Properties

Label 10.12.b.a
Level 10
Weight 12
Character orbit 10.b
Analytic conductor 7.683
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.68343180560\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{19}\cdot 5^{5} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{2} + ( \beta_{1} + 3 \beta_{2} ) q^{3} -1024 q^{4} + ( 88 + \beta_{1} - 63 \beta_{2} + \beta_{5} ) q^{5} + ( 2837 - \beta_{3} + \beta_{4} ) q^{6} + ( -1 - 80 \beta_{1} - 366 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{7} + 1024 \beta_{2} q^{8} + ( -82676 + 9 \beta_{1} + 9 \beta_{2} - 8 \beta_{3} - 19 \beta_{4} + 9 \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{2} + ( \beta_{1} + 3 \beta_{2} ) q^{3} -1024 q^{4} + ( 88 + \beta_{1} - 63 \beta_{2} + \beta_{5} ) q^{5} + ( 2837 - \beta_{3} + \beta_{4} ) q^{6} + ( -1 - 80 \beta_{1} - 366 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{7} + 1024 \beta_{2} q^{8} + ( -82676 + 9 \beta_{1} + 9 \beta_{2} - 8 \beta_{3} - 19 \beta_{4} + 9 \beta_{5} ) q^{9} + ( -64325 + 160 \beta_{1} - 39 \beta_{2} - 15 \beta_{3} - 17 \beta_{4} ) q^{10} + ( -107146 + 26 \beta_{1} + 26 \beta_{2} - 48 \beta_{3} - 30 \beta_{4} + 26 \beta_{5} ) q^{11} + ( -1024 \beta_{1} - 3072 \beta_{2} ) q^{12} + ( -39 - 1071 \beta_{1} + 14217 \beta_{2} + 39 \beta_{4} + 117 \beta_{5} ) q^{13} + ( -354325 + 32 \beta_{1} + 32 \beta_{2} + 33 \beta_{3} - 129 \beta_{4} + 32 \beta_{5} ) q^{14} + ( -116545 - 1700 \beta_{1} - 36546 \beta_{2} - 160 \beta_{3} + 277 \beta_{4} + 135 \beta_{5} ) q^{15} + 1048576 q^{16} + ( -78 + 8982 \beta_{1} - 34738 \beta_{2} + 78 \beta_{4} + 234 \beta_{5} ) q^{17} + ( 288 + 5344 \beta_{1} + 84369 \beta_{2} - 288 \beta_{4} - 864 \beta_{5} ) q^{18} + ( -4017594 - 366 \beta_{1} - 366 \beta_{2} + 1392 \beta_{3} - 294 \beta_{4} - 366 \beta_{5} ) q^{19} + ( -90112 - 1024 \beta_{1} + 64512 \beta_{2} - 1024 \beta_{5} ) q^{20} + ( 20912213 - 819 \beta_{1} - 819 \beta_{2} + 224 \beta_{3} + 2233 \beta_{4} - 819 \beta_{5} ) q^{21} + ( 832 - 10048 \beta_{1} + 106188 \beta_{2} - 832 \beta_{4} - 2496 \beta_{5} ) q^{22} + ( -771 - 10566 \beta_{1} - 484844 \beta_{2} + 771 \beta_{4} + 2313 \beta_{5} ) q^{23} + ( -2905088 + 1024 \beta_{3} - 1024 \beta_{4} ) q^{24} + ( -30287180 + 55415 \beta_{1} + 648495 \beta_{2} - 600 \beta_{3} + 1445 \beta_{4} + 1515 \beta_{5} ) q^{25} + ( 14874594 + 1248 \beta_{1} + 1248 \beta_{2} - 762 \beta_{3} - 2982 \beta_{4} + 1248 \beta_{5} ) q^{26} + ( -2394 - 44476 \beta_{1} - 1003584 \beta_{2} + 2394 \beta_{4} + 7182 \beta_{5} ) q^{27} + ( 1024 + 81920 \beta_{1} + 374784 \beta_{2} - 1024 \beta_{4} - 3072 \beta_{5} ) q^{28} + ( -42739110 + 2484 \beta_{1} + 2484 \beta_{2} - 9936 \beta_{3} + 2484 \beta_{4} + 2484 \beta_{5} ) q^{29} + ( -36920653 - 194656 \beta_{1} + 