Newspace parameters
Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 20.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(8.14757220122\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
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: | \( x^{20} + 94 x^{18} + 5343 x^{16} + 172772 x^{14} + 36131456 x^{12} + 3044563968 x^{10} + 147994443776 x^{8} + 2898633162752 x^{6} + \cdots + 11\!\cdots\!76 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{74}\cdot 3^{4}\cdot 5^{12} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{20} + 94 x^{18} + 5343 x^{16} + 172772 x^{14} + 36131456 x^{12} + 3044563968 x^{10} + 147994443776 x^{8} + 2898633162752 x^{6} + \cdots + 11\!\cdots\!76 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( 4\nu^{2} + 38 \) |
\(\beta_{3}\) | \(=\) | \( 8\nu^{3} + 76\nu \) |
\(\beta_{4}\) | \(=\) | \( ( - 3302597 \nu^{18} - 1391343702 \nu^{16} - 29876009883 \nu^{14} + 1448679820940 \nu^{12} - 186292651580544 \nu^{10} + \cdots - 37\!\cdots\!28 ) / 12\!\cdots\!32 \) |
\(\beta_{5}\) | \(=\) | \( ( 10120793 \nu^{19} - 1538239314 \nu^{17} - 72882621561 \nu^{15} + 834264549700 \nu^{13} + 124572188012160 \nu^{11} + \cdots - 47\!\cdots\!48 \nu ) / 16\!\cdots\!96 \) |
\(\beta_{6}\) | \(=\) | \( ( - 21623225 \nu^{19} + 457010706 \nu^{17} + 11425127385 \nu^{15} - 2821562731204 \nu^{13} - 540171803713152 \nu^{11} + \cdots + 20\!\cdots\!28 \nu ) / 32\!\cdots\!92 \) |
\(\beta_{7}\) | \(=\) | \( ( 11547117 \nu^{19} - 488855936 \nu^{18} + 1694508294 \nu^{17} - 65873566976 \nu^{16} + 36909113715 \nu^{15} - 1089638299264 \nu^{14} + \cdots - 16\!\cdots\!80 ) / 17\!\cdots\!40 \) |
\(\beta_{8}\) | \(=\) | \( ( 607811 \nu^{18} + 30995610 \nu^{16} + 223223901 \nu^{14} + 58863964588 \nu^{12} + 18819124901760 \nu^{10} + 857972427985920 \nu^{8} + \cdots + 60\!\cdots\!92 ) / 39\!\cdots\!76 \) |
\(\beta_{9}\) | \(=\) | \( ( 103924053 \nu^{19} + 6141831296 \nu^{18} + 15250574646 \nu^{17} - 1454773281024 \nu^{16} + 332182023435 \nu^{15} + \cdots - 41\!\cdots\!40 ) / 80\!\cdots\!80 \) |
\(\beta_{10}\) | \(=\) | \( ( - 103924053 \nu^{19} - 9822609536 \nu^{18} - 15250574646 \nu^{17} + 1108780126464 \nu^{16} - 332182023435 \nu^{15} + \cdots + 32\!\cdots\!60 ) / 80\!\cdots\!80 \) |
\(\beta_{11}\) | \(=\) | \( ( 11547117 \nu^{19} + 488855936 \nu^{18} + 1694508294 \nu^{17} + 65873566976 \nu^{16} + 36909113715 \nu^{15} + 1089638299264 \nu^{14} + \cdots + 16\!\cdots\!80 ) / 35\!\cdots\!88 \) |
\(\beta_{12}\) | \(=\) | \( ( 11250533 \nu^{18} + 356503830 \nu^{16} + 6095187195 \nu^{14} + 1380868386292 \nu^{12} + 320541156988032 \nu^{10} + \cdots + 16\!\cdots\!96 ) / 15\!\cdots\!04 \) |
\(\beta_{13}\) | \(=\) | \( ( 32459431 \nu^{19} - 1527461550 \nu^{17} - 42771655815 \nu^{15} + 3554605976252 \nu^{13} + 745081528209792 \nu^{11} + \cdots - 61\!\cdots\!48 \nu ) / 26\!\cdots\!16 \) |
\(\beta_{14}\) | \(=\) | \( ( - 4397900707 \nu^{19} - 16920576256 \nu^{18} - 1027539355098 \nu^{17} - 1855269723648 \nu^{16} - 17641443616509 \nu^{15} + \cdots - 15\!\cdots\!32 ) / 16\!\cdots\!60 \) |
\(\beta_{15}\) | \(=\) | \( ( - 4073259013 \nu^{19} - 4973631424 \nu^{18} - 608485077462 \nu^{17} + 44798068608 \nu^{16} - 10172775434331 \nu^{15} + \cdots - 24\!\cdots\!28 ) / 80\!\cdots\!80 \) |
