Properties

Label 688.6.a.h
Level 688
Weight 6
Character orbit 688.a
Self dual yes
Analytic conductor 110.344
Analytic rank 0
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 688.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(110.344068031\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -3 - \beta_{4} ) q^{3} + ( 14 + \beta_{7} ) q^{5} + ( -6 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{7} + ( 134 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{9} +O(q^{10})\) \( q + ( -3 - \beta_{4} ) q^{3} + ( 14 + \beta_{7} ) q^{5} + ( -6 + \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} ) q^{7} + ( 134 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{9} + ( -77 - 4 \beta_{1} - 2 \beta_{3} + 5 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{11} + ( 191 + \beta_{1} - 7 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 4 \beta_{9} ) q^{13} + ( -179 - 10 \beta_{1} + 5 \beta_{2} + 11 \beta_{3} - 25 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - 9 \beta_{8} - 2 \beta_{9} ) q^{15} + ( 411 + 10 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 27 \beta_{4} - 2 \beta_{6} - 5 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} ) q^{17} + ( 229 - 5 \beta_{1} - 4 \beta_{2} - \beta_{3} - 17 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} + 13 \beta_{7} - 10 \beta_{8} - 12 \beta_{9} ) q^{19} + ( -39 - 2 \beta_{1} + \beta_{2} - 19 \beta_{3} + \beta_{4} + 10 \beta_{5} - 5 \beta_{6} - 9 \beta_{7} - 20 \beta_{8} + \beta_{9} ) q^{21} + ( -168 - 2 \beta_{2} - 16 \beta_{3} - 12 \beta_{4} + 6 \beta_{5} - 2 \beta_{6} + 28 \beta_{7} + 16 \beta_{8} + 3 \beta_{9} ) q^{23} + ( 710 - 10 \beta_{1} + 20 \beta_{2} + \beta_{3} - 29 \beta_{4} - 11 \beta_{5} - 22 \beta_{6} + 57 \beta_{7} - 4 \beta_{8} - 7 \beta_{9} ) q^{25} + ( 214 - 20 \beta_{1} + 25 \beta_{2} - 6 \beta_{3} - 82 \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 12 \beta_{7} - 10 \beta_{8} + 24 \beta_{9} ) q^{27} + ( 728 - 4 \beta_{2} - 4 \beta_{3} + 48 \beta_{4} - 33 \beta_{5} - 9 \beta_{6} + 21 \beta_{7} + 45 \beta_{8} - 3 \beta_{9} ) q^{29} + ( 508 - 4 \beta_{1} - 10 \beta_{2} - 44 \beta_{3} + 62 \beta_{4} - 9 \beta_{5} - 11 \beta_{6} + 2 \beta_{7} + 27 \beta_{8} + 38 \beta_{9} ) q^{31} + ( -1577 + 21 \beta_{1} - 15 \beta_{2} - 80 \beta_{3} - 33 \beta_{4} - 3 \beta_{5} + 41 \beta_{6} + 6 \beta_{7} + 22 \beta_{8} + 20 \beta_{9} ) q^{33} + ( -664 + 5 \beta_{1} + 11 \beta_{2} - 12 \beta_{3} + 34 \beta_{4} + 40 \beta_{5} + 2 \beta_{6} + 66 \beta_{7} + 64 \beta_{8} - 3 \beta_{9} ) q^{35} + ( 163 + 41 \beta_{1} + 45 \beta_{2} + 16 \beta_{3} - 141 \beta_{4} + 32 \beta_{5} - 2 \beta_{6} + 109 \beta_{7} + 15 \beta_{8} - 29 \beta_{9} ) q^{37} + ( 2514 - 78 \beta_{1} + 43 \beta_{2} - 59 \beta_{3} - 244 \beta_{4} - 5 \beta_{5} - 36 \beta_{6} + 111 \beta_{7} - 90 \beta_{8} - 98 \beta_{9} ) q^{39} + ( 982 + 47 \beta_{1} - 76 \beta_{2} + 17 \beta_{3} - 276 \beta_{4} - 36 \beta_{5} + 42 \beta_{6} - 62 \beta_{7} - 42 \beta_{8} + 54 \beta_{9} ) q^{41} -1849 q^{43} + ( 7443 + 124 \beta_{1} + 61 \beta_{2} + 146 \beta_{3} - 79 \beta_{4} - 26 \beta_{5} - 92 \beta_{6} + 264 \beta_{7} - 95 \beta_{8} - 95 \beta_{9} ) q^{45} + ( -4822 + 21 \beta_{1} - 52 \beta_{2} + 139 \beta_{3} + 176 \beta_{4} - 58 \beta_{5} + 25 \beta_{6} - 12 \beta_{7} - 85 \beta_{8} - 30 \beta_{9} ) q^{47} + ( 2865 - 67 \beta_{1} - 63 \beta_{2} - 68 \beta_{3} - 250 \beta_{4} + 48 \beta_{5} + 74 \beta_{6} - 94 \beta_{7} + 98 \beta_{8} - 17 \beta_{9} ) q^{49} + ( -9281 + 45 \beta_{1} + 5 \beta_{2} + 48 \beta_{3} - 373 \beta_{4} + 80 \beta_{5} + 71 \beta_{6} - 30 \beta_{7} + 94 \beta_{8} + 64 \beta_{9} ) q^{51} + ( 12563 - 50 \beta_{1} + 105 \beta_{2} - 70 \beta_{3} - 417 \beta_{4} + 51 \beta_{5} - 13 \beta_{6} + 25 \beta_{7} - 16 \beta_{8} - 123 \beta_{9} ) q^{53} + ( -10472 + 361 \beta_{1} - 135 \beta_{2} + 111 \beta_{3} + 208 \beta_{4} + 61 \beta_{5} + 146 \beta_{6} - 219 \beta_{7} + 66 \beta_{8} - 26 \beta_{9} ) q^{55} + ( 3151 - 232 \beta_{1} + 35 \beta_{2} - 73 \beta_{3} - 829 \beta_{4} - 10 \beta_{5} + 88 \beta_{6} - 129 \beta_{7} - 54 \beta_{8} + 202 \beta_{9} ) q^{57} + ( -9921 + 144 \beta_{1} + 79 \beta_{2} - 11 \beta_{3} - 589 \beta_{4} - 220 \beta_{5} + 257 \beta_{6} + 133 \beta_{7} + 60 \beta_{8} - 15 \beta_{9} ) q^{59} + ( 1827 - 194 \beta_{1} - 141 \beta_{2} + 176 \beta_{3} + 75 \beta_{4} + 37 \beta_{5} + 125 \beta_{6} - 184 \beta_{7} + 258 \beta_{8} + 104 \beta_{9} ) q^{61} + ( -406 - 87 \beta_{1} - 117 \beta_{2} - 139 \beta_{3} - 782 \beta_{4} - 195 \beta_{5} - 62 \beta_{6} + 69 \beta_{7} + 86 \beta_{8} + 70 \beta_{9} ) q^{63} + ( 5485 + 66 \beta_{1} - 45 \beta_{2} + 208 \beta_{3} - 489 \beta_{4} + 219 \beta_{5} - 35 \beta_{6} + 44 \beta_{7} - 14 \beta_{8} + 168 \beta_{9} ) q^{65} + ( 484 + 307 \beta_{1} - 8 \beta_{2} + 593 \beta_{3} + 196 \beta_{4} - 63 \beta_{5} - 130 \beta_{6} - 290 \beta_{7} + 18 \beta_{8} + 22 \beta_{9} ) q^{67} + ( -1 - 172 \beta_{1} + 13 \beta_{2} + 98 \beta_{3} + 159 \beta_{4} + 88 \beta_{5} - 84 \beta_{6} - 516 \beta_{7} - 143 \beta_{8} + 29 \beta_{9} ) q^{69} + ( -1458 - 281 \beta_{1} + 162 \beta_{2} - 356 \beta_{3} - 664 \beta_{4} + 68 \beta_{5} - 318 \beta_{6} + 210 \beta_{7} - 32 \beta_{8} + 30 \beta_{9} ) q^{71} + ( 5257 + 29 \beta_{1} - 135 \beta_{2} + 112 \beta_{3} + 819 \beta_{4} - 625 \beta_{5} + 409 \beta_{6} - 44 \beta_{7} + 156 \beta_{8} - 74 \beta_{9} ) q^{73} + ( 3810 - 265 \beta_{1} + 250 \beta_{2} + 267 \beta_{3} - 2014 \beta_{4} + 433 \beta_{5} - 139 \beta_{6} - 21 \beta_{7} - 598 \beta_{8} + 16 \beta_{9} ) q^{75} + ( 9423 - 234 \beta_{1} - 129 \beta_{2} - 34 \beta_{3} - 903 \beta_{4} + 183 \beta_{5} - 493 \beta_{6} - 318 \beta_{7} - 16 \beta_{8} + 84 \beta_{9} ) q^{77} + ( 9043 - 94 \beta_{1} - 320 \beta_{2} - 214 \beta_{3} - 1383 \beta_{4} + 51 \beta_{5} - 141 \beta_{6} + 195 \beta_{7} + 430 \beta_{8} + 347 \beta_{9} ) q^{79} + ( -2674 + 329 \beta_{1} + 84 \beta_{2} + 271 \beta_{3} - 565 \beta_{4} - 312 \beta_{5} + 120 \beta_{6} + 561 \beta_{7} - 308 \beta_{8} - 461 \beta_{9} ) q^{81} + ( 10050 - 356 \beta_{1} + 15 \beta_{2} - 417 \beta_{3} - 1364 \beta_{4} - 431 \beta_{5} + 220 \beta_{6} + 636 \beta_{7} - 114 \beta_{8} - 117 \beta_{9} ) q^{83} + ( -8743 + 235 \beta_{1} - 81 \beta_{2} - 322 \beta_{3} - 133 \beta_{4} + 110 \beta_{5} - 342 \beta_{6} + 672 \beta_{7} - 157 \beta_{8} - 73 \beta_{9} ) q^{85} + ( -18324 + 69 \beta_{1} - 480 \beta_{2} - 327 \beta_{3} + 480 \beta_{4} + 1107 \beta_{5} + 498 \beta_{6} - 1692 \beta_{7} + 54 \beta_{8} + 285 \beta_{9} ) q^{87} + ( -6365 - 72 \beta_{1} + 133 \beta_{2} - 362 \beta_{3} - 1161 \beta_{4} + 363 \beta_{5} - 219 \beta_{6} + 210 \beta_{7} + 90 \beta_{8} + 678 \beta_{9} ) q^{89} + ( 29142 + 37 \beta_{1} + 231 \beta_{2} - 