71988 \beta_{2} - 775 \beta_{3} - 3545 \beta_{4} + 3744 \beta_{5} ) q^{30} + ( 76410748 + 5676 \beta_{1} + 5676 \beta_{2} - 2976 \beta_{3} - 14052 \beta_{4} + 5676 \beta_{5} ) q^{31} -1048576 \beta_{2} q^{32} + ( 252 - 446152 \beta_{1} + 2257584 \beta_{2} - 252 \beta_{4} - 756 \beta_{5} ) q^{33} + ( -37553080 + 2496 \beta_{1} + 2496 \beta_{2} - 12648 \beta_{3} + 5160 \beta_{4} + 2496 \beta_{5} ) q^{34} + ( -116181940 + 218525 \beta_{1} + 4117783 \beta_{2} + 13680 \beta_{3} - 18996 \beta_{4} - 8680 \beta_{5} ) q^{35} + ( 84660224 - 9216 \beta_{1} - 9216 \beta_{2} + 8192 \beta_{3} + 19456 \beta_{4} - 9216 \beta_{5} ) q^{36} + ( 12077 + 546097 \beta_{1} - 10293663 \beta_{2} - 12077 \beta_{4} - 36231 \beta_{5} ) q^{37} + ( -11712 + 874944 \beta_{1} + 4199412 \beta_{2} + 11712 \beta_{4} + 35136 \beta_{5} ) q^{38} + ( 219809052 - 13500 \beta_{1} - 13500 \beta_{2} + 14688 \beta_{3} + 25812 \beta_{4} - 13500 \beta_{5} ) q^{39} + ( 65868800 - 163840 \beta_{1} + 39936 \beta_{2} + 15360 \beta_{3} + 17408 \beta_{4} ) q^{40} + ( -27441925 - 22723 \beta_{1} - 22723 \beta_{2} + 35976 \beta_{3} + 32193 \beta_{4} - 22723 \beta_{5} ) q^{41} + ( -26208 - 1002400 \beta_{1} - 21184716 \beta_{2} + 26208 \beta_{4} + 78624 \beta_{5} ) q^{42} + ( 35314 - 564469 \beta_{1} + 18157245 \beta_{2} - 35314 \beta_{4} - 105942 \beta_{5} ) q^{43} + ( 109717504 - 26624 \beta_{1} - 26624 \beta_{2} + 49152 \beta_{3} + 30720 \beta_{4} - 26624 \beta_{5} ) q^{44} + ( 529005229 + 823108 \beta_{1} + 36897276 \beta_{2} - 33200 \beta_{3} + 36665 \beta_{4} - 15192 \beta_{5} ) q^{45} + ( -492718157 + 24672 \beta_{1} + 24672 \beta_{2} - 25671 \beta_{3} - 48345 \beta_{4} + 24672 \beta_{5} ) q^{46} + ( 25683 - 23832 \beta_{1} - 54935438 \beta_{2} - 25683 \beta_{4} - 77049 \beta_{5} ) q^{47} + ( 1048576 \beta_{1} + 3145728 \beta_{2} ) q^{48} + ( -112600188 + 67413 \beta_{1} + 67413 \beta_{2} + 23928 \beta_{3} - 226167 \beta_{4} + 67413 \beta_{5} ) q^{49} + ( 652007770 - 750560 \beta_{1} + 30122395 \beta_{2} - 80850 \beta_{3} + 32370 \beta_{4} + 27040 \beta_{5} ) q^{50} + ( -2176757620 + 73116 \beta_{1} + 73116 \beta_{2} - 156160 \beta_{3} - 63188 \beta_{4} + 73116 \beta_{5} ) q^{51} + ( 39936 + 1096704 \beta_{1} - 14558208 \beta_{2} - 39936 \beta_{4} - 119808 \beta_{5} ) q^{52} + ( -129273 - 2303649 \beta_{1} + 2542599 \beta_{2} + 129273 \beta_{4} + 387819 \beta_{5} ) q^{53} + ( -1013246510 + 76608 \beta_{1} + 76608 \beta_{2} - 68042 \beta_{3} - 161782 \beta_{4} + 76608 \beta_{5} ) q^{54} + ( 614698866 - 2405818 \beta_{1} + 104535414 \beta_{2} - 115200 \beta_{3} + 17690 \beta_{4} + 110382 \beta_{5} ) q^{55} + ( 362828800 - 32768 \beta_{1} - 32768 \beta_{2} - 33792 \beta_{3} + 132096 \beta_{4} - 32768 \beta_{5} ) q^{56} + ( -209844 - 1041216 \beta_{1} - 229586904 \beta_{2} + 209844 \beta_{4} + 629532 \beta_{5} ) q^{57} + ( 79488 - 6438528 \beta_{1} + 41390298 \beta_{2} - 79488 \beta_{4} - 238464 \beta_{5} ) q^{58} + ( 2944075010 - 152122 \beta_{1} - 152122 \beta_{2} + 91728 \beta_{3} + 364638 \beta_{4} - 152122 \beta_{5} ) q^{59} + ( 119342080 + 1740800 \beta_{1} + 37423104 \beta_{2} + 163840 \beta_{3} - 283648 \beta_{4} - 138240 \beta_{5} ) q^{60} + ( -836715361 - 115683 \beta_{1} - 115683 \beta_{2} + 134448 \beta_{3} + 212601 \beta_{4} - 115683 \beta_{5} ) q^{61} + ( 181632 + 5489280 \beta_{1} - 74856736 \beta_{2} - 181632 \beta_{4} - 544896 \beta_{5} ) q^{62} + ( 185859 + 20648158 \beta_{1} + 242532084 \beta_{2} - 185859 \beta_{4} - 557577 \beta_{5} ) q^{63} -1073741824 q^{64} + ( -3332618565 + 1464135 \beta_{1} + 86582199 \beta_{2} + 540840 \beta_{3} + 206577 \beta_{4} - 61905 \beta_{5} ) q^{65} + ( 2416193668 - 8064 \beta_{1} - 8064 \beta_{2} + 457996 \beta_{3} - 433804 \beta_{4} - 8064 \beta_{5} ) q^{66} + ( 276718 - 10748413 \beta_{1} - 331626195 \beta_{2} - 276718 \beta_{4} - 830154 \beta_{5} ) q^{67} + ( 79872 - 9197568 \beta_{1} + 35571712 \beta_{2} - 79872 \beta_{4} - 239616 \beta_{5} ) q^{68} + ( 3852695401 - 171423 \beta_{1} - 171423 \beta_{2} - 592208 \beta_{3} + 1106477 \beta_{4} - 171423 \beta_{5} ) q^{69} + ( 4159504269 + 15001088 \beta_{1} + 119588076 \beta_{2} - 70425 \beta_{3} + 348185 \beta_{4} - 170112 \beta_{5} ) q^{70} + ( 9465111096 - 215904 \beta_{1} - 215904 \beta_{2} + 907968 \beta_{3} - 260256 \beta_{4} - 215904 \beta_{5} ) q^{71} + ( -294912 - 5472256 \beta_{1} - 86393856 \beta_{2} + 294912 \beta_{4} + 884736 \beta_{5} ) q^{72} + ( -563408 - 7876564 \beta_{1} - 17959356 \beta_{2} + 563408 \beta_{4} + 1690224 \beta_{5} ) q^{73} + ( -10689079450 - 386464 \beta_{1} - 386464 \beta_{2} + 21522 \beta_{3} + 1137870 \beta_{4} - 386464 \beta_{5} ) q^{74} + ( -15916488920 - 26534115 \beta_{1} + 98914455 \beta_{2} - 118400 \beta_{3} - 1446520 \beta_{4} - 231840 \beta_{5} ) q^{75} + ( 4114016256 + 374784 \beta_{1} + 374784 \beta_{2} - 1425408 \beta_{3} + 301056 \beta_{4} + 374784 \beta_{5} ) q^{76} + ( -176448 + 29814732 \beta_{1} + 151026724 \beta_{2} + 176448 \beta_{4} + 529344 \beta_{5} ) q^{77} + ( -432000 - 5263488 \beta_{1} - 221716872 \beta_{2} + 432000 \beta_{4} + 1296000 \beta_{5} ) q^{78} + ( 433047348 + 1534068 \beta_{1} + 1534068 \beta_{2} - 599328 \beta_{3} - 4002876 \beta_{4} + 1534068 \beta_{5} ) q^{79} + ( 92274688 + 1048576 \beta_{1} - 66060288 \beta_{2} + 1048576 \beta_{5} ) q^{80} + ( -1174512118 + 957033 \beta_{1} + 957033 \beta_{2} - 2625784 \beta_{3} - 245315 \beta_{4} + 957033 \beta_{5} ) q^{81} + ( -727136 + 2664032 \beta_{1} + 26875254 \beta_{2} + 727136 \beta_{4} + 2181408 \beta_{5} ) q^{82} + ( 