Newspace parameters
Level: | \( N \) | \(=\) | \( 980 = 2^{2} \cdot 5 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 980.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(7.82533939809\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(8\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{16} - \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + 17766 x^{8} - 24680 x^{7} + 28682 x^{6} - 27248 x^{5} + 20861 x^{4} + \cdots + 196 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
Coefficient ring index: | \( 2^{10} \) |
Twist minimal: | no (minimal twist has level 140) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 802 x^{12} - 2264 x^{11} + 5402 x^{10} - 10642 x^{9} + 17766 x^{8} - 24680 x^{7} + 28682 x^{6} - 27248 x^{5} + 20861 x^{4} + \cdots + 196 \) :
\(\beta_{1}\) | \(=\) | \( ( - 103 \nu^{14} + 721 \nu^{13} - 4446 \nu^{12} + 17303 \nu^{11} - 57239 \nu^{10} + 144768 \nu^{9} - 310061 \nu^{8} + 532511 \nu^{7} - 766976 \nu^{6} + 885763 \nu^{5} + \cdots - 24976 ) / 2156 \) |
\(\beta_{2}\) | \(=\) | \( ( - 24 \nu^{14} + 168 \nu^{13} - 1045 \nu^{12} + 4086 \nu^{11} - 13636 \nu^{10} + 34729 \nu^{9} - 74976 \nu^{8} + 129612 \nu^{7} - 187089 \nu^{6} + 215748 \nu^{5} - 199886 \nu^{4} + \cdots - 3892 ) / 98 \) |
\(\beta_{3}\) | \(=\) | \( ( 921 \nu^{15} + 4134 \nu^{14} - 27981 \nu^{13} + 258387 \nu^{12} - 993848 \nu^{11} + 3534057 \nu^{10} - 8600435 \nu^{9} + 18275426 \nu^{8} - 28971201 \nu^{7} + \cdots - 627760 ) / 84868 \) |
\(\beta_{4}\) | \(=\) | \( ( 921 \nu^{15} - 17949 \nu^{14} + 126600 \nu^{13} - 689883 \nu^{12} + 2686219 \nu^{11} - 8556602 \nu^{10} + 21803093 \nu^{9} - 46150211 \nu^{8} + 80475178 \nu^{7} + \cdots - 3476900 ) / 84868 \) |
\(\beta_{5}\) | \(=\) | \( ( - 2249 \nu^{15} + 5826 \nu^{14} - 25703 \nu^{13} - 60501 \nu^{12} + 480220 \nu^{11} - 2553517 \nu^{10} + 7821237 \nu^{9} - 19570106 \nu^{8} + 37195221 \nu^{7} + \cdots - 1543192 ) / 84868 \) |
\(\beta_{6}\) | \(=\) | \( ( - 2249 \nu^{15} + 27909 \nu^{14} - 180284 \nu^{13} + 887769 \nu^{12} - 3199847 \nu^{11} + 9537142 \nu^{10} - 22582291 \nu^{9} + 44855531 \nu^{8} - 72251158 \nu^{7} + \cdots + 796740 ) / 84868 \) |
\(\beta_{7}\) | \(=\) | \( ( 4498 \nu^{15} - 33735 \nu^{14} + 205987 \nu^{13} - 827268 \nu^{12} + 2719627 \nu^{11} - 6983625 \nu^{10} + 14761054 \nu^{9} - 25285425 \nu^{8} + 35055937 \nu^{7} + \cdots + 661584 ) / 84868 \) |
\(\beta_{8}\) | \(=\) | \( ( 6283 \nu^{15} - 51669 \nu^{14} + 332274 \nu^{13} - 1453425 \nu^{12} + 5182367 \nu^{11} - 14821924 \nu^{10} + 35371231 \nu^{9} - 70552899 \nu^{8} + 117927236 \nu^{7} + \cdots - 4705876 ) / 84868 \) |
\(\beta_{9}\) | \(=\) | \( ( 37258 \nu^{15} - 235269 \nu^{14} + 1410813 \nu^{13} - 5023983 \nu^{12} + 15617521 \nu^{11} - 35160459 \nu^{10} + 66702361 \nu^{9} - 93259339 \nu^{8} + \cdots - 838334 ) / 466774 \) |