\(\beta_{16}\) | \(=\) | \( ( 875056263 \nu^{19} + 41392821248 \nu^{18} - 13654586990 \nu^{17} + 570874583040 \nu^{16} - 635936680551 \nu^{15} + \cdots - 42\!\cdots\!28 ) / 17\!\cdots\!40 \) |
\(\beta_{17}\) | \(=\) | \( ( 14557212835 \nu^{19} + 3251847424 \nu^{18} + 1888412158938 \nu^{17} + 1868182445568 \nu^{16} + 18352861149693 \nu^{15} + \cdots + 39\!\cdots\!04 ) / 16\!\cdots\!60 \) |
\(\beta_{18}\) | \(=\) | \( ( - 8062974223 \nu^{19} - 9608144128 \nu^{18} - 554664723330 \nu^{17} + 347685480960 \nu^{16} - 7477151383569 \nu^{15} + \cdots + 17\!\cdots\!68 ) / 80\!\cdots\!80 \) |
\(\beta_{19}\) | \(=\) | \( ( - 12462645249 \nu^{19} + 5208440704 \nu^{18} - 771214807134 \nu^{17} - 940547583744 \nu^{16} - 3157211583711 \nu^{15} + \cdots - 31\!\cdots\!88 ) / 80\!\cdots\!80 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{2} - 38 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{3} - 38\beta_1 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( - \beta_{19} + \beta_{18} - \beta_{17} - \beta_{16} - 2 \beta_{15} + \beta_{14} + \beta_{13} + 3 \beta_{12} - \beta_{11} + 3 \beta_{9} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 40 \beta_{2} + \beta _1 - 2943 ) / 16 \) |
\(\nu^{5}\) | \(=\) | \( ( - 7 \beta_{19} - \beta_{18} - 10 \beta_{17} - 2 \beta_{15} - 6 \beta_{14} - 27 \beta_{13} - 28 \beta_{11} - 132 \beta_{7} + 70 \beta_{6} + 7 \beta_{5} - 6 \beta_{3} - 729 \beta_1 ) / 8 \) |
\(\nu^{6}\) | \(=\) | \( ( - 9 \beta_{19} + 31 \beta_{18} - 9 \beta_{17} + 13 \beta_{16} - 40 \beta_{15} + 9 \beta_{14} + 9 \beta_{13} + 9 \beta_{12} - 51 \beta_{11} - 208 \beta_{10} - 379 \beta_{9} - 21 \beta_{8} + 619 \beta_{7} + 9 \beta_{6} - 9 \beta_{5} + 111 \beta_{4} + \cdots + 126275 ) / 8 \) |
\(\nu^{7}\) | \(=\) | \( ( - 274 \beta_{19} + 406 \beta_{18} + 108 \beta_{17} + 344 \beta_{15} - 76 \beta_{14} + 2082 \beta_{13} + 512 \beta_{11} - 302 \beta_{10} + 302 \beta_{9} + 1924 \beta_{7} + 24356 \beta_{6} - 13828 \beta_{5} + \cdots + 75386 \beta_1 ) / 8 \) |
\(\nu^{8}\) | \(=\) | \( ( 139 \beta_{19} - 2451 \beta_{18} + 139 \beta_{17} - 2173 \beta_{16} + 2590 \beta_{15} - 139 \beta_{14} - 139 \beta_{13} - 8073 \beta_{12} - 1869 \beta_{11} - 5952 \beta_{10} + 9703 \beta_{9} + 56461 \beta_{8} + \cdots - 213160723 ) / 16 \) |
\(\nu^{9}\) | \(=\) | \( ( 3349 \beta_{19} + 9643 \beta_{18} + 16430 \beta_{17} + 11374 \beta_{15} + 7170 \beta_{14} - 183799 \beta_{13} - 7196 \beta_{11} - 6732 \beta_{10} + 6732 \beta_{9} - 62460 \beta_{7} - 4571346 \beta_{6} + \cdots - 54768093 \beta_1 ) / 8 \) |
\(\nu^{10}\) | \(=\) | \( ( - 10014 \beta_{19} + 63685 \beta_{18} - 10014 \beta_{17} + 43657 \beta_{16} - 73699 \beta_{15} + 10014 \beta_{14} + 10014 \beta_{13} + 19965 \beta_{12} + 180161 \beta_{11} + 90488 \beta_{10} + \cdots + 75246103 ) / 4 \) |
\(\nu^{11}\) | \(=\) | \( ( 1140020 \beta_{19} + 97392 \beta_{18} + 497136 \beta_{17} - 1893956 \beta_{15} - 118064 \beta_{14} + 10633504 \beta_{13} - 4176584 \beta_{11} + 580346 \beta_{10} - 580346 \beta_{9} + \cdots + 97932580 \beta_1 ) / 8 \) |
