Newspace parameters
Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 270.m (of order \(12\), degree \(4\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.15596085457\) |
Analytic rank: | \(0\) |
Dimension: | \(16\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{12})\) |
Coefficient field: | 16.0.9349208943630483456.9 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 90) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \( x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25 \) :
\(\beta_{1}\) | \(=\) | \( ( - 4341 \nu^{15} + 25062 \nu^{14} - 134821 \nu^{13} + 417198 \nu^{12} - 1075015 \nu^{11} + 1783997 \nu^{10} - 1878159 \nu^{9} - 718793 \nu^{8} + 7025637 \nu^{7} + \cdots - 740270 ) / 17095 \) |
\(\beta_{2}\) | \(=\) | \( ( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} - 4666596 \nu^{10} + 9554736 \nu^{9} - 15945783 \nu^{8} + 21928484 \nu^{7} + \cdots - 142555 ) / 17095 \) |
\(\beta_{3}\) | \(=\) | \( ( - 5033 \nu^{15} + 21310 \nu^{14} - 104426 \nu^{13} + 187861 \nu^{12} - 172588 \nu^{11} - 1167140 \nu^{10} + 5261069 \nu^{9} - 15006590 \nu^{8} + 29556405 \nu^{7} + \cdots - 887140 ) / 17095 \) |
\(\beta_{4}\) | \(=\) | \( ( - 2793 \nu^{15} + 28180 \nu^{14} - 172311 \nu^{13} + 766484 \nu^{12} - 2581266 \nu^{11} + 7044831 \nu^{10} - 15479146 \nu^{9} + 28225918 \nu^{8} - 42069895 \nu^{7} + \cdots + 618310 ) / 17095 \) |
\(\beta_{5}\) | \(=\) | \( ( 15977 \nu^{15} - 141262 \nu^{14} + 856169 \nu^{13} - 3642449 \nu^{12} + 12106306 \nu^{11} - 32236863 \nu^{10} + 70027507 \nu^{9} - 125576015 \nu^{8} + \cdots - 1883725 ) / 17095 \) |
\(\beta_{6}\) | \(=\) | \( ( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} - 28666706 \nu^{10} + 56590650 \nu^{9} - 89528958 \nu^{8} + \cdots + 218885 ) / 17095 \) |
\(\beta_{7}\) | \(=\) | \( ( 26630 \nu^{15} - 210508 \nu^{14} + 1251886 \nu^{13} - 5050631 \nu^{12} + 16323908 \nu^{11} - 41702827 \nu^{10} + 87680143 \nu^{9} - 150958920 \nu^{8} + \cdots - 1160550 ) / 17095 \) |
\(\beta_{8}\) | \(=\) | \( ( - 26630 \nu^{15} + 212086 \nu^{14} - 1262932 \nu^{13} + 5122693 \nu^{12} - 16612682 \nu^{11} + 42720111 \nu^{10} - 90382731 \nu^{9} + 157187286 \nu^{8} + \cdots + 2084995 ) / 17095 \) |
\(\beta_{9}\) | \(=\) | \( ( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} - 44845414 \nu^{10} + 95362110 \nu^{9} - 166733397 \nu^{8} + \cdots - 2071845 ) / 17095 \) |
\(\beta_{10}\) | \(=\) | \( ( 36479 \nu^{15} - 258470 \nu^{14} + 1499462 \nu^{13} - 5670854 \nu^{12} + 17608509 \nu^{11} - 42377599 \nu^{10} + 84494792 \nu^{9} - 135892302 \nu^{8} + \cdots + 18365 ) / 17095 \) |
\(\beta_{11}\) | \(=\) | \( ( - 38207 \nu^{15} + 272482 \nu^{14} - 1582764 \nu^{13} + 6010234 \nu^{12} - 18706521 \nu^{11} + 45195343 \nu^{10} - 90418577 \nu^{9} + 146094250 \nu^{8} + \cdots + 68035 ) / 17095 \) |
\(\beta_{12}\) | \(=\) | \( ( - 35414 \nu^{15} + 286908 \nu^{14} - 1708695 \nu^{13} + 6974027 \nu^{12} - 22629771 \nu^{11} + 58331291 \nu^{10} - 123326697 \nu^{9} + 214327527 \nu^{8} + \cdots + 2511045 ) / 17095 \) |
