Properties

Label 1143.2.a.k
Level $1143$
Weight $2$
Character orbit 1143.a
Self dual yes
Analytic conductor $9.127$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1143.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.12690095103\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 26 x^{14} + 269 x^{12} - 1408 x^{10} + 3924 x^{8} - 5655 x^{6} + 3886 x^{4} - 1107 x^{2} + 108\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{14} q^{5} + ( 1 - \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} -\beta_{14} q^{5} + ( 1 - \beta_{6} ) q^{7} + ( \beta_{1} + \beta_{3} ) q^{8} + ( 1 - \beta_{5} ) q^{10} + ( \beta_{13} - \beta_{15} ) q^{11} + ( 1 - \beta_{10} ) q^{13} + ( \beta_{1} + \beta_{7} + \beta_{8} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{15} ) q^{14} + ( 1 + \beta_{2} + \beta_{4} + \beta_{6} + \beta_{10} ) q^{16} + ( \beta_{1} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{17} + ( 1 + \beta_{5} - \beta_{6} - \beta_{11} ) q^{19} + ( \beta_{1} - \beta_{8} + \beta_{12} + \beta_{15} ) q^{20} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{9} + \beta_{11} ) q^{22} + ( -\beta_{1} + \beta_{8} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{23} + ( 3 - \beta_{2} - \beta_{4} + \beta_{6} + \beta_{9} + \beta_{11} ) q^{25} + ( 2 \beta_{1} - \beta_{7} + \beta_{14} ) q^{26} + ( 1 + 2 \beta_{4} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} ) q^{28} + ( -\beta_{7} - \beta_{13} ) q^{29} + ( 2 + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{31} + ( \beta_{1} + \beta_{7} - \beta_{8} ) q^{32} + ( 1 - \beta_{4} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{34} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{7} + 2 \beta_{12} + \beta_{13} + \beta_{14} ) q^{35} + ( 1 - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} ) q^{37} + ( \beta_{1} - \beta_{3} + 2 \beta_{8} - \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{38} + ( -1 + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} ) q^{40} + ( -\beta_{1} + \beta_{7} - \beta_{8} - \beta_{12} - \beta_{14} ) q^{41} + ( 1 + 2 \beta_{4} - \beta_{5} - 2 \beta_{11} ) q^{43} + ( \beta_{3} - \beta_{7} - \beta_{8} + 3 \beta_{13} + 2 \beta_{14} - 2 \beta_{15} ) q^{44} + ( -2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{10} ) q^{46} + ( -2 \beta_{1} + 2 \beta_{3} + \beta_{7} - 2 \beta_{12} - \beta_{14} + \beta_{15} ) q^{47} + ( 2 - 2 \beta_{2} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{49} + ( 2 \beta_{1} - 2 \beta_{3} - \beta_{7} + \beta_{8} + \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{50} + ( 4 + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{52} + ( \beta_{7} - \beta_{13} ) q^{53} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{10} + 2 \beta_{11} ) q^{55} + ( 3 \beta_{1} + \beta_{7} + \beta_{8} + \beta_{12} - 3 \beta_{13} + \beta_{14} + \beta_{15} ) q^{56} + ( -1 - \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{58} + ( \beta_{1} + \beta_{8} + \beta_{12} + \beta_{14} - \beta_{15} ) q^{59} + ( 3 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{9} - 2 \beta_{10} ) q^{61} + ( \beta_{1} + \beta_{3} - \beta_{12} + 2 \beta_{13} + \beta_{14} - 2 \beta_{15} ) q^{62} + ( -\beta_{2} + \beta_{4} - \beta_{5} + \beta_{10} + \beta_{11} ) q^{64} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{7} - \beta_{13} - 2 \beta_{14} + 2 \beta_{15} ) q^{65} + ( 3 - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{67} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{7} + 2 \beta_{13} ) q^{68} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} + 2 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{70} + ( -2 \beta_{1} + \beta_{3} - 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{71} + ( 5 - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} ) q^{73} + ( -2 \beta_{1} + \beta_{3} - 3 \beta_{8} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{74} + ( -2 - \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{9} ) q^{76} + ( -\beta_{1} - 2 \beta_{3} + \beta_{8} + \beta_{12} + 2 \beta_{13} - \beta_{15} ) q^{77} + ( 1 - 3 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{9} + \beta_{10} + \beta_{11} ) q^{79} + ( -\beta_{1} + 2 \beta_{3} - \beta_{8} - \beta_{12} - \beta_{15} ) q^{80} + ( -1 - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{82} + ( -5 \beta_{1} + 2 \beta_{3} + \beta_{8} - \beta_{12} + 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{83} + ( 2 \beta_{2} - 2 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 4 \beta_{11} ) q^{85} + ( 3 \beta_{1} - \beta_{8} + \beta_{12} - 4 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{86} + ( 1 + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{9} - 3 \beta_{10} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{12} + \beta_{14} + 2 \beta_{15} ) q^{89} + ( \beta_{5} - 3 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{91} + ( -2 \beta_{1} - 4 \beta_{3} - 2 \beta_{7} + 2 \beta_{12} + 2 \beta_{14} - 2 \beta_{15} ) q^{92} + ( -2 + \beta_{2} + 2 \beta_{4} + \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{94} + ( -2 \beta_{1} - 2 \beta_{3} - 2 \beta_{8} + 2 \beta_{12} - 2 \beta_{13} ) q^{95} + ( 6 + 2 \beta_{2} + 2 \beta_{6} + 2 \beta_{10} ) q^{97} + ( 2 \beta_{1} + 4 \beta_{7} + \beta_{8} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} + 3 \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 20q^{4} + 10q^{7} + O(q^{10}) \) \( 16q + 20q^{4} + 10q^{7} + 14q^{10} + 20q^{13} + 28q^{16} + 12q^{19} + 18q^{22} + 52q^{25} + 42q^{28} + 18q^{31} + 10q^{34} + 16q^{37} + 6q^{40} + 26q^{43} - 24q^{46} + 54q^{49} + 52q^{52} + 20q^{55} - 14q^{58} + 36q^{61} - 4q^{64} + 26q^{67} + 36q^{70} + 60q^{73} - 20q^{76} + 12q^{79} - 20q^{82} - 12q^{85} + 8q^{88} - 24q^{91} - 26q^{94} + 