289 \beta_{3} - 666 \beta_{4} + 415 \beta_{5} - 268 \beta_{6} - 409 \beta_{7} - 12 \beta_{8} - 310 \beta_{9} ) q^{91} + ( -24536 - 245 \beta_{1} - 170 \beta_{2} - 942 \beta_{3} - 26 \beta_{4} + 262 \beta_{5} + 222 \beta_{6} - 897 \beta_{7} - 94 \beta_{8} - 318 \beta_{9} ) q^{93} + ( 30660 + 350 \beta_{1} - 39 \beta_{2} + 869 \beta_{3} + 188 \beta_{4} - 424 \beta_{5} - 473 \beta_{6} + 630 \beta_{7} - 385 \beta_{8} - 777 \beta_{9} ) q^{95} + ( 10615 - 395 \beta_{1} + 343 \beta_{2} - 286 \beta_{3} - 871 \beta_{4} + 296 \beta_{5} - 629 \beta_{6} + 975 \beta_{7} + 448 \beta_{8} - 503 \beta_{9} ) q^{97} + ( 27314 + 394 \beta_{1} - 95 \beta_{2} - 539 \beta_{3} + 1524 \beta_{4} - 395 \beta_{5} + 428 \beta_{6} - 810 \beta_{7} + 258 \beta_{8} - 37 \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 28q^{3} + 138q^{5} - 60q^{7} + 1356q^{9} + O(q^{10}) \) \( 10q - 28q^{3} + 138q^{5} - 60q^{7} + 1356q^{9} - 745q^{11} + 1917q^{13} - 1688q^{15} + 4017q^{17} + 2404q^{19} - 228q^{21} - 1733q^{23} + 7120q^{25} + 2324q^{27} + 6996q^{29} + 4899q^{31} - 15734q^{33} - 7084q^{35} + 1466q^{37} + 26542q^{39} + 10297q^{41} - 18490q^{43} + 73822q^{45} - 48592q^{47} + 29458q^{49} - 92972q^{51} + 127165q^{53} - 106672q^{55} + 34060q^{57} - 99372q^{59} + 17408q^{61} - 2244q^{63} + 54484q^{65} + 2021q^{67} + 1654q^{69} - 11286q^{71} + 49892q^{73} + 44662q^{75} + 98144q^{77} + 91524q^{79} - 26450q^{81} + 105203q^{83} - 87212q^{85} - 181200q^{87} - 62682q^{89} + 295304q^{91} - 238430q^{93} + 305340q^{95} + 108383q^{97} + 270499q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 8 \nu - 2 \)
\(\beta_{2}\)\(=\)\((\)\(-1300283 \nu^{9} + 26922273 \nu^{8} + 154132815 \nu^{7} - 5215675021 \nu^{6} - 1235557305 \nu^{5} + 318622394671 \nu^{4} - 177037969291 \nu^{3} - 6241757248231 \nu^{2} + 1758751326736 \nu + 27676046268420\)\()/ 197066179584 \)
\(\beta_{3}\)\(=\)\((\)\(-1397467 \nu^{9} - 1596255 \nu^{8} + 334667631 \nu^{7} + 479070067 \nu^{6} - 23691741465 \nu^{5} - 16891402705 \nu^{4} + 442286142037 \nu^{3} - 1609930003559 \nu^{2} - 2373830719216 \nu + 24355668723972\)\()/ 197066179584 \)
\(\beta_{4}\)\(=\)\((\)\(-474917 \nu^{9} - 1144977 \nu^{8} + 116809809 \nu^{7} + 421392509 \nu^{6} - 8979038055 \nu^{5} - 39727817087 \nu^{4} + 220849424171 \nu^{3} + 1003571811959 \nu^{2} - 1656152504528 \nu - 5839603776708\)\()/ 65688726528 \)
\(\beta_{5}\)\(=\)\((\)\(2282743 \nu^{9} - 7177205 \nu^{8} - 569012379 \nu^{7} + 1117284305 \nu^{6} + 46987493645 \nu^{5} - 48836433947 \nu^{4} - 1348681107321 \nu^{3} + 397753048627 \nu^{2} + 7380178820016 \nu + 15723616018348\)\()/ 197066179584 \)
\(\beta_{6}\)\(=\)\((\)\(-5726311 \nu^{9} - 8152363 \nu^{8} + 1437405579 \nu^{7} + 3217063567 \nu^{6} - 113680795613 \nu^{5} - 310996161541 \nu^{4} + 3011920741641 \nu^{3} + 7733935811885 \nu^{2} - 25558198123248 \nu - 40534043519788\)\()/ 197066179584 \)
\(\beta_{7}\)\(=\)\((\)\(-6235841 \nu^{9} - 47565 \nu^{8} + 1552893885 \nu^{7} + 1540088777 \nu^{6} - 122844007131 \nu^{5} - 194780377283 \nu^{4} + 3248986969295 \nu^{3} + 4539319373339 \nu^{2} - 25027510353488 \nu - 19301122595508\)\()/ 197066179584 \)
\(\beta_{8}\)\(=\)\((\)\(10838435 \nu^{9} + 23722567 \nu^{8} - 2828120727 \nu^{7} - 8018172347 \nu^{6} + 236840256305 \nu^{5} + 724329792617 \nu^{4} - 6867015632269 \nu^{3} - 17271162848625 \nu^{2} + 59454523056496 \nu + 81663371814556\)\()/ 197066179584 \)
