Properties

Label 10.14.b.a
Level 10
Weight 14
Character orbit 10.b
Analytic conductor 10.723
Analytic rank 0
Dimension 6
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 14 \)
Character orbit: \([\chi]\) = 10.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7230928952\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{21}\cdot 5^{5} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} -4096 q^{4} + ( -412 + 90 \beta_{1} + 15 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{5} + ( -3882 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{6} + ( 57 \beta_{1} - 163 \beta_{2} + 19 \beta_{4} - 19 \beta_{5} ) q^{7} -4096 \beta_{1} q^{8} + ( 710145 - 40 \beta_{1} + 26 \beta_{3} + 47 \beta_{4} + 27 \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + ( \beta_{1} - \beta_{2} ) q^{3} -4096 q^{4} + ( -412 + 90 \beta_{1} + 15 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{5} + ( -3882 - 2 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{6} + ( 57 \beta_{1} - 163 \beta_{2} + 19 \beta_{4} - 19 \beta_{5} ) q^{7} -4096 \beta_{1} q^{8} + ( 710145 - 40 \beta_{1} + 26 \beta_{3} + 47 \beta_{4} + 27 \beta_{5} ) q^{9} + ( -378274 - 495 \beta_{1} + 1280 \beta_{2} + 102 \beta_{3} + 52 \beta_{4} - 121 \beta_{5} ) q^{10} + ( 112504 + 208 \beta_{1} - 676 \beta_{3} + 26 \beta_{4} + 130 \beta_{5} ) q^{11} + ( -4096 \beta_{1} + 4096 \beta_{2} ) q^{12} + ( 136200 \beta_{1} + 16458 \beta_{2} + 819 \beta_{4} - 819 \beta_{5} ) q^{13} + ( -201326 + 586 \beta_{1} - 1846 \beta_{3} + 44 \beta_{4} + 337 \beta_{5} ) q^{14} + ( 13355196 - 264895 \beta_{1} + 21005 \beta_{2} - 1208 \beta_{3} + 367 \beta_{4} + 9 \beta_{5} ) q^{15} + 16777216 q^{16} + ( -438020 \beta_{1} + 11208 \beta_{2} + 6462 \beta_{4} - 6462 \beta_{5} ) q^{17} + ( 708439 \beta_{1} + 31232 \beta_{2} + 2112 \beta_{4} - 2112 \beta_{5} ) q^{18} + ( 23770128 - 8400 \beta_{1} - 7284 \beta_{3} + 16242 \beta_{4} + 12042 \beta_{5} ) q^{19} + ( 1687552 - 368640 \beta_{1} - 61440 \beta_{2} - 4096 \beta_{3} - 4096 \beta_{4} - 8192 \beta_{5} ) q^{20} + ( -154399074 - 5168 \beta_{1} + 17218 \beta_{3} - 857 \beta_{4} - 3441 \beta_{5} ) q^{21} + ( 100284 \beta_{1} + 252928 \beta_{2} - 28288 \beta_{4} + 28288 \beta_{5} ) q^{22} + ( 1992077 \beta_{1} - 466971 \beta_{2} - 12519 \beta_{4} + 12519 \beta_{5} ) q^{23} + ( 15900672 + 8192 \beta_{1} + 8192 \beta_{3} - 16384 \beta_{4} - 12288 \beta_{5} ) q^{24} + ( 303458265 - 1330800 \beta_{1} - 304550 \beta_{2} + 78780 \beta_{3} - 59095 \beta_{4} - 12065 \beta_{5} ) q^{25} + ( -561515148 + 72228 \beta_{1} - 32604 \beta_{3} - 92040 \beta_{4} - 55926 \beta_{5} ) q^{26} + ( -4146394 \beta_{1} - 1790138 \beta_{2} + 6006 \beta_{4} - 6006 \beta_{5} ) q^{27} + ( -233472 \beta_{1} + 667648 \beta_{2} - 77824 \beta_{4} + 77824 \beta_{5} ) q^{28} + ( 233699526 - 188784 \beta_{1} + 