Properties

Label 1024.4.b.j
Level 1024
Weight 4
Character orbit 1024.b
Analytic conductor 60.418
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1024.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(60.4179558459\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 36 x^{8} + 405 x^{6} + 1380 x^{4} + 420 x^{2} + 32\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} -\beta_{6} q^{5} + ( -3 - \beta_{2} ) q^{7} + ( -6 - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{7} q^{3} -\beta_{6} q^{5} + ( -3 - \beta_{2} ) q^{7} + ( -6 - \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{6} + \beta_{9} ) q^{11} + ( -\beta_{6} + 3 \beta_{7} + \beta_{8} - \beta_{9} ) q^{13} + ( 13 + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{15} + ( 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{17} + ( -\beta_{1} + 5 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{19} + ( \beta_{1} - 11 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{21} + ( -29 - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{23} + ( -6 - 3 \beta_{2} + 2 \beta_{3} - 6 \beta_{4} + \beta_{5} ) q^{25} + ( -5 \beta_{1} - 9 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{27} + ( 3 \beta_{1} - \beta_{6} + 18 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{29} + ( 36 - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{31} + ( -1 + 3 \beta_{2} + \beta_{3} - 8 \beta_{4} - 2 \beta_{5} ) q^{33} + ( -11 \beta_{1} + 10 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} ) q^{35} + ( 2 \beta_{1} + \beta_{6} - 29 \beta_{7} + \beta_{8} - \beta_{9} ) q^{37} + ( -73 - \beta_{2} + 2 \beta_{4} + 4 \beta_{5} ) q^{39} + ( -1 + \beta_{2} + 4 \beta_{3} - 14 \beta_{4} - \beta_{5} ) q^{41} + ( -20 \beta_{1} - 12 \beta_{6} + \beta_{7} - 4 \beta_{9} ) q^{43} + ( \beta_{1} + \beta_{6} + 41 \beta_{7} - \beta_{8} + 5 \beta_{9} ) q^{45} + ( 90 - 7 \beta_{2} - 10 \beta_{3} + 5 \beta_{4} + 3 \beta_{5} ) q^{47} + ( -11 - 4 \beta_{2} + 6 \beta_{3} - 16 \beta_{4} - 6 \beta_{5} ) q^{49} + ( -34 \beta_{1} + \beta_{6} + 7 \beta_{7} - 12 \beta_{8} - 5 \beta_{9} ) q^{51} + ( -3 \beta_{1} - \beta_{6} - 35 \beta_{7} - 13 \beta_{8} - 7 \beta_{9} ) q^{53} + ( -131 + 3 \beta_{2} + 12 \beta_{3} + 8 \beta_{4} - 6 \beta_{5} ) q^{55} + ( -13 + 15 \beta_{2} - 9 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{57} + ( -41 \beta_{1} + 18 \beta_{6} - 5 \beta_{7} - 14 \beta_{8} + 4 \beta_{9} ) q^{59} + ( -19 \beta_{1} + \beta_{6} + 41 \beta_{7} - 9 \beta_{8} - 11 \beta_{9} ) q^{61} + ( 263 + 2 \beta_{2} + 10 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} ) q^{63} + ( -57 - 13 \beta_{2} - 4 \beta_{3} - 18 \beta_{4} + 13 \beta_{5} ) q^{65} + ( -50 \beta_{1} - 11 \beta_{6} - 10 \beta_{7} + 12 \beta_{8} - \beta_{9} ) q^{67} + ( -13 \beta_{1} + 12 \beta_{6} - 55 \beta_{7} - \beta_{8} + 5 \beta_{9} ) q^{69} + ( -347 + 10 \beta_{2} + 2 \beta_{3} - 15 \beta_{4} - 3 \beta_{5} ) q^{71} + ( 29 + 11 \beta_{2} + \beta_{3} - 16 \beta_{4} + 6 \beta_{5} ) q^{73} + ( -71 \beta_{1} + 28 \beta_{6} - 7 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} ) q^{75} + ( -3 \beta_{1} + 12 \beta_{6} + 23 \beta_{7} - 15 \beta_{8} - 5 \beta_{9} ) q^{77} + ( 446 + 12 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} - 16 \beta_{5} ) q^{79} + ( -56 - 21 \beta_{2} - 9 \beta_{3} + 12 \beta_{5} ) q^{81} + ( -71 \beta_{1} - 48 \beta_{6} + 9 \beta_{7} + 22 \beta_{8} + 2 \beta_{9} ) q^{83} + ( -8 \beta_{1} - 3 \beta_{7} + 7 \beta_{8} + 17 \beta_{9} ) q^{85} + ( -617 - 2 \beta_{2} - 18 \beta_{3} - 29 \beta_{4} - 5 \beta_{5} ) q^{87} + ( -15 - 25 \beta_{2} + \beta_{3} - 8 \beta_{4} + 26 \beta_{5} ) q^{89} + ( -89 \beta_{1} + 4 \beta_{6} + 34 \beta_{8} + 2 \beta_{9} ) q^{91} + ( 48 \beta_{1} + 12 \beta_{6} + 6 \beta_{7} + 10 \beta_{8} + 14 \beta_{9} ) q^{93} + ( 701 + 15 \beta_{2} - 8 \beta_{3} + 56 \beta_{4} - 12 \beta_{5} ) q^{95} + ( 4 + 14 \beta_{2} - 3 \beta_{3} + 14 \beta_{4} - 29 \beta_{5} ) q^{97} + ( -125 \beta_{1} - 38 \beta_{6} + 17 \beta_{7} - 22 \beta_{8} + 4 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 28q^{7} - 54q^{9} + O(q^{10}) \) \( 10q - 28q^{7} - 54q^{9} + 124q^{15} + 4q^{17} - 276q^{23} - 50q^{25} + 368q^{31} - 4q^{33} - 732q^{39} + 944q^{47} - 94q^{49} - 1380q^{55} - 108q^{57} + 2628q^{63} - 492q^{65} - 3468q^{71} + 296q^{73} + 4416q^{79} - 482q^{81} - 6036q^{87} - 88q^{89} + 6900q^{95} - 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 36 x^{8} + 405 x^{6} + 1380 x^{4} + 420 x^{2} + 32\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 7 \nu^{9} + 250 \nu^{7} + 2823 \nu^{5} + 10042 \nu^{3} + 4528 \nu \)\()/208\)
\(\beta_{2}\)\(=\)\((\)\( 59 \nu^{8} + 2018 \nu^{6} + 21595 \nu^{4} + 70882 \nu^{2} + 17840 \)\()/416\)
\(\beta_{3}\)\(=\)\((\)\( 137 \nu^{8} + 4982 \nu^{6} + 55785 \nu^{4} + 180342 \nu^{2} + 24496 \)\()/416\)
\(\beta_{4}\)\(=\)\((\)\( -83 \nu^{8} - 2994 \nu^{6} - 33651 \nu^{4} - 112562 \nu^{2} - 18656 \)\()/208\)
\(\beta_{5}\)\(=\)\((\)\( 231 \nu^{8} + 8042 \nu^{6} + 86919 \nu^{4} + 277098 \nu^{2} + 35856 \)\()/416\)
\(\beta_{6}\)\(=\)\((\)\( 217 \nu^{9} + 7750 \nu^{7} + 85849 \nu^{5} + 278022 \nu^{3} + 18896 \nu \)\()/832\)
\(\beta_{7}\)\(=\)\((\)\( -157 \nu^{9} - 5622 \nu^{7} - 62573 \nu^{5} - 206166 \nu^{3} - 34432 \nu \)\()/416\)
\(\beta_{8}\)\(=\)\((\)\( 55 \nu^{9} + 1978 \nu^{7} + 22135 \nu^{5} + 73594 \nu^{3} + 13680 \nu \)\()/64\)
\(\beta_{9}\)\(=\)\((\)\( 1015 \nu^{9} + 36458 \nu^{7} + 407255 \nu^{5} + 1348970 \nu^{3} + 224336 \nu \)\()/832\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - \beta_{1}\)\()/32\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{5} + \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 113\)\()/16\)
\(\nu^{3}\)\(=\)\((\)\(26 \beta_{9} - 30 \beta_{8} + 2 \beta_{7} - 20 \beta_{6} + \beta_{1}\)\()/32\)
\(\nu^{4}\)\(=\)\((\)\(29 \beta_{5} - 25 \beta_{4} - 40 \beta_{3} - 91 \beta_{2} + 1516\)\()/16\)
\(\nu^{5}\)\(=\)\((\)\(-374 \beta_{9} + 454 \beta_{8} + 106 \beta_{7} + 384 \beta_{6} + 177 \beta_{1}\)\()/32\)