\(\beta_{10}\) | \(=\) | \( ( - 37258 \nu^{15} + 323601 \nu^{14} - 2029137 \nu^{13} + 8859497 \nu^{12} - 30592393 \nu^{11} + 84923417 \nu^{10} - 192984213 \nu^{9} + 363819389 \nu^{8} + \cdots + 6124398 ) / 466774 \) |
\(\beta_{11}\) | \(=\) | \( ( - 110763 \nu^{15} + 827042 \nu^{14} - 5299391 \nu^{13} + 21834951 \nu^{12} - 76233328 \nu^{11} + 206125891 \nu^{10} - 474572071 \nu^{9} + 888153026 \nu^{8} + \cdots + 29355256 ) / 933548 \) |
\(\beta_{12}\) | \(=\) | \( ( - 110763 \nu^{15} + 834403 \nu^{14} - 5350918 \nu^{13} + 22193475 \nu^{12} - 77714621 \nu^{11} + 211712024 \nu^{10} - 490152277 \nu^{9} + 926997889 \nu^{8} + \cdots + 36641780 ) / 933548 \) |
\(\beta_{13}\) | \(=\) | \( ( - 5508 \nu^{15} + 46506 \nu^{14} - 302999 \nu^{13} + 1332783 \nu^{12} - 4790008 \nu^{11} + 13638265 \nu^{10} - 32542271 \nu^{9} + 63996516 \nu^{8} + \cdots + 3098746 ) / 42434 \) |
\(\beta_{14}\) | \(=\) | \( ( - 12566 \nu^{15} + 94245 \nu^{14} - 600897 \nu^{13} + 2476448 \nu^{12} - 8609785 \nu^{11} + 23257531 \nu^{10} - 53380894 \nu^{9} + 99905415 \nu^{8} + \cdots + 4065068 ) / 84868 \) |
\(\beta_{15}\) | \(=\) | \( ( - 852 \nu^{15} + 6390 \nu^{14} - 40676 \nu^{13} + 167479 \nu^{12} - 580296 \nu^{11} + 1562572 \nu^{10} - 3560985 \nu^{9} + 6611661 \nu^{8} - 10337168 \nu^{7} + \cdots + 170743 ) / 4763 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{6} + \beta_{5} + 1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{7} + 2\beta_{5} - \beta_{4} + \beta_{3} - 5 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{15} - \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - 7\beta_{7} - 5\beta_{6} - 2\beta_{5} - 3\beta_{4} - 8 ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 2 \beta_{15} - 4 \beta_{12} - 4 \beta_{10} - 15 \beta_{7} - 2 \beta_{6} - 14 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} + 2 \beta _1 + 19 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 10 \beta_{15} + 5 \beta_{14} + 2 \beta_{12} + 12 \beta_{11} - 10 \beta_{9} + 36 \beta_{7} + 32 \beta_{6} - 3 \beta_{5} + 30 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} + 5 \beta _1 + 61 ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( - 35 \beta_{15} + 14 \beta_{14} + 36 \beta_{12} + 16 \beta_{11} + 36 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 146 \beta_{7} + 35 \beta_{6} + 93 \beta_{5} + 16 \beta_{4} + 74 \beta_{3} + 18 \beta_{2} - 25 \beta _1 - 70 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( 58 \beta_{15} - 49 \beta_{14} + 7 \beta_{13} + 19 \beta_{12} - 86 \beta_{11} + 60 \beta_{10} + 87 \beta_{9} - 7 \beta_{8} - 127 \beta_{7} - 211 \beta_{6} + 118 \beta_{5} - 224 \beta_{4} + 105 \beta_{3} + 84 \beta_{2} - 105 \beta _1 - 468 ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 400 \beta_{15} - 242 \beta_{14} + 28 \beta_{13} - 272 \beta_{12} - 248 \beta_{11} - 216 \beta_{10} + 88 \beta_{9} + 20 \beta_{8} - 