1933902 - 1064421 \beta_{1} - 140361035 \beta_{2} - 1933902 \beta_{4} - 5801706 \beta_{5} ) q^{83} + ( -21414106112 + 838656 \beta_{1} + 838656 \beta_{2} - 229376 \beta_{3} - 2286592 \beta_{4} + 838656 \beta_{5} ) q^{84} + ( -14171876510 - 535490 \beta_{1} - 237138522 \beta_{2} - 2146320 \beta_{3} + 1853254 \beta_{4} + 1377930 \beta_{5} ) q^{85} + ( 18667083169 - 1130048 \beta_{1} - 1130048 \beta_{2} + 2224227 \beta_{3} + 1165917 \beta_{4} - 1130048 \beta_{5} ) q^{86} + ( 1564920 - 61714386 \beta_{1} + 1471794138 \beta_{2} - 1564920 \beta_{4} - 4694760 \beta_{5} ) q^{87} + ( -851968 + 10289152 \beta_{1} - 108736512 \beta_{2} + 851968 \beta_{4} + 2555904 \beta_{5} ) q^{88} + ( 41577331750 - 2689220 \beta_{1} - 2689220 \beta_{2} + 5099760 \beta_{3} + 2967900 \beta_{4} - 2689220 \beta_{5} ) q^{89} + ( 37584330965 - 37979840 \beta_{1} - 537953577 \beta_{2} - 627745 \beta_{3} + 1113889 \beta_{4} + 110880 \beta_{5} ) q^{90} + ( -30740988072 + 235968 \beta_{1} + 235968 \beta_{2} + 654816 \beta_{3} - 1362720 \beta_{4} + 235968 \beta_{5} ) q^{91} + ( 789504 + 10819584 \beta_{1} + 496480256 \beta_{2} - 789504 \beta_{4} - 2368512 \beta_{5} ) q^{92} + ( -2105784 - 16406312 \beta_{1} - 1105985544 \beta_{2} + 2105784 \beta_{4} + 6317352 \beta_{5} ) q^{93} + ( -56290896089 - 821856 \beta_{1} - 821856 \beta_{2} + 1230933 \beta_{3} + 1234635 \beta_{4} - 821856 \beta_{5} ) q^{94} + ( 17486032170 + 169806390 \beta_{1} - 1302998490 \beta_{2} + 2176800 \beta_{3} + 2290290 \beta_{4} - 7729410 \beta_{5} ) q^{95} + ( 2974810112 - 1048576 \beta_{3} + 1048576 \beta_{4} ) q^{96} + ( -3426482 + 82860146 \beta_{1} + 1295726538 \beta_{2} + 3426482 \beta_{4} + 10279446 \beta_{5} ) q^{97} + ( 2157216 + 125891424 \beta_{1} + 144986229 \beta_{2} - 2157216 \beta_{4} - 6471648 \beta_{5} ) q^{98} + ( 86796227582 + 615402 \beta_{1} + 615402 \beta_{2} - 1264720 \beta_{3} - 581486 \beta_{4} + 615402 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6144q^{4} + 530q^{5} + 17024q^{6} - 496022q^{9} + O(q^{10}) \) \( 6q - 6144q^{4} + 530q^{5} + 17024q^{6} - 496022q^{9} - 385920q^{10} - 642728q^{11} - 2125952q^{14} - 698680q^{15} + 6291456q^{16} - 24109080q^{19} - 542720q^{20} + 125471192q^{21} - 17432576q^{24} - 181718850q^{25} + 89251584q^{26} - 256409820q^{29} - 221514880q^{30} + 458481792q^{31} - 225288192q^{34} - 697136360q^{35} + 507926528q^{36} + 1318797936q^{39} + 395182080q^{40} - 164768948q^{41} + 658153472q^{44} + 3174067390q^{45} - 2956208256q^{46} - 675514158q^{49} + 3912262400q^{50} - 13060087168q^{51} - 6079189760q^{54} + 3688644360q^{55} + 2176974848q^{56} + 17663962360q^{59} + 715448320q^{60} - 5020792428q^{61} - 6442450944q^{64} - 19996916880q^{65} + 14496229888q^{66} + 23117013976q^{69} + 24956826240q^{70} + 