Properties

Label 10.11.c.c
Level 10
Weight 11
Character orbit 10.c
Analytic conductor 6.354
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 11 \)
Character orbit: \([\chi]\) = 10.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.35357252674\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -16 - 16 \beta_{1} ) q^{2} + ( 21 - 21 \beta_{1} + \beta_{2} ) q^{3} + 512 \beta_{1} q^{4} + ( 910 - 896 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{5} + ( -672 - 16 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 2244 + 2244 \beta_{1} + 19 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{7} + ( 8192 - 8192 \beta_{1} ) q^{8} + ( -59175 \beta_{1} + 61 \beta_{2} - 61 \beta_{3} + 30 \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -16 - 16 \beta_{1} ) q^{2} + ( 21 - 21 \beta_{1} + \beta_{2} ) q^{3} + 512 \beta_{1} q^{4} + ( 910 - 896 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{5} + ( -672 - 16 \beta_{2} - 16 \beta_{3} ) q^{6} + ( 2244 + 2244 \beta_{1} + 19 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{7} + ( 8192 - 8192 \beta_{1} ) q^{8} + ( -59175 \beta_{1} + 61 \beta_{2} - 61 \beta_{3} + 30 \beta_{5} ) q^{9} + ( -28896 - 224 \beta_{1} + 48 \beta_{2} - 16 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} ) q^{10} + ( 108144 - 263 \beta_{2} - 263 \beta_{3} + 10 \beta_{4} ) q^{11} + ( 10752 + 10752 \beta_{1} + 512 \beta_{3} ) q^{12} + ( -123466 + 123466 \beta_{1} - 988 \beta_{2} - 65 \beta_{4} + 65 \beta_{5} ) q^{13} + ( -71808 \beta_{1} + 304 \beta_{2} - 304 \beta_{3} - 160 \beta_{5} ) q^{14} + ( 262272 + 52122 \beta_{1} + 2883 \beta_{2} - 776 \beta_{3} - 63 \beta_{4} - 27 \beta_{5} ) q^{15} -262144 q^{16} + ( -126945 - 126945 \beta_{1} + 3386 \beta_{3} - 110 \beta_{4} - 110 \beta_{5} ) q^{17} + ( -946800 + 946800 \beta_{1} - 1952 \beta_{2} + 480 \beta_{4} - 480 \beta_{5} ) q^{18} + ( -181986 \beta_{1} + 416 \beta_{2} - 416 \beta_{3} + 130 \beta_{5} ) q^{19} + ( 458752 + 465920 \beta_{1} - 1024 \beta_{2} - 512 \beta_{3} - 512 \beta_{5} ) q^{20} + ( 2050806 - 6661 \beta_{2} - 6661 \beta_{3} - 540 \beta_{4} ) q^{21} + ( -1730304 - 1730304 \beta_{1} + 8416 \beta_{3} - 160 \beta_{4} - 160 \beta_{5} ) q^{22} + ( -2511464 + 2511464 \beta_{1} + 8631 \beta_{2} - 735 \beta_{4} + 735 \beta_{5} ) q^{23} + ( -344064 \beta_{1} + 8192 \beta_{2} - 8192 \beta_{3} ) q^{24} + ( 7103915 - 1114930 \beta_{1} - 2485 \beta_{2} + 12895 \beta_{3} + 270 \beta_{4} + 1810 \beta_{5} ) q^{25} + ( 3950912 + 15808 \beta_{2} + 15808 \beta_{3} + 2080 \beta_{4} ) q^{26} + ( -7979328 - 7979328 \beta_{1} - 62996 \beta_{3} + 1920 \beta_{4} + 1920 \beta_{5} ) q^{27} + ( -1148928 + 1148928 \beta_{1} - 9728 \beta_{2} - 2560 \beta_{4} + 2560 \beta_{5} ) q^{28} + ( -3270462 \beta_{1} - 50938 \beta_{2} + 50938 \beta_{3} + 1630 \beta_{5} ) q^{29} + ( -3362400 - 5030304 \beta_{1} - 58544 \beta_{2} - 33712 \beta_{3} + 576 \beta_{4} + 1440 \beta_{5} ) q^{30} + ( 25623276 + 26719 \beta_{2} + 26719 \beta_{3} + 310 \beta_{4} ) q^{31} + ( 4194304 + 4194304 \beta_{1} ) q^{32} + ( -28862862 + 28862862 \beta_{1} + 67774 \beta_{2} + 7920 \beta_{4} - 7920 \beta_{5} ) q^{33} + ( 4062240 \beta_{1} + 54176 \beta_{2} - 54176 \beta_{3} + 3520 \beta_{5} ) q^{34} + ( -34852363 - 33003937 \beta_{1} + 38204 \beta_{2} + 49457 \beta_{3} - 1792 \beta_{4} - 13002 \beta_{5} ) q^{35} + ( 30297600 + 31232 \beta_{2} + 31232 \beta_{3} - 15360 \beta_{4} ) q^{36} + ( -5431860 - 5431860 \beta_{1} + 21066 \beta_{3} - 12735 \beta_{4} - 12735 \beta_{5} ) q^{37} + ( -2911776 + 2911776 \beta_{1} - 13312 \beta_{2} + 2080 \beta_{4} - 2080 \beta_{5} ) q^{38} + ( 117571248 \beta_{1} - 37341 \beta_{2} + 37341 \beta_{3} - 29250 \beta_{5} ) q^{39} + ( 114688 - 14794752 \beta_{1} + 8192 \beta_{2} + 24576 \beta_{3} - 8192 \beta_{4} + 8192 \beta_{5} ) q^{40} + ( 37820468 - 204319 \beta_{2} - 204319 \beta_{3} + 23870 \beta_{4} ) q^{41} + ( -32812896 - 32812896 \beta_{1} + 213152 \beta_{3} + 8640 \beta_{4} + 8640 \beta_{5} ) q^{42} + ( -10535633 + 10535633 \beta_{1} - 185267 \beta_{2} - 22210 \beta_{4} + 22210 \beta_{5} ) q^{43} + ( 55369728 \beta_{1} + 134656 \beta_{2} - 134656 \beta_{3} + 5120 \beta_{5} ) q^{44} + ( -29090754 - 289958130 \beta_{1} + 350233 \beta_{2} - 101096 \beta_{3} + 23010 \beta_{4} + 27549 \beta_{5} ) q^{45} + ( 80366848 - 138096 \beta_{2} - 138096 \beta_{3} + 23520 \beta_{4} ) q^{46} + ( -3421866 - 3421866 \beta_{1} + 383139 \beta_{3} + 32235 \beta_{4} + 32235 \beta_{5} ) q^{47} + ( -5505024 + 5505024 \beta_{1} - 262144 \beta_{2} ) q^{48} + ( 134255115 \beta_{1} + 287639 \beta_{2} - 287639 \beta_{3} + 123910 \beta_{5} ) q^{49} + ( -131501520 - 95823760 \beta_{1} + 246080 \beta_{2} - 166560 \beta_{3} + 24640 \beta_{4} - 33280 \beta_{5} ) q^{50} + ( 397993002 + 221125 \beta_{2} + 221125 \beta_{3} - 102240 \beta_{4} ) q^{51} + ( -63214592 - 63214592 \beta_{1} - 505856 \beta_{3} - 33280 \beta_{4} - 33280 \beta_{5} ) q^{52} + ( -158256646 + 158256646 \beta_{1} - 264378 \beta_{2} + 33795 \beta_{4} - 33795 \beta_{5} ) q^{53} + ( 255338496 \beta_{1} - 1007936 \beta_{2} + 1007936 \beta_{3} - 61440 \beta_{5} ) q^{54} + ( -22013600 - 190298682 \beta_{1} - 1059677 \beta_{2} - 351921 \beta_{3} - 120142 \beta_{4} + 14170 \beta_{5} ) q^{55} + ( 36765696 + 155648 \beta_{2} + 155648 \beta_{3} + 81920 \beta_{4} ) q^{56} + ( -56184198 - 56184198 \beta_{1} - 466556 \beta_{3} + 12870 \beta_{4} + 12870 \beta_{5} ) q^{57} + ( -52327392 + 52327392 \beta_{1} + 1630016 \beta_{2} + 26080 \beta_{4} - 26080 \beta_{5} ) q^{58} + ( -1396934 \beta_{1} + 605484 \beta_{2} - 605484 \beta_{3} - 230490 \beta_{5} ) q^{59} + ( -26686464 + 134283264 \beta_{1} + 397312 \beta_{2} + 1476096 \beta_{3} + 13824 \beta_{4} - 32256 \beta_{5} ) q^{60} + ( 266705180 + 892993 \beta_{2} + 892993 \beta_{3} + 90670 \beta_{4} ) q^{61} + ( -409972416 - 409972416 \beta_{1} - 855008 \beta_{3} - 4960 \beta_{4} - 4960 \beta_{5} ) q^{62} + ( -856315332 + 856315332 \beta_{1} + 1439815 \beta_{2} - 97035 \beta_{4} + 97035 \beta_{5} ) q^{63} -134217728 \beta_{1} q^{64} + ( 233605367 - 349511323 \beta_{1} - 2857977 \beta_{2} + 835469 \beta_{3} + 294442 \beta_{4} - 82022 \beta_{5} ) q^{65} + ( 923611584 - 1084384 \beta_{2} - 1084384 \beta_{3} - 253440 \beta_{4} ) q^{66} + ( 417379583 + 417379583 \beta_{1} - 530403 \beta_{3} - 284490 \beta_{4} - 284490 \beta_{5} ) q^{67} + ( 64995840 - 64995840 \beta_{1} - 1733632 \beta_{2} + 56320 \beta_{4} - 56320 \beta_{5} ) q^{68} + ( -947419494 \beta_{1} - 745469 \beta_{2} + 745469 \beta_{3} + 263340 \beta_{5} ) q^{69} + ( 29574816 + 1085700800 \beta_{1} + 180048 \beta_{2} - 1402576 \beta_{3} - 179360 \beta_{4} + 236704 \beta_{5} ) q^{70} + ( -1190882180 + 1849663 \beta_{2} + 1849663 \beta_{3} + 180310 \beta_{4} ) q^{71} + ( -484761600 - 484761600 \beta_{1} - 999424 \beta_{3} + 245760 \beta_{4} + 245760 \beta_{5} ) q^{72} + ( 384852279 - 384852279 \beta_{1} - 2445988 \beta_{2} + 120370 \beta_{4} - 120370 \beta_{5} ) q^{73} + ( 173819520 \beta_{1} + 337056 \beta_{2} - 337056 \beta_{3} + 407520 \beta_{5} ) q^{74} + ( 1582122255 + 76966365 \beta_{1} + 6998405 \beta_{2} - 3998460 \beta_{3} - 380610 \beta_{4} - 69930 \beta_{5} ) q^{75} + ( 93176832 + 212992 \beta_{2} + 212992 \beta_{3} - 66560 \beta_{4} ) q^{76} + ( 97585072 + 97585072 \beta_{1} + 5376026 \beta_{3} + 853690 \beta_{4} + 853690 \beta_{5} ) q^{77} + ( 1881139968 - 1881139968 \beta_{1} + 1194912 \beta_{2} - 468000 \beta_{4} + 468000 \beta_{5} ) q^{78} + ( -948328216 \beta_{1} + 195656 \beta_{2} - 195656 \beta_{3} - 193640 \beta_{5} ) q^{79} + ( -238551040 + 234881024 \beta_{1} + 262144 \beta_{2} - 524288 \beta_{3} + 262144 \beta_{4} ) q^{80} + ( -4337792793 - 10608539 \beta_{2} - 10608539 \beta_{3} + 129930 \beta_{4} ) q^{81} + ( -605127488 - 605127488 \beta_{1} + 6538208 \beta_{3} - 381920 \beta_{4} - 381920 \beta_{5} ) q^{82} + ( -12859987 + 12859987 \beta_{1} - 5994475 \beta_{2} + 484120 \beta_{4} - 484120 \beta_{5} ) q^{83} + ( 1050012672 \beta_{1} + 3410432 \beta_{2} - 