\(\beta_{13}\) | \(=\) | \( ( - 38207 \nu^{15} + 300623 \nu^{14} - 1779751 \nu^{13} + 7154021 \nu^{12} - 23008412 \nu^{11} + 58552587 \nu^{10} - 122465653 \nu^{9} + 210008510 \nu^{8} + \cdots + 1923500 ) / 17095 \) |
\(\beta_{14}\) | \(=\) | \( ( 39935 \nu^{15} - 314635 \nu^{14} + 1863053 \nu^{13} - 7493401 \nu^{12} + 24106424 \nu^{11} - 61370331 \nu^{10} + 128389438 \nu^{9} - 220210458 \nu^{8} + \cdots - 2009900 ) / 17095 \) |
\(\beta_{15}\) | \(=\) | \( ( 54184 \nu^{15} - 399016 \nu^{14} + 2335837 \nu^{13} - 9056462 \nu^{12} + 28575749 \nu^{11} - 70523459 \nu^{10} + 143951776 \nu^{9} - 239041170 \nu^{8} + \cdots - 1312670 ) / 17095 \) |
\(\nu\) | \(=\) | \( \beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} \) |
\(\nu^{2}\) | \(=\) | \( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} - 2 \beta_{6} - \beta_{5} + \beta_{4} - 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{15} - 3 \beta_{14} - 5 \beta_{13} + 3 \beta_{12} - 3 \beta_{10} + 3 \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{3} + 2 \beta_{2} - \beta _1 + 1 \) |
\(\nu^{4}\) | \(=\) | \( - 8 \beta_{15} - 6 \beta_{14} - 8 \beta_{13} - 2 \beta_{12} - 6 \beta_{11} - 8 \beta_{10} + 5 \beta_{9} + 6 \beta_{8} - 6 \beta_{7} + 13 \beta_{6} + 8 \beta_{5} - 8 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 2 \beta _1 + 5 \) |
\(\nu^{5}\) | \(=\) | \( - 17 \beta_{15} + 13 \beta_{14} + 27 \beta_{13} - 20 \beta_{12} - 21 \beta_{11} + 8 \beta_{10} - 21 \beta_{9} - 11 \beta_{7} + 15 \beta_{6} + 15 \beta_{5} - 10 \beta_{4} + 20 \beta_{3} - 14 \beta_{2} + 8 \beta _1 - 4 \) |
\(\nu^{6}\) | \(=\) | \( 43 \beta_{15} + 40 \beta_{14} + 64 \beta_{13} - 2 \beta_{12} + 11 \beta_{11} + 59 \beta_{10} - 63 \beta_{9} - 44 \beta_{8} + 34 \beta_{7} - 81 \beta_{6} - 49 \beta_{5} + 43 \beta_{4} + 75 \beta_{3} - 35 \beta_{2} + 29 \beta _1 - 32 \) |
\(\nu^{7}\) | \(=\) | \( 170 \beta_{15} - 68 \beta_{14} - 154 \beta_{13} + 133 \beta_{12} + 208 \beta_{11} + 16 \beta_{10} + 104 \beta_{9} - 21 \beta_{8} + 103 \beta_{7} - 167 \beta_{6} - 155 \beta_{5} + 119 \beta_{4} - 77 \beta_{3} + 93 \beta_{2} - 39 \beta _1 + 9 \) |
\(\nu^{8}\) | \(=\) | \( - 160 \beta_{15} - 316 \beta_{14} - 556 \beta_{13} + 108 \beta_{12} + 196 \beta_{11} - 388 \beta_{10} + 590 \beta_{9} + 324 \beta_{8} - 172 \beta_{7} + 472 \beta_{6} + 248 \beta_{5} - 172 \beta_{4} - 672 \beta_{3} + 385 \beta_{2} + \cdots + 231 \) |
\(\nu^{9}\) | \(=\) | \( - 1442 \beta_{15} + 303 \beta_{14} + 759 \beta_{13} - 888 \beta_{12} - 1474 \beta_{11} - 546 \beta_{10} - 254 \beta_{9} + 396 \beta_{8} - 906 \beta_{7} + 1654 \beta_{6} + 1414 \beta_{5} - 1044 \beta_{4} - 96 \beta_{3} + \cdots + 116 \) |
\(\nu^{10}\) | \(=\) | \( - 163 \beta_{15} + 2609 \beta_{14} + 4787 \beta_{13} - 1507 \beta_{12} - 3305 \beta_{11} + 2222 \beta_{10} - 4829 \beta_{9} - 2201 \beta_{8} + 557 \beta_{7} - 2176 \beta_{6} - 679 \beta_{5} + 158 \beta_{4} + \cdots - 1611 \) |