\(\nu^{6}\)\(=\)\((\)\(-446 \beta_{5} + 423 \beta_{4} + 678 \beta_{3} + 1362 \beta_{2} - 21951\)\()/16\)
\(\nu^{7}\)\(=\)\((\)\(5622 \beta_{9} - 6754 \beta_{8} - 2066 \beta_{7} - 6460 \beta_{6} - 4433 \beta_{1}\)\()/32\)
\(\nu^{8}\)\(=\)\((\)\(7043 \beta_{5} - 6519 \beta_{4} - 10952 \beta_{3} - 20373 \beta_{2} + 326836\)\()/16\)
\(\nu^{9}\)\(=\)\((\)\(-85962 \beta_{9} + 99866 \beta_{8} + 29462 \beta_{7} + 104544 \beta_{6} + 87103 \beta_{1}\)\()/32\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1024\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(1023\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
513.1
2.34476i
0.357936i
0.446984i
3.82089i
3.94652i
3.94652i
3.82089i
0.446984i
0.357936i
2.34476i
0 8.43597i 0 12.2748i 0 −1.63924 0 −44.1656 0
513.2 0 7.77277i 0 6.59550i 0 −24.8965 0 −33.4160 0
513.3 0 4.62644i 0 17.8826i 0 −13.8754 0 5.59607 0
513.4 0 2.80518i 0 0.844070i 0 29.0828 0 19.1310 0
513.5 0 1.07024i 0 11.6331i 0 −2.67171 0 25.8546 0
513.6 0 1.07024i 0 11.6331i 0 −2.67171 0 25.8546 0
513.7 0 2.80518i 0 0.844070i 0 29.0828 0 19.1310 0
513.8 0 4.62644i 0 17.8826i 0 −13.8754 0 5.59607 0
513.9 0 7.77277i 0 6.59550i 0 −24.8965 0 −33.4160 0
513.10 0 8.43597i 0 12.2748i 0 −1.63924 0 −44.1656 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 513.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1024.4.b.j 10
4.b odd 2 1 1024.4.b.k 10
8.b even 2 1 inner 1024.4.b.j 10
8.d odd 2 1 1024.4.b.k 10
16.e even 4 2 1024.4.a.n 10
16.f odd 4 2 1024.4.a.m 10
32.g even 8 2 16.4.e.a 10
32.g even 8 2 128.4.e.b 10
32.h odd 8 2 64.4.e.a 10
32.h odd 8 2 128.4.e.a 10
96.o even 8 2 576.4.k.a 10
96.p odd 8 2 144.4.k.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.4.e.a 10 32.g even 8 2
64.4.e.a 10 32.h odd 8 2
128.4.e.a 10 32.h odd 8 2
128.4.e.b 10 32.g even 8 2
144.4.k.a 10 96.p odd 8 2
576.4.k.a 10 96.o even 8 2
1024.4.a.m 10 16.f odd 4 2
1024.4.a.n 10 16.e even 4 2
1024.4.b.j 10 1.a even 1 1 trivial
1024.4.b.j 10 8.b even 2 1 inner
1024.4.b.k 10 4.b odd 2 1
1024.4.b.k 10 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1024, [\chi])\):

\( T_{3}^{10} + 162 T_{3}^{8} + 8504 T_{3}^{6} + 157552 T_{3}^{4} + 893712 T_{3}^{2} + 829472 \)
\( T_{5}^{10} + 650 T_{5}^{8} + 138664 T_{5}^{6} + 11484496 T_{5}^{4} + 291758672 T_{5}^{2} + 202085408 \)
\( T_{7}^{5} + 14 T_{7}^{4} - 736 T_{7}^{3} - 13376 T_{7}^{2} - 46736 T_{7} - 44000 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 108 T^{2} + 6317 T^{4} - 275312 T^{6} + 9907770 T^{8} - 295366792 T^{10} + 7222764330 T^{12} - 146312084592 T^{14} + 2447335229013 T^{16} - 30502389939948 T^{18} + 205891132094649 T^{20} \)
$5$ \( 1 - 600 T^{2} + 191789 T^{4} - 42513504 T^{6} + 7224666922 T^{8} - 994659832592 T^{10} + 112885420656250 T^{12} - 10379273437500000 T^{14} + 731616973876953125 T^{16} - 35762786865234375000 T^{18} + \)\(93\!\cdots\!25\)\( T^{20} \)
$7$ \( ( 1 + 14 T + 979 T^{2} + 5832 T^{3} + 372410 T^{4} + 662580 T^{5} + 127736630 T^{6} + 686128968 T^{7} + 39506181253 T^{8} + 193778020814 T^{9} + 4747561509943 T^{10} )^{2} \)
