Newspace parameters
Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 570.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.55147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Relative dimension: | \(10\) over \(\Q(i)\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{20} + \cdots)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{20} + 153x^{16} + 6416x^{12} + 78648x^{8} + 19120x^{4} + 16 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
Coefficient ring index: | \( 2^{10} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) 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 \( x^{20} + 153x^{16} + 6416x^{12} + 78648x^{8} + 19120x^{4} + 16 \) :
\(\beta_{1}\) | \(=\) | \( ( -16765\nu^{16} - 2625903\nu^{12} - 115242314\nu^{8} - 1515721480\nu^{4} - 599186688 ) / 174141704 \) |
\(\beta_{2}\) | \(=\) | \( ( 806911\nu^{17} + 123405389\nu^{13} + 5168924442\nu^{9} + 63121710948\nu^{5} + 12328816792\nu ) / 2089700448 \) |
\(\beta_{3}\) | \(=\) | \( ( -604825\nu^{17} - 92597906\nu^{13} - 3887923677\nu^{9} - 47719026672\nu^{5} - 10577293948\nu ) / 522425112 \) |
\(\beta_{4}\) | \(=\) | \( ( 54129\nu^{18} + 8282406\nu^{14} + 347394797\nu^{10} + 4261561356\nu^{6} + 1096207364\nu^{2} ) / 31662128 \) |
\(\beta_{5}\) | \(=\) | \( ( 11884471 \nu^{18} - 808344 \nu^{16} + 1818129965 \nu^{14} - 123229932 \nu^{12} + 76223127354 \nu^{10} - 5124003060 \nu^{8} + 933673009644 \nu^{6} + \cdots - 7006091376 ) / 4179400896 \) |
\(\beta_{6}\) | \(=\) | \( ( - 11884471 \nu^{18} - 808344 \nu^{16} - 1818129965 \nu^{14} - 123229932 \nu^{12} - 76223127354 \nu^{10} - 5124003060 \nu^{8} + \cdots - 7006091376 ) / 4179400896 \) |
\(\beta_{7}\) | \(=\) | \( ( 13498293 \nu^{18} - 604298 \nu^{16} + 2064940743 \nu^{14} - 92070010 \nu^{12} + 86560976238 \nu^{10} - 3830938056 \nu^{8} + 1059916431540 \nu^{6} + \cdots - 6281901056 ) / 4179400896 \) |
\(\beta_{8}\) | \(=\) | \( ( - 12734103 \nu^{18} + 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} + 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + \cdots - 7190240256 ) / 4179400896 \) |
\(\beta_{9}\) | \(=\) | \( ( - 13498293 \nu^{18} - 604298 \nu^{16} - 2064940743 \nu^{14} - 92070010 \nu^{12} - 86560976238 \nu^{10} - 3830938056 \nu^{8} + \cdots - 6281901056 ) / 4179400896 \) |
\(\beta_{10}\) | \(=\) | \( ( - 12734103 \nu^{18} - 1699264 \nu^{17} - 201180 \nu^{16} - 1948517589 \nu^{14} - 260775248 \nu^{13} - 31510836 \nu^{12} - 81728941602 \nu^{10} + \cdots - 7190240256 ) / 4179400896 \) |
\(\beta_{11}\) | \(=\) | \( ( - 45261895 \nu^{19} - 6924865889 \nu^{15} - 290369158398 \nu^{11} - 3558464452956 \nu^{7} - 850240722328 \nu^{3} ) / 4179400896 \) |
\(\beta_{12}\) | \(=\) | \( ( - 21963356 \nu^{19} + 518651 \nu^{17} - 3360431092 \nu^{15} + 79345865 \nu^{13} - 140922857856 \nu^{11} + 3327761602 \nu^{9} + \cdots + 13651993928 \nu ) / 1393133632 \) |
\(\beta_{13}\) | \(=\) | \( ( 21963356 \nu^{19} + 518651 \nu^{17} + 3360431092 \nu^{15} + 79345865 \nu^{13} + 140922857856 \nu^{11} + 3327761602 \nu^{9} + \cdots + 13651993928 \nu ) / 1393133632 \) |