\(\nu^{11}\) | \(=\) | \( 10909 \beta_{15} - 253 \beta_{14} - 1879 \beta_{13} + 5489 \beta_{12} + 8068 \beta_{11} + 6380 \beta_{10} - 2613 \beta_{9} - 4950 \beta_{8} + 7438 \beta_{7} - 14737 \beta_{6} - 11667 \beta_{5} + 7920 \beta_{4} + \cdots - 2314 \) |
\(\nu^{12}\) | \(=\) | \( 11866 \beta_{15} - 20444 \beta_{14} - 38548 \beta_{13} + 16364 \beta_{12} + 34826 \beta_{11} - 9970 \beta_{10} + 35121 \beta_{9} + 12788 \beta_{8} + 2360 \beta_{7} + 3142 \beta_{6} - 5568 \beta_{5} + \cdots + 10052 \) |
\(\nu^{13}\) | \(=\) | \( - 72687 \beta_{15} - 16328 \beta_{14} - 20114 \beta_{13} - 27664 \beta_{12} - 26597 \beta_{11} - 58357 \beta_{10} + 54341 \beta_{9} + 50414 \beta_{8} - 55565 \beta_{7} + 117537 \beta_{6} + \cdots + 27130 \) |
\(\nu^{14}\) | \(=\) | \( - 163323 \beta_{15} + 144445 \beta_{14} + 280994 \beta_{13} - 152817 \beta_{12} - 300247 \beta_{11} + 16587 \beta_{10} - 220979 \beta_{9} - 52573 \beta_{8} - 70669 \beta_{7} + 88965 \beta_{6} + \cdots - 51022 \) |
\(\nu^{15}\) | \(=\) | \( 402584 \beta_{15} + 264336 \beta_{14} + 422112 \beta_{13} + 70486 \beta_{12} - 100959 \beta_{11} + 463038 \beta_{10} - 640520 \beta_{9} - 444317 \beta_{8} + 365424 \beta_{7} - 828756 \beta_{6} + \cdots - 258101 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(217\) |
\(\chi(n)\) | \(-\beta_{2}\) | \(-\beta_{6} - \beta_{9}\) |
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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17.1 |
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−0.965926 | − | 0.258819i | 0 | 0.866025 | + | 0.500000i | 0.847015 | − | 2.06944i | 0 | 0.686453 | − | 2.56188i | −0.707107 | − | 0.707107i | 0 | −1.35376 | + | 1.77970i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
17.2 | −0.965926 | − | 0.258819i | 0 | 0.866025 | + | 0.500000i | 2.22612 | + | 0.210717i | 0 | −0.521929 | + | 1.94786i | −0.707107 | − | 0.707107i | 0 | −2.09573 | − | 0.779698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
17.3 | 0.965926 | + | 0.258819i | 0 | 0.866025 | + | 0.500000i | 0.139908 | + | 2.23169i | 0 | −0.622279 | + | 2.32238i | 0.707107 | + | 0.707107i | 0 | −0.442462 | + | 2.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
17.4 | 0.965926 | + | 0.258819i | 0 | 0.866025 | + | 0.500000i | 1.51901 | − | 1.64092i | 0 | −1.00635 | + | 3.75574i | 0.707107 | + | 0.707107i | 0 | 1.89195 | − | 1.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.1 | −0.965926 | + | 0.258819i | 0 | 0.866025 | − | 0.500000i | 0.847015 | + | 2.06944i | 0 | 0.686453 | + | 2.56188i | −0.707107 | + | 0.707107i | 0 | −1.35376 | − | 1.77970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.2 | −0.965926 | + | 0.258819i | 0 | 0.866025 | − | 0.500000i | 2.22612 | − | 0.210717i | 0 | −0.521929 | − | 1.94786i | −0.707107 | + | 0.707107i | 0 | −2.09573 | + | 0.779698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.3 | 0.965926 | − | 0.258819i | 0 | 0.866025 | − | 0.500000i | 0.139908 | − | 2.23169i | 0 | −0.622279 | − | 2.32238i | 0.707107 | − | 0.707107i | 0 | −0.442462 | − | 2.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