3410432 \beta_{3} - 276480 \beta_{5} ) q^{84} + ( 284105332 + 1721564848 \beta_{1} + 2583799 \beta_{2} + 9721417 \beta_{3} + 299593 \beta_{4} + 203853 \beta_{5} ) q^{85} + ( 337140256 + 2964272 \beta_{2} + 2964272 \beta_{3} + 710720 \beta_{4} ) q^{86} + ( 5863997874 + 5863997874 \beta_{1} - 2346212 \beta_{3} - 1523250 \beta_{4} - 1523250 \beta_{5} ) q^{87} + ( 885915648 - 885915648 \beta_{1} - 4308992 \beta_{2} + 81920 \beta_{4} - 81920 \beta_{5} ) q^{88} + ( -4350280900 \beta_{1} + 1011780 \beta_{2} - 1011780 \beta_{3} - 882300 \beta_{5} ) q^{89} + ( -4173878016 + 5104782144 \beta_{1} - 7221264 \beta_{2} - 3986192 \beta_{3} + 72624 \beta_{4} - 808944 \beta_{5} ) q^{90} + ( -7290495682 + 7179181 \beta_{2} + 7179181 \beta_{3} - 1998020 \beta_{4} ) q^{91} + ( -1285869568 - 1285869568 \beta_{1} + 4419072 \beta_{3} - 376320 \beta_{4} - 376320 \beta_{5} ) q^{92} + ( 3664888554 - 3664888554 \beta_{1} + 27161566 \beta_{2} - 800640 \beta_{4} + 800640 \beta_{5} ) q^{93} + ( 109499712 \beta_{1} + 6130224 \beta_{2} - 6130224 \beta_{3} - 1031520 \beta_{5} ) q^{94} + ( -5971980 - 1214821790 \beta_{1} + 1675620 \beta_{2} - 1073840 \beta_{3} + 86060 \beta_{4} + 44030 \beta_{5} ) q^{95} + ( 176160768 + 4194304 \beta_{2} + 4194304 \beta_{3} ) q^{96} + ( 5553402337 + 5553402337 \beta_{1} - 21140480 \beta_{3} + 1477760 \beta_{4} + 1477760 \beta_{5} ) q^{97} + ( 2148081840 - 2148081840 \beta_{1} - 9204448 \beta_{2} + 1982560 \beta_{4} - 1982560 \beta_{5} ) q^{98} + ( 77635512 \beta_{1} - 25931375 \beta_{2} + 25931375 \beta_{3} + 2576190 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 96q^{2} + 128q^{3} + 5460q^{5} - 4096q^{6} + 13512q^{7} + 49152q^{8} + O(q^{10}) \) \( 6q - 96q^{2} + 128q^{3} + 5460q^{5} - 4096q^{6} + 13512q^{7} + 49152q^{8} - 173280q^{10} + 647832q^{11} + 65536q^{12} - 742902q^{13} + 1577720q^{15} - 1572864q^{16} - 755118q^{17} - 5683744q^{18} + 2749440q^{20} + 12277112q^{21} - 10365312q^{22} - 15052992q^{23} + 42644850q^{25} + 23772864q^{26} - 47998120q^{27} - 6918144q^{28} - 20357760q^{30} + 153847152q^{31} + 25165824q^{32} - 173025784q^{33} - 208942440q^{35} + 181879808q^{36} - 32574498q^{37} - 17493120q^{38} + 737280q^{40} + 226153272q^{41} - 196433792q^{42} - 63628752q^{43} - 174000230q^{45} + 481695744q^{46} - 19700448q^{47} - 33554432q^{48} - 788800800q^{50} + 2388638032q^{51} - 380365824q^{52} - 950001042q^{53} - 135145080q^{55} + 221380608q^{56} - 338012560q^{57} - 310652160q^{58} - 156344320q^{60} + 1603984392q^{61} - 2461554432q^{62} - 5135206432q^{63} + 1398176070q^{65} + 5536825088q^{66} + 2502647712q^{67} + 386620416q^{68} + 174645120q^{70} - 7137533808q^{71} - 2910076928q^{72} + 