\(\beta_{14}\) | \(=\) | \( ( 24345032 \nu^{19} - 518651 \nu^{17} + 3724856956 \nu^{15} - 79345865 \nu^{13} + 156208228924 \nu^{11} - 3327761602 \nu^{9} + \cdots - 12258860296 \nu ) / 1393133632 \) |
\(\beta_{15}\) | \(=\) | \( ( - 24345032 \nu^{19} - 518651 \nu^{17} - 3724856956 \nu^{15} - 79345865 \nu^{13} - 156208228924 \nu^{11} - 3327761602 \nu^{9} + \cdots - 12258860296 \nu ) / 1393133632 \) |
\(\beta_{16}\) | \(=\) | \( ( - 77025497 \nu^{19} - 2419300 \nu^{17} + 1726694 \nu^{16} - 11784791035 \nu^{15} - 370391624 \nu^{13} + 264708058 \nu^{12} + \cdots + 11065189808 ) / 4179400896 \) |
\(\beta_{17}\) | \(=\) | \( ( - 77025497 \nu^{19} - 22270858 \nu^{18} - 2419300 \nu^{17} - 11784791035 \nu^{15} - 3406995518 \nu^{14} - 370391624 \nu^{13} + \cdots - 42309175792 \nu ) / 4179400896 \) |
\(\beta_{18}\) | \(=\) | \( ( 113542400 \nu^{19} + 12734103 \nu^{18} + 201180 \nu^{16} + 17372793436 \nu^{15} + 1948517589 \nu^{14} + 31510836 \nu^{12} + 728609156748 \nu^{11} + \cdots + 7190240256 ) / 4179400896 \) |
\(\beta_{19}\) | \(=\) | \( ( 77025497 \nu^{19} + 11784791035 \nu^{15} + 494177340558 \nu^{11} + 6057012474372 \nu^{7} + 1459603333880 \nu^{3} ) / 2089700448 \) |
\(\nu\) | \(=\) | \( ( \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( - \beta_{19} - 2 \beta_{17} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} + 8 \beta_{4} + \beta_{3} - 2 \beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( 2 \beta_{19} - 2 \beta_{18} - 7 \beta_{15} + 7 \beta_{14} - 7 \beta_{13} + 7 \beta_{12} + 4 \beta_{11} - \beta_{10} - \beta_{8} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( -9\beta_{19} - 18\beta_{16} - 3\beta_{9} - 3\beta_{7} - 13\beta_{6} - 13\beta_{5} + 9\beta_{3} - 16\beta _1 - 60 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( - 51 \beta_{15} - 51 \beta_{14} - 59 \beta_{13} - 59 \beta_{12} - 13 \beta_{10} + 13 \beta_{8} + 28 \beta_{3} + 72 \beta_{2} ) / 2 \) |
\(\nu^{6}\) | \(=\) | \( ( 79 \beta_{19} + 158 \beta_{17} - 136 \beta_{10} - 143 \beta_{9} - 136 \beta_{8} + 143 \beta_{7} + 11 \beta_{6} - 11 \beta_{5} - 496 \beta_{4} - 79 \beta_{3} + 136 \beta_1 ) / 2 \) |
\(\nu^{7}\) | \(=\) | \( ( - 312 \beta_{19} + 294 \beta_{18} + 531 \beta_{15} - 531 \beta_{14} + 395 \beta_{13} - 395 \beta_{12} - 888 \beta_{11} + 147 \beta_{10} + 147 \beta_{8} ) / 2 \) |
\(\nu^{8}\) | \(=\) | \( ( 707 \beta_{19} + 1414 \beta_{16} - 343 \beta_{9} - 343 \beta_{7} + 1463 \beta_{6} + 1463 \beta_{5} - 707 \beta_{3} + 1220 \beta _1 + 4328 ) / 2 \) |
\(\nu^{9}\) | \(=\) | \( ( 3235 \beta_{15} + 3235 \beta_{14} + 4947 \beta_{13} + 4947 \beta_{12} + 1563 \beta_{10} - 1563 \beta_{8} - 3220 \beta_{3} - 9664 \beta_{2} ) / 2 \) |
\(\nu^{10}\) | \(=\) | \( ( - 6455 \beta_{19} - 12910 \beta_{17} + 11308 \beta_{10} + 14507 \beta_{9} + 11308 \beta_{8} - 14507 \beta_{7} - 4723 \beta_{6} + 4723 \beta_{5} + 39168 \beta_{4} + 6455 \beta_{3} - 11308 \beta_1 ) / 2 \) |
\(\nu^{11}\) | \(=\) | \( ( 32140 \beta_{19} - 32062 \beta_{18} - 46923 \beta_{15} + 46923 \beta_{14} - 27771 \beta_{13} + 27771 \beta_{12} + 99536 \beta_{11} - 16031 \beta_{10} - 16031 \beta_{8} ) / 2 \) |