143.4 | 0.965926 | − | 0.258819i | 0 | 0.866025 | − | 0.500000i | 1.51901 | + | 1.64092i | 0 | −1.00635 | − | 3.75574i | 0.707107 | − | 0.707107i | 0 | 1.89195 | + | 1.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
197.1 | −0.258819 | − | 0.965926i | 0 | −0.866025 | + | 0.500000i | −1.36868 | − | 1.76825i | 0 | −2.56188 | + | 0.686453i | 0.707107 | + | 0.707107i | 0 | −1.35376 | + | 1.77970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
197.2 | −0.258819 | − | 0.965926i | 0 | −0.866025 | + | 0.500000i | 1.29554 | − | 1.82252i | 0 | 1.94786 | − | 0.521929i | 0.707107 | + | 0.707107i | 0 | −2.09573 | − | 0.779698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
197.3 | 0.258819 | + | 0.965926i | 0 | −0.866025 | + | 0.500000i | −0.661570 | − | 2.13596i | 0 | 3.75574 | − | 1.00635i | −0.707107 | − | 0.707107i | 0 | 1.89195 | − | 1.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
197.4 | 0.258819 | + | 0.965926i | 0 | −0.866025 | + | 0.500000i | 2.00265 | + | 0.994679i | 0 | 2.32238 | − | 0.622279i | −0.707107 | − | 0.707107i | 0 | −0.442462 | + | 2.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
233.1 | −0.258819 | + | 0.965926i | 0 | −0.866025 | − | 0.500000i | −1.36868 | + | 1.76825i | 0 | −2.56188 | − | 0.686453i | 0.707107 | − | 0.707107i | 0 | −1.35376 | − | 1.77970i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
233.2 | −0.258819 | + | 0.965926i | 0 | −0.866025 | − | 0.500000i | 1.29554 | + | 1.82252i | 0 | 1.94786 | + | 0.521929i | 0.707107 | − | 0.707107i | 0 | −2.09573 | + | 0.779698i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
233.3 | 0.258819 | − | 0.965926i | 0 | −0.866025 | − | 0.500000i | −0.661570 | + | 2.13596i | 0 | 3.75574 | + | 1.00635i | −0.707107 | + | 0.707107i | 0 | 1.89195 | + | 1.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
233.4 | 0.258819 | − | 0.965926i | 0 | −0.866025 | − | 0.500000i | 2.00265 | − | 0.994679i | 0 | 2.32238 | + | 0.622279i | −0.707107 | + | 0.707107i | 0 | −0.442462 | − | 2.19185i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 270.2.m.b | 16 | |
3.b | odd | 2 | 1 | 90.2.l.b | ✓ | 16 | |
5.b | even | 2 | 1 | 1350.2.q.h | 16 | ||
5.c | odd | 4 | 1 | inner | 270.2.m.b | 16 | |
5.c | odd | 4 | 1 | 1350.2.q.h | 16 | ||
9.c | even | 3 | 1 | 90.2.l.b | ✓ | 16 | |
9.c | even | 3 | 1 | 810.2.f.c | 16 | ||
9.d | odd | 6 | 1 | inner | 270.2.m.b | 16 | |
9.d | odd | 6 | 1 | 810.2.f.c | 16 | ||
12.b | even | 2 | 1 | 720.2.cu.b | 16 | ||
15.d | odd | 2 | 1 | 450.2.p.h | 16 | ||
15.e | even | 4 | 1 | 90.2.l.b | ✓ | 16 | |
15.e | even | 4 | 1 | 450.2.p.h | 16 | ||
36.f | odd | 6 | 1 | 720.2.cu.b | 16 | ||
45.h | odd | 6 | 1 | 1350.2.q.h | 16 | ||
45.j | even | 6 | 1 | 450.2.p.h | 16 | ||
45.k | odd | 12 | 1 | 90.2.l.b | ✓ | 16 | |
45.k | odd | 12 | 1 | 450.2.p.h | 16 | ||
45.k | odd | 12 | 1 | 810.2.f.c | 16 | ||
45.l | even | 12 | 1 | inner | 270.2.m.b | 16 | |
45.l | even | 12 | 1 | 810.2.f.c | 16 | ||
45.l | even | 12 | 1 | 1350.2.q.h | 16 | ||