2304462438q^{73} + 9497972200q^{75} + 559779840q^{76} + 597969864q^{77} + 11288293632q^{78} - 1431306240q^{80} - 26068931054q^{81} - 3618452352q^{82} - 88180632q^{83} + 1729841610q^{85} + 2036120064q^{86} + 35176248320q^{87} + 5307039744q^{88} - 25065537760q^{90} - 43718253408q^{91} - 7707131904q^{92} + 22042053176q^{93} - 34456200q^{95} + 1073741824q^{96} + 33281088582q^{97} + 12874047264q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 1148 x^{3} + 68121 x^{2} - 299628 x + 658952\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 68121 \nu^{5} + 149814 \nu^{4} + 329476 \nu^{3} - 39101454 \nu^{2} + 4554477405 \nu - 10394598718 \)\()/ 10016360270 \)
\(\beta_{2}\)\(=\)\((\)\( 60546049 \nu^{5} - 58729384 \nu^{4} + 638377544 \nu^{3} - 84835233476 \nu^{2} + 4158168070345 \nu - 18141291569772 \)\()/ 30049080810 \)
\(\beta_{3}\)\(=\)\((\)\( -60844529 \nu^{5} - 325695636 \nu^{4} - 716280824 \nu^{3} + 85006560996 \nu^{2} - 4158168070345 \nu + 822290385952 \)\()/ 30049080810 \)
\(\beta_{4}\)\(=\)\((\)\( -31840 \nu^{5} - 738610 \nu^{4} - 8310240 \nu^{3} + 18276160 \nu^{2} - 20819537159 \)\()/10470063\)
\(\beta_{5}\)\(=\)\((\)\( -517677579 \nu^{5} - 1138493986 \nu^{4} + 16684615276 \nu^{3} + 797964943846 \nu^{2} - 29603038676095 \nu + 67978379651882 \)\()/ 30049080810 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4} - 10 \beta_{3} + 3 \beta_{1} + 3\)\()/400\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{5} + 25 \beta_{3} - 25 \beta_{2} + 17383 \beta_{1}\)\()/100\)
\(\nu^{3}\)\(=\)\((\)\(261 \beta_{5} - 261 \beta_{4} + 2610 \beta_{2} - 228817 \beta_{1} + 228817\)\()/400\)
\(\nu^{4}\)\(=\)\((\)\(13 \beta_{4} - 3980 \beta_{3} - 3980 \beta_{2} - 2268051\)\()/50\)
\(\nu^{5}\)\(=\)\((\)\(-13165 \beta_{5} - 13165 \beta_{4} + 159202 \beta_{3} + 19926521 \beta_{1} + 19926521\)\()/80\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

\(n\) \(7\)
\(\chi(n)\) \(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
2.29143 2.29143i
10.1043 10.1043i
−12.3957 + 12.3957i
2.29143 + 2.29143i
10.1043 + 10.1043i
−12.3957 12.3957i
−16.0000 + 16.0000i −266.441 266.441i 512.000i 2583.39 + 1758.32i 8526.10 −13021.6 + 13021.6i 8192.00 + 8192.00i 82932.3i −69467.5 + 13201.2i
3.2 −16.0000 + 16.0000i 4.29207 + 4.29207i 512.000i −2978.33 + 946.124i −137.346 21284.7 21284.7i 8192.00 + 8192.00i 59012.2i 32515.4 62791.3i
3.3 −16.0000 + 16.0000i 326.149 + 326.149i 512.000i 3124.94 19.4459i −10436.8 −1507.13 + 1507.13i 8192.00 + 8192.00i 153697.i −49687.9 + 50310.2i
7.1 −16.0000 16.0000i −266.441 + 266.441i 512.000i 2583.39 1758.32i 8526.10 −13021.6 13021.6i 8192.00 8192.00i 82932.3i −69467.5 13201.2i