\(\nu^{12}\) | \(=\) | \( ( 14017923 \beta_{19} - 21976067 \beta_{18} + 14017923 \beta_{17} + 6059779 \beta_{16} + 35993990 \beta_{15} - 14017923 \beta_{14} - 14017923 \beta_{13} - 36358857 \beta_{12} + \cdots + 213177952749 ) / 16 \) |
\(\nu^{13}\) | \(=\) | \( ( 66676517 \beta_{19} + 105062355 \beta_{18} + 125825694 \beta_{17} + 44106694 \beta_{15} + 43848850 \beta_{14} + 109690273 \beta_{13} + 798523348 \beta_{11} + \cdots + 50771922467 \beta_1 ) / 8 \) |
\(\nu^{14}\) | \(=\) | \( ( 344100561 \beta_{19} - 769664947 \beta_{18} + 344100561 \beta_{17} - 81463825 \beta_{16} + 1113765508 \beta_{15} - 344100561 \beta_{14} - 344100561 \beta_{13} + \cdots + 2477743922529 ) / 8 \) |
\(\nu^{15}\) | \(=\) | \( ( 9684699242 \beta_{19} - 12949656150 \beta_{18} - 2590174476 \beta_{17} - 9848016752 \beta_{15} + 3398721388 \beta_{14} - 28392840962 \beta_{13} + \cdots + 985218925214 \beta_1 ) / 8 \) |
\(\nu^{16}\) | \(=\) | \( ( - 3613266745 \beta_{19} + 38623462337 \beta_{18} - 3613266745 \beta_{17} + 31396928847 \beta_{16} - 42236729082 \beta_{15} + 3613266745 \beta_{14} + \cdots + 14\!\cdots\!21 ) / 16 \) |
\(\nu^{17}\) | \(=\) | \( ( - 264166320887 \beta_{19} - 247665438841 \beta_{18} - 506965570042 \beta_{17} - 145026223306 \beta_{15} - 222830891798 \beta_{14} + \cdots + 402665339208175 \beta_1 ) / 8 \) |
\(\nu^{18}\) | \(=\) | \( ( 206826061833 \beta_{19} - 1356326570262 \beta_{18} + 206826061833 \beta_{17} - 942674446596 \beta_{16} + 1563152632095 \beta_{15} - 206826061833 \beta_{14} + \cdots + 21\!\cdots\!92 ) / 4 \) |
\(\nu^{19}\) | \(=\) | \( ( - 28979466545816 \beta_{19} - 15746025254036 \beta_{18} - 15226592799096 \beta_{17} + 47319711735396 \beta_{15} + 6286559214840 \beta_{14} + \cdots + 20\!\cdots\!24 \beta_1 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
\(n\) | \(11\) | \(17\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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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19.1 |
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−14.9261 | − | 5.76290i | −76.0331 | 189.578 | + | 172.035i | 291.455 | + | 552.882i | 1134.88 | + | 438.171i | −269.408 | −1838.24 | − | 3660.34i | −779.974 | −1164.08 | − | 9932.01i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.2 | −14.9261 | + | 5.76290i | −76.0331 | 189.578 | − | 172.035i | 291.455 | − | 552.882i | 1134.88 | − | 438.171i | −269.408 | −1838.24 | + | 3660.34i | −779.974 | −1164.08 | + | 9932.01i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.3 | −14.3016 | − | 7.17394i | 42.6663 | 153.069 | + | 205.197i | −560.298 | − | 276.932i | −610.194 | − | 306.085i | 869.685 | −717.054 | − | 4032.75i | −4740.59 | 6026.43 | + | 7980.11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.4 | −14.3016 | + | 7.17394i | 42.6663 | 153.069 | − | 205.197i | −560.298 | + | 276.932i | −610.194 | + | 306.085i | 869.685 | −717.054 | + | 4032.75i | −4740.59 | 6026.43 | − | 7980.11i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.5 | −7.23243 | − | 14.2721i | 44.3270 | −151.384 | + | 206.443i | 530.961 | − | 329.705i | −320.592 | − | 632.638i | 2826.75 | 4041.25 | + | 667.476i | −4596.11 | −8545.71 | − | 5193.35i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.6 | −7.23243 | + | 14.2721i | 44.3270 | −151.384 | − | 206.443i | 530.961 | + | 329.705i | −320.592 | + | 632.638i | 2826.75 | 4041.25 | − | 667.476i | −4596.11 | −8545.71 | + | 5193.35i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.7 | −7.06329 | − | 14.3565i | −134.970 | −156.220 | + | 202.809i | −416.235 | − | 466.234i | 953.329 | + | 1937.69i | −1863.96 | 4015.06 | + | 810.276i | 11655.8 | −3753.51 | + | 9268.83i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.8 | −7.06329 | + | 14.3565i | −134.970 | −156.220 | − | 202.809i | −416.235 | + | 466.234i | 953.329 | − | 1937.69i | −1863.96 | 4015.06 | − | 810.276i | 11655.8 | −3753.51 | − | 9268.83i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.9 | −4.05935 | − | 15.4765i | 75.2390 | −223.043 | + | 125.649i | −200.884 | + | 591.837i | −305.422 | − | 1164.44i | −4411.35 | 2850.02 | + | 2941.87i | −900.091 | 9975.01 | + | 706.503i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.10 | −4.05935 | + | 15.4765i | 75.2390 | −223.043 | − | 125.649i | −200.884 | − | 591.837i | −305.422 | + | 1164.44i | −4411.35 | 2850.02 | − | 2941.87i | −900.091 | 9975.01 | − | 706.503i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.11 | 4.05935 | − | 15.4765i | −75.2390 | −223.043 | − | 125.649i | −200.884 | + | 591.837i | −305.422 | + | 1164.44i | 4411.35 | −2850.02 | + | 2941.87i | −900.091 | 8344.09 | + | 5511.45i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.12 | 4.05935 | + | 15.4765i | −75.2390 | −223.043 | + | 125.649i | −200.884 | − | 591.837i | −305.422 | − | 1164.44i | 4411.35 | −2850.02 | − | 2941.87i | −900.091 | 8344.09 | − | 5511.45i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.13 | 7.06329 | − | 14.3565i | 134.970 | −156.220 | − | 202.809i | −416.235 | − | 466.234i | 953.329 | − | 1937.69i | 1863.96 | −4015.06 | + | 810.276i | 11655.8 | −9633.48 | + | 2682.54i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.14 | 7.06329 | + | 14.3565i | 134.970 | −156.220 | + | 202.809i | −416.235 | + | 466.234i | 953.329 | + | 1937.69i | 1863.96 | −4015.06 | − | 810.276i | 11655.8 | −9633.48 | − | 2682.54i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.15 | 7.23243 | − | 14.2721i | −44.3270 | −151.384 | − | 206.443i | 530.961 | − | 329.705i | −320.592 | + | 632.638i | −2826.75 | −4041.25 | + | 667.476i | −4596.11 | −865.429 | − | 9962.48i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.16 | 7.23243 | + | 14.2721i | −44.3270 | −151.384 | + | 206.443i | 530.961 | + | 329.705i | −320.592 | − | 632.638i | −2826.75 | −4041.25 | − | 667.476i | −4596.11 | −865.429 | + | 9962.48i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.17 | 14.3016 | − | 7.17394i | −42.6663 | 153.069 | − | 205.197i | −560.298 | − | 276.932i | −610.194 | + | 306.085i | −869.685 | 717.054 | − | 4032.75i | −4740.59 | −9999.83 | + | 58.9826i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.18 | 14.3016 | + | 7.17394i | −42.6663 | 153.069 | + | 205.197i | −560.298 | + | 276.932i | −610.194 | − | 306.085i | −869.685 | 717.054 | + | 4032.75i | −4740.59 | −9999.83 | − | 58.9826i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.19 | 14.9261 | − | 5.76290i | 76.0331 | 189.578 | − | 172.035i | 291.455 | + | 552.882i | 1134.88 | − | 438.171i | 269.408 | 1838.24 | − | 3660.34i | −779.974 | 7536.50 | + | 6572.76i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