$11$ \( 1 - 7260 T^{2} + 24476093 T^{4} - 51189622576 T^{6} + 78543614693626 T^{8} - 105919424238644520 T^{10} + \)\(13\!\cdots\!86\)\( T^{12} - \)\(16\!\cdots\!96\)\( T^{14} + \)\(13\!\cdots\!33\)\( T^{16} - \)\(71\!\cdots\!60\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 - 12168 T^{2} + 76341149 T^{4} - 324341675296 T^{6} + 1028186938895082 T^{8} - 2540163312339385904 T^{10} + \)\(49\!\cdots\!38\)\( T^{12} - \)\(75\!\cdots\!76\)\( T^{14} + \)\(85\!\cdots\!21\)\( T^{16} - \)\(66\!\cdots\!48\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{20} \)
$17$ \( ( 1 - 2 T + 12653 T^{2} - 102520 T^{3} + 98460610 T^{4} - 354493580 T^{5} + 483736976930 T^{6} - 2474583573880 T^{7} + 1500492401316541 T^{8} - 1165244474459522 T^{9} + 2862423051509815793 T^{10} )^{2} \)
$19$ \( 1 - 33948 T^{2} + 631492045 T^{4} - 8119820572464 T^{6} + 79259035180229178 T^{8} - \)\(60\!\cdots\!76\)\( T^{10} + \)\(37\!\cdots\!18\)\( T^{12} - \)\(17\!\cdots\!04\)\( T^{14} + \)\(65\!\cdots\!45\)\( T^{16} - \)\(16\!\cdots\!08\)\( T^{18} + \)\(23\!\cdots\!01\)\( T^{20} \)
$23$ \( ( 1 + 138 T + 47715 T^{2} + 4085400 T^{3} + 882303386 T^{4} + 56970028764 T^{5} + 10734985297462 T^{6} + 604785820920600 T^{7} + 85941999241707045 T^{8} + 3024218171618804298 T^{9} + \)\(26\!\cdots\!07\)\( T^{10} )^{2} \)
$29$ \( 1 - 154168 T^{2} + 11039253245 T^{4} - 493373494152672 T^{6} + 16008852482124057194 T^{8} - \)\(42\!\cdots\!28\)\( T^{10} + \)\(95\!\cdots\!74\)\( T^{12} - \)\(17\!\cdots\!52\)\( T^{14} + \)\(23\!\cdots\!45\)\( T^{16} - \)\(19\!\cdots\!08\)\( T^{18} + \)\(74\!\cdots\!01\)\( T^{20} \)
$31$ \( ( 1 - 184 T + 134043 T^{2} - 19809056 T^{3} + 7638677322 T^{4} - 852982867024 T^{5} + 227563836099702 T^{6} - 17580610117135136 T^{7} + 3544046273282822853 T^{8} - \)\(14\!\cdots\!24\)\( T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \)
$37$ \( 1 - 361064 T^{2} + 63572869101 T^{4} - 7160954906480288 T^{6} + \)\(56\!\cdots\!06\)\( T^{8} - \)\(33\!\cdots\!92\)\( T^{10} + \)\(14\!\cdots\!54\)\( T^{12} - \)\(47\!\cdots\!28\)\( T^{14} + \)\(10\!\cdots\!29\)\( T^{16} - \)\(15\!\cdots\!04\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \)
$41$ \( ( 1 + 220509 T^{2} - 10715136 T^{3} + 24270968490 T^{4} - 1235215122432 T^{5} + 1672779419299290 T^{6} - 50898012956491776 T^{7} + 72190662971277946149 T^{8} + \)\(15\!\cdots\!01\)\( T^{10} )^{2} \)
$43$ \( 1 - 551948 T^{2} + 145009588605 T^{4} - 24152295654927088 T^{6} + \)\(28\!\cdots\!30\)\( T^{8} - \)\(25\!\cdots\!12\)\( T^{10} + \)\(18\!\cdots\!70\)\( T^{12} - \)\(96\!\cdots\!88\)\( T^{14} + \)\(36\!\cdots\!45\)\( T^{16} - \)\(88\!\cdots\!48\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \)
$47$ \( ( 1 - 472 T + 462219 T^{2} - 171516064 T^{3} + 90105579914 T^{4} - 25593405310224 T^{5} + 9355031623411222 T^{6} - 1848808586238545056 T^{7} + \)\(51\!\cdots\!73\)\( T^{8} - \)\(54\!\cdots\!52\)\( T^{9} + \)\(12\!\cdots\!43\)\( T^{10} )^{2} \)