7.2 −16.0000 16.0000i 4.29207 4.29207i 512.000i −2978.33 946.124i −137.346 21284.7 + 21284.7i 8192.00 8192.00i 59012.2i 32515.4 + 62791.3i
7.3 −16.0000 16.0000i 326.149 326.149i 512.000i 3124.94 + 19.4459i −10436.8 −1507.13 1507.13i 8192.00 8192.00i 153697.i −49687.9 50310.2i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.11.c.c 6
3.b odd 2 1 90.11.g.c 6
4.b odd 2 1 80.11.p.c 6
5.b even 2 1 50.11.c.e 6
5.c odd 4 1 inner 10.11.c.c 6
5.c odd 4 1 50.11.c.e 6
15.e even 4 1 90.11.g.c 6
20.e even 4 1 80.11.p.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.11.c.c 6 1.a even 1 1 trivial
10.11.c.c 6 5.c odd 4 1 inner
50.11.c.e 6 5.b even 2 1
50.11.c.e 6 5.c odd 4 1
80.11.p.c 6 4.b odd 2 1
80.11.p.c 6 20.e even 4 1
90.11.g.c 6 3.b odd 2 1
90.11.g.c 6 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 128 T_{3}^{5} + 8192 T_{3}^{4} + 20688696 T_{3}^{3} + 30028037796 T_{3}^{2} - 258527462832 T_{3} + \)\(11\!\cdots\!72\)\( \) acting on \(S_{11}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 32 T + 512 T^{2} )^{3} \)
$3$ \( 1 - 128 T + 8192 T^{2} + 13130424 T^{3} - 441069057 T^{4} + 422886187368 T^{5} + 35687822941728 T^{6} + 24971006477893032 T^{7} - 1537912707711379857 T^{8} + \)\(27\!\cdots\!76\)\( T^{9} + \)\(99\!\cdots\!92\)\( T^{10} - \)\(91\!\cdots\!72\)\( T^{11} + \)\(42\!\cdots\!01\)\( T^{12} \)
$5$ \( 1 - 5460 T - 6416625 T^{2} + 85710975000 T^{3} - 62662353515625 T^{4} - 520706176757812500 T^{5} + \)\(93\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 13512 T + 91287072 T^{2} + 5680562811056 T^{3} - 79588467032526417 T^{4} - \)\(39\!\cdots\!08\)\( T^{5} + \)\(28\!\cdots\!88\)\( T^{6} - \)\(11\!\cdots\!92\)\( T^{7} - \)\(63\!\cdots\!17\)\( T^{8} + \)\(12\!\cdots\!44\)\( T^{9} + \)\(58\!\cdots\!72\)\( T^{10} - \)\(24\!\cdots\!88\)\( T^{11} + \)\(50\!\cdots\!01\)\( T^{12} \)
$11$ \( ( 1 - 323916 T + 86871291855 T^{2} - 14136588581335480 T^{3} + \)\(22\!\cdots\!55\)\( T^{4} - \)\(21\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!01\)\( T^{6} )^{2} \)
$13$ \( 1 + 742902 T + 275951690802 T^{2} + 104807015418799014 T^{3} - \)\(65\!\cdots\!77\)\( T^{4} - \)\(19\!\cdots\!52\)\( T^{5} - \)\(72\!\cdots\!52\)\( T^{6} - \)\(26\!\cdots\!48\)\( T^{7} - \)\(12\!\cdots\!77\)\( T^{8} + \)\(27\!\cdots\!86\)\( T^{9} + \)\(99\!\cdots\!02\)\( T^{10} + \)\(36\!\cdots\!98\)\( T^{11} + \)\(68\!\cdots\!01\)\( T^{12} \)
$17$ \( 1 + 755118 T + 285101596962 T^{2} + 2224663782493800046 T^{3} - \)\(91\!\cdots\!97\)\( T^{4} - \)\(64\!\cdots\!28\)\( T^{5} - \)\(21\!\cdots\!32\)\( T^{6} - \)\(12\!\cdots\!72\)\( T^{7} - \)\(37\!\cdots\!97\)\( T^{8} + \)\(18\!\cdots\!54\)\( T^{9} + \)\(47\!\cdots\!62\)\( T^{10} + \)\(25\!\cdots\!82\)\( T^{11} + \)\(67\!\cdots\!01\)\( T^{12} \)
