Properties

Label 19.7.b.b
Level $19$
Weight $7$
Character orbit 19.b
Analytic conductor $4.371$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 19.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.37102758878\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 483 x^{6} + 75582 x^{4} + 4242376 x^{2} + 71047680\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3}\cdot 29 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -57 + \beta_{3} - \beta_{4} ) q^{4} + ( 14 - \beta_{6} ) q^{5} + ( -48 + 2 \beta_{3} + 7 \beta_{4} + \beta_{6} ) q^{6} + ( -16 - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{7} + ( -39 \beta_{1} + 7 \beta_{2} + \beta_{5} ) q^{8} + ( -134 + 8 \beta_{4} + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{2} q^{3} + ( -57 + \beta_{3} - \beta_{4} ) q^{4} + ( 14 - \beta_{6} ) q^{5} + ( -48 + 2 \beta_{3} + 7 \beta_{4} + \beta_{6} ) q^{6} + ( -16 - 3 \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{7} + ( -39 \beta_{1} + 7 \beta_{2} + \beta_{5} ) q^{8} + ( -134 + 8 \beta_{4} + \beta_{6} ) q^{9} + ( 8 \beta_{1} - 27 \beta_{2} - \beta_{7} ) q^{10} + ( -259 + 13 \beta_{3} - 2 \beta_{4} ) q^{11} + ( -71 \beta_{1} + 49 \beta_{2} + 2 \beta_{5} + \beta_{7} ) q^{12} + ( 42 \beta_{1} + 9 \beta_{2} - \beta_{5} + \beta_{7} ) q^{13} + ( 109 \beta_{1} - 56 \beta_{2} - 3 \beta_{5} - \beta_{7} ) q^{14} + ( 182 \beta_{1} - 28 \beta_{2} - 5 \beta_{5} + \beta_{7} ) q^{15} + ( 1492 - 66 \beta_{3} - 41 \beta_{4} - 11 \beta_{6} ) q^{16} + ( 771 + 28 \beta_{3} - 112 \beta_{4} - 12 \beta_{6} ) q^{17} + ( -72 \beta_{1} + 91 \beta_{2} + \beta_{7} ) q^{18} + ( 2607 - 118 \beta_{1} - 35 \beta_{2} - 41 \beta_{3} - 94 \beta_{4} + 5 \beta_{5} + 12 \beta_{6} - \beta_{7} ) q^{19} + ( -1481 + 75 \beta_{3} + 230 \beta_{4} + 89 \beta_{6} ) q^{20} + ( 448 \beta_{1} + 21 \beta_{2} - 10 \beta_{5} - 2 \beta_{7} ) q^{21} + ( -780 \beta_{1} + 179 \beta_{2} + 13 \beta_{5} ) q^{22} + ( -6185 - 10 \beta_{3} - 392 \beta_{4} + 13 \beta_{6} ) q^{23} + ( 8154 - 208 \beta_{3} + 193 \beta_{4} - 119 \beta_{6} ) q^{24} + ( 9721 - 100 \beta_{3} + 480 \beta_{4} + 41 \beta_{6} ) q^{25} + ( -4622 + 88 \beta_{3} - 187 \beta_{4} - 131 \beta_{6} ) q^{26} + ( -606 \beta_{1} - 541 \beta_{2} - 3 \beta_{5} - \beta_{7} ) q^{27} + ( -17269 + 273 \beta_{3} + 361 \beta_{4} + 130 \beta_{6} ) q^{28} + ( 588 \beta_{1} + 437 \beta_{2} + 8 \beta_{5} + 12 \beta_{7} ) q^{29} + ( -23678 + 610 \beta_{3} - 200 \beta_{4} - 78 \beta_{6} ) q^{30} + ( 1622 \beta_{1} + 302 \beta_{2} - \beta_{5} - 11 \beta_{7} ) q^{31} + ( 1217 \beta_{1} - 1167 \beta_{2} - 2 \beta_{5} - 11 \beta_{7} ) q^{32} + ( -1506 \beta_{1} + 210 \beta_{2} + 15 \beta_{5} + 13 \beta_{7} ) q^{33} + ( -1177 \beta_{1} - 800 \beta_{2} + 28 \beta_{5} - 12 \beta_{7} ) q^{34} + ( 26585 - 365 \beta_{3} - 190 \beta_{4} - 10 \beta_{6} ) q^{35} + ( 4617 - 267 \beta_{3} - 102 \beta_{4} - 153 \beta_{6} ) q^{36} + ( -2294 \beta_{1} + 1218 \beta_{2} - 7 \beta_{5} - 13 \beta_{7} ) q^{37} + ( 12910 + 3620 \beta_{1} - 1043 \beta_{2} - 420 \beta_{3} + 577 \beta_{4} - 41 \beta_{5} + 141 \beta_{6} + 12 \beta_{7} ) q^{38} + ( 5503 - 380 \beta_{3} + 172 \beta_{4} + 219 \beta_{6} ) q^{39} + ( -1750 \beta_{1} + 3640 \beta_{2} + 75 \beta_{5} + 25 \beta_{7} ) q^{40} + ( -1858 \beta_{1} + 274 \beta_{2} - 109 \beta_{5} + \beta_{7} ) q^{41} + ( -54276 + 1202 \beta_{3} - 827 \beta_{4} + 271 \beta_{6} ) q^{42} + ( 32595 - 415 \beta_{3} + 806 \beta_{4} + 22 \beta_{6} ) q^{43} + ( 87501 - 1307 \beta_{3} - 172 \beta_{4} - 231 \beta_{6} ) q^{44} + ( -23906 + 420 \beta_{3} - 640 \beta_{4} - 41 \beta_{6} ) q^{45} + ( -8461 \beta_{1} - 2935 \beta_{2} - 10 \beta_{5} + 13 \beta_{7} ) q^{46} + ( -12179 + 793 \beta_{3} - 2946 \beta_{4} - 24 \beta_{6} ) q^{47} + ( 12359 \beta_{1} - 1653 \beta_{2} - 80 \beta_{5} - 55 \beta_{7} ) q^{48} + ( -36944 - 640 \beta_{3} - 1464 \beta_{4} - 208 \beta_{6} ) q^{49} + ( 17227 \beta_{1} + 3447 \beta_{2} - 100 \beta_{5} + 41 \beta_{7} ) q^{50} + ( 4648 \beta_{1} - 1583 \beta_{2} + 80 \beta_{5} + 40 \beta_{7} ) q^{51} + ( -7461 \beta_{1} - 3137 \beta_{2} + 24 \beta_{5} - 67 \beta_{7} ) q^{52} + ( -2376 \beta_{1} - 1605 \beta_{2} + 6 \beta_{5} + 54 \beta_{7} ) q^{53} + ( 46990 + 720 \beta_{3} + 4343 \beta_{4} + 679 \beta_{6} ) q^{54} + ( -8215 + 1415 \beta_{3} + 2770 \beta_{4} + 510 \beta_{6} ) q^{55} + ( -17633 \beta_{1} + 6909 \beta_{2} + 81 \beta_{5} + 66 \beta_{7} ) q^{56} + ( -23329 + 7882 \beta_{1} - 2786 \beta_{2} - 292 \beta_{3} + 1592 \beta_{4} + 113 \beta_{5} - 561 \beta_{6} - 53 \beta_{7} ) q^{57} + ( -48136 - 1058 \beta_{3} - 3971 \beta_{4} - 1981 \beta_{6} ) q^{58} + ( 12502 \beta_{1} + 861 \beta_{2} - 125 \beta_{5} - 119 \beta_{7} ) q^{59} + ( -37688 \beta_{1} + 3652 \beta_{2} + 290 \beta_{5} - 14 \beta_{7} ) q^{60} + ( -8614 + 648 \beta_{3} + 7032 \beta_{4} - 573 \beta_{6} ) q^{61} + ( -183094 + 1238 \beta_{3} - 3230 \beta_{4} + 1088 \beta_{6} ) q^{62} + ( -17441 + 1197 \beta_{3} - 1738 \beta_{4} + 202 \beta_{6} ) q^{63} + ( -109198 - 376 \beta_{3} + 4801 \beta_{4} + 1857 \beta_{6} ) q^{64} + ( -29012 \beta_{1} - 7692 \beta_{2} - 30 \beta_{5} - 66 \beta_{7} ) q^{65} + ( 195050 - 3250 \beta_{3} - 106 \beta_{4} - 1908 \beta_{6} ) q^{66} + ( 20556 \beta_{1} + 4475 \beta_{2} + 298 \beta_{5} + 102 \beta_{7} ) q^{67} + ( 154385 + 215 \beta_{3} + 1121 \beta_{4} + 1432 \beta_{6} ) q^{68} + ( 19650 \beta_{1} + 4437 \beta_{2} + 447 \beta_{5} - 23 \beta_{7} ) q^{69} + ( 39430 \beta_{1} - 7265 \beta_{2} - 365 \beta_{5} - 10 \beta_{7} ) q^{70} + ( -6700 \beta_{1} + 3294 \beta_{2} - 266 \beta_{5} + 154 \beta_{7} ) q^{71} + ( 8790 \beta_{1} - 3128 \beta_{2} - 267 \beta_{5} - 89 \beta_{7} ) q^{72} + ( 62485 + 1720 \beta_{3} - 8912 \beta_{4} - 2242 \beta_{6} ) q^{73} + ( 333974 - 4022 \beta_{3} - 5826 \beta_{4} + 448 \beta_{6} ) q^{74} + ( -20502 \beta_{1} - 6447 \beta_{2} - 375 \beta_{5} - 141 \beta_{7} ) q^{75} + ( -323365 + 26623 \beta_{1} - 117 \beta_{2} + 6083 \beta_{3} - 4276 \beta_{4} - 100 \beta_{5} + 463 \beta_{6} + 77 \beta_{7} ) q^{76} + ( -222276 + 2752 \beta_{3} + 3336 \beta_{4} + 1307 \beta_{6} ) q^{77} + ( 22841 \beta_{1} + 1589 \beta_{2} - 380 \beta_{5} + 219 \beta_{7} ) q^{78} + ( -52694 \beta_{1} + 2046 \beta_{2} - 75 \beta_{5} + 79 \beta_{7} ) q^{79} + ( 300886 - 10330 \beta_{3} - 7760 \beta_{4} - 1394 \beta_{6} ) q^{80} + ( -536197 - 228 \beta_{3} + 3880 \beta_{4} + 770 \beta_{6} ) q^{81} + ( 228818 + 5974 \beta_{3} - 3706 \beta_{4} + 36 \beta_{6} ) q^{82} + ( 64654 - 3238 \beta_{3} - 8396 \beta_{4} - 1112 \beta_{6} ) q^{83} + ( -76645 \beta_{1} + 20075 \beta_{2} + 562 \beta_{5} + 143 \beta_{7} ) q^{84} + ( 260710 - 2460 \beta_{3} + 13880 \beta_{4} + 2045 \beta_{6} ) q^{85} + ( 54554 \beta_{1} + 817 \beta_{2} - 415 \beta_{5} + 22 \beta_{7} ) q^{86} + ( 350751 - 6992 \beta_{3} + 10460 \beta_{4} + 923 \beta_{6} ) q^{87} + ( 85964 \beta_{1} - 15762 \beta_{2} - 475 \beta_{5} - 231 \beta_{7} ) q^{88} + ( 6838 \beta_{1} + 13272 \beta_{2} + 515 \beta_{5} - 295 \beta_{7} ) q^{89} + ( -45012 \beta_{1} + 73 \beta_{2} + 420 \beta_{5} - 41 \beta_{7} ) q^{90} + ( -15670 \beta_{1} - 29 \beta_{2} + 141 \beta_{5} - 89 \beta_{7} ) q^{91} + ( 487680 - 2630 \beta_{3} + 2951 \beta_{4} + 2169 \beta_{6} ) q^{92} + ( 182606 + 9908 \beta_{3} + 3224 \beta_{4} + 150 \beta_{6} ) q^{93} + ( -63872 \beta_{1} - 12321 \beta_{2} + 793 \beta_{5} - 24 \beta_{7} ) q^{94} + ( -289949 + 32480 \beta_{1} + 18580 \beta_{2} - 7275 \beta_{3} - 12490 \beta_{4} + 60 \beta_{5} - 2554 \beta_{6} + 140 \beta_{7} ) q^{95} + ( -1065942 + 9228 \beta_{3} + 11619 \beta_{4} + 1287 \beta_{6} ) q^{96} + ( -29696 \beta_{1} - 8078 \beta_{2} + 1432 \beta_{5} + 152 \beta_{7} ) q^{97} + ( -23480 \beta_{1} - 26928 \beta_{2} - 640 \beta_{5} - 208 \beta_{7} ) q^{98} + ( 68231 - 3207 \beta_{3} + 286 \beta_{4} - 1342 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 454q^{4} + 108q^{5} - 358q^{6} - 140q^{7} - 1052q^{9} + O(q^{10}) \) \( 8q - 454q^{4} + 108q^{5} - 358q^{6} - 140q^{7} - 1052q^{9} - 2024q^{11} + 11546q^{16} + 6008q^{17} + 20552q^{19} - 10732q^{20} - 50252q^{23} + 64310q^{24} + 78492q^{25} - 37522q^{26} - 135818q^{28} - 187696q^{30} + 210800q^{35} + 35052q^{36} + 103318q^{38} + 43724q^{39} - 429970q^{42} + 260800q^{43} + 693512q^{44} - 191012q^{45} - 100248q^{47} - 301872q^{49} + 390202q^{54} - 52480q^{55} - 186860q^{57} - 405186q^{58} - 54548q^{61} - 1461908q^{62} - 137408q^{63} - 858058q^{64} + 1539556q^{66} + 1243910q^{68} + 479968q^{73} + 2645844q^{74} - 2569288q^{76} - 1755300q^{77} + 2344672q^{80} - 4279648q^{81} + 1847172q^{82} + 483040q^{83} + 2111780q^{85} + 2802652q^{87} + 3905498q^{92} + 1507528q^{93} - 2383888q^{95} - 8462238q^{96} + 528224q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 483 x^{6} + 75582 x^{4} + 4242376 x^{2} + 71047680\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -11 \nu^{7} - 5377 \nu^{5} - 753578 \nu^{3} - 25301016 \nu \)\()/1200192\)