\(\nu^{12}\) | \(=\) | \( ( - 59911 \beta_{19} - 119822 \beta_{16} + 54059 \beta_{9} + 54059 \beta_{7} - 141819 \beta_{6} - 141819 \beta_{5} + 59911 \beta_{3} - 106756 \beta _1 - 363056 ) / 2 \) |
\(\nu^{13}\) | \(=\) | \( ( - 247291 \beta_{15} - 247291 \beta_{14} - 449099 \beta_{13} - 449099 \beta_{12} - 160815 \beta_{10} + 160815 \beta_{8} + 315700 \beta_{3} + 997024 \beta_{2} ) / 2 \) |
\(\nu^{14}\) | \(=\) | \( ( 562991 \beta_{19} + 1125982 \beta_{17} - 1018020 \beta_{10} - 1377203 \beta_{9} - 1018020 \beta_{8} + 1377203 \beta_{7} + 572851 \beta_{6} - 572851 \beta_{5} - 3416960 \beta_{4} + \cdots + 1018020 \beta_1 ) / 2 \) |
\(\nu^{15}\) | \(=\) | \( ( - 3076036 \beta_{19} + 3181742 \beta_{18} + 4317371 \beta_{15} - 4317371 \beta_{14} + 2261611 \beta_{13} - 2261611 \beta_{12} - 9836256 \beta_{11} + 1590871 \beta_{10} + \cdots + 1590871 \beta_{8} ) / 2 \) |
\(\nu^{16}\) | \(=\) | \( ( 5337647 \beta_{19} + 10675294 \beta_{16} - 5838259 \beta_{9} - 5838259 \beta_{7} + 13331811 \beta_{6} + 13331811 \beta_{5} - 5337647 \beta_{3} + 9760724 \beta _1 + 32468000 ) / 2 \) |
\(\nu^{17}\) | \(=\) | \( ( 21071035 \beta_{15} + 21071035 \beta_{14} + 41593707 \beta_{13} + 41593707 \beta_{12} + 15598983 \beta_{10} - 15598983 \beta_{8} - 29845364 \beta_{3} - 96201728 \beta_{2} ) / 2 \) |
\(\nu^{18}\) | \(=\) | \( ( - 50916399 \beta_{19} - 101832798 \beta_{17} + 93862708 \beta_{10} + 128862627 \beta_{9} + 93862708 \beta_{8} - 128862627 \beta_{7} - 58227795 \beta_{6} + \cdots - 93862708 \beta_1 ) / 2 \) |
\(\nu^{19}\) | \(=\) | \( ( 288923220 \beta_{19} - 304181006 \beta_{18} - 401128059 \beta_{15} + 401128059 \beta_{14} - 198779851 \beta_{13} + 198779851 \beta_{12} + 936087104 \beta_{11} + \cdots - 152090503 \beta_{8} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
\(n\) | \(191\) | \(211\) | \(457\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
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−0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | −2.23502 | + | 0.0685835i | 1.00000 | −2.16643 | − | 2.16643i | 0.707107 | + | 0.707107i | 1.00000i | 1.53190 | − | 1.62889i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.2 | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | −1.66396 | + | 1.49373i | 1.00000 | −0.170229 | − | 0.170229i | 0.707107 | + | 0.707107i | 1.00000i | 0.120370 | − | 2.23283i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.3 | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | −0.253765 | − | 2.22162i | 1.00000 | −2.47539 | − | 2.47539i | 0.707107 | + | 0.707107i | 1.00000i | 1.75036 | + | 1.39149i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.4 | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | 1.25884 | + | 1.84806i | 1.00000 | 3.10690 | + | 3.10690i | 0.707107 | + | 0.707107i | 1.00000i | −2.19691 | − | 0.416642i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.5 | −0.707107 | + | 0.707107i | −0.707107 | − | 0.707107i | − | 1.00000i | 1.89390 | − | 1.18875i | 1.00000 | 0.705149 | + | 0.705149i | 0.707107 | + | 0.707107i | 1.00000i | −0.498616 | + | 2.17977i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.6 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | −2.23502 | + | 0.0685835i | 1.00000 | −2.16643 | − | 2.16643i | −0.707107 | − | 0.707107i | 1.00000i | −1.53190 | + | 1.62889i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.7 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | −1.66396 | + | 1.49373i | 1.00000 | −0.170229 | − | 0.170229i | −0.707107 | − | 0.707107i | 1.00000i | −0.120370 | + | 2.23283i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.8 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | −0.253765 | − | 2.22162i | 1.00000 | −2.47539 | − | 2.47539i | −0.707107 | − | 0.707107i | 1.00000i | −1.75036 | − | 1.39149i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.9 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | 1.25884 | + | 1.84806i | 1.00000 | 3.10690 | + | 3.10690i | −0.707107 | − | 0.707107i | 1.00000i | 2.19691 | + | 0.416642i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
37.10 | 0.707107 | − | 0.707107i | 0.707107 | + | 0.707107i | − | 1.00000i | 1.89390 | − | 1.18875i | 1.00000 | 0.705149 | + | 0.705149i | −0.707107 | − | 0.707107i | 1.00000i | 0.498616 | − | 2.17977i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.1 | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | −2.23502 | − | 0.0685835i | 1.00000 | −2.16643 | + | 2.16643i | 0.707107 | − | 0.707107i | − | 1.00000i | 1.53190 | + | 1.62889i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.2 | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | −1.66396 | − | 1.49373i | 1.00000 | −0.170229 | + | 0.170229i | 0.707107 | − | 0.707107i | − | 1.00000i | 0.120370 | + | 2.23283i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.3 | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | −0.253765 | + | 2.22162i | 1.00000 | −2.47539 | + | 2.47539i | 0.707107 | − | 0.707107i | − | 1.00000i | 1.75036 | − | 1.39149i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.4 | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | 1.25884 | − | 1.84806i | 1.00000 | 3.10690 | − | 3.10690i | 0.707107 | − | 0.707107i | − | 1.00000i | −2.19691 | + | 0.416642i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.5 | −0.707107 | − | 0.707107i | −0.707107 | + | 0.707107i | 1.00000i | 1.89390 | + | 1.18875i | 1.00000 | 0.705149 | − | 0.705149i | 0.707107 | − | 0.707107i | − | 1.00000i | −0.498616 | − | 2.17977i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.6 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | −2.23502 | − | 0.0685835i | 1.00000 | −2.16643 | + | 2.16643i | −0.707107 | + | 0.707107i | − | 1.00000i | −1.53190 | − | 1.62889i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.7 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | −1.66396 | − | 1.49373i | 1.00000 | −0.170229 | + | 0.170229i | −0.707107 | + | 0.707107i | − | 1.00000i | −0.120370 | − | 2.23283i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.8 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | −0.253765 | + | 2.22162i | 1.00000 | −2.47539 | + | 2.47539i | −0.707107 | + | 0.707107i | − | 1.00000i | −1.75036 | + | 1.39149i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.9 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | 1.25884 | − | 1.84806i | 1.00000 | 3.10690 | − | 3.10690i | −0.707107 | + | 0.707107i | − | 1.00000i | 2.19691 | − | 0.416642i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
