \beta_{6} - \beta_{4} - 5 \beta_{2} + 10 \beta_{1}\)\()/16\)
\(\nu^{4}\)\(=\)\((\)\(169 \beta_{7} + 29 \beta_{5} + 283 \beta_{3} + 1157 \beta_{2} - 338 \beta_{1} + 121760\)\()/640\)
\(\nu^{5}\)\(=\)\((\)\(-3016 \beta_{7} - 10296 \beta_{6} + 3443 \beta_{4} + 8227 \beta_{2} - 14120 \beta_{1}\)\()/1280\)
\(\nu^{6}\)\(=\)\((\)\(-53 \beta_{7} - 16 \beta_{5} - 71 \beta_{3} - 352 \beta_{2} + 106 \beta_{1} - 30496\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(57832 \beta_{7} + 224376 \beta_{6} - 101633 \beta_{4} - 166417 \beta_{2} + 282728 \beta_{1}\)\()/1280\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/80\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.64654i
0.0965878i
4.73066i
3.98753i
3.98753i
4.73066i
0.0965878i
1.64654i
0 28.9338i 0 −13.1588 + 54.3309i 0 146.828i 0 −594.165 0
49.2 0 24.1383i 0 46.7401 30.6653i 0 179.876i 0 −339.657 0
49.3 0 5.49000i 0 −53.0051 17.7613i 0 188.968i 0 212.860 0
49.4 0 4.69449i 0 23.4238 50.7575i 0 10.2635i 0 220.962 0
49.5 0 4.69449i 0 23.4238 + 50.7575i 0 10.2635i 0 220.962 0
49.6 0 5.49000i 0 −53.0051 + 17.7613i 0 188.968i 0 212.860 0
49.7 0 24.1383i 0 46.7401 + 30.6653i 0 179.876i 0 −339.657 0
49.8 0 28.9338i 0 −13.1588 54.3309i 0 146.828i 0 −594.165 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.6.c.d 8
3.b odd 2 1 720.6.f.n 8
4.b odd 2 1 40.6.c.a 8
5.b even 2 1 inner 80.6.c.d 8
5.c odd 4 1 400.6.a.z 4
5.c odd 4 1 400.6.a.ba 4
8.b even 2 1 320.6.c.i 8
8.d odd 2 1 320.6.c.j 8
12.b even 2 1 360.6.f.b 8
15.d odd 2 1 720.6.f.n 8
20.d odd 2 1 40.6.c.a 8
20.e even 4 1 200.6.a.j 4
20.e even 4 1 200.6.a.k 4
40.e odd 2 1 320.6.c.j 8
40.f even 2 1 320.6.c.i 8
60.h even 2 1 360.6.f.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.6.c.a 8 4.b odd 2 1
40.6.c.a 8 20.d odd 2 1
80.6.c.d 8 1.a even 1 1 trivial
80.6.c.d 8 5.b even 2 1 inner
200.6.a.j 4 20.e even 4 1
200.6.a.k 4 20.e even 4 1
320.6.c.i 8 8.b even 2 1
320.6.c.i 8 40.f even 2 1
320.6.c.j 8 8.d odd 2 1
320.6.c.j 8 40.e odd 2 1
360.6.f.b 8 12.b even 2 1
360.6.f.b 8 60.h even 2 1
400.6.a.z 4 5.c odd 4 1
400.6.a.ba 4 5.c odd 4 1
720.6.f.n 8 3.b odd 2 1
720.6.f.n 8 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 1472 T_{3}^{6} + 562528 T_{3}^{4} + 26394624 T_{3}^{2} + 324000000 \) acting on \(S_{6}^{\mathrm{new}}(80, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 472 T^{2} + 69724 T^{4} - 20119464 T^{6} + 8439593958 T^{8} - 1188034229736 T^{10} + 243112555575324 T^{12} - 97180614348674328 T^{14} + 12157665459056928801 T^{16} \)
$5$ \( 1 - 8 T + 1100 T^{2} + 113000 T^{3} - 438250 T^{4} + 353125000 T^{5} + 10742187500 T^{6} - 244140625000 T^{7} + 95367431640625 T^{8} \)
$7$ \( 1 - 44728 T^{2} + 1493128636 T^{4} - 37445733732616 T^{6} + 682894235558230726 T^{8} - \)\(10\!\cdots\!84\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(10\!\cdots\!72\)\( T^{14} + \)\(63\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 - 368 T + 176204 T^{2} - 51158896 T^{3} + 42277949270 T^{4} - 8239191359696 T^{5} + 4570277964394604 T^{6} - 1537227326344959568 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \)
$13$ \( 1 - 1578472 T^{2} + 1074189263356 T^{4} - 437549632721743384 T^{6} + \)\(15\!\cdots\!86\)\( T^{8} - \)\(60\!\cdots\!16\)\( T^{10} + \)\(20\!\cdots\!56\)\( T^{12} - \)\(41\!\cdots\!28\)\( T^{14} + \)\(36\!\cdots\!01\)\( T^{16} \)
$17$ \( 1 - 6260872 T^{2} + 17547242668444 T^{4} - 31297293718759478968 T^{6} + \)\(45\!\cdots\!30\)\( T^{8} - \)\(63\!\cdots\!32\)\( T^{10} + \)\(71\!\cdots\!44\)\( T^{12} - \)\(51\!\cdots\!28\)\( T^{14} + \)\(16\!\cdots\!01\)\( T^{16} \)
$19$ \( ( 1 + 688 T + 5308396 T^{2} + 6058136368 T^{3} + 15069081422710 T^{4} + 15000545402668432 T^{5} + 32546127598645797196 T^{6} + \)\(10\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \)
$23$ \( 1 - 34675896 T^{2} + 569897415616828 T^{4} - \)\(59\!\cdots\!20\)\( T^{6} + \)\(44\!\cdots\!82\)\( T^{8} - \)\(24\!\cdots\!80\)\( T^{10} + \)\(97\!\cdots\!28\)\( T^{12} - \)\(24\!\cdots\!04\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \)
