Properties

Label 38.7.b.a
Level 38
Weight 7
Character orbit 38.b
Analytic conductor 8.742
Analytic rank 0
Dimension 10
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 38.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.74205517755\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 5050 x^{8} + 7354489 x^{6} + 2475755792 x^{4} + 232626987584 x^{2} + 2900002611200\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{11} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{6} q^{2} + ( \beta_{5} + \beta_{6} ) q^{3} -32 q^{4} + ( -11 - \beta_{1} ) q^{5} + ( 16 - \beta_{1} + \beta_{3} ) q^{6} + ( -23 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + 32 \beta_{6} q^{8} + ( -288 - \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{9} +O(q^{10})\) \( q -\beta_{6} q^{2} + ( \beta_{5} + \beta_{6} ) q^{3} -32 q^{4} + ( -11 - \beta_{1} ) q^{5} + ( 16 - \beta_{1} + \beta_{3} ) q^{6} + ( -23 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{7} + 32 \beta_{6} q^{8} + ( -288 - \beta_{1} + \beta_{2} - 4 \beta_{3} - 3 \beta_{4} ) q^{9} + ( -10 \beta_{5} + 6 \beta_{6} + \beta_{9} ) q^{10} + ( 364 + 6 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + \beta_{4} ) q^{11} + ( -32 \beta_{5} - 32 \beta_{6} ) q^{12} + ( 7 \beta_{5} - 19 \beta_{6} - \beta_{7} + 3 \beta_{8} + 3 \beta_{9} ) q^{13} + ( -5 \beta_{5} + 20 \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{14} + ( -45 \beta_{5} + 272 \beta_{6} - 5 \beta_{8} + 2 \beta_{9} ) q^{15} + 1024 q^{16} + ( -1048 + 18 \beta_{1} + \beta_{2} + 12 \beta_{3} - 15 \beta_{4} ) q^{17} + ( 33 \beta_{5} + 301 \beta_{6} - \beta_{7} - 12 \beta_{8} + 9 \beta_{9} ) q^{18} + ( -1719 - 20 \beta_{1} + 15 \beta_{2} - 4 \beta_{4} - 7 \beta_{5} + 72 \beta_{6} - 3 \beta_{7} - 7 \beta_{8} + 4 \beta_{9} ) q^{19} + ( 352 + 32 \beta_{1} ) q^{20} + ( 34 \beta_{5} + 187 \beta_{6} - \beta_{7} + 10 \beta_{8} - \beta_{9} ) q^{21} + ( 144 \beta_{5} - 294 \beta_{6} - 6 \beta_{7} + 4 \beta_{8} + 3 \beta_{9} ) q^{22} + ( 3785 - 98 \beta_{1} + 7 \beta_{2} + 29 \beta_{3} + 13 \beta_{4} ) q^{23} + ( -512 + 32 \beta_{1} - 32 \beta_{3} ) q^{24} + ( -5204 + 11 \beta_{1} + 30 \beta_{2} - 30 \beta_{3} + 10 \beta_{4} ) q^{25} + ( -728 + 61 \beta_{1} - 32 \beta_{2} + 27 \beta_{3} - 24 \beta_{4} ) q^{26} + ( -702 \beta_{5} - 1345 \beta_{6} + 16 \beta_{7} + 21 \beta_{8} - 12 \beta_{9} ) q^{27} + ( 736 - 64 \beta_{1} - 32 \beta_{2} - 32 \beta_{3} ) q^{28} + ( 216 \beta_{5} + 1497 \beta_{6} + 15 \beta_{7} - 36 \beta_{8} + 9 \beta_{9} ) q^{29} + ( 9464 + 44 \beta_{1} - 20 \beta_{3} + 40 \beta_{4} ) q^{30} + ( 315 \beta_{5} + 370 \beta_{6} + 10 \beta_{7} + 