60.l | odd | 4 | 1 | 720.2.cu.b | 16 | ||
180.x | even | 12 | 1 | 720.2.cu.b | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.2.l.b | ✓ | 16 | 3.b | odd | 2 | 1 | |
90.2.l.b | ✓ | 16 | 9.c | even | 3 | 1 | |
90.2.l.b | ✓ | 16 | 15.e | even | 4 | 1 | |
90.2.l.b | ✓ | 16 | 45.k | odd | 12 | 1 | |
270.2.m.b | 16 | 1.a | even | 1 | 1 | trivial | |
270.2.m.b | 16 | 5.c | odd | 4 | 1 | inner | |
270.2.m.b | 16 | 9.d | odd | 6 | 1 | inner | |
270.2.m.b | 16 | 45.l | even | 12 | 1 | inner | |
450.2.p.h | 16 | 15.d | odd | 2 | 1 | ||
450.2.p.h | 16 | 15.e | even | 4 | 1 | ||
450.2.p.h | 16 | 45.j | even | 6 | 1 | ||
450.2.p.h | 16 | 45.k | odd | 12 | 1 | ||
720.2.cu.b | 16 | 12.b | even | 2 | 1 | ||
720.2.cu.b | 16 | 36.f | odd | 6 | 1 | ||
720.2.cu.b | 16 | 60.l | odd | 4 | 1 | ||
720.2.cu.b | 16 | 180.x | even | 12 | 1 | ||
810.2.f.c | 16 | 9.c | even | 3 | 1 | ||
810.2.f.c | 16 | 9.d | odd | 6 | 1 | ||
810.2.f.c | 16 | 45.k | odd | 12 | 1 | ||
810.2.f.c | 16 | 45.l | even | 12 | 1 | ||
1350.2.q.h | 16 | 5.b | even | 2 | 1 | ||
1350.2.q.h | 16 | 5.c | odd | 4 | 1 | ||
1350.2.q.h | 16 | 45.h | odd | 6 | 1 | ||
1350.2.q.h | 16 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} - 8 T_{7}^{15} + 32 T_{7}^{14} - 144 T_{7}^{13} + 476 T_{7}^{12} - 592 T_{7}^{11} - 128 T_{7}^{10} + 4656 T_{7}^{9} - 20276 T_{7}^{8} + 26016 T_{7}^{7} + 6400 T_{7}^{6} - 207200 T_{7}^{5} + 1088624 T_{7}^{4} + \cdots + 6250000 \)
acting on \(S_{2}^{\mathrm{new}}(270, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{8} - T^{4} + 1)^{2} \)
$3$
\( T^{16} \)
$5$
\( T^{16} - 12 T^{15} + 80 T^{14} + \cdots + 390625 \)
$7$
\( T^{16} - 8 T^{15} + 32 T^{14} + \cdots + 6250000 \)
$11$
\( (T^{8} - 22 T^{6} + 441 T^{4} - 528 T^{3} + \cdots + 1849)^{2} \)
$13$
\( T^{16} + 48 T^{13} - 156 T^{12} + \cdots + 1296 \)
$17$
\( T^{16} + 4132 T^{12} + 4269606 T^{8} + \cdots + 390625 \)
$19$
\( (T^{8} + 52 T^{6} + 858 T^{4} + \cdots + 10201)^{2} \)
$23$
\( T^{16} - 24 T^{15} + \cdots + 82538991616 \)
$29$
\( T^{16} + 80 T^{14} + 4960 T^{12} + \cdots + 40960000 \)
$31$
\( (T^{8} + 4 T^{7} + 88 T^{6} - 32 T^{5} + \cdots + 448900)^{2} \)
$37$
\( (T^{8} - 576 T^{5} + 9792 T^{4} + \cdots + 82944)^{2} \)
$41$
\( (T^{8} + 12 T^{7} - 10 T^{6} + \cdots + 555025)^{2} \)
$43$
\( T^{16} + 96 T^{13} + \cdots + 35806100625 \)
$47$
\( T^{16} + 48 T^{15} + \cdots + 2981133747216 \)
$53$
\( T^{16} + 10552 T^{12} + \cdots + 3906250000 \)
$59$
\( T^{16} + 44 T^{14} + 1462 T^{12} + \cdots + 625 \)
$61$
\( (T^{8} + 12 T^{7} + 252 T^{6} + \cdots + 45887076)^{2} \)
$67$
\( T^{16} + 16 T^{15} + 128 T^{14} + \cdots + 3418801 \)
$71$
\( (T^{8} + 272 T^{6} + 25980 T^{4} + \cdots + 14032516)^{2} \)
$73$
\( (T^{8} - 8 T^{7} + 32 T^{6} - 152 T^{5} + \cdots + 966289)^{2} \)
$79$
\( T^{16} - 432 T^{14} + \cdots + 17\!\cdots\!00 \)
$83$
\( T^{16} + 48 T^{15} + \cdots + 21743271936 \)
$89$
\( (T^{4} - 28 T^{2} + 100)^{4} \)
$97$
\( T^{16} + 48 T^{15} + \cdots + 19559470366881 \)
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