\(\beta_{3}\)\(=\)\((\)\( -11 \nu^{6} - 5377 \nu^{4} - 603554 \nu^{2} - 6097944 \)\()/150024\)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{6} - 5377 \nu^{4} - 753578 \nu^{2} - 24250848 \)\()/150024\)
\(\beta_{5}\)\(=\)\((\)\( 11 \nu^{7} + 5377 \nu^{5} + 925034 \nu^{3} + 53934168 \nu \)\()/171456\)
\(\beta_{6}\)\(=\)\((\)\( 107 \nu^{6} + 38665 \nu^{4} + 3811514 \nu^{2} + 91462416 \)\()/150024\)
\(\beta_{7}\)\(=\)\((\)\( 1153 \nu^{7} + 454499 \nu^{5} + 50838718 \nu^{3} + 1407625608 \nu \)\()/1200192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{4} + \beta_{3} - 121\)
\(\nu^{3}\)\(=\)\(\beta_{5} + 7 \beta_{2} - 167 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-11 \beta_{6} + 151 \beta_{4} - 258 \beta_{3} + 20628\)
\(\nu^{5}\)\(=\)\(-11 \beta_{7} - 258 \beta_{5} - 2959 \beta_{2} + 31681 \beta_{1}\)
\(\nu^{6}\)\(=\)\(5377 \beta_{6} - 18943 \beta_{4} + 57608 \beta_{3} - 3998606\)
\(\nu^{7}\)\(=\)\(5377 \beta_{7} + 57608 \beta_{5} + 857755 \beta_{2} - 6345657 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
14.9269i
12.8592i
8.08057i
5.43437i
5.43437i
8.08057i
12.8592i
14.9269i
14.9269i 33.0827i −158.814 159.622 −493.824 445.794 1415.28i −365.466 2382.67i
18.2 12.8592i 21.6433i −101.358 −216.848 278.315 −134.238 480.398i 260.567 2788.49i
18.3 8.08057i 27.2115i −1.29562 162.986 219.884 −68.0674 506.687i −11.4646 1317.02i
18.4 5.43437i 33.7437i 34.4677 −51.7597 −183.376 −313.488 535.109i −409.636 281.281i
18.5 5.43437i 33.7437i 34.4677 −51.7597 −183.376 −313.488 535.109i −409.636 281.281i
18.6 8.08057i 27.2115i −1.29562 162.986 219.884 −68.0674 506.687i −11.4646 1317.02i
18.7 12.8592i 21.6433i −101.358 −216.848 278.315 −134.238 480.398i 260.567 2788.49i
18.8 14.9269i 33.0827i −158.814 159.622 −493.824 445.794 1415.28i −365.466 2382.67i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 18.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 19.7.b.b 8
3.b odd 2 1 171.7.c.d 8
4.b odd 2 1 304.7.e.d 8
19.b odd 2 1 inner 19.7.b.b 8
57.d even 2 1 171.7.c.d 8
76.d even 2 1 304.7.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.7.b.b 8 1.a even 1 1 trivial
19.7.b.b 8 19.b odd 2 1 inner
171.7.c.d 8 3.b odd 2 1
171.7.c.d 8 57.d even 2 1
304.7.e.d 8 4.b odd 2 1
304.7.e.d 8 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 483 T_{2}^{6} + 75582 T_{2}^{4} + 4242376 T_{2}^{2} + 71047680 \) acting on \(S_{7}^{\mathrm{new}}(19, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 71047680 + 4242376 T^{2} + 75582 T^{4} + 483 T^{6} + T^{8} \)