$19$ \( 1 - 36166867207206 T^{2} + \)\(54\!\cdots\!15\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{6} + \)\(20\!\cdots\!15\)\( T^{8} - \)\(51\!\cdots\!06\)\( T^{10} + \)\(53\!\cdots\!01\)\( T^{12} \)
$23$ \( 1 + 15052992 T + 113296284076032 T^{2} + \)\(91\!\cdots\!84\)\( T^{3} + \)\(71\!\cdots\!63\)\( T^{4} + \)\(42\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!12\)\( T^{7} + \)\(12\!\cdots\!63\)\( T^{8} + \)\(65\!\cdots\!16\)\( T^{9} + \)\(33\!\cdots\!32\)\( T^{10} + \)\(18\!\cdots\!08\)\( T^{11} + \)\(50\!\cdots\!01\)\( T^{12} \)
$29$ \( 1 - 634112330650806 T^{2} + \)\(25\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(45\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(55\!\cdots\!01\)\( T^{12} \)
$31$ \( ( 1 - 76923576 T + 4180287219854295 T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(34\!\cdots\!95\)\( T^{4} - \)\(51\!\cdots\!76\)\( T^{5} + \)\(55\!\cdots\!01\)\( T^{6} )^{2} \)
$37$ \( 1 + 32574498 T + 530548959976002 T^{2} - \)\(35\!\cdots\!14\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} + \)\(17\!\cdots\!52\)\( T^{5} + \)\(50\!\cdots\!48\)\( T^{6} + \)\(84\!\cdots\!48\)\( T^{7} + \)\(30\!\cdots\!23\)\( T^{8} - \)\(39\!\cdots\!86\)\( T^{9} + \)\(28\!\cdots\!02\)\( T^{10} + \)\(83\!\cdots\!02\)\( T^{11} + \)\(12\!\cdots\!01\)\( T^{12} \)
$41$ \( ( 1 - 113076636 T + 22611520248205335 T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!35\)\( T^{4} - \)\(20\!\cdots\!36\)\( T^{5} + \)\(24\!\cdots\!01\)\( T^{6} )^{2} \)
$43$ \( 1 + 63628752 T + 2024309040538752 T^{2} + \)\(11\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!23\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(20\!\cdots\!48\)\( T^{6} - \)\(26\!\cdots\!48\)\( T^{7} + \)\(13\!\cdots\!23\)\( T^{8} + \)\(11\!\cdots\!36\)\( T^{9} + \)\(44\!\cdots\!52\)\( T^{10} + \)\(29\!\cdots\!48\)\( T^{11} + \)\(10\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 + 19700448 T + 194053825700352 T^{2} - \)\(41\!\cdots\!64\)\( T^{3} + \)\(61\!\cdots\!23\)\( T^{4} + \)\(52\!\cdots\!52\)\( T^{5} + \)\(18\!\cdots\!48\)\( T^{6} + \)\(27\!\cdots\!48\)\( T^{7} + \)\(16\!\cdots\!23\)\( T^{8} - \)\(59\!\cdots\!36\)\( T^{9} + \)\(14\!\cdots\!52\)\( T^{10} + \)\(79\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!01\)\( T^{12} \)
$53$ \( 1 + 950001042 T + 451250989900542882 T^{2} + \)\(26\!\cdots\!34\)\( T^{3} + \)\(18\!\cdots\!63\)\( T^{4} + \)\(79\!\cdots\!88\)\( T^{5} + \)\(28\!\cdots\!08\)\( T^{6} + \)\(13\!\cdots\!12\)\( T^{7} + \)\(55\!\cdots\!63\)\( T^{8} + \)\(14\!\cdots\!66\)\( T^{9} + \)\(42\!\cdots\!82\)\( T^{10} + \)\(15\!\cdots\!58\)\( T^{11} + \)\(28\!\cdots\!01\)\( T^{12} \)
