Newspace parameters
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.m (of order \(11\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.50967773947\) |
Analytic rank: | \(0\) |
Dimension: | \(20\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{11})\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{20} - \cdots)\) |
Defining polynomial: |
\( x^{20} - 4 x^{19} - 3 x^{18} + 66 x^{17} - 163 x^{16} - 52 x^{15} + 1567 x^{14} - 6182 x^{13} + 17043 x^{12} - 35832 x^{11} + 60906 x^{10} - 87666 x^{9} + 106197 x^{8} - 102542 x^{7} + \cdots + 529 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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
\( x^{20} - 4 x^{19} - 3 x^{18} + 66 x^{17} - 163 x^{16} - 52 x^{15} + 1567 x^{14} - 6182 x^{13} + 17043 x^{12} - 35832 x^{11} + 60906 x^{10} - 87666 x^{9} + 106197 x^{8} - 102542 x^{7} + \cdots + 529 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 77\!\cdots\!35 \nu^{19} + \cdots - 13\!\cdots\!32 ) / 73\!\cdots\!31 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 47\!\cdots\!30 \nu^{19} + \cdots - 57\!\cdots\!72 ) / 73\!\cdots\!31 \)
|
\(\beta_{4}\) | \(=\) |
\( ( - 55\!\cdots\!96 \nu^{19} + \cdots - 18\!\cdots\!58 ) / 73\!\cdots\!31 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 56\!\cdots\!12 \nu^{19} + \cdots + 20\!\cdots\!88 ) / 73\!\cdots\!31 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 73\!\cdots\!18 \nu^{19} + \cdots - 63\!\cdots\!04 ) / 73\!\cdots\!31 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 82\!\cdots\!88 \nu^{19} + \cdots + 25\!\cdots\!38 ) / 73\!\cdots\!31 \)
|
\(\beta_{8}\) | \(=\) |
\( ( - 87\!\cdots\!48 \nu^{19} + \cdots - 10\!\cdots\!60 ) / 73\!\cdots\!31 \)
|
\(\beta_{9}\) | \(=\) |
\( ( 10\!\cdots\!68 \nu^{19} + \cdots + 12\!\cdots\!95 ) / 73\!\cdots\!31 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 11\!\cdots\!76 \nu^{19} + \cdots + 12\!\cdots\!77 ) / 73\!\cdots\!31 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 14\!\cdots\!02 \nu^{19} + \cdots + 11\!\cdots\!84 ) / 73\!\cdots\!31 \)
|
\(\beta_{12}\) | \(=\) |
\( ( 26\!\cdots\!08 \nu^{19} + \cdots + 10\!\cdots\!01 ) / 73\!\cdots\!31 \)
|
\(\beta_{13}\) | \(=\) |
\( ( 31\!\cdots\!52 \nu^{19} + \cdots - 20\!\cdots\!18 ) / 73\!\cdots\!31 \)
|
\(\beta_{14}\) | \(=\) |
\( ( - 34\!\cdots\!02 \nu^{19} + \cdots - 18\!\cdots\!88 ) / 73\!\cdots\!31 \)
|
\(\beta_{15}\) | \(=\) |
\( ( 38\!\cdots\!72 \nu^{19} + \cdots + 92\!\cdots\!94 ) / 73\!\cdots\!31 \)
|
\(\beta_{16}\) | \(=\) |
\( ( 47\!\cdots\!22 \nu^{19} + \cdots + 55\!\cdots\!36 ) / 73\!\cdots\!31 \)
|
\(\beta_{17}\) | \(=\) |
\( ( - 47\!\cdots\!34 \nu^{19} + \cdots - 78\!\cdots\!96 ) / 73\!\cdots\!31 \)
|
\(\beta_{18}\) | \(=\) |
\( ( - 66\!\cdots\!69 \nu^{19} + \cdots + 20\!\cdots\!21 ) / 73\!\cdots\!31 \)
|
\(\beta_{19}\) | \(=\) |
\( ( 74\!\cdots\!15 \nu^{19} + \cdots + 17\!\cdots\!51 ) / 73\!\cdots\!31 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{19} - \beta_{18} - 2 \beta_{17} - 2 \beta_{16} - \beta_{15} - \beta_{14} - \beta_{13} - 4 \beta_{12} - \beta_{11} + \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \beta_{5} + 1 \)
|
\(\nu^{3}\) | \(=\) |
\( - 2 \beta_{19} + 4 \beta_{17} + 4 \beta_{16} + \beta_{15} - \beta_{14} + 2 \beta_{12} - \beta_{10} + 5 \beta_{9} + 3 \beta_{8} + \beta_{7} - \beta_{3} - 6 \beta_{2} + 2 \beta _1 - 2 \)
|
\(\nu^{4}\) | \(=\) |
\( 7 \beta_{19} - 2 \beta_{18} - 49 \beta_{17} - 34 \beta_{16} - 38 \beta_{15} - 21 \beta_{14} - 13 \beta_{13} - \beta_{12} - 21 \beta_{11} + 18 \beta_{10} - 49 \beta_{9} - 47 \beta_{8} + 2 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} + 7 \beta_{2} - 13 \beta _1 - 19 \)