56788418832q^{71} - 64135292672q^{74} - 95499160400q^{75} + 24687697920q^{76} + 2602550880q^{79} + 555745280q^{80} - 7039907074q^{81} - 128482500608q^{84} - 85024210560q^{85} + 111995790464q^{86} + 249448412540q^{89} + 225507463040q^{90} - 184446766128q^{91} - 337749482112q^{94} + 104896380600q^{95} + 17850957824q^{96} + 520781125736q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 198 x^{3} + 3568321 x^{2} - 6762620 x + 6408200\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 807955 \nu^{5} + 842118 \nu^{4} - 3200275588 \nu^{3} + 4808106038 \nu^{2} - 3161951077723 \nu + 2990639720130 \)\()/ 604523721605 \)
\(\beta_{2}\)\(=\)\((\)\( -5715200 \nu^{5} + 6014576 \nu^{4} - 315216 \nu^{3} - 11372932784 \nu^{2} - 20394204345776 \nu + 19324355382560 \)\()/ 604523721605 \)
\(\beta_{3}\)\(=\)\((\)\(-4247878015 \nu^{5} - 14980310918 \nu^{4} + 509621899488 \nu^{3} + 3127575282762 \nu^{2} - 16157580597638917 \nu - 31530915177973145\)\()/ 5440713494445 \)
\(\beta_{4}\)\(=\)\((\)\(-4735302175 \nu^{5} - 13979944358 \nu^{4} - 412097187072 \nu^{3} + 3079320290922 \nu^{2} - 16155835619146117 \nu - 29393614134689930\)\()/ 5440713494445 \)
\(\beta_{5}\)\(=\)\((\)\( -2690537304 \nu^{5} + 4096032176 \nu^{4} + 26667276564 \nu^{3} + 1873448147736 \nu^{2} - 9651607974962172 \nu + 12125790194228963 \)\()/ 1088142698889 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} - \beta_{3} - 14 \beta_{2} - 32 \beta_{1} + 213\)\()/640\)
\(\nu^{2}\)\(=\)\((\)\(54 \beta_{5} + 18 \beta_{4} - 15781 \beta_{2} - 6 \beta_{1} - 18\)\()/400\)
\(\nu^{3}\)\(=\)\((\)\(576 \beta_{5} - 9523 \beta_{4} + 9235 \beta_{3} - 79334 \beta_{2} - 302144 \beta_{1} - 316047\)\()/3200\)
\(\nu^{4}\)\(=\)\((\)\(34038 \beta_{5} - 52599 \beta_{4} - 49515 \beta_{3} + 34038 \beta_{2} + 34038 \beta_{1} - 951686251\)\()/400\)
\(\nu^{5}\)\(=\)\((\)\(-114624 \beta_{5} - 3714187 \beta_{4} + 3484939 \beta_{3} + 32565482 \beta_{2} + 114269024 \beta_{1} - 198490599\)\()/640\)

Character values

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

\(n\) \(7\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−30.7073 + 30.7073i
0.947541 0.947541i
30.7598 30.7598i
30.7598 + 30.7598i
0.947541 + 0.947541i
−30.7073 30.7073i
32.0000i 532.146i −1024.00 1594.62 6803.33i −17028.7 24477.0i 32768.0i −106033. −217707. 51027.9i
9.2 32.0000i 100.951i −1024.00 2827.02 + 6390.31i 3230.43 15908.8i 32768.0i 166956. 204490. 90464.5i
9.3 32.0000i 697.195i −1024.00 −4156.64 5616.98i 22310.3 73603.8i 32768.0i −308934. −179743. + 133012.i
9.4 32.0000i 697.195i −1024.00 −4156.64 + 5616.98i 22310.3 73603.8i 32768.0i −308934. −179743. 133012.i
9.5 32.0000i 100.951i −1024.00 2827.02 6390.31i 3230.43 15908.8i 32768.0i 166956. 204490. + 90464.5i