$59$ \( 1 - 1661016969037924806 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(96\!\cdots\!20\)\( T^{6} + \)\(40\!\cdots\!15\)\( T^{8} - \)\(11\!\cdots\!06\)\( T^{10} + \)\(17\!\cdots\!01\)\( T^{12} \)
$61$ \( ( 1 - 801992196 T + 1994865806990658975 T^{2} - \)\(98\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!75\)\( T^{4} - \)\(40\!\cdots\!96\)\( T^{5} + \)\(36\!\cdots\!01\)\( T^{6} )^{2} \)
$67$ \( 1 - 2502647712 T + 3131622785189417472 T^{2} - \)\(49\!\cdots\!44\)\( T^{3} + \)\(47\!\cdots\!83\)\( T^{4} - \)\(16\!\cdots\!08\)\( T^{5} + \)\(15\!\cdots\!88\)\( T^{6} - \)\(29\!\cdots\!92\)\( T^{7} + \)\(15\!\cdots\!83\)\( T^{8} - \)\(30\!\cdots\!56\)\( T^{9} + \)\(34\!\cdots\!72\)\( T^{10} - \)\(50\!\cdots\!88\)\( T^{11} + \)\(36\!\cdots\!01\)\( T^{12} \)
$71$ \( ( 1 + 3568766904 T + 12497161629854598375 T^{2} + \)\(23\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!75\)\( T^{4} + \)\(37\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!01\)\( T^{6} )^{2} \)
$73$ \( 1 - 2304462438 T + 2655273564076451922 T^{2} - \)\(10\!\cdots\!06\)\( T^{3} + \)\(46\!\cdots\!83\)\( T^{4} - \)\(57\!\cdots\!92\)\( T^{5} + \)\(64\!\cdots\!88\)\( T^{6} - \)\(24\!\cdots\!08\)\( T^{7} + \)\(85\!\cdots\!83\)\( T^{8} - \)\(82\!\cdots\!94\)\( T^{9} + \)\(90\!\cdots\!22\)\( T^{10} - \)\(33\!\cdots\!62\)\( T^{11} + \)\(63\!\cdots\!01\)\( T^{12} \)
$79$ \( 1 - 53253337181723476806 T^{2} + \)\(12\!\cdots\!15\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!15\)\( T^{8} - \)\(42\!\cdots\!06\)\( T^{10} + \)\(72\!\cdots\!01\)\( T^{12} \)
$83$ \( 1 + 88180632 T + 3887911929959712 T^{2} - \)\(16\!\cdots\!96\)\( T^{3} + \)\(11\!\cdots\!03\)\( T^{4} + \)\(59\!\cdots\!28\)\( T^{5} + \)\(18\!\cdots\!68\)\( T^{6} + \)\(92\!\cdots\!72\)\( T^{7} + \)\(28\!\cdots\!03\)\( T^{8} - \)\(60\!\cdots\!04\)\( T^{9} + \)\(22\!\cdots\!12\)\( T^{10} + \)\(79\!\cdots\!68\)\( T^{11} + \)\(13\!\cdots\!01\)\( T^{12} \)
$89$ \( 1 - \)\(11\!\cdots\!06\)\( T^{2} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(22\!\cdots\!20\)\( T^{6} + \)\(60\!\cdots\!15\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(91\!\cdots\!01\)\( T^{12} \)
$97$ \( 1 - 33281088582 T + \)\(55\!\cdots\!62\)\( T^{2} - \)\(56\!\cdots\!54\)\( T^{3} + \)\(34\!\cdots\!03\)\( T^{4} - \)\(76\!\cdots\!28\)\( T^{5} - \)\(25\!\cdots\!32\)\( T^{6} - \)\(56\!\cdots\!72\)\( T^{7} + \)\(18\!\cdots\!03\)\( T^{8} - \)\(22\!\cdots\!46\)\( T^{9} + \)\(16\!\cdots\!62\)\( T^{10} - \)\(72\!\cdots\!18\)\( T^{11} + \)\(16\!\cdots\!01\)\( T^{12} \)
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