|
\(\nu^{5}\) | \(=\) |
\( - 71 \beta_{19} + 13 \beta_{18} + 134 \beta_{17} + 66 \beta_{16} + 114 \beta_{15} - 13 \beta_{14} + 48 \beta_{13} + 74 \beta_{12} + 48 \beta_{11} + 37 \beta_{10} + 123 \beta_{9} + 136 \beta_{8} - 13 \beta_{7} - 18 \beta_{6} - 18 \beta_{5} + \cdots - 2 \)
|
\(\nu^{6}\) | \(=\) |
\( 151 \beta_{19} - 4 \beta_{18} - 568 \beta_{17} - 238 \beta_{16} - 505 \beta_{15} - 151 \beta_{13} - 48 \beta_{12} - 94 \beta_{11} - 44 \beta_{10} - 501 \beta_{9} - 572 \beta_{8} + 43 \beta_{7} + 135 \beta_{6} + 43 \beta_{5} + 177 \beta_{4} + \cdots - 234 \)
|
\(\nu^{7}\) | \(=\) |
\( - 702 \beta_{19} + 2126 \beta_{17} + 408 \beta_{16} + 2126 \beta_{15} - 38 \beta_{14} + 877 \beta_{13} + 294 \beta_{12} + 664 \beta_{11} + 702 \beta_{10} + 1767 \beta_{9} + 2268 \beta_{8} - 290 \beta_{7} - 438 \beta_{6} + \cdots + 1065 \)
|
\(\nu^{8}\) | \(=\) |
\( 2390 \beta_{19} + 132 \beta_{18} - 6136 \beta_{17} - 7012 \beta_{15} + 2086 \beta_{14} - 2444 \beta_{13} - 383 \beta_{12} - 934 \beta_{11} - 3900 \beta_{10} - 4451 \beta_{9} - 7012 \beta_{8} + 1692 \beta_{7} + 2390 \beta_{6} + \cdots - 3746 \)
|
\(\nu^{9}\) | \(=\) |
\( - 5111 \beta_{19} - 2040 \beta_{18} + 17477 \beta_{17} - 6362 \beta_{16} + 22925 \beta_{15} - 6791 \beta_{14} + 8259 \beta_{13} - 4107 \beta_{12} + 1897 \beta_{11} + 15050 \beta_{10} + 10299 \beta_{9} + \cdots + 17813 \)
|
\(\nu^{10}\) | \(=\) |
\( 14021 \beta_{19} + 9229 \beta_{18} - 34089 \beta_{17} + 49307 \beta_{16} - 64960 \beta_{15} + 38468 \beta_{14} - 24860 \beta_{13} + 20295 \beta_{12} + 2741 \beta_{11} - 71749 \beta_{10} - 6488 \beta_{9} + \cdots - 60168 \)
|
\(\nu^{11}\) | \(=\) |
\( - 39811 \beta_{18} - 23464 \beta_{17} - 258961 \beta_{16} + 124682 \beta_{15} - 159691 \beta_{14} + 43390 \beta_{13} - 121103 \beta_{12} - 63275 \beta_{11} + 258961 \beta_{10} - 119880 \beta_{9} + \cdots + 194703 \)
|
\(\nu^{12}\) | \(=\) |
\( - 194262 \beta_{19} + 194262 \beta_{18} + 590249 \beta_{17} + 1222410 \beta_{16} + 590249 \beta_{14} + 612124 \beta_{12} + 374692 \beta_{11} - 907833 \beta_{10} + 907833 \beta_{9} + \cdots - 612124 \)
|
\(\nu^{13}\) | \(=\) |
\( 1291028 \beta_{19} - 672687 \beta_{18} - 4241226 \beta_{17} - 5090636 \beta_{16} - 1882408 \beta_{15} - 2176925 \beta_{14} - 672687 \beta_{13} - 2413435 \beta_{12} - 1882408 \beta_{11} + \cdots + 1291028 \)
|
\(\nu^{14}\) | \(=\) |
\( - 7306375 \beta_{19} + 2462944 \beta_{18} + 22504950 \beta_{17} + 19715461 \beta_{16} + 14176470 \beta_{15} + 6870095 \beta_{14} + 5252433 \beta_{13} + 9769319 \beta_{12} + \cdots - 1185708 \)
|
\(\nu^{15}\) | \(=\) |
\( 33609406 \beta_{19} - 8199524 \beta_{18} - 102255210 \beta_{17} - 69885095 \beta_{16} - 76091057 \beta_{15} - 18895885 \beta_{14} - 27803501 \beta_{13} - 35108969 \beta_{12} + \cdots - 10696361 \)
|
\(\nu^{16}\) | \(=\) |
\( - 140271748 \beta_{19} + 20954963 \beta_{18} + 427756633 \beta_{17} + 223899277 \beta_{16} + 353063932 \beta_{15} + 38521255 \beta_{14} + 129164655 \beta_{13} + \cdots + 100976707 \)
|
\(\nu^{17}\) | \(=\) |
\( 546164663 \beta_{19} - 45887093 \beta_{18} - 1632307100 \beta_{17} - 619032881 \beta_{16} - 1487267752 \beta_{15} - 546164663 \beta_{13} - 324068924 \beta_{12} + \cdots - 573145788 \)
|
\(\nu^{18}\) | \(=\) |
\( - 1926997917 \beta_{19} + 5726696450 \beta_{17} + 1243247780 \beta_{16} + 5726696450 \beta_{15} - 571490131 \beta_{14} + 2094830388 \beta_{13} + 683750137 \beta_{12} + \cdots + 2764584594 \)
|
\(\nu^{19}\) | \(=\) |