$29$ \( ( 1 - 2936 T + 58625996 T^{2} - 77951973928 T^{3} + 1466411094282230 T^{4} - 1598884552081323272 T^{5} + \)\(24\!\cdots\!96\)\( T^{6} - \)\(25\!\cdots\!64\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} )^{2} \)
$31$ \( ( 1 + 2112 T + 80187004 T^{2} + 163080265536 T^{3} + 3196344720873606 T^{4} + 4668849547150239936 T^{5} + \)\(65\!\cdots\!04\)\( T^{6} + \)\(49\!\cdots\!12\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} )^{2} \)
$37$ \( 1 - 251774632 T^{2} + 38631208311838780 T^{4} - \)\(41\!\cdots\!52\)\( T^{6} + \)\(32\!\cdots\!34\)\( T^{8} - \)\(19\!\cdots\!48\)\( T^{10} + \)\(89\!\cdots\!80\)\( T^{12} - \)\(27\!\cdots\!68\)\( T^{14} + \)\(53\!\cdots\!01\)\( T^{16} \)
$41$ \( ( 1 - 11800 T + 337909340 T^{2} - 2943020124776 T^{3} + 51155654972384870 T^{4} - \)\(34\!\cdots\!76\)\( T^{5} + \)\(45\!\cdots\!40\)\( T^{6} - \)\(18\!\cdots\!00\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \)
$43$ \( 1 - 283211672 T^{2} + 48021567531024796 T^{4} - \)\(54\!\cdots\!84\)\( T^{6} + \)\(43\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!16\)\( T^{10} + \)\(22\!\cdots\!96\)\( T^{12} - \)\(28\!\cdots\!28\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 963352312 T^{2} + 473832864723586300 T^{4} - \)\(15\!\cdots\!72\)\( T^{6} + \)\(40\!\cdots\!54\)\( T^{8} - \)\(83\!\cdots\!28\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{12} - \)\(14\!\cdots\!88\)\( T^{14} + \)\(76\!\cdots\!01\)\( T^{16} \)
$53$ \( 1 - 1183385640 T^{2} + 874785099161623996 T^{4} - \)\(49\!\cdots\!80\)\( T^{6} + \)\(23\!\cdots\!06\)\( T^{8} - \)\(86\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!96\)\( T^{12} - \)\(63\!\cdots\!60\)\( T^{14} + \)\(93\!\cdots\!01\)\( T^{16} \)
$59$ \( ( 1 + 45840 T + 3064286732 T^{2} + 94721285480976 T^{3} + 3348109683185502486 T^{4} + \)\(67\!\cdots\!24\)\( T^{5} + \)\(15\!\cdots\!32\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} )^{2} \)
$61$ \( ( 1 - 61928 T + 3903014764 T^{2} - 145287706763384 T^{3} + 5198153942066716726 T^{4} - \)\(12\!\cdots\!84\)\( T^{5} + \)\(27\!\cdots\!64\)\( T^{6} - \)\(37\!\cdots\!28\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} )^{2} \)
$67$ \( 1 - 9281919064 T^{2} + 39492482666681482588 T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(16\!\cdots\!62\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(56\!\cdots\!36\)\( T^{14} + \)\(11\!\cdots\!01\)\( T^{16} \)
$71$ \( ( 1 - 62816 T + 3398787356 T^{2} - 184024084124896 T^{3} + 8353296562609817510 T^{4} - \)\(33\!\cdots\!96\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} - \)\(36\!\cdots\!16\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \)
$73$ \( 1 - 9140679496 T^{2} + 46078306824990298588 T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(37\!\cdots\!02\)\( T^{8} - \)\(66\!\cdots\!40\)\( T^{10} + \)\(85\!\cdots\!88\)\( T^{12} - \)\(72\!\cdots\!04\)\( T^{14} + \)\(34\!\cdots\!01\)\( T^{16} \)
$79$ \( ( 1 + 21632 T + 7152876604 T^{2} + 332616618908288 T^{3} + 24121899620797566790 T^{4} + \)\(10\!\cdots\!12\)\( T^{5} + \)\(67\!\cdots\!04\)\( T^{6} + \)\(63\!\cdots\!68\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} )^{2} \)
$83$ \( 1 - 6759897816 T^{2} + 40943120759345365468 T^{4} - \)\(86\!\cdots\!80\)\( T^{6} + \)\(39\!\cdots\!42\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(98\!\cdots\!68\)\( T^{12} - \)\(25\!\cdots\!84\)\( T^{14} + \)\(57\!\cdots\!01\)\( T^{16} \)
$89$ \( ( 1 + 20952 T + 16118164796 T^{2} + 497915996461992 T^{3} + \)\(11\!\cdots\!30\)\( T^{4} + \)\(27\!\cdots\!08\)\( T^{5} + \)\(50\!\cdots\!96\)\( T^{6} + \)\(36\!\cdots\!48\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \)
$97$ \( 1 - 45263915272 T^{2} + \)\(10\!\cdots\!40\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{6} + \)\(15\!\cdots\!94\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(55\!\cdots\!40\)\( T^{12} - \)\(18\!\cdots\!28\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \)
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