493.10 | 0.707107 | + | 0.707107i | 0.707107 | − | 0.707107i | 1.00000i | 1.89390 | + | 1.18875i | 1.00000 | 0.705149 | − | 0.705149i | −0.707107 | + | 0.707107i | − | 1.00000i | 0.498616 | + | 2.17977i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.b | odd | 2 | 1 | inner |
95.g | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 570.2.m.a | ✓ | 20 |
3.b | odd | 2 | 1 | 1710.2.p.d | 20 | ||
5.c | odd | 4 | 1 | inner | 570.2.m.a | ✓ | 20 |
15.e | even | 4 | 1 | 1710.2.p.d | 20 | ||
19.b | odd | 2 | 1 | inner | 570.2.m.a | ✓ | 20 |
57.d | even | 2 | 1 | 1710.2.p.d | 20 | ||
95.g | even | 4 | 1 | inner | 570.2.m.a | ✓ | 20 |
285.j | odd | 4 | 1 | 1710.2.p.d | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
570.2.m.a | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
570.2.m.a | ✓ | 20 | 5.c | odd | 4 | 1 | inner |
570.2.m.a | ✓ | 20 | 19.b | odd | 2 | 1 | inner |
570.2.m.a | ✓ | 20 | 95.g | even | 4 | 1 | inner |
1710.2.p.d | 20 | 3.b | odd | 2 | 1 | ||
1710.2.p.d | 20 | 15.e | even | 4 | 1 | ||
1710.2.p.d | 20 | 57.d | even | 2 | 1 | ||
1710.2.p.d | 20 | 285.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{10} + 2 T_{7}^{9} + 2 T_{7}^{8} + 8 T_{7}^{7} + 320 T_{7}^{6} + 864 T_{7}^{5} + 1120 T_{7}^{4} - 1600 T_{7}^{3} + 1600 T_{7}^{2} + 640 T_{7} + 128 \)
acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 1)^{5} \)
$3$
\( (T^{4} + 1)^{5} \)
$5$
\( (T^{10} + 2 T^{9} + T^{8} + 8 T^{7} + 22 T^{6} + \cdots + 3125)^{2} \)
$7$
\( (T^{10} + 2 T^{9} + 2 T^{8} + 8 T^{7} + \cdots + 128)^{2} \)
$11$
\( (T^{5} + 2 T^{4} - 32 T^{3} - 8 T^{2} + \cdots - 272)^{4} \)
$13$
\( T^{20} + 2656 T^{16} + 1634400 T^{12} + \cdots + 4096 \)
$17$
\( (T^{10} - 2 T^{9} + 2 T^{8} + 16 T^{7} + \cdots + 86528)^{2} \)
$19$
\( T^{20} + 54 T^{18} + \cdots + 6131066257801 \)
$23$
\( (T^{10} - 22 T^{9} + 242 T^{8} - 1464 T^{7} + \cdots + 128)^{2} \)
$29$
\( (T^{10} - 136 T^{8} + 7088 T^{6} + \cdots - 9193472)^{2} \)
$31$
\( (T^{10} + 136 T^{8} + 7088 T^{6} + \cdots + 9193472)^{2} \)
$37$
\( T^{20} + 21472 T^{16} + \cdots + 50\!\cdots\!76 \)
$41$
\( (T^{10} + 216 T^{8} + 13864 T^{6} + \cdots + 700928)^{2} \)
$43$
\( (T^{10} - 26 T^{9} + 338 T^{8} + \cdots + 12500000)^{2} \)
$47$
\( (T^{10} - 2 T^{9} + 2 T^{8} - 968 T^{7} + \cdots + 1520768)^{2} \)
$53$
\( T^{20} + 33488 T^{16} + \cdots + 77\!\cdots\!76 \)
$59$
\( (T^{10} - 144 T^{8} + 6008 T^{6} + \cdots - 2196608)^{2} \)
$61$
\( (T^{5} - 8 T^{4} - 16 T^{3} + 112 T^{2} + \cdots - 320)^{4} \)
$67$
\( T^{20} + 39168 T^{16} + \cdots + 17592186044416 \)
$71$
\( (T^{10} + 232 T^{8} + 19456 T^{6} + \cdots + 22151168)^{2} \)
$73$
\( (T^{10} + 10 T^{9} + 50 T^{8} + \cdots + 22957088)^{2} \)
$79$
\( (T^{10} - 184 T^{8} + 9968 T^{6} + \cdots - 8192)^{2} \)
$83$
\( (T^{10} + 58 T^{9} + 1682 T^{8} + \cdots + 14623232)^{2} \)
$89$
\( (T^{10} - 472 T^{8} + 63560 T^{6} + \cdots - 62720000)^{2} \)
$97$
\( T^{20} + 94128 T^{16} + \cdots + 43\!\cdots\!96 \)
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