$3$ \( 432254085120 + 2281096296 T^{2} + 4292649 T^{4} + 3442 T^{6} + T^{8} \)
$5$ \( ( 292006000 + 3367200 T - 49415 T^{2} - 54 T^{3} + T^{4} )^{2} \)
$7$ \( ( -1276939885 - 29481334 T - 157380 T^{2} + 70 T^{3} + T^{4} )^{2} \)
$11$ \( ( 401057409740 - 456204412 T - 1577997 T^{2} + 1012 T^{3} + T^{4} )^{2} \)
$13$ \( \)\(25\!\cdots\!80\)\( + 9427729558012913056 T^{2} + 44831312024289 T^{4} + 14425362 T^{6} + T^{8} \)
$17$ \( ( -21923949558015 + 54981558852 T - 27529338 T^{2} - 3004 T^{3} + T^{4} )^{2} \)
$19$ \( \)\(48\!\cdots\!21\)\( - \)\(21\!\cdots\!32\)\( T + \)\(31\!\cdots\!56\)\( T^{2} + 4770870492621376968 T^{3} - 5869847316568962 T^{4} + 101408888328 T^{5} + 144430096 T^{6} - 20552 T^{7} + T^{8} \)
$23$ \( ( -7335568675200580 - 2996194244464 T - 56925063 T^{2} + 25126 T^{3} + T^{4} )^{2} \)
$29$ \( \)\(79\!\cdots\!00\)\( + \)\(17\!\cdots\!40\)\( T^{2} + 1722370266650575953 T^{4} + 2760432378 T^{6} + T^{8} \)
$31$ \( \)\(18\!\cdots\!00\)\( + \)\(42\!\cdots\!40\)\( T^{2} + 2319750675886043088 T^{4} + 2963275728 T^{6} + T^{8} \)
$37$ \( \)\(21\!\cdots\!20\)\( + \)\(49\!\cdots\!84\)\( T^{2} + 37605062047833382608 T^{4} + 10947702288 T^{6} + T^{8} \)
$41$ \( \)\(15\!\cdots\!00\)\( + \)\(72\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!12\)\( T^{4} + 22588953936 T^{6} + T^{8} \)
$43$ \( ( -1733938754887040560 + 57659975836784 T + 3772254615 T^{2} - 130400 T^{3} + T^{4} )^{2} \)
$47$ \( ( -36330072548540490980 - 1650087163302204 T - 17694876341 T^{2} + 50124 T^{3} + T^{4} )^{2} \)
$53$ \( \)\(44\!\cdots\!20\)\( + \)\(30\!\cdots\!96\)\( T^{2} + \)\(65\!\cdots\!09\)\( T^{4} + 47671898922 T^{6} + T^{8} \)
$59$ \( \)\(41\!\cdots\!00\)\( + \)\(82\!\cdots\!40\)\( T^{2} + \)\(26\!\cdots\!77\)\( T^{4} + 296237152074 T^{6} + T^{8} \)
$61$ \( ( -\)\(10\!\cdots\!80\)\( - 10167316552637744 T - 121704664935 T^{2} + 27274 T^{3} + T^{4} )^{2} \)
$67$ \( \)\(86\!\cdots\!80\)\( + \)\(53\!\cdots\!04\)\( T^{2} + \)\(92\!\cdots\!49\)\( T^{4} + 555571931250 T^{6} + T^{8} \)
$71$ \( \)\(87\!\cdots\!00\)\( + \)\(41\!\cdots\!60\)\( T^{2} + \)\(68\!\cdots\!72\)\( T^{4} + 448388036040 T^{6} + T^{8} \)
$73$ \( ( -\)\(18\!\cdots\!15\)\( + 160405740236954736 T - 346509736338 T^{2} - 239984 T^{3} + T^{4} )^{2} \)
$79$ \( \)\(59\!\cdots\!00\)\( + \)\(10\!\cdots\!40\)\( T^{2} + \)\(65\!\cdots\!08\)\( T^{4} + 1476186591600 T^{6} + T^{8} \)
$83$ \( ( \)\(12\!\cdots\!60\)\( - 3762431272496032 T - 321702333684 T^{2} - 241520 T^{3} + T^{4} )^{2} \)
$89$ \( \)\(10\!\cdots\!00\)\( + \)\(40\!\cdots\!60\)\( T^{2} + \)\(49\!\cdots\!48\)\( T^{4} + 2032849953192 T^{6} + T^{8} \)
$97$ \( \)\(14\!\cdots\!00\)\( + \)\(80\!\cdots\!00\)\( T^{2} + \)\(53\!\cdots\!60\)\( T^{4} + 4640339680008 T^{6} + T^{8} \)
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