Properties

Label 1045.2.b.c
Level $1045$
Weight $2$
Character orbit 1045.b
Analytic conductor $8.344$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.34436701122\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \( x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + 2753 x^{4} + 276 x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{11} q^{5} + (\beta_{12} + \beta_{2} - 1) q^{6} + \beta_{3} q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + (\beta_{8} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{7} q^{3} + (\beta_{2} - 1) q^{4} - \beta_{11} q^{5} + (\beta_{12} + \beta_{2} - 1) q^{6} + \beta_{3} q^{7} + ( - \beta_{7} + \beta_{6}) q^{8} + (\beta_{8} - 1) q^{9} + ( - \beta_{17} - \beta_{14} - \beta_{7} + \beta_{3} + \beta_1) q^{10} + q^{11} + (\beta_{4} - \beta_1) q^{12} + (\beta_{19} - \beta_{7} + \beta_{3}) q^{13} + (\beta_{18} - \beta_{15} + \beta_{14} - \beta_{8} - \beta_{2} + 1) q^{14} + ( - \beta_{18} + \beta_{15} + \beta_{11} + \beta_{8} + \beta_{7} + \beta_{4}) q^{15} + (\beta_{18} + \beta_{16} - \beta_{12} + \beta_{5} - \beta_{2}) q^{16} + ( - \beta_{13} - \beta_{11} + \beta_{10} - \beta_{4} - \beta_1) q^{17} + ( - \beta_{19} - \beta_{17} - \beta_{6} + \beta_{4}) q^{18} + q^{19} + ( - \beta_{16} - \beta_{15} + \beta_{14} - \beta_{8} + \beta_{3}) q^{20} + (\beta_{18} - \beta_{15} + \beta_{14} - \beta_{12} - \beta_{2} - 2) q^{21} + \beta_1 q^{22} + ( - \beta_{15} - \beta_{14} - \beta_{13} - \beta_{11} + \beta_{10} - 2 \beta_{7} - \beta_{6} - \beta_{4} + \cdots + 2 \beta_1) q^{23}+ \cdots + (\beta_{8} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 12 q^{4} - 8 q^{6} - 10 q^{9} - 6 q^{10} + 20 q^{11} + 24 q^{14} - 6 q^{15} - 4 q^{16} + 20 q^{19} - 6 q^{20} - 30 q^{21} + 38 q^{24} + 2 q^{25} + 8 q^{26} + 50 q^{29} - 20 q^{30} - 50 q^{31} + 28 q^{34} + 6 q^{35} - 12 q^{36} + 48 q^{39} + 12 q^{40} - 34 q^{41} - 12 q^{44} - 18 q^{45} - 36 q^{46} - 6 q^{49} + 26 q^{50} - 40 q^{51} - 6 q^{54} - 40 q^{56} + 30 q^{59} - 30 q^{60} - 14 q^{61} + 36 q^{64} + 30 q^{65} - 8 q^{66} - 12 q^{69} - 54 q^{70} - 40 q^{71} + 50 q^{74} - 8 q^{75} - 12 q^{76} + 106 q^{79} + 8 q^{80} - 30 q^{84} - 22 q^{85} + 56 q^{86} + 36 q^{89} - 64 q^{90} - 56 q^{91} + 28 q^{94} + 66 q^{96} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 26 x^{18} + 281 x^{16} + 1640 x^{14} + 5623 x^{12} + 11551 x^{10} + 13894 x^{8} + 9095 x^{6} + 2753 x^{4} + 276 x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61 \nu^{19} + 1299 \nu^{17} + 10262 \nu^{15} + 33228 \nu^{13} + 5459 \nu^{11} - 240718 \nu^{9} - 592650 \nu^{7} - 522039 \nu^{5} - 131958 \nu^{3} + 1604 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 57 \nu^{19} + 1205 \nu^{17} + 9104 \nu^{15} + 23376 \nu^{13} - 49969 \nu^{11} - 428112 \nu^{9} - 946122 \nu^{7} - 866947 \nu^{5} - 288912 \nu^{3} - 27024 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 127 \nu^{18} - 3119 \nu^{16} - 31252 \nu^{14} - 164582 \nu^{12} - 489083 \nu^{10} - 819526 \nu^{8} - 735502 \nu^{6} - 319949 \nu^{4} - 59110 \nu^{2} - 1602 ) / 538 