108q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 26 x^{14} + 269 x^{12} - 1408 x^{10} + 3924 x^{8} - 5655 x^{6} + 3886 x^{4} - 1107 x^{2} + 108\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{14} - 127 \nu^{12} + 1288 \nu^{10} - 6632 \nu^{8} + 18204 \nu^{6} - 25503 \nu^{4} + 15677 \nu^{2} - 2676 \)\()/48\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{14} - 25 \nu^{12} + 248 \nu^{10} - 1240 \nu^{8} + 3276 \nu^{6} - 4363 \nu^{4} + 2499 \nu^{2} - 388 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 7 \nu^{14} - 185 \nu^{12} + 1928 \nu^{10} - 10024 \nu^{8} + 27060 \nu^{6} - 35781 \nu^{4} + 19675 \nu^{2} - 2892 \)\()/48\)
\(\beta_{7}\)\(=\)\((\)\( 5 \nu^{15} - 127 \nu^{13} + 1276 \nu^{11} - 6416 \nu^{9} + 16812 \nu^{7} - 21711 \nu^{5} + 11813 \nu^{3} - 1728 \nu \)\()/24\)
\(\beta_{8}\)\(=\)\((\)\( 5 \nu^{15} - 127 \nu^{13} + 1276 \nu^{11} - 6416 \nu^{9} + 16812 \nu^{7} - 21735 \nu^{5} + 12005 \nu^{3} - 1992 \nu \)\()/24\)
\(\beta_{9}\)\(=\)\((\)\( \nu^{14} - 25 \nu^{12} + 248 \nu^{10} - 1236 \nu^{8} + 3224 \nu^{6} - 4163 \nu^{4} + 2287 \nu^{2} - 360 \)\()/4\)
\(\beta_{10}\)\(=\)\((\)\( -\nu^{14} + 26 \nu^{12} - 268 \nu^{10} + 1388 \nu^{8} - 3772 \nu^{6} + 5111 \nu^{4} - 2974 \nu^{2} + 488 \)\()/4\)
\(\beta_{11}\)\(=\)\((\)\( 13 \nu^{14} - 335 \nu^{12} + 3416 \nu^{10} - 17464 \nu^{8} + 46764 \nu^{6} - 62487 \nu^{4} + 36205 \nu^{2} - 6036 \)\()/48\)
\(\beta_{12}\)\(=\)\((\)\( 25 \nu^{15} - 641 \nu^{13} + 6500 \nu^{11} - 32968 \nu^{9} + 87012 \nu^{7} - 112827 \nu^{5} + 61555 \nu^{3} - 9504 \nu \)\()/72\)
\(\beta_{13}\)\(=\)\((\)\( 59 \nu^{15} - 1525 \nu^{13} + 15592 \nu^{11} - 79832 \nu^{9} + 213444 \nu^{7} - 282849 \nu^{5} + 160703 \nu^{3} - 26028 \nu \)\()/144\)
\(\beta_{14}\)\(=\)\((\)\( 11 \nu^{15} - 283 \nu^{13} + 2884 \nu^{11} - 14744 \nu^{9} + 39444 \nu^{7} - 52377 \nu^{5} + 29657 \nu^{3} - 4680 \nu \)\()/24\)
\(\beta_{15}\)\(=\)\((\)\( 47 \nu^{15} - 1213 \nu^{13} + 12400 \nu^{11} - 63584 \nu^{9} + 170604 \nu^{7} - 227373 \nu^{5} + 129911 \nu^{3} - 21060 \nu \)\()/72\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10} + \beta_{6} + \beta_{4} + 7 \beta_{2} + 15\)
\(\nu^{5}\)\(=\)\(-\beta_{8} + \beta_{7} + 8 \beta_{3} + 29 \beta_{1}\)
\(\nu^{6}\)\(=\)\(\beta_{11} + 11 \beta_{10} + 10 \beta_{6} - \beta_{5} + 11 \beta_{4} + 45 \beta_{2} + 86\)
\(\nu^{7}\)\(=\)\(2 \beta_{15} - 3 \beta_{14} - \beta_{12} - 11 \beta_{8} + 13 \beta_{7} + 57 \beta_{3} + 176 \beta_{1}\)
\(\nu^{8}\)\(=\)\(13 \beta_{11} + 93 \beta_{10} + \beta_{9} + 80 \beta_{6} - 15 \beta_{5} + 93 \beta_{4} + 288 \beta_{2} + 520\)
\(\nu^{9}\)\(=\)\(29 \beta_{15} - 43 \beta_{14} - 2 \beta_{13} - 11 \beta_{12} - 93 \beta_{8} + 119 \beta_{7} + 393 \beta_{3} + 1098 \beta_{1}\)
\(\nu^{10}\)\(=\)\(117 \beta_{11} + 712 \beta_{10} + 17 \beta_{9} + 594 \beta_{6} - 154 \beta_{5} + 717 \beta_{4} + 1855 \beta_{2} + 3236\)
