Newspace parameters
Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 570.q (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.55147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
Defining polynomial: |
\( x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} - 11968 x^{8} + 10368 x^{7} + 9344 x^{6} + 18176 x^{5} + 56320 x^{4} + 28160 x^{3} + \cdots + 1024 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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
\( x^{20} - 49 x^{16} - 8 x^{15} + 72 x^{13} + 2145 x^{12} - 648 x^{11} + 32 x^{10} - 7056 x^{9} - 11968 x^{8} + 10368 x^{7} + 9344 x^{6} + 18176 x^{5} + 56320 x^{4} + 28160 x^{3} + \cdots + 1024 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 12\!\cdots\!99 \nu^{19} + \cdots + 78\!\cdots\!92 ) / 12\!\cdots\!52 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 33\!\cdots\!57 \nu^{19} + \cdots - 93\!\cdots\!36 ) / 10\!\cdots\!76 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 42\!\cdots\!01 \nu^{19} + \cdots + 27\!\cdots\!92 ) / 10\!\cdots\!76 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 31\!\cdots\!63 \nu^{19} + \cdots + 22\!\cdots\!44 ) / 54\!\cdots\!88 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 16\!\cdots\!58 \nu^{19} + \cdots - 52\!\cdots\!24 ) / 27\!\cdots\!44 \)
|
\(\beta_{7}\) | \(=\) |
\( ( - 10\!\cdots\!81 \nu^{19} + \cdots - 15\!\cdots\!92 ) / 10\!\cdots\!76 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 43\!\cdots\!88 \nu^{19} + \cdots - 16\!\cdots\!96 ) / 27\!\cdots\!44 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 6169005809303 \nu^{19} + 3083823703934 \nu^{18} - 1600056848304 \nu^{17} + 905444482148 \nu^{16} + 301739777308423 \nu^{15} + \cdots - 21\!\cdots\!64 ) / 34\!\cdots\!32 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 11\!\cdots\!77 \nu^{19} + \cdots + 22\!\cdots\!04 ) / 27\!\cdots\!44 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 31\!\cdots\!33 \nu^{19} + \cdots + 65\!\cdots\!88 ) / 54\!\cdots\!88 \)
|
\(\beta_{12}\) | \(=\) |
\( ( - 20734554827161 \nu^{19} + 6169005809303 \nu^{18} - 3083823703934 \nu^{17} + 1600056848304 \nu^{16} + \cdots - 42\!\cdots\!20 ) / 34\!\cdots\!32 \)
|
\(\beta_{13}\) | \(=\) |
\( ( - 36\!\cdots\!23 \nu^{19} + \cdots - 17\!\cdots\!48 ) / 49\!\cdots\!08 \)
|
\(\beta_{14}\) | \(=\) |
\( ( - 46\!\cdots\!25 \nu^{19} + \cdots - 10\!\cdots\!08 ) / 54\!\cdots\!88 \)
|
\(\beta_{15}\) | \(=\) |
\( ( 10\!\cdots\!39 \nu^{19} + \cdots + 14\!\cdots\!28 ) / 10\!\cdots\!76 \)
|
\(\beta_{16}\) | \(=\) |
\( ( - 59\!\cdots\!77 \nu^{19} + \cdots - 11\!\cdots\!84 ) / 54\!\cdots\!88 \)
|
\(\beta_{17}\) | \(=\) |
\( ( - 15\!\cdots\!31 \nu^{19} + \cdots - 33\!\cdots\!52 ) / 10\!\cdots\!76 \)
|
\(\beta_{18}\) | \(=\) |
\( ( - 40\!\cdots\!87 \nu^{19} + \cdots - 82\!\cdots\!84 ) / 27\!\cdots\!44 \)
|
\(\beta_{19}\) | \(=\) |
\( ( 22\!\cdots\!27 \nu^{19} + \cdots + 21\!\cdots\!60 ) / 13\!\cdots\!72 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( -\beta_{16} + \beta_{14} - 4\beta_{5} \)