55 \beta_{8} - 64 \beta_{9} ) q^{31} -1024 \beta_{6} q^{32} + ( 273 \beta_{5} - 4258 \beta_{6} + 103 \beta_{8} - 70 \beta_{9} ) q^{33} + ( -297 \beta_{5} + 884 \beta_{6} - \beta_{7} - 60 \beta_{8} - 14 \beta_{9} ) q^{34} + ( -16160 - 60 \beta_{1} - 70 \beta_{2} + 15 \beta_{4} ) q^{35} + ( 9216 + 32 \beta_{1} - 32 \beta_{2} + 128 \beta_{3} + 96 \beta_{4} ) q^{36} + ( 1247 \beta_{5} - 320 \beta_{6} + 16 \beta_{7} - 39 \beta_{8} + 120 \beta_{9} ) q^{37} + ( 2520 - 43 \beta_{1} - 96 \beta_{2} + 19 \beta_{3} + 56 \beta_{4} - 301 \beta_{5} + 1557 \beta_{6} - 15 \beta_{7} - 16 \beta_{8} + 39 \beta_{9} ) q^{38} + ( -7815 - 300 \beta_{1} - 69 \beta_{2} - 63 \beta_{3} - 45 \beta_{4} ) q^{39} + ( 320 \beta_{5} - 192 \beta_{6} - 32 \beta_{9} ) q^{40} + ( 843 \beta_{5} - 998 \beta_{6} - 14 \beta_{7} - 77 \beta_{8} - 112 \beta_{9} ) q^{41} + ( 5376 + 9 \beta_{1} - 32 \beta_{2} + 7 \beta_{3} - 80 \beta_{4} ) q^{42} + ( 550 - 12 \beta_{1} - 40 \beta_{2} + 416 \beta_{3} + 55 \beta_{4} ) q^{43} + ( -11648 - 192 \beta_{1} - 192 \beta_{2} + 128 \beta_{3} - 32 \beta_{4} ) q^{44} + ( 31029 + 229 \beta_{1} + 140 \beta_{2} - 470 \beta_{3} + 120 \beta_{4} ) q^{45} + ( -1457 \beta_{5} - 4504 \beta_{6} - 7 \beta_{7} + 52 \beta_{8} + 63 \beta_{9} ) q^{46} + ( 32100 + 464 \beta_{1} + 146 \beta_{2} + 146 \beta_{3} + 169 \beta_{4} ) q^{47} + ( 1024 \beta_{5} + 1024 \beta_{6} ) q^{48} + ( -23603 + 270 \beta_{1} - 119 \beta_{2} - 140 \beta_{3} - 283 \beta_{4} ) q^{49} + ( 820 \beta_{5} + 5609 \beta_{6} - 30 \beta_{7} + 40 \beta_{8} + 39 \beta_{9} ) q^{50} + ( -4101 \beta_{5} + 6725 \beta_{6} + 48 \beta_{7} - 32 \beta_{8} + 134 \beta_{9} ) q^{51} + ( -224 \beta_{5} + 608 \beta_{6} + 32 \beta_{7} - 96 \beta_{8} - 96 \beta_{9} ) q^{52} + ( 24 \beta_{5} - 3061 \beta_{6} - 97 \beta_{7} + 74 \beta_{8} - 83 \beta_{9} ) q^{53} + ( -32232 + 1027 \beta_{1} + 512 \beta_{2} - 731 \beta_{3} - 168 \beta_{4} ) q^{54} + ( -37440 - 1570 \beta_{1} + 70 \beta_{2} - 170 \beta_{3} - 225 \beta_{4} ) q^{55} + ( 160 \beta_{5} - 640 \beta_{6} + 32 \beta_{7} + 64 \beta_{9} ) q^{56} + ( 2385 + 987 \beta_{1} + 243 \beta_{2} - 798 \beta_{3} - 213 \beta_{4} - 4453 \beta_{5} + 5696 \beta_{6} + 16 \beta_{7} - 7 \beta_{8} + 4 \beta_{9} ) q^{57} + ( 44496 + 33 \beta_{1} + 480 \beta_{2} + 447 \beta_{3} + 288 \beta_{4} ) q^{58} + ( 1458 \beta_{5} + 11195 \beta_{6} - 64 \beta_{7} + 11 \beta_{8} + 358 \beta_{9} ) q^{59} + ( 1440 \beta_{5} - 8704 \beta_{6} + 160 \beta_{8} - 64 \beta_{9} ) q^{60} + ( 42425 - 463 \beta_{1} + 20 \beta_{2} + 1490 \beta_{3} + 140 \beta_{4} ) q^{61} + ( 6200 - 978 \beta_{1} + 320 \beta_{2} - 310 \beta_{3} - 440 \beta_{4} ) q^{62} + ( -52062 + 454 \beta_{1} + 416 \beta_{2} + 898 \beta_{3} - 117 \beta_{4} ) q^{63} -32768 q^{64} + ( -2940 \beta_{5} - 23852 \beta_{6} - 110 \beta_{7} - 230 \beta_{8} + 518 \beta_{9} ) q^{65} + ( -141448 - 886 \beta_{1} - 530 \beta_{3} - 824 \beta_{4} ) q^{66} + ( 6373 \beta_{5} + 7785 \beta_{6} - 54 \beta_{7} + 274 \beta_{8} - 478 \beta_{9} ) q^{67} + ( 33536 - 576 \beta_{1} - 32 \beta_{2} - 384 \beta_{3} + 480 \beta_{4} ) q^{68} + ( 5151 \beta_{5} + 49523 \beta_{6} - 81 \beta_{7} - 605 \beta_{8} + 293 \beta_{9} ) q^{69} + ( -180 \beta_{5} + 16120 \beta_{6} + 70 \beta_{7} + 60 \beta_{8} - 25 \beta_{9} ) q^{70} + ( 66 \beta_{5} + 1214 \beta_{6} - 10 \beta_{7} - 754 \beta_{8} + 94 \beta_{9} ) q^{71} + ( -1056 \beta_{5} - 9632 \beta_{6} + 32 \beta_{7} + 384 \beta_{8} - 288 \beta_{9} ) q^{72} + ( -78458 + 1134 \beta_{1} - 1129 \beta_{2} - 1882 \beta_{3} + 595 \beta_{4} ) q^{73} + ( -30136 + 1442 \beta_{1} + 512 \beta_{2} + 2598 \beta_{3} + 312 \beta_{4} ) q^{74} + ( -5410 \beta_{5} - 31597 \beta_{6} - 10 \beta_{7} + 505 \beta_{8} - 412 \beta_{9} ) q^{75} + ( 55008 + 640 \beta_{1} - 480 \beta_{2} + 128 \beta_{4} + 224 \beta_{5} - 2304 \beta_{6} + 96 \beta_{7} + 224 \beta_{8} - 128 \beta_{9} ) q^{76} + ( 230075 + 1945 \beta_{1} - 1020 \beta_{2} - 462 \beta_{3} - 540 \beta_{4} ) q^{77} + ( -2037 \beta_{5} + 6786 \beta_{6} + 69 \beta_{7} - 180 \beta_{8} + 339 \beta_{9} ) q^{78} + ( 395 \beta_{5} - 62530 \beta_{6} + 128 \beta_{7} + 325 \beta_{8} - 226 \beta_{9} ) q^{79} + ( -11264 - 1024 \beta_{1} ) q^{80} + ( 526143 - 3478 \beta_{1} - 674 \beta_{2} + 3308 \beta_{3} + 1146 \beta_{4} ) q^{81} + ( -44584 - 4350 \beta_{1} - 448 \beta_{2} - 298 \beta_{3} + 616 \beta_{4} ) q^{82} + ( -9862 + 1798 \beta_{1} - 974 \beta_{2} - 3238 \beta_{3} - 752 \beta_{4} ) q^{83} + ( -1088 \beta_{5} - 5984 \beta_{6} + 32 \beta_{7} - 320 \beta_{8} + 32 \beta_{9} ) q^{84} + ( -126575 + 3245 \beta_{1} - 1030 \beta_{2} - 1060 \beta_{3} + 1310 \beta_{4} ) q^{85} + ( -8382 \beta_{5} - 4666 \beta_{6} + 40 \beta_{7} + 220 \beta_{8} - 499 \beta_{9} ) q^{86} + ( -246447 + 5034 \beta_{1} + 1599 \beta_{2} - 3165 \beta_{3} + 315 \beta_{4} ) q^{87} + ( -4608 \beta_{5} + 9408 \beta_{6} + 192 \beta_{7} - 128 \beta_{8} - 96 \beta_{9} ) q^{88} + ( 5667 \beta_{5} + 39092 \beta_{6} - 316 \beta_{7} + 11 \beta_{8} - 470 \beta_{9} ) q^{89} + ( 13890 \beta_{5} - 24034 \beta_{6} - 140 \beta_{7} + 480 \beta_{8} + 261 \beta_{9} ) q^{90} + ( 8644 \beta_{5} + 18579 \beta_{6} + 582 \beta_{7} + 157 \beta_{8} - 274 \beta_{9} ) q^{91} + ( -121120 + 3136 \beta_{1} - 224 \beta_{2} - 928 \beta_{3} - 416 \beta_{4} ) q^{92} + ( -282150 - 7722 \beta_{1} - 2070 \beta_{2} + 4956 \beta_{3} - 450 \beta_{4} ) q^{93} + ( 3356 \beta_{5} - 30326 \beta_{6} - 146 \beta_{7} + 676 \beta_{8} - 633 \beta_{9} ) q^{94} + ( 311046 + 6 \beta_{1} + 860 \beta_{2} - 2090 \beta_{3} + 575 \beta_{4} + 7680 \beta_{5} - 3890 \beta_{6} - 20 \beta_{7} + 460 \beta_{8} + 470 \beta_{9} ) q^{95} + ( 16384 - 1024 \beta_{1} + 1024 \beta_{3} ) q^{96} + ( -11734 \beta_{5} - 66262 \beta_{6} + 356 \beta_{7} - 488 \beta_{8} - 376 \beta_{9} ) q^{97} + ( 2175 \beta_{5} + 24467 \beta_{6} + 119 \beta_{7} - 1132 \beta_{8} + 34 \beta_{9} ) q^{98} + ( 107058 - 13226 \beta_{1} + 548 \beta_{2} + 8650 \beta_{3} - 69 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 320q^{4} - 112q^{5} + 160q^{6} - 224q^{7} - 2890q^{9} + O(q^{10}) \) \( 10q - 320q^{4} - 112q^{5} + 160q^{6} - 224q^{7} - 2890q^{9} + 3644q^{11} + 10240q^{16} - 10420q^{17} - 17230q^{19} + 3584q^{20} + 37712q^{23} - 5120q^{24} - 52078q^{25} - 7104q^{26} + 7168q^{28} + 94688q^{30} - 161720q^{35} + 92480q^{36} + 25152q^{38} - 78876q^{39} + 53792q^{42} + 6308q^{43} - 116608q^{44} + 309808q^{45} + 322220q^{47} - 235770q^{49} - 321728q^{54} - 377880q^{55} + 24228q^{57} + 445920q^{58} + 426304q^{61} + 59424q^{62} - 517916q^{63} - 327680q^{64} - 1417312q^{66} + 333440q^{68} - 786076q^{73} - 293280q^{74} + 551360q^{76} + 2303716q^{77} - 114688q^{80} + 5261090q^{81} - 455136q^{82} - 101500q^{83} - 1261380q^{85} - 2460732q^{87} - 1206784q^{92} - 2827032q^{93} + 3106292q^{95} + 163840q^{96} + 1061428q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 5050 x^{8} + 7354489 x^{6} + 2475755792 x^{4} + 232626987584 x^{2} + 2900002611200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -1003849 \nu^{8} - 4834130050 \nu^{6} - 6581721953521 \nu^{4} - 1845042240099528 \nu^{2} - 102497988859192320 \)\()/ 486640529977344 \)
\(\beta_{2}\)\(=\)\((\)\( -3036701 \nu^{8} - 14052901482 \nu^{6} - 16755790470373 \nu^{4} - 1011447157780712 \nu^{2} + 325693833756348416 \)\()/ 1155771258696192 \)
\(\beta_{3}\)\(=\)\((\)\( -80563009 \nu^{8} - 413005498610 \nu^{6} - 584839838800809 \nu^{4} - 143692636451786888 \nu^{2} - 5761655028472529920 \)\()/ 27738510208708608 \)
\(\beta_{4}\)\(=\)\((\)\( 94415663 \nu^{8} + 484279228654 \nu^{6} + 700898949271623 \nu^{4} + 196466494243753720 \nu^{2} + 2932743674976963584 \)\()/ 27738510208708608 \)
\(\beta_{5}\)\(=\)\((\)\( -128881 \nu^{9} - 758922410 \nu^{7} - 1584249446409 \nu^{5} - 1289835989850272 \nu^{3} + 48500090563401536 \nu \)\()/ 256840585421679360 \)
\(\beta_{6}\)\(=\)\((\)\( 128881 \nu^{9} + 758922410 \nu^{7} + 1584249446409 \nu^{5} + 1289835989850272 \nu^{3} + 208340494858277824 \nu \)\()/ 128420292710839680 \)