\(\nu^{11}\)\(=\)\(294 \beta_{15} - 426 \beta_{14} - 40 \beta_{13} - 86 \beta_{12} - 714 \beta_{8} + 952 \beta_{7} + 2672 \beta_{3} + 6985 \beta_{1}\)
\(\nu^{12}\)\(=\)\(912 \beta_{11} + 5188 \beta_{10} + 192 \beta_{9} + 4252 \beta_{6} - 1348 \beta_{5} + 5284 \beta_{4} + 12035 \beta_{2} + 20521\)
\(\nu^{13}\)\(=\)\(2572 \beta_{15} - 3632 \beta_{14} - 504 \beta_{13} - 596 \beta_{12} - 5216 \beta_{8} + 7132 \beta_{7} + 18039 \beta_{3} + 45071 \beta_{1}\)
\(\nu^{14}\)\(=\)\(6628 \beta_{11} + 36771 \beta_{10} + 1824 \beta_{9} + 29791 \beta_{6} - 10824 \beta_{5} + 37931 \beta_{4} + 78577 \beta_{2} + 131897\)
\(\nu^{15}\)\(=\)\(20788 \beta_{15} - 28628 \beta_{14} - 5160 \beta_{13} - 3944 \beta_{12} - 36967 \beta_{8} + 51539 \beta_{7} + 121312 \beta_{3} + 293859 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.59858
−2.57927
−2.28010
−2.01982
−1.36017
−0.979723
−0.518678
−0.487100
0.487100
0.518678
0.979723
1.36017
2.01982
2.28010
2.57927
2.59858
−2.59858 0 4.75262 −0.560562 0 4.70134 −7.15292 0 1.45667
1.2 −2.57927 0 4.65263 −1.53253 0 −2.06948 −6.84184 0 3.95280
1.3 −2.28010 0 3.19885 3.42195 0 0.843402 −2.73351 0 −7.80238
1.4 −2.01982 0 2.07968 −3.33517 0 −0.226097 −0.160938 0 6.73645
1.5 −1.36017 0 −0.149936 −3.81085 0 4.20131 2.92428 0 5.18341
1.6 −0.979723 0 −1.04014 1.37010 0 3.49660 2.97850 0 −1.34231
1.7 −0.518678 0 −1.73097 4.37543 0 −5.00960 1.93517 0 −2.26944
1.8 −0.487100 0 −1.76273 −2.22708 0 −0.937481 1.83283 0 1.08481
1.9 0.487100 0 −1.76273 2.22708 0 −0.937481 −1.83283 0 1.08481
1.10 0.518678 0 −1.73097 −4.37543 0 −5.00960 −1.93517 0 −2.26944
1.11 0.979723 0 −1.04014 −1.37010 0 3.49660 −2.97850 0 −1.34231
1.12 1.36017 0 −0.149936 3.81085 0 4.20131 −2.92428 0 5.18341
1.13 2.01982 0 2.07968 3.33517 0 −0.226097 0.160938 0 6.73645
1.14 2.28010 0 3.19885 −3.42195 0 0.843402 2.73351 0 −7.80238
1.15 2.57927 0 4.65263 1.53253 0 −2.06948 6.84184 0 3.95280
1.16 2.59858 0 4.75262 0.560562 0 4.70134 7.15292 0 1.45667
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(127\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1143.2.a.k 16
3.b odd 2 1 inner 1143.2.a.k 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1143.2.a.k 16 1.a even 1 1 trivial
1143.2.a.k 16 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1143))\):

\(T_{2}^{16} - \cdots\)
\(T_{5}^{16} - \cdots\)
\(T_{7}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 108 - 1107 T^{2} + 3886 T^{4} - 5655 T^{6} + 3924 T^{8} - 1408 T^{10} + 269 T^{12} - 26 T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( 248832 - 1154304 T^{2} + 1351552 T^{4} - 682800 T^{6} + 173128 T^{8} - 23541 T^{10} + 1742 T^{12} - 66 T^{14} + T^{16} \)
$7$ \( ( 128 + 544 T - 288 T^{2} - 824 T^{3} + 112 T^{4} + 158 T^{5} - 29 T^{6} - 5 T^{7} + T^{8} )^{2} \)
$11$ \( 101059248 - 122364816 T^{2} + 55175557 T^{4} - 12414755 T^{6} + 1552162 T^{8} - 112855 T^{10} + 4746 T^{12} - 107 T^{14} + T^{16} \)