9.6 32.0000i 532.146i −1024.00 1594.62 + 6803.33i −17028.7 24477.0i 32768.0i −106033. −217707. + 51027.9i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.6
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 10.12.b.a 6
3.b odd 2 1 90.12.c.b 6
4.b odd 2 1 80.12.c.c 6
5.b even 2 1 inner 10.12.b.a 6
5.c odd 4 1 50.12.a.i 3
5.c odd 4 1 50.12.a.j 3
15.d odd 2 1 90.12.c.b 6
20.d odd 2 1 80.12.c.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.12.b.a 6 1.a even 1 1 trivial
10.12.b.a 6 5.b even 2 1 inner
50.12.a.i 3 5.c odd 4 1
50.12.a.j 3 5.c odd 4 1
80.12.c.c 6 4.b odd 2 1
80.12.c.c 6 20.d odd 2 1
90.12.c.b 6 3.b odd 2 1
90.12.c.b 6 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 1024 T^{2} )^{3} \)
$3$ \( 1 - 283430 T^{2} + 63893461527 T^{4} - 14563740202522740 T^{6} + \)\(20\!\cdots\!43\)\( T^{8} - \)\(27\!\cdots\!30\)\( T^{10} + \)\(30\!\cdots\!29\)\( T^{12} \)
$5$ \( 1 - 530 T + 90999875 T^{2} + 98147812500 T^{3} + 4443353271484375 T^{4} - 1263618469238281250 T^{5} + \)\(11\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 5594223150 T^{2} + 13826563370852398047 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!03\)\( T^{8} - \)\(85\!\cdots\!50\)\( T^{10} + \)\(59\!\cdots\!49\)\( T^{12} \)
$11$ \( ( 1 + 321364 T + 451069377065 T^{2} + 68139027321485880 T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(26\!\cdots\!44\)\( T^{5} + \)\(23\!\cdots\!31\)\( T^{6} )^{2} \)
$13$ \( 1 - 7510240213710 T^{2} + \)\(27\!\cdots\!07\)\( T^{4} - \)\(60\!\cdots\!80\)\( T^{6} + \)\(87\!\cdots\!83\)\( T^{8} - \)\(77\!\cdots\!10\)\( T^{10} + \)\(33\!\cdots\!09\)\( T^{12} \)
$17$ \( 1 - 132432185976870 T^{2} + \)\(83\!\cdots\!67\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{6} + \)\(97\!\cdots\!63\)\( T^{8} - \)\(18\!\cdots\!70\)\( T^{10} + \)\(16\!\cdots\!69\)\( T^{12} \)
$19$ \( ( 1 + 12054540 T + 30008106286257 T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(34\!\cdots\!83\)\( T^{4} + \)\(16\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} )^{2} \)
$23$ \( 1 - 4235045141210190 T^{2} + \)\(84\!\cdots\!87\)\( T^{4} - \)\(10\!\cdots\!20\)\( T^{6} + \)\(76\!\cdots\!23\)\( T^{8} - \)\(34\!\cdots\!90\)\( T^{10} + \)\(74\!\cdots\!89\)\( T^{12} \)
$29$ \( ( 1 + 128204910 T + 22822131113381787 T^{2} + \)\(33\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!23\)\( T^{4} + \)\(19\!\cdots\!10\)\( T^{5} + \)\(18\!\cdots\!89\)\( T^{6} )^{2} \)
$31$ \( ( 1 - 229240896 T + 64386954291598365 T^{2} - \)\(90\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!15\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} )^{2} \)
$37$ \( 1 - 345211813058554110 T^{2} + \)\(91\!\cdots\!07\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!83\)\( T^{8} - \)\(34\!\cdots\!10\)\( T^{10} + \)\(31\!\cdots\!09\)\( T^{12} \)