|
\(\nu^{3}\) | \(=\) |
\( 7\beta_{17} - 6\beta_{16} - \beta_{15} - \beta_{13} + 7\beta_{10} - \beta_{8} - 6\beta_{7} + \beta_{6} + 6\beta _1 + 1 \)
|
\(\nu^{4}\) | \(=\) |
\( 7 \beta_{19} - 2 \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 2 \beta_{13} + 13 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 9 \beta_{3} - 4 \beta _1 + 9 \)
|
\(\nu^{5}\) | \(=\) |
\( - 9 \beta_{17} + 18 \beta_{16} + 9 \beta_{13} - 4 \beta_{12} + 9 \beta_{11} - 47 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} + 4 \beta_{5} + 9 \beta_{3} + 9 \beta_{2} + 38 \beta _1 - 5 \)
|
\(\nu^{6}\) | \(=\) |
\( - 47 \beta_{18} - 51 \beta_{17} + 22 \beta_{16} + 47 \beta_{14} - 87 \beta_{11} - 22 \beta_{10} + 4 \beta_{9} + 47 \beta_{8} + 22 \beta_{7} - 134 \beta_{5} + 65 \beta_{2} - 4 \beta_1 \)
|
\(\nu^{7}\) | \(=\) |
\( - 4 \beta_{19} + 4 \beta_{18} + 250 \beta_{17} - 250 \beta_{16} - 65 \beta_{15} - 48 \beta_{12} - 17 \beta_{11} + 181 \beta_{10} - 315 \beta_{8} - 181 \beta_{7} + 69 \beta_{3} - 69 \beta_{2} + 181 \beta _1 - 4 \)
|
\(\nu^{8}\) | \(=\) |
\( 311 \beta_{19} + 319 \beta_{17} - 319 \beta_{16} - 441 \beta_{15} - 189 \beta_{13} + 413 \beta_{12} + 319 \beta_{10} + 635 \beta_{9} - 125 \beta_{8} - 319 \beta_{7} + 449 \beta_{6} + 311 \beta_{4} - 449 \beta_{3} - 186 \beta _1 - 94 \)
|
\(\nu^{9}\) | \(=\) |
\( 64 \beta_{18} - 8 \beta_{17} + 505 \beta_{16} + 505 \beta_{15} - 64 \beta_{14} + 505 \beta_{13} + 428 \beta_{11} - 64 \beta_{10} - 2119 \beta_{9} + 441 \beta_{8} + 505 \beta_{7} - 513 \beta_{6} + 492 \beta_{5} - 64 \beta_{4} - 72 \beta_{2} + \cdots - 13 \)
|
\(\nu^{10}\) | \(=\) |
\( - 2047 \beta_{18} - 2747 \beta_{17} + 3629 \beta_{16} - 3383 \beta_{11} - 1438 \beta_{10} + 2747 \beta_{8} + 556 \beta_{7} + 882 \beta_{5} + 2191 \beta_{2} \)
|
\(\nu^{11}\) | \(=\) |
\( - 700 \beta_{19} - 3629 \beta_{17} + 844 \beta_{15} + 700 \beta_{14} + 3485 \beta_{13} - 3288 \beta_{12} - 2929 \beta_{11} - 6558 \beta_{10} - 988 \beta_{9} - 9994 \beta_{8} + 3773 \beta_{7} - 3773 \beta_{6} + \cdots - 341 \)
|
\(\nu^{12}\) | \(=\) |
\( 25857 \beta_{17} - 25857 \beta_{16} - 15167 \beta_{15} - 15167 \beta_{13} + 25857 \beta_{10} + 25857 \beta_{9} - 15167 \beta_{8} - 25857 \beta_{7} + 21025 \beta_{6} + 13479 \beta_{4} + 10690 \beta _1 - 21751 \)
|
\(\nu^{13}\) | \(=\) |
\( 6520 \beta_{19} + 6520 \beta_{18} + 24169 \beta_{17} - 25857 \beta_{16} + 17649 \beta_{15} + 1688 \beta_{13} + 24524 \beta_{12} + 6875 \beta_{11} + 1688 \beta_{9} - 4832 \beta_{8} + 1688 \beta_{7} - 1688 \beta_{5} + \cdots + 8208 \)
|
\(\nu^{14}\) | \(=\) |
\( 105487 \beta_{16} - 89071 \beta_{14} + 16416 \beta_{11} + 16416 \beta_{10} - 55668 \beta_{9} + 55668 \beta_{8} - 38674 \beta_{7} + 285688 \beta_{5} - 22258 \beta_{2} + 55668 \beta_1 \)
|
\(\nu^{15}\) | \(=\) |
\( - 55668 \beta_{18} - 719327 \beta_{17} + 552330 \beta_{16} + 183413 \beta_{15} + 55668 \beta_{14} + 183413 \beta_{13} - 195104 \beta_{11} - 680075 \beta_{10} - 72084 \beta_{9} + 239081 \beta_{8} + \cdots + 67359 \)