\( 6182769136 \beta_{19} + 669331509 \beta_{18} - 18145320837 \beta_{17} - 20325168529 \beta_{15} + 4210658628 \beta_{14} - 7449454423 \beta_{13} - 286561532 \beta_{12} + \cdots - 11962551701 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/690\mathbb{Z}\right)^\times\).
\(n\) | \(277\) | \(461\) | \(511\) |
\(\chi(n)\) | \(1\) | \(1\) | \(\beta_{16}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 |
|
−0.142315 | + | 0.989821i | −0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | 0.959493 | − | 0.281733i | −1.32376 | − | 0.850727i | 0.415415 | − | 0.909632i | −0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
31.2 | −0.142315 | + | 0.989821i | −0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | 0.959493 | − | 0.281733i | 1.99973 | + | 1.28515i | 0.415415 | − | 0.909632i | −0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
121.1 | 0.415415 | − | 0.909632i | 0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | 0.654861 | − | 0.755750i | −0.668052 | + | 4.64640i | −0.959493 | + | 0.281733i | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
121.2 | 0.415415 | − | 0.909632i | 0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | 0.654861 | − | 0.755750i | 0.0903198 | − | 0.628188i | −0.959493 | + | 0.281733i | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
151.1 | −0.654861 | − | 0.755750i | −0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | 0.142315 | + | 0.989821i | 0.170792 | + | 0.0501489i | 0.841254 | − | 0.540641i | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
151.2 | −0.654861 | − | 0.755750i | −0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | 0.142315 | + | 0.989821i | 2.42713 | + | 0.712670i | 0.841254 | − | 0.540641i | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
211.1 | 0.415415 | + | 0.909632i | 0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | 0.654861 | + | 0.755750i | −0.668052 | − | 4.64640i | −0.959493 | − | 0.281733i | 0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
211.2 | 0.415415 | + | 0.909632i | 0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | 0.654861 | + | 0.755750i | 0.0903198 | + | 0.628188i | −0.959493 | − | 0.281733i | 0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.1 | −0.959493 | − | 0.281733i | 0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | −0.841254 | + | 0.540641i | −0.838100 | − | 1.83518i | −0.654861 | − | 0.755750i | −0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
271.2 | −0.959493 | − | 0.281733i | 0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | −0.841254 | + | 0.540641i | −0.113936 | − | 0.249486i | −0.654861 | − | 0.755750i | −0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
301.1 | 0.841254 | + | 0.540641i | 0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | −0.415415 | + | 0.909632i | −2.29325 | + | 2.64655i | −0.142315 | + | 0.989821i | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
301.2 | 0.841254 | + | 0.540641i | 0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | −0.415415 | + | 0.909632i | 1.54913 | − | 1.78779i | −0.142315 | + | 0.989821i | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
331.1 | −0.959493 | + | 0.281733i | 0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i | −0.142315 | − | 0.989821i | −0.841254 | − | 0.540641i | −0.838100 | + | 1.83518i | −0.654861 | + | 0.755750i | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
331.2 | −0.959493 | + | 0.281733i | 0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i | −0.142315 | − | 0.989821i | −0.841254 | − | 0.540641i | −0.113936 | + | 0.249486i | −0.654861 | + | 0.755750i | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