\(\beta_{7}\)\(=\)\((\)\(2748818487 \nu^{9} + 13702415612990 \nu^{7} + 19467457749725199 \nu^{5} + 6337401708404491000 \nu^{3} + 1269965551955694627328 \nu\)\()/ 5798889661965027328 \)
\(\beta_{8}\)\(=\)\((\)\(-83926954043 \nu^{9} - 412231451711590 \nu^{7} - 565091498731173747 \nu^{5} - 143486961555925440376 \nu^{3} - 6054794255099484799232 \nu\)\()/ 65237508697106557440 \)
\(\beta_{9}\)\(=\)\((\)\(-97505079905 \nu^{9} - 489205973118514 \nu^{7} - 699063050265817545 \nu^{5} - 213332869018261197448 \nu^{3} - 14636876727188374780928 \nu\)\()/ 52190006957685245952 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + 2 \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\(-3 \beta_{4} - 3 \beta_{3} + \beta_{2} - 2 \beta_{1} - 1009\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{9} + 6 \beta_{8} + 29 \beta_{7} - 2556 \beta_{6} - 4269 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\(7515 \beta_{4} + 7854 \beta_{3} - 1885 \beta_{2} + 3743 \beta_{1} + 2156385\)
\(\nu^{5}\)\(=\)\((\)\(-1113 \beta_{9} - 23766 \beta_{8} - 73943 \beta_{7} + 7288856 \beta_{6} + 9801079 \beta_{5}\)\()/2\)
\(\nu^{6}\)\(=\)\(-17515695 \beta_{4} - 19389633 \beta_{3} + 4436769 \beta_{2} - 7253208 \beta_{1} - 4953548237\)
\(\nu^{7}\)\(=\)\((\)\(-19152537 \beta_{9} + 85880598 \beta_{8} + 174468057 \beta_{7} - 19310070620 \beta_{6} - 22676527321 \beta_{5}\)\()/2\)
\(\nu^{8}\)\(=\)\(40590401331 \beta_{4} + 47391878700 \beta_{3} - 10844673557 \beta_{2} + 13578744841 \beta_{1} + 11468377737353\)
\(\nu^{9}\)\(=\)\((\)\(96438196491 \beta_{9} - 273620269878 \beta_{8} - 408662294491 \beta_{7} + 50067893094144 \beta_{6} + 52544592576699 \beta_{5}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\) \(21\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
47.4187i
16.6843i
3.82791i
11.5377i
48.7374i
48.7374i
11.5377i
3.82791i
16.6843i
47.4187i
5.65685i 44.5903i −32.0000 −145.013 −252.241 126.579 181.019i −1259.29 820.316i
37.2 5.65685i 13.8558i −32.0000 −9.64927 −78.3804 −472.908 181.019i 537.016 54.5845i
37.3 5.65685i 6.65634i −32.0000 146.942 37.6539 −183.624 181.019i 684.693 831.227i
37.4 5.65685i 14.3661i −32.0000 −88.1981 81.2670 443.109 181.019i 522.615 498.924i
37.5 5.65685i 51.5658i −32.0000 39.9185 291.700 −25.1565 181.019i −1930.03 225.813i
37.6 5.65685i 51.5658i −32.0000 39.9185 291.700 −25.1565 181.019i −1930.03 225.813i
37.7 5.65685i 14.3661i −32.0000 −88.1981 81.2670 443.109 181.019i 522.615 498.924i
37.8 5.65685i 6.65634i −32.0000 146.942 37.6539 −183.624 181.019i 684.693 831.227i
37.9 5.65685i 13.8558i −32.0000 −9.64927 −78.3804 −472.908 181.019i 537.016 54.5845i