\(\beta_{9}\)\(=\)\((\)\(-17656675 \nu^{9} - 6000871 \nu^{8} + 4377440727 \nu^{7} + 6072341467 \nu^{6} - 345300273137 \nu^{5} - 699986633545 \nu^{4} + 9030426367437 \nu^{3} + 17296111236689 \nu^{2} - 62809921330800 \nu - 87327465633820\)\()/ 197066179584 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 2\)\()/8\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{9} + \beta_{7} + \beta_{6} - 4 \beta_{5} + 9 \beta_{4} + \beta_{3} + \beta_{1} + 415\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-18 \beta_{9} - 4 \beta_{8} + 27 \beta_{7} + 3 \beta_{6} - 8 \beta_{5} + 43 \beta_{4} + 3 \beta_{3} + 4 \beta_{2} + 84 \beta_{1} + 639\)\()/8\)
\(\nu^{4}\)\(=\)\((\)\(-67 \beta_{9} - 7 \beta_{8} + 53 \beta_{7} + 7 \beta_{6} - 113 \beta_{5} + 291 \beta_{4} + 43 \beta_{3} + 3 \beta_{2} + 57 \beta_{1} + 9105\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-2584 \beta_{9} - 184 \beta_{8} + 4448 \beta_{7} + 176 \beta_{6} - 1480 \beta_{5} + 7504 \beta_{4} + 368 \beta_{3} + 232 \beta_{2} + 8329 \beta_{1} + 126802\)\()/8\)
\(\nu^{6}\)\(=\)\((\)\(-33266 \beta_{9} - 4416 \beta_{8} + 36193 \beta_{7} - 5727 \beta_{6} - 48164 \beta_{5} + 145961 \beta_{4} + 18945 \beta_{3} + 1712 \beta_{2} + 35005 \beta_{1} + 3762135\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(-339754 \beta_{9} - 5108 \beta_{8} + 639047 \beta_{7} - 29553 \beta_{6} - 228008 \beta_{5} + 1088727 \beta_{4} + 38895 \beta_{3} + 1380 \beta_{2} + 877968 \beta_{1} + 19803219\)\()/8\)
\(\nu^{8}\)\(=\)\((\)\(-1042617 \beta_{9} - 131727 \beta_{8} + 1430958 \beta_{7} - 383568 \beta_{6} - 1322709 \beta_{5} + 4578420 \beta_{4} + 468636 \beta_{3} + 44007 \beta_{2} + 1204965 \beta_{1} + 104680478\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-44351112 \beta_{9} + 56616 \beta_{8} + 88331664 \beta_{7} - 10812960 \beta_{6} - 32440968 \beta_{5} + 151802112 \beta_{4} + 4019520 \beta_{3} - 2095944 \beta_{2} + 96526153 \beta_{1} + 2824172738\)\()/8\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
11.5305
−4.38824
−9.70631
3.50018
−6.91219
9.86547
−8.57770
2.86024
5.31531
−1.48720
0 −27.4953 0 86.8464 0 19.8137 0 512.989 0
1.2 0 −25.0462 0 −0.456695 0 −166.517 0 384.312 0
1.3 0 −18.3440 0 −72.1865 0 96.4803 0 93.5034 0
1.4 0 −16.8892 0 47.4635 0 −67.4603 0 42.2439 0
1.5 0 −12.8799 0 79.5677 0 172.354 0 −77.1083 0
1.6 0 −1.50169 0 −37.8251 0 124.747 0 −240.745 0
1.7 0 7.84343 0 28.1028 0 −195.604 0 −181.481 0
1.8 0 14.8716 0 −42.2365 0 −202.971 0 −21.8357 0
1.9 0 23.8469 0 −52.5837 0 174.859 0 325.673 0
1.10 0 27.5943 0 101.308 0 −15.7005 0 518.448 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.6.a.h 10
4.b odd 2 1 43.6.a.b 10
12.b even 2 1 387.6.a.e 10
20.d odd 2 1 1075.6.a.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.b 10 4.b odd 2 1
387.6.a.e 10 12.b even 2 1
688.6.a.h 10 1.a even 1 1 trivial
1075.6.a.b 10 20.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(43\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{10} + \cdots\) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(688))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 28 T + 929 T^{2} + 15652 T^{3} + 384094 T^{4} + 5703294 T^{5} + 127217574 T^{6} + 1773759114 T^{7} + 38322042597 T^{8} + 511917861924 T^{9} + 10155906521514 T^{10} + 124396040447532 T^{11} + 2262878293310253 T^{12} + 25451504567188398 T^{13} + 443580252556263174 T^{14} + 4832336042504605242 T^{15} + 79081548490762113006 T^{16} + \)\(78\!\cdots\!64\)\( T^{17} + \)\(11\!\cdots\!29\)\( T^{18} + \)\(82\!\cdots\!04\)\( T^{19} + \)\(71\!\cdots\!49\)\( T^{20} \)