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 424 \nu^{19} + 10233 \nu^{17} + 99883 \nu^{15} + 504698 \nu^{13} + 1398132 \nu^{11} + 2051391 \nu^{9} + 1360162 \nu^{7} + 189834 \nu^{5} - 78267 \nu^{3} - 8898 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 424 \nu^{19} + 10233 \nu^{17} + 99883 \nu^{15} + 504698 \nu^{13} + 1398132 \nu^{11} + 2051391 \nu^{9} + 1360162 \nu^{7} + 189834 \nu^{5} - 79343 \nu^{3} - 13202 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 119 \nu^{18} - 2931 \nu^{16} - 29474 \nu^{14} - 155907 \nu^{12} - 465652 \nu^{10} - 783140 \nu^{8} - 697023 \nu^{6} - 282189 \nu^{4} - 36260 \nu^{2} - 29 ) / 269 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 284 \nu^{18} - 6943 \nu^{16} - 69037 \nu^{14} - 358938 \nu^{12} - 1043638 \nu^{10} - 1682291 \nu^{8} - 1400840 \nu^{6} - 516802 \nu^{4} - 68197 \nu^{2} - 1100 ) / 538 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 169 \nu^{19} + 4644 \nu^{17} + 53633 \nu^{15} + 338506 \nu^{13} + 1269599 \nu^{11} + 2874857 \nu^{9} + 3799744 \nu^{7} + 2655663 \nu^{5} + 772083 \nu^{3} + 44494 \nu ) / 1076 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 563 \nu^{19} - 1152 \nu^{18} - 13634 \nu^{17} - 28148 \nu^{16} - 133533 \nu^{15} - 279704 \nu^{14} - 676778 \nu^{13} - 1452564 \nu^{12} - 1878649 \nu^{11} - 4211192 \nu^{10} + \cdots + 2676 ) / 2152 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 791 \nu^{18} - 19261 \nu^{16} - 190662 \nu^{14} - 986020 \nu^{12} - 2846233 \nu^{10} - 4530894 \nu^{8} - 3666446 \nu^{6} - 1246615 \nu^{4} - 131302 \nu^{2} + \cdots - 3848 ) / 1076 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 563 \nu^{19} + 1152 \nu^{18} - 13634 \nu^{17} + 28148 \nu^{16} - 133533 \nu^{15} + 279704 \nu^{14} - 676778 \nu^{13} + 1452564 \nu^{12} - 1878649 \nu^{11} + 4211192 \nu^{10} + \cdots - 2676 ) / 2152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 269 \nu^{19} + 502 \nu^{18} + 6994 \nu^{17} + 12335 \nu^{16} + 75589 \nu^{15} + 123271 \nu^{14} + 441160 \nu^{13} + 643550 \nu^{12} + 1512587 \nu^{11} + 1873190 \nu^{10} + \cdots + 1126 ) / 1076 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 269 \nu^{19} - 502 \nu^{18} + 6994 \nu^{17} - 12335 \nu^{16} + 75589 \nu^{15} - 123271 \nu^{14} + 441160 \nu^{13} - 643550 \nu^{12} + 1512587 \nu^{11} - 1873190 \nu^{10} + \cdots - 1126 ) / 1076 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 615 \nu^{18} + 15125 \nu^{16} + 151546 \nu^{14} + 795708 \nu^{12} + 2343125 \nu^{10} + 3838002 \nu^{8} + 3254612 \nu^{6} + 1214287 \nu^{4} + 153152 \nu^{2} + 3674 ) / 538 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 604 \nu^{19} - 15001 \nu^{17} - 152531 \nu^{15} - 819456 \nu^{13} - 2505428 \nu^{11} - 4382125 \nu^{9} - 4207528 \nu^{7} - 2033120 \nu^{5} - 434513 \nu^{3} + \cdots - 28512 \nu ) / 538 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 1767 \nu^{18} - 43273 \nu^{16} - 431250 \nu^{14} - 2248272 \nu^{12} - 6554317 \nu^{10} - 10567846 \nu^{8} - 8704666 \nu^{6} - 3034215 \nu^{4} - 311854 \nu^{2} + \cdots - 460 ) / 1076 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 