37.10 5.65685i 44.5903i −32.0000 −145.013 −252.241 126.579 181.019i −1259.29 820.316i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.7.b.a 10
3.b odd 2 1 342.7.d.a 10
4.b odd 2 1 304.7.e.e 10
19.b odd 2 1 inner 38.7.b.a 10
57.d even 2 1 342.7.d.a 10
76.d even 2 1 304.7.e.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.7.b.a 10 1.a even 1 1 trivial
38.7.b.a 10 19.b odd 2 1 inner
304.7.e.e 10 4.b odd 2 1
304.7.e.e 10 76.d even 2 1
342.7.d.a 10 3.b odd 2 1
342.7.d.a 10 57.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 32 T^{2} )^{5} \)
$3$ \( 1 - 2200 T^{2} + 1631430 T^{4} - 515174238 T^{6} + 586589648553 T^{8} - 731587903612596 T^{10} + 311737789416654873 T^{12} - \)\(14\!\cdots\!78\)\( T^{14} + \)\(24\!\cdots\!30\)\( T^{16} - \)\(17\!\cdots\!00\)\( T^{18} + \)\(42\!\cdots\!01\)\( T^{20} \)
$5$ \( ( 1 + 56 T + 53650 T^{2} + 2237550 T^{3} + 1359300625 T^{4} + 43303587500 T^{5} + 21239072265625 T^{6} + 546276855468750 T^{7} + 204658508300781250 T^{8} + 3337860107421875000 T^{9} + \)\(93\!\cdots\!25\)\( T^{10} )^{2} \)
$7$ \( ( 1 + 112 T + 359337 T^{2} + 34246958 T^{3} + 62172914657 T^{4} + 5080318669710 T^{5} + 7314581236481393 T^{6} + 474021981438584558 T^{7} + \)\(58\!\cdots\!13\)\( T^{8} + \)\(21\!\cdots\!12\)\( T^{9} + \)\(22\!\cdots\!49\)\( T^{10} )^{2} \)
$11$ \( ( 1 - 1822 T + 3854200 T^{2} - 5223454506 T^{3} + 11147752733431 T^{4} - 14402631038211088 T^{5} + 19748923980189755791 T^{6} - \)\(16\!\cdots\!26\)\( T^{7} + \)\(21\!\cdots\!00\)\( T^{8} - \)\(17\!\cdots\!02\)\( T^{9} + \)\(17\!\cdots\!01\)\( T^{10} )^{2} \)
$13$ \( 1 - 17666128 T^{2} + 150473464968294 T^{4} - \)\(75\!\cdots\!02\)\( T^{6} + \)\(23\!\cdots\!57\)\( T^{8} - \)\(72\!\cdots\!08\)\( T^{10} + \)\(54\!\cdots\!17\)\( T^{12} - \)\(41\!\cdots\!22\)\( T^{14} + \)\(19\!\cdots\!54\)\( T^{16} - \)\(52\!\cdots\!88\)\( T^{18} + \)\(68\!\cdots\!01\)\( T^{20} \)
$17$ \( ( 1 + 5210 T + 51971035 T^{2} + 61430519544 T^{3} + 668715425929933 T^{4} - 1842471159207853078 T^{5} + \)\(16\!\cdots\!77\)\( T^{6} + \)\(35\!\cdots\!84\)\( T^{7} + \)\(73\!\cdots\!15\)\( T^{8} + \)\(17\!\cdots\!10\)\( T^{9} + \)\(81\!\cdots\!49\)\( T^{10} )^{2} \)
$19$ \( 1 + 17230 T + 148537497 T^{2} + 766830204512 T^{3} + 3600408038311166 T^{4} + 18553772959394864484 T^{5} + \)\(16\!\cdots\!46\)\( T^{6} + \)\(16\!\cdots\!32\)\( T^{7} + \)\(15\!\cdots\!77\)\( T^{8} + \)\(84\!\cdots\!30\)\( T^{9} + \)\(23\!\cdots\!01\)\( T^{10} \)
$23$ \( ( 1 - 18856 T + 571101934 T^{2} - 9065914883226 T^{3} + 164009711653394737 T^{4} - \)\(18\!\cdots\!76\)\( T^{5} + \)\(24\!\cdots\!93\)\( T^{6} - \)\(19\!\cdots\!46\)\( T^{7} + \)\(18\!\cdots\!46\)\( T^{8} - \)\(90\!\cdots\!96\)\( T^{9} + \)\(71\!\cdots\!49\)\( T^{10} )^{2} \)