$5$ \( 1 - 138 T + 21587 T^{2} - 2230608 T^{3} + 229459202 T^{4} - 19157670180 T^{5} + 1558167162738 T^{6} - 109829488059312 T^{7} + 7514260269046033 T^{8} - 457924168440376602 T^{9} + 27012319396517106654 T^{10} - \)\(14\!\cdots\!50\)\( T^{11} + \)\(73\!\cdots\!25\)\( T^{12} - \)\(33\!\cdots\!00\)\( T^{13} + \)\(14\!\cdots\!50\)\( T^{14} - \)\(57\!\cdots\!00\)\( T^{15} + \)\(21\!\cdots\!50\)\( T^{16} - \)\(64\!\cdots\!00\)\( T^{17} + \)\(19\!\cdots\!75\)\( T^{18} - \)\(39\!\cdots\!50\)\( T^{19} + \)\(88\!\cdots\!25\)\( T^{20} \)
$7$ \( 1 + 60 T + 71106 T^{2} + 5672772 T^{3} + 2998880769 T^{4} + 247503266272 T^{5} + 92903828588736 T^{6} + 7232195163049824 T^{7} + 2203025939434531374 T^{8} + \)\(15\!\cdots\!56\)\( T^{9} + \)\(41\!\cdots\!04\)\( T^{10} + \)\(26\!\cdots\!92\)\( T^{11} + \)\(62\!\cdots\!26\)\( T^{12} + \)\(34\!\cdots\!32\)\( T^{13} + \)\(74\!\cdots\!36\)\( T^{14} + \)\(33\!\cdots\!04\)\( T^{15} + \)\(67\!\cdots\!81\)\( T^{16} + \)\(21\!\cdots\!96\)\( T^{17} + \)\(45\!\cdots\!06\)\( T^{18} + \)\(64\!\cdots\!20\)\( T^{19} + \)\(17\!\cdots\!49\)\( T^{20} \)
$11$ \( 1 + 745 T + 1092200 T^{2} + 566790565 T^{3} + 495359905314 T^{4} + 194173124255379 T^{5} + 135383021483146674 T^{6} + 43098674076488688185 T^{7} + \)\(27\!\cdots\!81\)\( T^{8} + \)\(77\!\cdots\!58\)\( T^{9} + \)\(47\!\cdots\!16\)\( T^{10} + \)\(12\!\cdots\!58\)\( T^{11} + \)\(71\!\cdots\!81\)\( T^{12} + \)\(18\!\cdots\!35\)\( T^{13} + \)\(91\!\cdots\!74\)\( T^{14} + \)\(21\!\cdots\!29\)\( T^{15} + \)\(86\!\cdots\!14\)\( T^{16} + \)\(15\!\cdots\!15\)\( T^{17} + \)\(49\!\cdots\!00\)\( T^{18} + \)\(54\!\cdots\!95\)\( T^{19} + \)\(11\!\cdots\!01\)\( T^{20} \)
$13$ \( 1 - 1917 T + 3889346 T^{2} - 4848596699 T^{3} + 5963954609742 T^{4} - 5730569373162787 T^{5} + 5360926546177278684 T^{6} - \)\(42\!\cdots\!79\)\( T^{7} + \)\(32\!\cdots\!21\)\( T^{8} - \)\(21\!\cdots\!94\)\( T^{9} + \)\(14\!\cdots\!96\)\( T^{10} - \)\(81\!\cdots\!42\)\( T^{11} + \)\(44\!\cdots\!29\)\( T^{12} - \)\(21\!\cdots\!03\)\( T^{13} + \)\(10\!\cdots\!84\)\( T^{14} - \)\(40\!\cdots\!91\)\( T^{15} + \)\(15\!\cdots\!58\)\( T^{16} - \)\(47\!\cdots\!43\)\( T^{17} + \)\(14\!\cdots\!46\)\( T^{18} - \)\(25\!\cdots\!81\)\( T^{19} + \)\(49\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 - 4017 T + 16731071 T^{2} - 43371481422 T^{3} + 109028323359989 T^{4} - 214433083727073801 T^{5} + \)\(40\!\cdots\!11\)\( T^{6} - \)\(64\!\cdots\!84\)\( T^{7} + \)\(98\!\cdots\!87\)\( T^{8} - \)\(13\!\cdots\!91\)\( T^{9} + \)\(16\!\cdots\!00\)\( T^{10} - \)\(18\!\cdots\!87\)\( T^{11} + \)\(19\!\cdots\!63\)\( T^{12} - \)\(18\!\cdots\!12\)\( T^{13} + \)\(16\!\cdots\!11\)\( T^{14} - \)\(12\!\cdots\!57\)\( T^{15} + \)\(89\!\cdots\!61\)\( T^{16} - \)\(50\!\cdots\!46\)\( T^{17} + \)\(27\!\cdots\!71\)\( T^{18} - \)\(94\!\cdots\!69\)\( T^{19} + \)\(33\!\cdots\!49\)\( T^{20} \)
$19$ \( 1 - 2404 T + 15222557 T^{2} - 35388594664 T^{3} + 123780260921510 T^{4} - 260083719347815286 T^{5} + \)\(66\!\cdots\!54\)\( T^{6} - \)\(12\!\cdots\!42\)\( T^{7} + \)\(25\!\cdots\!37\)\( T^{8} - \)\(41\!\cdots\!16\)\( T^{9} + \)\(73\!\cdots\!66\)\( T^{10} - \)\(10\!\cdots\!84\)\( T^{11} + \)\(15\!\cdots\!37\)\( T^{12} - \)\(18\!\cdots\!58\)\( T^{13} + \)\(25\!\cdots\!54\)\( T^{14} - \)\(24\!\cdots\!14\)\( T^{15} + \)\(28\!\cdots\!10\)\( T^{16} - \)\(20\!\cdots\!36\)\( T^{17} + \)\(21\!\cdots\!57\)\( T^{18} - \)\(84\!\cdots\!96\)\( T^{19} + \)\(86\!\cdots\!01\)\( T^{20} \)