1317 \nu^{19} + 32698 \nu^{17} + 332179 \nu^{15} + 1781218 \nu^{13} + 5425363 \nu^{11} + 9417307 \nu^{9} + 8896864 \nu^{7} + 4138215 \nu^{5} + 803421 \nu^{3} + \cdots + 40090 \nu ) / 1076 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{18} + \beta_{16} - \beta_{12} + \beta_{5} - 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{19} - 2\beta_{17} + \beta_{13} + \beta_{11} - \beta_{10} + 7\beta_{7} - 8\beta_{6} + \beta_{4} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -10\beta_{18} - 9\beta_{16} - 2\beta_{13} + 8\beta_{12} + 2\beta_{11} - 8\beta_{5} + 43\beta_{2} - 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{19} + 24 \beta_{17} + \beta_{15} + \beta_{14} - 11 \beta_{13} - 11 \beta_{11} + 10 \beta_{10} - 41 \beta_{7} + 54 \beta_{6} - 12 \beta_{4} - \beta_{3} - 111 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 75 \beta_{18} + 66 \beta_{16} + 2 \beta_{15} - 2 \beta_{14} + 24 \beta_{13} - 51 \beta_{12} - 24 \beta_{11} + 3 \beta_{8} + 52 \beta_{5} - 257 \beta_{2} + 389 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 101 \beta_{19} - 206 \beta_{17} - 15 \beta_{15} - 15 \beta_{14} + 88 \beta_{13} + 88 \beta_{11} - 75 \beta_{10} + 226 \beta_{7} - 350 \beta_{6} + 102 \beta_{4} + 20 \beta_{3} + 651 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 509 \beta_{18} - 455 \beta_{16} - 33 \beta_{15} + 33 \beta_{14} - 206 \beta_{13} + 303 \beta_{12} + 206 \beta_{11} + \beta_{9} - 47 \beta_{8} - 324 \beta_{5} + 1533 \beta_{2} - 2175 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 741 \beta_{19} + 1554 \beta_{17} + 151 \beta_{15} + 151 \beta_{14} - 631 \beta_{13} - 631 \beta_{11} + 510 \beta_{10} - 1211 \beta_{7} + 2241 \beta_{6} - 764 \beta_{4} - 238 \beta_{3} - 3940 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 3303 \beta_{18} + 3054 \beta_{16} + 359 \beta_{15} - 359 \beta_{14} + 1554 \beta_{13} - 1747 \beta_{12} - 1554 \beta_{11} - 23 \beta_{9} + 492 \beta_{8} + 2010 \beta_{5} - 9190 \beta_{2} + 12486 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5088 \beta_{19} - 11001 \beta_{17} - 1282 \beta_{15} - 1282 \beta_{14} + 4316 \beta_{13} + 4316 \beta_{11} - 3326 \beta_{10} + 6386 \beta_{7} - 14292 \beta_{6} + 5397 \beta_{4} + 2261 \beta_{3} + \cdots + 24301 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 20961 \beta_{18} - 20205 \beta_{16} - 3251 \beta_{15} + 3251 \beta_{14} - 11001 \beta_{13} + 9919 \beta_{12} + 11001 \beta_{11} + 309 \beta_{9} - 4332 \beta_{8} - 12530 \beta_{5} + 55455 \beta_{2} + \cdots - 73183 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 33724 \beta_{19} + 75175 \beta_{17} + 9936 \beta_{15} + 9936 \beta_{14} - 28837 \beta_{13} - 28837 \beta_{11} + 21270 \beta_{10} - 33281 \beta_{7} + 91034 \beta_{6} - 36953 \beta_{4} - 19008 \beta_{3} + \cdots - 151663 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 131518 \beta_{18} + 132485 \beta_{16} + 26575 \beta_{15} - 26575 \beta_{14} + 75175 \beta_{13} - 55820 \beta_{12} - 75175 \beta_{11} - 3229 \beta_{9} + 34691 \beta_{8} + 78580 \beta_{5} - 336819 \beta_{2} + \cdots + 436023 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 219147 \beta_{19} - 502949 \beta_{17} - 72913 \beta_{15} - 72913 \beta_{14} + 190159 \beta_{13} + 190159 \beta_{11} - 134747 \beta_{10} + 171462 \beta_{7} - 579693 \beta_{6} + \cdots + 953706 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 820724 \beta_{18} - 863495 \beta_{16} - 203712 \beta_{15} + 203712 \beta_{14} - 502949 \beta_{13} + 312428 \beta_{12} + 502949 \beta_{11} + 29218 \beta_{9} - 261918 \beta_{8} + \cdots - 2631488 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 1408073 \beta_{19} + 3320237 \beta_{17} + 516502 \beta_{15} + 516502 \beta_{14} - 1243813 \beta_{13} - 1243813 \beta_{11} + 849942 \beta_{10} - 871422 \beta_{7} + 3691782 \beta_{6} + \cdots - 6026704 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1045\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(761\) \(837\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
419.1
2.52711i
2.33380i
2.04146i
2.00304i
1.51095i
1.23709i
1.22468i
0.712399i
0.386431i
0.131596i
0.131596i
0.386431i
0.712399i
1.22468i
1.23709i
1.51095i
2.00304i
2.04146i
2.33380i
2.52711i
2.52711i 0.473551i −4.38628 2.21995 0.267979i −1.19672 1.17543i 6.03039i 2.77575 −0.677212 5.61006i
419.2 2.33380i 1.97214i −3.44661 −1.75107 1.39059i −4.60257 4.79282i 3.37609i −0.889329 −3.24536 + 4.08665i
419.3 2.04146i 1.84076i −2.16755 −2.00596 0.987985i 3.75784 2.14151i 0.342047i −0.388411 −2.01693 + 4.09509i
419.4 2.00304i 2.62978i −2.01218 0.741511 + 2.10954i −5.26757 4.24477i 0.0243959i −3.91575 4.22550 1.48528i
419.5 1.51095i 0.900785i −0.282982 0.672383 2.13258i 1.36104 0.603482i 2.59434i 2.18859 −3.22223 1.01594i
419.6 1.23709i 2.07277i 0.469598 2.13362 0.669071i −2.56421 0.314194i 3.05513i −1.29637 −0.827703 2.63949i
419.7 1.22468i 1.51450i 0.500160 −1.12399 + 1.93304i 1.85478 0.966830i 3.06189i 0.706295 2.36736 + 1.37652i
419.8 0.712399i 3.32227i 1.49249 1.31011 + 1.81207i 2.36678 2.70450i 2.48804i −8.03747 1.29092 0.933320i
419.9 0.386431i 0.261531i 1.85067 0.0276061 2.23590i 0.101064 4.13791i 1.48802i 2.93160 −0.864019 0.0106678i
419.10 0.131596i 1.44045i 1.98268 −2.22416 0.230433i 0.189557 0.456875i 0.524103i 0.925103 −0.0303239 + 0.292690i
419.11 0.131596i 1.44045i 1.98268 −2.22416 + 0.230433i 0.189557 0.456875i 0.524103i 0.925103 −0.0303239 0.292690i
419.12 0.386431i 0.261531i 1.85067 0.0276061 + 2.23590i 0.101064 4.13791i 1.48802i 2.93160 −0.864019 + 0.0106678i
419.13 0.712399i 3.32227i 1.49249 1.31011 1.81207i 2.36678 2.70450i 2.48804i −8.03747 1.29092 + 0.933320i
419.14 1.22468i 1.51450i 0.500160 −1.12399 1.93304i 1.85478 0.966830i 3.06189i 0.706295 2.36736 1.37652i
419.15 1.23709i 2.07277i 0.469598 2.13362 + 0.669071i −2.56421 0.314194i 3.05513i −1.29637 −0.827703 + 2.63949i
419.16 1.51095i 0.900785i −0.282982 0.672383 + 2.13258i 1.36104 0.603482i 2.59434i 2.18859 −3.22223 + 1.01594i
419.17 2.00304i 2.62978i −2.01218 0.741511 2.10954i −5.26757 4.24477i 0.0243959i −3.91575 4.22550 + 1.48528i
419.18 2.04146i 1.84076i −2.16755 −2.00596 + 0.987985i 3.75784 2.14151i 0.342047i −0.388411 −2.01693 4.09509i