1225 \beta_{7} - 440 \beta_{6} - 568 \beta_{5} - 385 \beta_{4} - 525 \beta_{3} - 70 \beta_{2} + 120 \beta _1 + 89 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( - 78 \beta_{15} + 192 \beta_{14} - 48 \beta_{13} - 371 \beta_{12} + 409 \beta_{11} - 807 \beta_{10} - 693 \beta_{9} + 132 \beta_{8} - 224 \beta_{7} + 1246 \beta_{6} - 1457 \beta_{5} + 1386 \beta_{4} - 1371 \beta_{3} + \cdots + 3481 ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 3645 \beta_{15} + 2608 \beta_{14} - 450 \beta_{13} + 1922 \beta_{12} + 2552 \beta_{11} + 722 \beta_{10} - 1288 \beta_{9} - 34 \beta_{8} + 9166 \beta_{7} + 4575 \beta_{6} + 2909 \beta_{5} + 4570 \beta_{4} + \cdots + 2494 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( - 3305 \beta_{15} + 1309 \beta_{14} - 176 \beta_{13} + 4638 \beta_{12} - 312 \beta_{11} + 7536 \beta_{10} + 4828 \beta_{9} - 1474 \beta_{8} + 10625 \beta_{7} - 5493 \beta_{6} + 14142 \beta_{5} - 6413 \beta_{4} + \cdots - 24056 ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( ( 27668 \beta_{15} - 21904 \beta_{14} + 4356 \beta_{13} - 11876 \beta_{12} - 21172 \beta_{11} + 3216 \beta_{10} + 15040 \beta_{9} - 1634 \beta_{8} - 60753 \beta_{7} - 41348 \beta_{6} - 8804 \beta_{5} + \cdots - 41671 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( 57555 \beta_{15} - 35867 \beta_{14} + 7475 \beta_{13} - 48811 \beta_{12} - 21576 \beta_{11} - 57400 \beta_{10} - 26853 \beta_{9} + 11973 \beta_{8} - 141868 \beta_{7} + 3476 \beta_{6} - 119673 \beta_{5} + \cdots + 146745 ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( - 170623 \beta_{15} + 148774 \beta_{14} - 30758 \beta_{13} + 53306 \beta_{12} + 147184 \beta_{11} - 91176 \beta_{10} - 149788 \beta_{9} + 29644 \beta_{8} + 337546 \beta_{7} + 331357 \beta_{6} + \cdots + 465346 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( - 657499 \beta_{15} + 463584 \beta_{14} - 103744 \beta_{13} + 451091 \beta_{12} + 335591 \beta_{11} + 361643 \beta_{10} + 80673 \beta_{9} - 71496 \beta_{8} + 1444309 \beta_{7} + \cdots - 702512 ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/980\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(197\) | \(491\) |
\(\chi(n)\) | \(-1\) | \(-\beta_{7}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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97.1 |
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0 | −2.28727 | − | 2.28727i | 0 | −1.22200 | − | 1.87262i | 0 | 0 | 0 | 7.46321i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.2 | 0 | −1.77536 | − | 1.77536i | 0 | 1.45225 | − | 1.70029i | 0 | 0 | 0 | 3.30382i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.3 | 0 | −1.11777 | − | 1.11777i | 0 | −0.524151 | + | 2.17377i | 0 | 0 | 0 | − | 0.501168i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.4 | 0 | −0.605864 | − | 0.605864i | 0 | 2.15010 | + | 0.614051i | 0 | 0 | 0 | − | 2.26586i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.5 | 0 | 0.605864 | + | 0.605864i | 0 | −2.15010 | − | 0.614051i | 0 | 0 | 0 | − | 2.26586i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.6 | 0 | 1.11777 | + | 1.11777i | 0 | 0.524151 | − | 2.17377i | 0 | 0 | 0 | − | 0.501168i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.7 | 0 | 1.77536 | + | 1.77536i | 0 | −1.45225 | + | 1.70029i | 0 | 0 | 0 | 3.30382i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
97.8 | 0 | 2.28727 | + | 2.28727i | 0 | 1.22200 | + | 1.87262i | 0 | 0 | 0 | 7.46321i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.1 | 0 | −2.28727 | + | 2.28727i | 0 | −1.22200 | + | 1.87262i | 0 | 0 | 0 | − | 7.46321i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.2 | 0 | −1.77536 | + | 1.77536i | 0 | 1.45225 | + | 1.70029i | 0 | 0 | 0 | − | 3.30382i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.3 | 0 | −1.11777 | + | 1.11777i | 0 | −0.524151 | − | 2.17377i | 0 | 0 | 0 | 0.501168i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.4 | 0 | −0.605864 | + | 0.605864i | 0 | 2.15010 | − | 0.614051i | 0 | 0 | 0 | 2.26586i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.5 | 0 | 0.605864 | − | 0.605864i | 0 | −2.15010 | + | 0.614051i | 0 | 0 | 0 | 2.26586i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.6 | 0 | 1.11777 | − | 1.11777i | 0 | 0.524151 | + | 2.17377i | 0 | 0 | 0 | 0.501168i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.7 | 0 | 1.77536 | − | 1.77536i | 0 | −1.45225 | − | 1.70029i | 0 | 0 | 0 | − | 3.30382i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
293.8 | 0 | 2.28727 | − | 2.28727i | 0 | 1.22200 | − | 1.87262i | 0 | 0 | 0 | − | 7.46321i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
35.f | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 980.2.m.a | 16 | |
5.c | odd | 4 | 1 | inner | 980.2.m.a | 16 | |
7.b | odd | 2 | 1 | inner | 980.2.m.a | 16 | |
7.c | even | 3 | 1 | 140.2.u.a | ✓ | 16 | |
7.c | even | 3 | 1 | 980.2.v.a | 16 | ||
7.d | odd | 6 | 1 | 140.2.u.a | ✓ | 16 | |
7.d | odd | 6 | 1 | 980.2.v.a | 16 | ||
21.g | even | 6 | 1 | 1260.2.dq.a | 16 | ||
21.h | odd | 6 | 1 | 1260.2.dq.a | 16 | ||
28.f | even | 6 | 1 | 560.2.ci.d | 16 | ||
28.g | odd | 6 | 1 | 560.2.ci.d | 16 | ||
35.f | even | 4 | 1 | inner | 980.2.m.a | 16 | |
35.i | odd | 6 | 1 | 700.2.bc.b | 16 | ||
35.j | even | 6 | 1 | 700.2.bc.b | 16 | ||
35.k | even | 12 | 1 | 140.2.u.a | ✓ | 16 | |
35.k | even | 12 | 1 | 700.2.bc.b | 16 | ||
35.k | even | 12 | 1 | 980.2.v.a | 16 | ||
35.l | odd | 12 | 1 | 140.2.u.a | ✓ | 16 | |
35.l | odd | 12 | 1 | 700.2.bc.b | 16 | ||
35.l | odd | 12 | 1 | 980.2.v.a | 16 | ||
105.w | odd | 12 | 1 | 1260.2.dq.a | 16 | ||
105.x | even | 12 | 1 | 1260.2.dq.a | 16 | ||
140.w | even | 12 | 1 | 560.2.ci.d | 16 | ||