$13$ \( ( -6902 + 457 T + 5944 T^{2} - 1781 T^{3} - 719 T^{4} + 319 T^{5} - 5 T^{6} - 10 T^{7} + T^{8} )^{2} \)
$17$ \( 212487168 - 433528128 T^{2} + 290537248 T^{4} - 77114408 T^{6} + 8465409 T^{8} - 457766 T^{10} + 12927 T^{12} - 182 T^{14} + T^{16} \)
$19$ \( ( -2624 + 15520 T - 3404 T^{2} - 6634 T^{3} + 1337 T^{4} + 538 T^{5} - 91 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$23$ \( 28390195200 - 34180890624 T^{2} + 7411290112 T^{4} - 721507584 T^{6} + 38723520 T^{8} - 1226272 T^{10} + 22880 T^{12} - 233 T^{14} + T^{16} \)
$29$ \( 1108992 - 94894848 T^{2} + 119604928 T^{4} - 49477968 T^{6} + 7777844 T^{8} - 530381 T^{10} + 16230 T^{12} - 214 T^{14} + T^{16} \)
$31$ \( ( -74656 + 26020 T + 30386 T^{2} - 15897 T^{3} + 435 T^{4} + 739 T^{5} - 69 T^{6} - 9 T^{7} + T^{8} )^{2} \)
$37$ \( ( 4799818 + 725129 T - 410874 T^{2} - 51715 T^{3} + 12887 T^{4} + 1145 T^{5} - 183 T^{6} - 8 T^{7} + T^{8} )^{2} \)
$41$ \( 256633651200 - 177261086976 T^{2} + 42610370704 T^{4} - 4370811012 T^{6} + 208919265 T^{8} - 5045746 T^{10} + 63727 T^{12} - 402 T^{14} + T^{16} \)
$43$ \( ( 538624 + 679712 T - 87712 T^{2} - 157112 T^{3} + 12224 T^{4} + 2810 T^{5} - 215 T^{6} - 13 T^{7} + T^{8} )^{2} \)
$47$ \( 4140380534832 - 1197296430480 T^{2} + 140001608797 T^{4} - 8600164887 T^{6} + 302113442 T^{8} - 6195907 T^{10} + 72090 T^{12} - 431 T^{14} + T^{16} \)
$53$ \( 40368000000 - 27543360000 T^{2} + 7093403200 T^{4} - 889950688 T^{6} + 59759420 T^{8} - 2183525 T^{10} + 41294 T^{12} - 350 T^{14} + T^{16} \)
$59$ \( 50331648 - 3230466048 T^{2} + 5123657728 T^{4} - 1226862848 T^{6} + 103417088 T^{8} - 3712864 T^{10} + 60188 T^{12} - 421 T^{14} + T^{16} \)
$61$ \( ( 532930 - 1349477 T + 632752 T^{2} - 52907 T^{3} - 21519 T^{4} + 4305 T^{5} - 133 T^{6} - 18 T^{7} + T^{8} )^{2} \)
$67$ \( ( 304768 + 72736 T - 147360 T^{2} - 45208 T^{3} + 8000 T^{4} + 1622 T^{5} - 147 T^{6} - 13 T^{7} + T^{8} )^{2} \)
$71$ \( 220283977728 - 189843224940 T^{2} + 46274433385 T^{4} - 4980719683 T^{6} + 265718230 T^{8} - 7109911 T^{10} + 90238 T^{12} - 511 T^{14} + T^{16} \)
$73$ \( ( 55534 + 67655 T - 76616 T^{2} - 78319 T^{3} - 8279 T^{4} + 2457 T^{5} + 123 T^{6} - 30 T^{7} + T^{8} )^{2} \)
$79$ \( ( -798208 + 1535392 T - 173684 T^{2} - 120446 T^{3} + 17961 T^{4} + 1602 T^{5} - 263 T^{6} - 6 T^{7} + T^{8} )^{2} \)
$83$ \( 6491311177728 - 6856221904896 T^{2} + 2142042686464 T^{4} - 200033593088 T^{6} + 5239407232 T^{8} - 59503728 T^{10} + 335204 T^{12} - 925 T^{14} + T^{16} \)
$89$ \( 2174768483328 - 3318295530240 T^{2} + 1342420507840 T^{4} - 99030253136 T^{6} + 2986727828 T^{8} - 42645005 T^{10} + 288158 T^{12} - 886 T^{14} + T^{16} \)
$97$ \( ( -12139520 + 13035264 T - 5290944 T^{2} + 966624 T^{3} - 59104 T^{4} - 5248 T^{5} + 1000 T^{6} - 54 T^{7} + T^{8} )^{2} \)
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