19.20 | 14.9261 | + | 5.76290i | 76.0331 | 189.578 | + | 172.035i | 291.455 | − | 552.882i | 1134.88 | + | 438.171i | 269.408 | 1838.24 | + | 3660.34i | −779.974 | 7536.50 | − | 6572.76i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 20.9.d.c | ✓ | 20 |
4.b | odd | 2 | 1 | inner | 20.9.d.c | ✓ | 20 |
5.b | even | 2 | 1 | inner | 20.9.d.c | ✓ | 20 |
5.c | odd | 4 | 2 | 100.9.b.g | 20 | ||
8.b | even | 2 | 1 | 320.9.h.g | 20 | ||
8.d | odd | 2 | 1 | 320.9.h.g | 20 | ||
20.d | odd | 2 | 1 | inner | 20.9.d.c | ✓ | 20 |
20.e | even | 4 | 2 | 100.9.b.g | 20 | ||
40.e | odd | 2 | 1 | 320.9.h.g | 20 | ||
40.f | even | 2 | 1 | 320.9.h.g | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.9.d.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
20.9.d.c | ✓ | 20 | 4.b | odd | 2 | 1 | inner |
20.9.d.c | ✓ | 20 | 5.b | even | 2 | 1 | inner |
20.9.d.c | ✓ | 20 | 20.d | odd | 2 | 1 | inner |
100.9.b.g | 20 | 5.c | odd | 4 | 2 | ||
100.9.b.g | 20 | 20.e | even | 4 | 2 | ||
320.9.h.g | 20 | 8.b | even | 2 | 1 | ||
320.9.h.g | 20 | 8.d | odd | 2 | 1 | ||
320.9.h.g | 20 | 40.e | odd | 2 | 1 | ||
320.9.h.g | 20 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{10} - 33444 T_{3}^{8} + 357004800 T_{3}^{6} - 1615110935040 T_{3}^{4} + \cdots - 21\!\cdots\!00 \)
acting on \(S_{9}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{20} + 376 T^{18} + \cdots + 12\!\cdots\!76 \)
$3$
\( (T^{10} - 33444 T^{8} + \cdots - 21\!\cdots\!00)^{2} \)
$5$
\( (T^{10} + 710 T^{9} + \cdots + 90\!\cdots\!25)^{2} \)
$7$
\( (T^{10} - 31753764 T^{8} + \cdots - 29\!\cdots\!00)^{2} \)
$11$
\( (T^{10} + 999897600 T^{8} + \cdots + 15\!\cdots\!00)^{2} \)
$13$
\( (T^{10} + 4307527296 T^{8} + \cdots + 15\!\cdots\!00)^{2} \)
$17$
\( (T^{10} + 39851142656 T^{8} + \cdots + 57\!\cdots\!00)^{2} \)
$19$
\( (T^{10} + 92184752640 T^{8} + \cdots + 54\!\cdots\!00)^{2} \)
$23$
\( (T^{10} - 510426064164 T^{8} + \cdots - 72\!\cdots\!00)^{2} \)
$29$
\( (T^{5} - 165034 T^{4} + \cdots - 30\!\cdots\!52)^{4} \)
$31$
\( (T^{10} + 5638508605440 T^{8} + \cdots + 51\!\cdots\!00)^{2} \)
$37$
\( (T^{10} + 11893689000576 T^{8} + \cdots + 30\!\cdots\!00)^{2} \)
$41$
\( (T^{5} - 1767130 T^{4} + \cdots + 51\!\cdots\!28)^{4} \)
$43$
\( (T^{10} - 38122870151844 T^{8} + \cdots - 13\!\cdots\!00)^{2} \)
$47$
\( (T^{10} - 29539975189284 T^{8} + \cdots - 25\!\cdots\!00)^{2} \)
$53$
\( (T^{10} + 171214156558976 T^{8} + \cdots + 74\!\cdots\!00)^{2} \)
$59$
\( (T^{10} + \cdots + 53\!\cdots\!00)^{2} \)
$61$
\( (T^{5} + 4415110 T^{4} + \cdots + 50\!\cdots\!68)^{4} \)
$67$
\( (T^{10} + \cdots - 19\!\cdots\!00)^{2} \)
$71$
\( (T^{10} + \cdots + 92\!\cdots\!00)^{2} \)
$73$
\( (T^{10} + \cdots + 86\!\cdots\!00)^{2} \)
$79$
\( (T^{10} + \cdots + 74\!\cdots\!00)^{2} \)
$83$
\( (T^{10} + \cdots - 78\!\cdots\!00)^{2} \)
$89$
\( (T^{5} - 26251594 T^{4} + \cdots + 38\!\cdots\!28)^{4} \)
$97$
\( (T^{10} + \cdots + 34\!\cdots\!00)^{2} \)
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