361.1 | −0.654861 | + | 0.755750i | −0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | 0.142315 | − | 0.989821i | 0.170792 | − | 0.0501489i | 0.841254 | + | 0.540641i | 0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
361.2 | −0.654861 | + | 0.755750i | −0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | 0.142315 | − | 0.989821i | 2.42713 | − | 0.712670i | 0.841254 | + | 0.540641i | 0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
541.1 | 0.841254 | − | 0.540641i | 0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i | −0.959493 | + | 0.281733i | −0.415415 | − | 0.909632i | −2.29325 | − | 2.64655i | −0.142315 | − | 0.989821i | −0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
541.2 | 0.841254 | − | 0.540641i | 0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i | −0.959493 | + | 0.281733i | −0.415415 | − | 0.909632i | 1.54913 | + | 1.78779i | −0.142315 | − | 0.989821i | −0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
601.1 | −0.142315 | − | 0.989821i | −0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | 0.959493 | + | 0.281733i | −1.32376 | + | 0.850727i | 0.415415 | + | 0.909632i | −0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
601.2 | −0.142315 | − | 0.989821i | −0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | 0.959493 | + | 0.281733i | 1.99973 | − | 1.28515i | 0.415415 | + | 0.909632i | −0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.2.m.c | ✓ | 20 |
23.c | even | 11 | 1 | inner | 690.2.m.c | ✓ | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.2.m.c | ✓ | 20 | 1.a | even | 1 | 1 | trivial |
690.2.m.c | ✓ | 20 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{20} - 2 T_{7}^{19} + 20 T_{7}^{18} - 94 T_{7}^{17} + 200 T_{7}^{16} - 705 T_{7}^{15} + 3796 T_{7}^{14} - 9544 T_{7}^{13} + 11360 T_{7}^{12} - 17376 T_{7}^{11} + 61478 T_{7}^{10} - 66922 T_{7}^{9} - 30534 T_{7}^{8} + \cdots + 529 \)
acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
$3$
\( (T^{10} - T^{9} + T^{8} - T^{7} + T^{6} - T^{5} + T^{4} + \cdots + 1)^{2} \)
$5$
\( (T^{10} + T^{9} + T^{8} + T^{7} + T^{6} + T^{5} + T^{4} + \cdots + 1)^{2} \)
$7$
\( T^{20} - 2 T^{19} + 20 T^{18} - 94 T^{17} + \cdots + 529 \)
$11$
\( T^{20} + 24 T^{19} + 273 T^{18} + \cdots + 4489 \)
$13$
\( T^{20} + 4 T^{19} + 23 T^{18} + 197 T^{17} + \cdots + 529 \)
$17$
\( T^{20} - 16 T^{19} + \cdots + 3500852224 \)
$19$
\( T^{20} - 14 T^{19} + 147 T^{18} + \cdots + 59274601 \)
$23$
\( T^{20} + 2 T^{19} + \cdots + 41426511213649 \)
$29$
\( T^{20} - 18 T^{19} + \cdots + 59067463607296 \)
$31$
\( T^{20} + \cdots + 152659665224704 \)
$37$
\( T^{20} + 16 T^{19} + \cdots + 42213400681 \)
$41$
\( T^{20} - 29 T^{19} + \cdots + 35\!\cdots\!81 \)
$43$
\( T^{20} + 22 T^{19} + \cdots + 51\!\cdots\!36 \)
$47$
\( (T^{10} + 47 T^{9} + 812 T^{8} + \cdots + 1159181)^{2} \)
$53$
\( T^{20} - 58 T^{19} + \cdots + 43\!\cdots\!89 \)
$59$
\( T^{20} - 45 T^{19} + \cdots + 10\!\cdots\!81 \)
$61$
\( T^{20} + \cdots + 554879105680384 \)
$67$
\( T^{20} - 16 T^{19} + \cdots + 15\!\cdots\!64 \)
$71$
\( T^{20} - 59 T^{19} + \cdots + 59348755456 \)
$73$
\( T^{20} - 3 T^{19} + \cdots + 20\!\cdots\!44 \)
$79$
\( T^{20} + 20 T^{19} + \cdots + 23\!\cdots\!96 \)
$83$
\( T^{20} - 13 T^{19} + \cdots + 10300915926016 \)
$89$
\( T^{20} + 97 T^{19} + \cdots + 10\!\cdots\!29 \)
$97$
\( T^{20} + 17 T^{19} + \cdots + 61671996335104 \)
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