$23$ \( 1 + 1733 T + 47557957 T^{2} + 72067471074 T^{3} + 1063199488621929 T^{4} + 1395811354288602239 T^{5} + \)\(14\!\cdots\!11\)\( T^{6} + \)\(16\!\cdots\!50\)\( T^{7} + \)\(14\!\cdots\!39\)\( T^{8} + \)\(14\!\cdots\!37\)\( T^{9} + \)\(10\!\cdots\!02\)\( T^{10} + \)\(93\!\cdots\!91\)\( T^{11} + \)\(61\!\cdots\!11\)\( T^{12} + \)\(45\!\cdots\!50\)\( T^{13} + \)\(25\!\cdots\!11\)\( T^{14} + \)\(15\!\cdots\!77\)\( T^{15} + \)\(75\!\cdots\!21\)\( T^{16} + \)\(32\!\cdots\!18\)\( T^{17} + \)\(14\!\cdots\!57\)\( T^{18} + \)\(32\!\cdots\!19\)\( T^{19} + \)\(12\!\cdots\!49\)\( T^{20} \)
$29$ \( 1 - 6996 T + 88174667 T^{2} - 354425388876 T^{3} + 2685110189553942 T^{4} - 3595496283776034360 T^{5} + \)\(29\!\cdots\!74\)\( T^{6} + \)\(15\!\cdots\!88\)\( T^{7} - \)\(13\!\cdots\!47\)\( T^{8} + \)\(63\!\cdots\!00\)\( T^{9} - \)\(82\!\cdots\!38\)\( T^{10} + \)\(13\!\cdots\!00\)\( T^{11} - \)\(58\!\cdots\!47\)\( T^{12} + \)\(13\!\cdots\!12\)\( T^{13} + \)\(52\!\cdots\!74\)\( T^{14} - \)\(13\!\cdots\!40\)\( T^{15} + \)\(19\!\cdots\!42\)\( T^{16} - \)\(54\!\cdots\!24\)\( T^{17} + \)\(27\!\cdots\!67\)\( T^{18} - \)\(44\!\cdots\!04\)\( T^{19} + \)\(13\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - 4899 T + 169335305 T^{2} - 687234491898 T^{3} + 14314848605964873 T^{4} - 50115254632047551877 T^{5} + \)\(80\!\cdots\!39\)\( T^{6} - \)\(24\!\cdots\!98\)\( T^{7} + \)\(33\!\cdots\!11\)\( T^{8} - \)\(91\!\cdots\!15\)\( T^{9} + \)\(10\!\cdots\!46\)\( T^{10} - \)\(26\!\cdots\!65\)\( T^{11} + \)\(27\!\cdots\!11\)\( T^{12} - \)\(58\!\cdots\!98\)\( T^{13} + \)\(54\!\cdots\!39\)\( T^{14} - \)\(96\!\cdots\!27\)\( T^{15} + \)\(78\!\cdots\!73\)\( T^{16} - \)\(10\!\cdots\!98\)\( T^{17} + \)\(76\!\cdots\!05\)\( T^{18} - \)\(63\!\cdots\!49\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 - 1466 T + 314595515 T^{2} - 1000159162152 T^{3} + 48977417173952218 T^{4} - \)\(27\!\cdots\!48\)\( T^{5} + \)\(50\!\cdots\!42\)\( T^{6} - \)\(41\!\cdots\!16\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} - \)\(41\!\cdots\!94\)\( T^{9} + \)\(29\!\cdots\!30\)\( T^{10} - \)\(28\!\cdots\!58\)\( T^{11} + \)\(19\!\cdots\!49\)\( T^{12} - \)\(13\!\cdots\!88\)\( T^{13} + \)\(11\!\cdots\!42\)\( T^{14} - \)\(43\!\cdots\!36\)\( T^{15} + \)\(54\!\cdots\!82\)\( T^{16} - \)\(77\!\cdots\!36\)\( T^{17} + \)\(16\!\cdots\!15\)\( T^{18} - \)\(54\!\cdots\!62\)\( T^{19} + \)\(25\!\cdots\!49\)\( T^{20} \)
$41$ \( 1 - 10297 T + 512459663 T^{2} - 5996761727014 T^{3} + 151679237600092005 T^{4} - \)\(17\!\cdots\!49\)\( T^{5} + \)\(31\!\cdots\!39\)\( T^{6} - \)\(34\!\cdots\!68\)\( T^{7} + \)\(50\!\cdots\!19\)\( T^{8} - \)\(50\!\cdots\!39\)\( T^{9} + \)\(65\!\cdots\!96\)\( T^{10} - \)\(58\!\cdots\!39\)\( T^{11} + \)\(68\!\cdots\!19\)\( T^{12} - \)\(52\!\cdots\!68\)\( T^{13} + \)\(57\!\cdots\!39\)\( T^{14} - \)\(36\!\cdots\!49\)\( T^{15} + \)\(36\!\cdots\!05\)\( T^{16} - \)\(16\!\cdots\!14\)\( T^{17} + \)\(16\!\cdots\!63\)\( T^{18} - \)\(38\!\cdots\!97\)\( T^{19} + \)\(43\!\cdots\!01\)\( T^{20} \)
$43$ \( ( 1 + 1849 T )^{10} \)