$29$ \( 1 - 2295730648 T^{2} + 3022757912409924870 T^{4} - \)\(26\!\cdots\!30\)\( T^{6} + \)\(18\!\cdots\!13\)\( T^{8} - \)\(11\!\cdots\!52\)\( T^{10} + \)\(65\!\cdots\!33\)\( T^{12} - \)\(33\!\cdots\!30\)\( T^{14} + \)\(13\!\cdots\!70\)\( T^{16} - \)\(35\!\cdots\!28\)\( T^{18} + \)\(55\!\cdots\!01\)\( T^{20} \)
$31$ \( 1 - 2408322202 T^{2} + 3352416717263589885 T^{4} - \)\(38\!\cdots\!88\)\( T^{6} + \)\(45\!\cdots\!50\)\( T^{8} - \)\(46\!\cdots\!52\)\( T^{10} + \)\(35\!\cdots\!50\)\( T^{12} - \)\(24\!\cdots\!48\)\( T^{14} + \)\(16\!\cdots\!85\)\( T^{16} - \)\(92\!\cdots\!82\)\( T^{18} + \)\(30\!\cdots\!01\)\( T^{20} \)
$37$ \( 1 + 2697991190 T^{2} + 11569645850506827021 T^{4} - \)\(95\!\cdots\!16\)\( T^{6} - \)\(55\!\cdots\!38\)\( T^{8} - \)\(35\!\cdots\!32\)\( T^{10} - \)\(36\!\cdots\!78\)\( T^{12} - \)\(41\!\cdots\!76\)\( T^{14} + \)\(33\!\cdots\!61\)\( T^{16} + \)\(50\!\cdots\!90\)\( T^{18} + \)\(12\!\cdots\!01\)\( T^{20} \)
$41$ \( 1 - 13733644282 T^{2} + 81015381804592963677 T^{4} - \)\(19\!\cdots\!72\)\( T^{6} - \)\(42\!\cdots\!54\)\( T^{8} + \)\(45\!\cdots\!60\)\( T^{10} - \)\(95\!\cdots\!74\)\( T^{12} - \)\(98\!\cdots\!92\)\( T^{14} + \)\(93\!\cdots\!57\)\( T^{16} - \)\(35\!\cdots\!22\)\( T^{18} + \)\(58\!\cdots\!01\)\( T^{20} \)
$43$ \( ( 1 - 3154 T + 21492915636 T^{2} - 397463731677654 T^{3} + \)\(21\!\cdots\!67\)\( T^{4} - \)\(45\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!83\)\( T^{6} - \)\(15\!\cdots\!54\)\( T^{7} + \)\(54\!\cdots\!64\)\( T^{8} - \)\(50\!\cdots\!54\)\( T^{9} + \)\(10\!\cdots\!49\)\( T^{10} )^{2} \)
$47$ \( ( 1 - 161110 T + 48913956136 T^{2} - 5932854681494826 T^{3} + \)\(98\!\cdots\!11\)\( T^{4} - \)\(90\!\cdots\!28\)\( T^{5} + \)\(10\!\cdots\!19\)\( T^{6} - \)\(68\!\cdots\!66\)\( T^{7} + \)\(61\!\cdots\!04\)\( T^{8} - \)\(21\!\cdots\!10\)\( T^{9} + \)\(14\!\cdots\!49\)\( T^{10} )^{2} \)
$53$ \( 1 - 128223896920 T^{2} + \)\(80\!\cdots\!82\)\( T^{4} - \)\(33\!\cdots\!10\)\( T^{6} + \)\(10\!\cdots\!21\)\( T^{8} - \)\(26\!\cdots\!80\)\( T^{10} + \)\(51\!\cdots\!61\)\( T^{12} - \)\(81\!\cdots\!10\)\( T^{14} + \)\(96\!\cdots\!22\)\( T^{16} - \)\(74\!\cdots\!20\)\( T^{18} + \)\(28\!\cdots\!01\)\( T^{20} \)
$59$ \( 1 - 227814608464 T^{2} + \)\(27\!\cdots\!38\)\( T^{4} - \)\(22\!\cdots\!38\)\( T^{6} + \)\(13\!\cdots\!09\)\( T^{8} - \)\(65\!\cdots\!72\)\( T^{10} + \)\(24\!\cdots\!29\)\( T^{12} - \)\(70\!\cdots\!18\)\( T^{14} + \)\(15\!\cdots\!58\)\( T^{16} - \)\(22\!\cdots\!44\)\( T^{18} + \)\(17\!\cdots\!01\)\( T^{20} \)