$53$ \( 1 - 525160 T^{2} + 112387404877 T^{4} - 12025532417207968 T^{6} + \)\(67\!\cdots\!30\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{10} + \)\(15\!\cdots\!70\)\( T^{12} - \)\(59\!\cdots\!88\)\( T^{14} + \)\(12\!\cdots\!53\)\( T^{16} - \)\(12\!\cdots\!60\)\( T^{18} + \)\(53\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 - 958284 T^{2} + 532967592733 T^{4} - 207330855222256112 T^{6} + \)\(61\!\cdots\!62\)\( T^{8} - \)\(14\!\cdots\!28\)\( T^{10} + \)\(25\!\cdots\!42\)\( T^{12} - \)\(36\!\cdots\!72\)\( T^{14} + \)\(39\!\cdots\!93\)\( T^{16} - \)\(30\!\cdots\!24\)\( T^{18} + \)\(13\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 - 1146824 T^{2} + 509607070653 T^{4} - 80080659050426912 T^{6} - \)\(14\!\cdots\!54\)\( T^{8} + \)\(80\!\cdots\!72\)\( T^{10} - \)\(77\!\cdots\!94\)\( T^{12} - \)\(21\!\cdots\!52\)\( T^{14} + \)\(69\!\cdots\!93\)\( T^{16} - \)\(80\!\cdots\!84\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 - 2130652 T^{2} + 2127009110445 T^{4} - 1334413095892987440 T^{6} + \)\(59\!\cdots\!58\)\( T^{8} - \)\(20\!\cdots\!52\)\( T^{10} + \)\(53\!\cdots\!02\)\( T^{12} - \)\(10\!\cdots\!40\)\( T^{14} + \)\(15\!\cdots\!05\)\( T^{16} - \)\(14\!\cdots\!92\)\( T^{18} + \)\(60\!\cdots\!49\)\( T^{20} \)
$71$ \( ( 1 + 1734 T + 2753587 T^{2} + 2611066280 T^{3} + 2272726706522 T^{4} + 1414434499369348 T^{5} + 813433888257995542 T^{6} + \)\(33\!\cdots\!80\)\( T^{7} + \)\(12\!\cdots\!97\)\( T^{8} + \)\(28\!\cdots\!94\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{2} \)
$73$ \( ( 1 - 148 T + 1578093 T^{2} - 253463696 T^{3} + 1105903969594 T^{4} - 150341055768952 T^{5} + 430215444539549098 T^{6} - 38357732326510304144 T^{7} + \)\(92\!\cdots\!09\)\( T^{8} - \)\(33\!\cdots\!08\)\( T^{9} + \)\(89\!\cdots\!57\)\( T^{10} )^{2} \)
$79$ \( ( 1 - 2208 T + 3816107 T^{2} - 4320867712 T^{3} + 4245684014154 T^{4} - 3176789940661184 T^{5} + 2093287800654474006 T^{6} - \)\(10\!\cdots\!52\)\( T^{7} + \)\(45\!\cdots\!33\)\( T^{8} - \)\(13\!\cdots\!28\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \)
$83$ \( 1 - 2541516 T^{2} + 3317268679373 T^{4} - 2806354404035135728 T^{6} + \)\(18\!\cdots\!66\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{10} + \)\(59\!\cdots\!54\)\( T^{12} - \)\(29\!\cdots\!08\)\( T^{14} + \)\(11\!\cdots\!57\)\( T^{16} - \)\(29\!\cdots\!36\)\( T^{18} + \)\(37\!\cdots\!49\)\( T^{20} \)
$89$ \( ( 1 + 44 T + 1822557 T^{2} + 625587056 T^{3} + 1619388326906 T^{4} + 839474353817096 T^{5} + 1141618569430595914 T^{6} + \)\(31\!\cdots\!16\)\( T^{7} + \)\(63\!\cdots\!13\)\( T^{8} + \)\(10\!\cdots\!24\)\( T^{9} + \)\(17\!\cdots\!49\)\( T^{10} )^{2} \)
$97$ \( ( 1 + 2 T + 2565789 T^{2} - 723000584 T^{3} + 3231145430658 T^{4} - 1257303020547316 T^{5} + 2948979193634928834 T^{6} - \)\(60\!\cdots\!36\)\( T^{7} + \)\(19\!\cdots\!13\)\( T^{8} + \)\(13\!\cdots\!82\)\( T^{9} + \)\(63\!\cdots\!93\)\( T^{10} )^{2} \)
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