$41$ \( ( 1 + 82384474 T + 1346507617380213815 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(74\!\cdots\!15\)\( T^{4} + \)\(24\!\cdots\!94\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} )^{2} \)
$43$ \( 1 - 2824235125166049750 T^{2} + \)\(42\!\cdots\!47\)\( T^{4} - \)\(44\!\cdots\!00\)\( T^{6} + \)\(36\!\cdots\!03\)\( T^{8} - \)\(21\!\cdots\!50\)\( T^{10} + \)\(64\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - 4774521242059704030 T^{2} + \)\(20\!\cdots\!27\)\( T^{4} - \)\(47\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!43\)\( T^{8} - \)\(17\!\cdots\!30\)\( T^{10} + \)\(22\!\cdots\!29\)\( T^{12} \)
$53$ \( 1 - 32371227025452806430 T^{2} + \)\(55\!\cdots\!27\)\( T^{4} - \)\(63\!\cdots\!40\)\( T^{6} + \)\(48\!\cdots\!43\)\( T^{8} - \)\(23\!\cdots\!30\)\( T^{10} + \)\(63\!\cdots\!29\)\( T^{12} \)
$59$ \( ( 1 - 8831981180 T + 96619062073692665177 T^{2} - \)\(50\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!43\)\( T^{4} - \)\(80\!\cdots\!80\)\( T^{5} + \)\(27\!\cdots\!79\)\( T^{6} )^{2} \)
$61$ \( ( 1 + 2510396214 T + \)\(12\!\cdots\!15\)\( T^{2} + \)\(21\!\cdots\!80\)\( T^{3} + \)\(54\!\cdots\!15\)\( T^{4} + \)\(47\!\cdots\!94\)\( T^{5} + \)\(82\!\cdots\!81\)\( T^{6} )^{2} \)
$67$ \( 1 - \)\(21\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!67\)\( T^{4} - \)\(49\!\cdots\!60\)\( T^{6} + \)\(47\!\cdots\!63\)\( T^{8} - \)\(48\!\cdots\!70\)\( T^{10} + \)\(33\!\cdots\!69\)\( T^{12} \)
$71$ \( ( 1 - 28394209416 T + \)\(79\!\cdots\!65\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(18\!\cdots\!15\)\( T^{4} - \)\(15\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!11\)\( T^{6} )^{2} \)
$73$ \( 1 - \)\(14\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!87\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(98\!\cdots\!23\)\( T^{8} - \)\(14\!\cdots\!90\)\( T^{10} + \)\(95\!\cdots\!89\)\( T^{12} \)
$79$ \( ( 1 - 1301275440 T - \)\(13\!\cdots\!63\)\( T^{2} + \)\(24\!\cdots\!80\)\( T^{3} - \)\(10\!\cdots\!77\)\( T^{4} - \)\(72\!\cdots\!40\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} )^{2} \)
$83$ \( 1 - \)\(32\!\cdots\!70\)\( T^{2} + \)\(56\!\cdots\!67\)\( T^{4} - \)\(71\!\cdots\!60\)\( T^{6} + \)\(94\!\cdots\!63\)\( T^{8} - \)\(90\!\cdots\!70\)\( T^{10} + \)\(45\!\cdots\!69\)\( T^{12} \)
$89$ \( ( 1 - 124724206270 T + \)\(86\!\cdots\!67\)\( T^{2} - \)\(49\!\cdots\!60\)\( T^{3} + \)\(24\!\cdots\!63\)\( T^{4} - \)\(96\!\cdots\!70\)\( T^{5} + \)\(21\!\cdots\!69\)\( T^{6} )^{2} \)
$97$ \( 1 - \)\(18\!\cdots\!30\)\( T^{2} + \)\(17\!\cdots\!27\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{6} + \)\(91\!\cdots\!43\)\( T^{8} - \)\(48\!\cdots\!30\)\( T^{10} + \)\(13\!\cdots\!29\)\( T^{12} \)
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