$61$ \( ( 1 - 213152 T + 149383123554 T^{2} - 43098614383216666 T^{3} + \)\(11\!\cdots\!33\)\( T^{4} - \)\(31\!\cdots\!88\)\( T^{5} + \)\(61\!\cdots\!13\)\( T^{6} - \)\(11\!\cdots\!86\)\( T^{7} + \)\(20\!\cdots\!74\)\( T^{8} - \)\(15\!\cdots\!32\)\( T^{9} + \)\(36\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( 1 - 396287882632 T^{2} + \)\(73\!\cdots\!06\)\( T^{4} - \)\(82\!\cdots\!98\)\( T^{6} + \)\(68\!\cdots\!13\)\( T^{8} - \)\(55\!\cdots\!12\)\( T^{10} + \)\(56\!\cdots\!93\)\( T^{12} - \)\(55\!\cdots\!58\)\( T^{14} + \)\(40\!\cdots\!86\)\( T^{16} - \)\(17\!\cdots\!12\)\( T^{18} + \)\(36\!\cdots\!01\)\( T^{20} \)
$71$ \( 1 - 706078923010 T^{2} + \)\(26\!\cdots\!37\)\( T^{4} - \)\(68\!\cdots\!92\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(18\!\cdots\!64\)\( T^{10} + \)\(21\!\cdots\!26\)\( T^{12} - \)\(18\!\cdots\!52\)\( T^{14} + \)\(11\!\cdots\!77\)\( T^{16} - \)\(51\!\cdots\!10\)\( T^{18} + \)\(11\!\cdots\!01\)\( T^{20} \)
$73$ \( ( 1 + 393038 T + 262845231891 T^{2} + 80521825338052188 T^{3} + \)\(48\!\cdots\!17\)\( T^{4} + \)\(19\!\cdots\!18\)\( T^{5} + \)\(73\!\cdots\!13\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(91\!\cdots\!79\)\( T^{8} + \)\(20\!\cdots\!58\)\( T^{9} + \)\(79\!\cdots\!49\)\( T^{10} )^{2} \)
$79$ \( 1 - 1563370816666 T^{2} + \)\(12\!\cdots\!89\)\( T^{4} - \)\(62\!\cdots\!48\)\( T^{6} + \)\(23\!\cdots\!98\)\( T^{8} - \)\(64\!\cdots\!48\)\( T^{10} + \)\(13\!\cdots\!18\)\( T^{12} - \)\(21\!\cdots\!88\)\( T^{14} + \)\(25\!\cdots\!69\)\( T^{16} - \)\(19\!\cdots\!26\)\( T^{18} + \)\(72\!\cdots\!01\)\( T^{20} \)
$83$ \( ( 1 + 50750 T + 830672751385 T^{2} + 45811815263701056 T^{3} + \)\(33\!\cdots\!98\)\( T^{4} + \)\(24\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!62\)\( T^{6} + \)\(48\!\cdots\!16\)\( T^{7} + \)\(29\!\cdots\!65\)\( T^{8} + \)\(57\!\cdots\!50\)\( T^{9} + \)\(37\!\cdots\!49\)\( T^{10} )^{2} \)
$89$ \( 1 - 3396603770530 T^{2} + \)\(57\!\cdots\!37\)\( T^{4} - \)\(61\!\cdots\!32\)\( T^{6} + \)\(47\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!84\)\( T^{10} + \)\(11\!\cdots\!06\)\( T^{12} - \)\(37\!\cdots\!12\)\( T^{14} + \)\(86\!\cdots\!57\)\( T^{16} - \)\(12\!\cdots\!30\)\( T^{18} + \)\(91\!\cdots\!01\)\( T^{20} \)
$97$ \( 1 - 5659092169474 T^{2} + \)\(15\!\cdots\!21\)\( T^{4} - \)\(28\!\cdots\!80\)\( T^{6} + \)\(37\!\cdots\!34\)\( T^{8} - \)\(35\!\cdots\!52\)\( T^{10} + \)\(25\!\cdots\!94\)\( T^{12} - \)\(13\!\cdots\!80\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} - \)\(13\!\cdots\!14\)\( T^{18} + \)\(16\!\cdots\!01\)\( T^{20} \)
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