$47$ \( 1 + 48592 T + 2515660075 T^{2} + 84810946501416 T^{3} + 2666904651640316270 T^{4} + \)\(68\!\cdots\!52\)\( T^{5} + \)\(16\!\cdots\!42\)\( T^{6} + \)\(33\!\cdots\!20\)\( T^{7} + \)\(64\!\cdots\!57\)\( T^{8} + \)\(11\!\cdots\!92\)\( T^{9} + \)\(17\!\cdots\!66\)\( T^{10} + \)\(25\!\cdots\!44\)\( T^{11} + \)\(34\!\cdots\!93\)\( T^{12} + \)\(40\!\cdots\!60\)\( T^{13} + \)\(45\!\cdots\!42\)\( T^{14} + \)\(43\!\cdots\!64\)\( T^{15} + \)\(38\!\cdots\!30\)\( T^{16} + \)\(28\!\cdots\!88\)\( T^{17} + \)\(19\!\cdots\!75\)\( T^{18} + \)\(85\!\cdots\!44\)\( T^{19} + \)\(40\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 - 127165 T + 9906770390 T^{2} - 550920294329671 T^{3} + 24581308628857110150 T^{4} - \)\(91\!\cdots\!35\)\( T^{5} + \)\(29\!\cdots\!20\)\( T^{6} - \)\(83\!\cdots\!31\)\( T^{7} + \)\(21\!\cdots\!05\)\( T^{8} - \)\(49\!\cdots\!78\)\( T^{9} + \)\(10\!\cdots\!76\)\( T^{10} - \)\(20\!\cdots\!54\)\( T^{11} + \)\(37\!\cdots\!45\)\( T^{12} - \)\(61\!\cdots\!67\)\( T^{13} + \)\(90\!\cdots\!20\)\( T^{14} - \)\(11\!\cdots\!55\)\( T^{15} + \)\(13\!\cdots\!50\)\( T^{16} - \)\(12\!\cdots\!47\)\( T^{17} + \)\(92\!\cdots\!90\)\( T^{18} - \)\(49\!\cdots\!45\)\( T^{19} + \)\(16\!\cdots\!49\)\( T^{20} \)
$59$ \( 1 + 99372 T + 7563799766 T^{2} + 407898209580180 T^{3} + 19000827281933813957 T^{4} + \)\(74\!\cdots\!72\)\( T^{5} + \)\(26\!\cdots\!44\)\( T^{6} + \)\(87\!\cdots\!68\)\( T^{7} + \)\(26\!\cdots\!22\)\( T^{8} + \)\(76\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!96\)\( T^{10} + \)\(54\!\cdots\!24\)\( T^{11} + \)\(13\!\cdots\!22\)\( T^{12} + \)\(31\!\cdots\!32\)\( T^{13} + \)\(70\!\cdots\!44\)\( T^{14} + \)\(13\!\cdots\!28\)\( T^{15} + \)\(25\!\cdots\!57\)\( T^{16} + \)\(38\!\cdots\!20\)\( T^{17} + \)\(51\!\cdots\!66\)\( T^{18} + \)\(48\!\cdots\!28\)\( T^{19} + \)\(34\!\cdots\!01\)\( T^{20} \)
$61$ \( 1 - 17408 T + 3107203610 T^{2} - 59824183442672 T^{3} + 5726081070022748105 T^{4} - \)\(10\!\cdots\!64\)\( T^{5} + \)\(76\!\cdots\!32\)\( T^{6} - \)\(13\!\cdots\!44\)\( T^{7} + \)\(84\!\cdots\!70\)\( T^{8} - \)\(13\!\cdots\!80\)\( T^{9} + \)\(77\!\cdots\!00\)\( T^{10} - \)\(11\!\cdots\!80\)\( T^{11} + \)\(60\!\cdots\!70\)\( T^{12} - \)\(78\!\cdots\!44\)\( T^{13} + \)\(38\!\cdots\!32\)\( T^{14} - \)\(44\!\cdots\!64\)\( T^{15} + \)\(20\!\cdots\!05\)\( T^{16} - \)\(18\!\cdots\!72\)\( T^{17} + \)\(80\!\cdots\!10\)\( T^{18} - \)\(38\!\cdots\!08\)\( T^{19} + \)\(18\!\cdots\!01\)\( T^{20} \)
$67$ \( 1 - 2021 T + 3389673908 T^{2} + 95129046044495 T^{3} + 5951903574650911354 T^{4} + \)\(26\!\cdots\!45\)\( T^{5} + \)\(15\!\cdots\!46\)\( T^{6} + \)\(38\!\cdots\!75\)\( T^{7} + \)\(27\!\cdots\!93\)\( T^{8} + \)\(78\!\cdots\!74\)\( T^{9} + \)\(36\!\cdots\!28\)\( T^{10} + \)\(10\!\cdots\!18\)\( T^{11} + \)\(50\!\cdots\!57\)\( T^{12} + \)\(94\!\cdots\!25\)\( T^{13} + \)\(51\!\cdots\!46\)\( T^{14} + \)\(11\!\cdots\!15\)\( T^{15} + \)\(36\!\cdots\!46\)\( T^{16} + \)\(77\!\cdots\!85\)\( T^{17} + \)\(37\!\cdots\!08\)\( T^{18} - \)\(30\!\cdots\!47\)\( T^{19} + \)\(20\!\cdots\!49\)\( T^{20} \)
$71$ \( 1 + 11286 T + 12540359910 T^{2} + 44404438984650 T^{3} + 71277357843109037437 T^{4} - \)\(33\!\cdots\!36\)\( T^{5} + \)\(24\!\cdots\!24\)\( T^{6} - \)\(30\!\cdots\!96\)\( T^{7} + \)\(61\!\cdots\!22\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{9} + \)\(12\!\cdots\!84\)\( T^{10} - \)\(18\!\cdots\!68\)\( T^{11} + \)\(20\!\cdots\!22\)\( T^{12} - \)\(17\!\cdots\!96\)\( T^{13} + \)\(26\!\cdots\!24\)\( T^{14} - \)\(63\!\cdots\!36\)\( T^{15} + \)\(24\!\cdots\!37\)\( T^{16} + \)\(27\!\cdots\!50\)\( T^{17} + \)\(14\!\cdots\!10\)\( T^{18} + \)\(22\!\cdots\!86\)\( T^{19} + \)\(36\!\cdots\!01\)\( T^{20} \)