419.19 2.33380i 1.97214i −3.44661 −1.75107 + 1.39059i −4.60257 4.79282i 3.37609i −0.889329 −3.24536 4.08665i
419.20 2.52711i 0.473551i −4.38628 2.21995 + 0.267979i −1.19672 1.17543i 6.03039i 2.77575 −0.677212 + 5.61006i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 419.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1045.2.b.c 20
5.b even 2 1 inner 1045.2.b.c 20
5.c odd 4 2 5225.2.a.ba 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.b.c 20 1.a even 1 1 trivial
1045.2.b.c 20 5.b even 2 1 inner
5225.2.a.ba 20 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} + 26 T_{2}^{18} + 281 T_{2}^{16} + 1640 T_{2}^{14} + 5623 T_{2}^{12} + 11551 T_{2}^{10} + 13894 T_{2}^{8} + 9095 T_{2}^{6} + 2753 T_{2}^{4} + 276 T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1045, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 26 T^{18} + 281 T^{16} + 1640 T^{14} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{20} + 35 T^{18} + 500 T^{16} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{20} - T^{18} + 6 T^{17} + \cdots + 9765625 \) Copy content Toggle raw display
$7$ \( T^{20} + 73 T^{18} + 2053 T^{16} + \cdots + 2304 \) Copy content Toggle raw display
$11$ \( (T - 1)^{20} \) Copy content Toggle raw display
$13$ \( T^{20} + 137 T^{18} + \cdots + 33732864 \) Copy content Toggle raw display
$17$ \( T^{20} + 157 T^{18} + 9767 T^{16} + \cdots + 7054336 \) Copy content Toggle raw display
$19$ \( (T - 1)^{20} \) Copy content Toggle raw display
$23$ \( T^{20} + 220 T^{18} + \cdots + 172764736 \) Copy content Toggle raw display
$29$ \( (T^{10} - 25 T^{9} + 153 T^{8} + \cdots + 4194192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + 25 T^{9} + 124 T^{8} - 1350 T^{7} + \cdots - 7456)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + 411 T^{18} + \cdots + 61309721664 \) Copy content Toggle raw display
$41$ \( (T^{10} + 17 T^{9} - 44 T^{8} + \cdots + 166848)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + 379 T^{18} + \cdots + 7783474176 \) Copy content Toggle raw display
$47$ \( T^{20} + 606 T^{18} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + 641 T^{18} + \cdots + 58\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{10} - 15 T^{9} - 296 T^{8} + \cdots + 86423136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + 7 T^{9} - 203 T^{8} + \cdots - 26701648)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 345472702914816 \) Copy content Toggle raw display
$71$ \( (T^{10} + 20 T^{9} - 73 T^{8} + \cdots - 400032)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + 964 T^{18} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{10} - 53 T^{9} + 990 T^{8} + \cdots - 204893824)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + 425 T^{18} + \cdots + 5917659525376 \) Copy content Toggle raw display
$89$ \( (T^{10} - 18 T^{9} - 234 T^{8} + \cdots + 14189256)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + 1248 T^{18} + \cdots + 19285658703936 \) Copy content Toggle raw display
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