140.x | odd | 12 | 1 | 560.2.ci.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
140.2.u.a | ✓ | 16 | 7.c | even | 3 | 1 | |
140.2.u.a | ✓ | 16 | 7.d | odd | 6 | 1 | |
140.2.u.a | ✓ | 16 | 35.k | even | 12 | 1 | |
140.2.u.a | ✓ | 16 | 35.l | odd | 12 | 1 | |
560.2.ci.d | 16 | 28.f | even | 6 | 1 | ||
560.2.ci.d | 16 | 28.g | odd | 6 | 1 | ||
560.2.ci.d | 16 | 140.w | even | 12 | 1 | ||
560.2.ci.d | 16 | 140.x | odd | 12 | 1 | ||
700.2.bc.b | 16 | 35.i | odd | 6 | 1 | ||
700.2.bc.b | 16 | 35.j | even | 6 | 1 | ||
700.2.bc.b | 16 | 35.k | even | 12 | 1 | ||
700.2.bc.b | 16 | 35.l | odd | 12 | 1 | ||
980.2.m.a | 16 | 1.a | even | 1 | 1 | trivial | |
980.2.m.a | 16 | 5.c | odd | 4 | 1 | inner | |
980.2.m.a | 16 | 7.b | odd | 2 | 1 | inner | |
980.2.m.a | 16 | 35.f | even | 4 | 1 | inner | |
980.2.v.a | 16 | 7.c | even | 3 | 1 | ||
980.2.v.a | 16 | 7.d | odd | 6 | 1 | ||
980.2.v.a | 16 | 35.k | even | 12 | 1 | ||
980.2.v.a | 16 | 35.l | odd | 12 | 1 | ||
1260.2.dq.a | 16 | 21.g | even | 6 | 1 | ||
1260.2.dq.a | 16 | 21.h | odd | 6 | 1 | ||
1260.2.dq.a | 16 | 105.w | odd | 12 | 1 | ||
1260.2.dq.a | 16 | 105.x | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} + 156T_{3}^{12} + 5366T_{3}^{8} + 30012T_{3}^{4} + 14641 \)
acting on \(S_{2}^{\mathrm{new}}(980, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{16} \)
$3$
\( T^{16} + 156 T^{12} + 5366 T^{8} + \cdots + 14641 \)
$5$
\( T^{16} + 6 T^{14} + 33 T^{12} + \cdots + 390625 \)
$7$
\( T^{16} \)
$11$
\( (T^{4} - 13 T^{2} - 6 T + 10)^{4} \)
$13$
\( T^{16} + 1800 T^{12} + \cdots + 268435456 \)
$17$
\( T^{16} + 2286 T^{12} + 20417 T^{8} + \cdots + 256 \)
$19$
\( (T^{8} - 74 T^{6} + 1853 T^{4} + \cdots + 68644)^{2} \)
$23$
\( (T^{8} - 16 T^{7} + 128 T^{6} - 540 T^{5} + \cdots + 2809)^{2} \)
$29$
\( (T^{8} + 162 T^{6} + 7073 T^{4} + \cdots + 16384)^{2} \)
$31$
\( (T^{8} + 134 T^{6} + 5345 T^{4} + \cdots + 414736)^{2} \)
$37$
\( (T^{8} - 14 T^{7} + 98 T^{6} - 114 T^{5} + \cdots + 64)^{2} \)
$41$
\( (T^{8} + 170 T^{6} + 7001 T^{4} + \cdots + 12544)^{2} \)
$43$
\( (T^{8} - 14 T^{7} + 98 T^{6} + \cdots + 10432900)^{2} \)
$47$
\( T^{16} + 3318 T^{12} + 2247521 T^{8} + \cdots + 10000 \)
$53$
\( (T^{8} - 10 T^{7} + 50 T^{6} + \cdots + 188356)^{2} \)
$59$
\( (T^{8} - 266 T^{6} + 19181 T^{4} + \cdots + 2502724)^{2} \)
$61$
\( (T^{8} + 224 T^{6} + 16718 T^{4} + \cdots + 4092529)^{2} \)
$67$
\( (T^{8} + 8 T^{7} + 32 T^{6} - 4 T^{5} + \cdots + 49)^{2} \)
$71$
\( (T^{4} + 2 T^{3} - 142 T^{2} - 560 T + 448)^{4} \)
$73$
\( T^{16} + 48846 T^{12} + \cdots + 3841600000000 \)
$79$
\( (T^{8} + 270 T^{6} + 15989 T^{4} + \cdots + 1909924)^{2} \)
$83$
\( T^{16} + 25542 T^{12} + \cdots + 5006411536 \)
$89$
\( (T^{8} - 184 T^{6} + 7142 T^{4} + \cdots + 182329)^{2} \)
$97$
\( T^{16} + 68112 T^{12} + \cdots + 47698139955456 \)
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