|
\(\nu^{16}\) | \(=\) |
\( - 591575 \beta_{19} + 561930 \beta_{17} - 561930 \beta_{16} + 111322 \beta_{15} - 255490 \beta_{13} - 818429 \beta_{12} + 561930 \beta_{10} - 255490 \beta_{9} - 706098 \beta_{8} + \cdots - 847065 \)
|
\(\nu^{17}\) | \(=\) |
\( 450608 \beta_{19} + 1297673 \beta_{17} - 2595346 \beta_{16} - 594776 \beta_{15} + 450608 \beta_{14} - 1153505 \beta_{13} + 1286132 \beta_{12} - 847065 \beta_{11} + 450608 \beta_{10} + \cdots + 11541 \)
|
\(\nu^{18}\) | \(=\) |
\( 3951199 \beta_{18} + 7472267 \beta_{17} - 4025646 \beta_{16} - 3951199 \beta_{14} + 7347879 \beta_{11} + 5215198 \beta_{10} - 3521068 \beta_{9} - 3951199 \beta_{8} - 4025646 \beta_{7} + \cdots + 3521068 \beta_1 \)
|
\(\nu^{19}\) | \(=\) |
\( 3521068 \beta_{19} - 3521068 \beta_{18} - 25606226 \beta_{17} + 26795778 \beta_{16} + 4455777 \beta_{15} + 1189552 \beta_{13} + 9193256 \beta_{12} - 4737479 \beta_{11} - 17629381 \beta_{10} + \cdots + 4710620 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(211\) | \(457\) |
\(\chi(n)\) | \(1\) | \(-1 + \beta_{12}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
−0.866025 | − | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −2.22489 | + | 0.223342i | −0.500000 | − | 0.866025i | − | 1.07560i | − | 1.00000i | 0.500000 | + | 0.866025i | 2.03848 | + | 0.919023i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.2 | −0.866025 | − | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | −1.34502 | − | 1.78632i | −0.500000 | − | 0.866025i | 4.03495i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.271659 | + | 2.21950i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.3 | −0.866025 | − | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 0.384547 | + | 2.20275i | −0.500000 | − | 0.866025i | 2.51805i | − | 1.00000i | 0.500000 | + | 0.866025i | 0.768349 | − | 2.09991i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.4 | −0.866025 | − | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 1.99313 | − | 1.01362i | −0.500000 | − | 0.866025i | − | 2.79875i | − | 1.00000i | 0.500000 | + | 0.866025i | −2.23291 | − | 0.118742i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.5 | −0.866025 | − | 0.500000i | 0.866025 | + | 0.500000i | 0.500000 | + | 0.866025i | 2.05825 | + | 0.873846i | −0.500000 | − | 0.866025i | 2.32136i | − | 1.00000i | 0.500000 | + | 0.866025i | −1.34557 | − | 1.78590i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.6 | 0.866025 | + | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −2.09991 | + | 0.768349i | −0.500000 | − | 0.866025i | − | 2.51805i | 1.00000i | 0.500000 | + | 0.866025i | −2.20275 | − | 0.384547i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.7 | 0.866025 | + | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −1.78590 | − | 1.34557i | −0.500000 | − | 0.866025i | − | 2.32136i | 1.00000i | 0.500000 | + | 0.866025i | −0.873846 | − | 2.05825i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.8 | 0.866025 | + | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | −0.118742 | − | 2.23291i | −0.500000 | − | 0.866025i | 2.79875i | 1.00000i | 0.500000 | + | 0.866025i | 1.01362 | − | 1.99313i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.9 | 0.866025 | + | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 0.919023 | + | 2.03848i | −0.500000 | − | 0.866025i | 1.07560i | 1.00000i | 0.500000 | + | 0.866025i | −0.223342 | + | 2.22489i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
49.10 | 0.866025 | + | 0.500000i | −0.866025 | − | 0.500000i | 0.500000 | + | 0.866025i | 2.21950 | + | 0.271659i | −0.500000 | − | 0.866025i | − | 4.03495i | 1.00000i | 0.500000 | + | 0.866025i | 1.78632 | + | 1.34502i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.1 | −0.866025 | + | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −2.22489 | − | 0.223342i | −0.500000 | + | 0.866025i | 1.07560i | 1.00000i | 0.500000 | − | 0.866025i | 2.03848 | − | 0.919023i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.2 | −0.866025 | + | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | −1.34502 | + | 1.78632i | −0.500000 | + | 0.866025i | − | 4.03495i | 1.00000i | 0.500000 | − | 0.866025i | 0.271659 | − | 2.21950i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.3 | −0.866025 | + | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 0.384547 | − | 2.20275i | −0.500000 | + | 0.866025i | − | 2.51805i | 1.00000i | 0.500000 | − | 0.866025i | 0.768349 | + | 2.09991i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.4 | −0.866025 | + | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 1.99313 | + | 1.01362i | −0.500000 | + | 0.866025i | 2.79875i | 1.00000i | 0.500000 | − | 0.866025i | −2.23291 | + | 0.118742i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.5 | −0.866025 | + | 0.500000i | 0.866025 | − | 0.500000i | 0.500000 | − | 0.866025i | 2.05825 | − | 0.873846i | −0.500000 | + | 0.866025i | − | 2.32136i | 1.00000i | 0.500000 | − | 0.866025i | −1.34557 | + | 1.78590i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.6 | 0.866025 | − | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −2.09991 | − | 0.768349i | −0.500000 | + | 0.866025i | 2.51805i | − | 1.00000i | 0.500000 | − | 0.866025i | −2.20275 | + | 0.384547i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.7 | 0.866025 | − | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −1.78590 | + | 1.34557i | −0.500000 | + | 0.866025i | 2.32136i | − | 1.00000i | 0.500000 | − | 0.866025i | −0.873846 | + | 2.05825i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.8 | 0.866025 | − | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | −0.118742 | + | 2.23291i | −0.500000 | + | 0.866025i | − | 2.79875i | − | 1.00000i | 0.500000 | − | 0.866025i | 1.01362 | + | 1.99313i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.9 | 0.866025 | − | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 0.919023 | − | 2.03848i | −0.500000 | + | 0.866025i | − | 1.07560i | − | 1.00000i | 0.500000 | − | 0.866025i | −0.223342 | − | 2.22489i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