$73$ \( 1 - 49892 T + 9676771082 T^{2} - 435365481214596 T^{3} + 51788415419782211185 T^{4} - \)\(22\!\cdots\!88\)\( T^{5} + \)\(19\!\cdots\!88\)\( T^{6} - \)\(79\!\cdots\!24\)\( T^{7} + \)\(57\!\cdots\!14\)\( T^{8} - \)\(21\!\cdots\!32\)\( T^{9} + \)\(13\!\cdots\!60\)\( T^{10} - \)\(44\!\cdots\!76\)\( T^{11} + \)\(24\!\cdots\!86\)\( T^{12} - \)\(71\!\cdots\!68\)\( T^{13} + \)\(36\!\cdots\!88\)\( T^{14} - \)\(85\!\cdots\!84\)\( T^{15} + \)\(41\!\cdots\!65\)\( T^{16} - \)\(71\!\cdots\!72\)\( T^{17} + \)\(33\!\cdots\!82\)\( T^{18} - \)\(35\!\cdots\!56\)\( T^{19} + \)\(14\!\cdots\!49\)\( T^{20} \)
$79$ \( 1 - 91524 T + 16857154847 T^{2} - 842558519651240 T^{3} + 99200238774687370926 T^{4} - \)\(23\!\cdots\!36\)\( T^{5} + \)\(37\!\cdots\!90\)\( T^{6} - \)\(59\!\cdots\!56\)\( T^{7} + \)\(16\!\cdots\!17\)\( T^{8} - \)\(38\!\cdots\!04\)\( T^{9} + \)\(63\!\cdots\!18\)\( T^{10} - \)\(11\!\cdots\!96\)\( T^{11} + \)\(15\!\cdots\!17\)\( T^{12} - \)\(17\!\cdots\!44\)\( T^{13} + \)\(33\!\cdots\!90\)\( T^{14} - \)\(65\!\cdots\!64\)\( T^{15} + \)\(84\!\cdots\!26\)\( T^{16} - \)\(22\!\cdots\!60\)\( T^{17} + \)\(13\!\cdots\!47\)\( T^{18} - \)\(22\!\cdots\!76\)\( T^{19} + \)\(76\!\cdots\!01\)\( T^{20} \)
$83$ \( 1 - 105203 T + 28621889044 T^{2} - 2443253098090515 T^{3} + \)\(38\!\cdots\!90\)\( T^{4} - \)\(27\!\cdots\!61\)\( T^{5} + \)\(31\!\cdots\!66\)\( T^{6} - \)\(19\!\cdots\!11\)\( T^{7} + \)\(18\!\cdots\!57\)\( T^{8} - \)\(99\!\cdots\!22\)\( T^{9} + \)\(83\!\cdots\!32\)\( T^{10} - \)\(39\!\cdots\!46\)\( T^{11} + \)\(29\!\cdots\!93\)\( T^{12} - \)\(11\!\cdots\!77\)\( T^{13} + \)\(76\!\cdots\!66\)\( T^{14} - \)\(25\!\cdots\!23\)\( T^{15} + \)\(14\!\cdots\!10\)\( T^{16} - \)\(35\!\cdots\!05\)\( T^{17} + \)\(16\!\cdots\!44\)\( T^{18} - \)\(24\!\cdots\!29\)\( T^{19} + \)\(89\!\cdots\!49\)\( T^{20} \)
$89$ \( 1 + 62682 T + 34400806898 T^{2} + 1591902274833594 T^{3} + \)\(57\!\cdots\!17\)\( T^{4} + \)\(20\!\cdots\!96\)\( T^{5} + \)\(63\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!12\)\( T^{7} + \)\(50\!\cdots\!94\)\( T^{8} + \)\(10\!\cdots\!68\)\( T^{9} + \)\(31\!\cdots\!32\)\( T^{10} + \)\(60\!\cdots\!32\)\( T^{11} + \)\(15\!\cdots\!94\)\( T^{12} + \)\(29\!\cdots\!88\)\( T^{13} + \)\(61\!\cdots\!68\)\( T^{14} + \)\(10\!\cdots\!04\)\( T^{15} + \)\(17\!\cdots\!17\)\( T^{16} + \)\(26\!\cdots\!06\)\( T^{17} + \)\(32\!\cdots\!98\)\( T^{18} + \)\(33\!\cdots\!18\)\( T^{19} + \)\(29\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - 108383 T + 55022053811 T^{2} - 4727913772355050 T^{3} + \)\(13\!\cdots\!17\)\( T^{4} - \)\(99\!\cdots\!31\)\( T^{5} + \)\(22\!\cdots\!35\)\( T^{6} - \)\(13\!\cdots\!96\)\( T^{7} + \)\(26\!\cdots\!83\)\( T^{8} - \)\(14\!\cdots\!33\)\( T^{9} + \)\(25\!\cdots\!28\)\( T^{10} - \)\(12\!\cdots\!81\)\( T^{11} + \)\(19\!\cdots\!67\)\( T^{12} - \)\(88\!\cdots\!28\)\( T^{13} + \)\(12\!\cdots\!35\)\( T^{14} - \)\(46\!\cdots\!67\)\( T^{15} + \)\(55\!\cdots\!33\)\( T^{16} - \)\(16\!\cdots\!50\)\( T^{17} + \)\(16\!\cdots\!11\)\( T^{18} - \)\(27\!\cdots\!31\)\( T^{19} + \)\(21\!\cdots\!49\)\( T^{20} \)
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