349.10 | 0.866025 | − | 0.500000i | −0.866025 | + | 0.500000i | 0.500000 | − | 0.866025i | 2.21950 | − | 0.271659i | −0.500000 | + | 0.866025i | 4.03495i | − | 1.00000i | 0.500000 | − | 0.866025i | 1.78632 | − | 1.34502i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
95.i | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 570.2.q.c | ✓ | 20 |
3.b | odd | 2 | 1 | 1710.2.t.c | 20 | ||
5.b | even | 2 | 1 | inner | 570.2.q.c | ✓ | 20 |
15.d | odd | 2 | 1 | 1710.2.t.c | 20 | ||
19.c | even | 3 | 1 | inner | 570.2.q.c | ✓ | 20 |
57.h | odd | 6 | 1 | 1710.2.t.c | 20 | ||
95.i | even | 6 | 1 | inner | 570.2.q.c | ✓ | 20 |
285.n | odd | 6 | 1 | 1710.2.t.c | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
570.2.q.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
570.2.q.c | ✓ | 20 | 5.b | even | 2 | 1 | inner |
570.2.q.c | ✓ | 20 | 19.c | even | 3 | 1 | inner |
570.2.q.c | ✓ | 20 | 95.i | even | 6 | 1 | inner |
1710.2.t.c | 20 | 3.b | odd | 2 | 1 | ||
1710.2.t.c | 20 | 15.d | odd | 2 | 1 | ||
1710.2.t.c | 20 | 57.h | odd | 6 | 1 | ||
1710.2.t.c | 20 | 285.n | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{10} + 37T_{7}^{8} + 486T_{7}^{6} + 2834T_{7}^{4} + 7041T_{7}^{2} + 5041 \)
acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} - T^{2} + 1)^{5} \)
$3$
\( (T^{4} - T^{2} + 1)^{5} \)
$5$
\( T^{20} - 7 T^{18} + 5 T^{16} + \cdots + 9765625 \)
$7$
\( (T^{10} + 37 T^{8} + 486 T^{6} + 2834 T^{4} + \cdots + 5041)^{2} \)
$11$
\( (T^{5} - 3 T^{4} - 13 T^{3} + 27 T^{2} + \cdots - 20)^{4} \)
$13$
\( T^{20} - 98 T^{18} + 6123 T^{16} + \cdots + 12960000 \)
$17$
\( T^{20} - 92 T^{18} + 5992 T^{16} + \cdots + 65536 \)
$19$
\( (T^{10} - 3 T^{9} - 10 T^{8} + \cdots + 2476099)^{2} \)
$23$
\( T^{20} - 127 T^{18} + \cdots + 283982410000 \)
$29$
\( (T^{10} - 4 T^{9} + 70 T^{8} - 596 T^{7} + \cdots + 501264)^{2} \)
$31$
\( (T^{5} - 10 T^{4} - 63 T^{3} + 880 T^{2} + \cdots - 1648)^{4} \)
$37$
\( (T^{10} + 81 T^{8} + 1358 T^{6} + \cdots + 18769)^{2} \)
$41$
\( (T^{10} + 7 T^{9} + 154 T^{8} + \cdots + 244421956)^{2} \)
$43$
\( T^{20} - 134 T^{18} + \cdots + 3110228525056 \)
$47$
\( T^{20} - 140 T^{18} + \cdots + 72699496960000 \)
$53$
\( T^{20} - 343 T^{18} + \cdots + 34\!\cdots\!76 \)
$59$
\( (T^{10} - 4 T^{9} + 76 T^{8} + 136 T^{7} + \cdots + 6400)^{2} \)
$61$
\( (T^{10} - 8 T^{9} + 303 T^{8} + \cdots + 313219204)^{2} \)
$67$
\( T^{20} - 382 T^{18} + \cdots + 10\!\cdots\!00 \)
$71$
\( (T^{10} + 2 T^{9} + 216 T^{8} + \cdots + 593994384)^{2} \)
$73$
\( T^{20} - 542 T^{18} + \cdots + 31\!\cdots\!00 \)
$79$
\( (T^{10} - 4 T^{9} + 183 T^{8} + \cdots + 263997504)^{2} \)
$83$
\( (T^{10} + 356 T^{8} + 45124 T^{6} + \cdots + 184090624)^{2} \)
$89$
\( (T^{10} + T^{9} + 116 T^{8} + \cdots + 13468900)^{2} \)
$97$
\( T^{20} - 368 T^{18} + \cdots + 18\!\cdots\!16 \)
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