85752 \beta_{3} + 240300 \beta_{4} + 145908 \beta_{5} ) q^{29} + ( 1081840242 + 13385310 \beta_{1} + 501760 \beta_{2} + 21034 \beta_{3} - 141716 \beta_{4} - 15807 \beta_{5} ) q^{30} + ( -3711799064 - 174048 \beta_{1} + 163848 \beta_{3} + 179148 \beta_{4} + 92124 \beta_{5} ) q^{31} + 16777216 \beta_{1} q^{32} + ( -58026460 \beta_{1} - 2519324 \beta_{2} + 265668 \beta_{4} - 265668 \beta_{5} ) q^{33} + ( 1790800880 + 332592 \beta_{1} - 494544 \beta_{3} - 251616 \beta_{4} - 85320 \beta_{5} ) q^{34} + ( 6818593048 + 53691365 \beta_{1} + 2731815 \beta_{2} - 427204 \beta_{3} + 678896 \beta_{4} - 133508 \beta_{5} ) q^{35} + ( -2908753920 + 163840 \beta_{1} - 106496 \beta_{3} - 192512 \beta_{4} - 110592 \beta_{5} ) q^{36} + ( -63677700 \beta_{1} + 12825518 \beta_{2} + 237607 \beta_{4} - 237607 \beta_{5} ) q^{37} + ( 22914852 \beta_{1} + 16346112 \beta_{2} + 35712 \beta_{4} - 35712 \beta_{5} ) q^{38} + ( 13498407216 + 402144 \beta_{1} - 182856 \beta_{3} - 511788 \beta_{4} - 310716 \beta_{5} ) q^{39} + ( 1549410304 + 2027520 \beta_{1} - 5242880 \beta_{2} - 417792 \beta_{3} - 212992 \beta_{4} + 495616 \beta_{5} ) q^{40} + ( -26591361812 + 197320 \beta_{1} - 1328878 \beta_{3} + 368459 \beta_{4} + 467119 \beta_{5} ) q^{41} + ( -154078996 \beta_{1} - 6608384 \beta_{2} + 716352 \beta_{4} - 716352 \beta_{5} ) q^{42} + ( 444976377 \beta_{1} - 21436373 \beta_{2} - 82654 \beta_{4} + 82654 \beta_{5} ) q^{43} + ( -460816384 - 851968 \beta_{1} + 2768896 \beta_{3} - 106496 \beta_{4} - 532480 \beta_{5} ) q^{44} + ( 18715854906 + 43737330 \beta_{1} + 6771305 \beta_{2} + 1712837 \beta_{3} + 12962 \beta_{4} + 1290549 \beta_{5} ) q^{45} + ( -8057812862 - 1534854 \beta_{1} + 67578 \beta_{3} + 2268492 \beta_{4} + 1501065 \beta_{5} ) q^{46} + ( -851657179 \beta_{1} - 18028263 \beta_{2} - 2540817 \beta_{4} + 2540817 \beta_{5} ) q^{47} + ( 16777216 \beta_{1} - 16777216 \beta_{2} ) q^{48} + ( -23764806435 + 3048792 \beta_{1} + 2414274 \beta_{3} - 5780325 \beta_{4} - 4255929 \beta_{5} ) q^{49} + ( 5507141780 + 304549025 \beta_{1} - 56601600 \beta_{2} + 1333060 \beta_{3} + 4613560 \beta_{4} - 1510630 \beta_{5} ) q^{50} + ( 7931178312 + 1689664 \beta_{1} + 4904680 \beta_{3} - 4986836 \beta_{4} - 4142004 \beta_{5} ) q^{51} + ( -557875200 \beta_{1} - 67411968 \beta_{2} - 3354624 \beta_{4} + 3354624 \beta_{5} ) q^{52} + ( 894558024 \beta_{1} - 31252362 \beta_{2} - 7323147 \beta_{4} + 7323147 \beta_{5} ) q^{53} + ( 17365854492 - 3291988 \beta_{1} - 4060756 \beta_{3} + 6968360 \beta_{4} + 5322366 \beta_{5} ) q^{54} + ( -77615610844 + 2531182080 \beta_{1} + 145438180 \beta_{2} - 9234288 \beta_{3} - 13137538 \beta_{4} + 644674 \beta_{5} ) q^{55} + ( 824631296 - 2400256 \beta_{1} + 7561216 \beta_{3} - 180224 \beta_{4} - 1380352 \beta_{5} ) q^{56} + ( -3491439444 \beta_{1} + 90547452 \beta_{2} + 8257716 \beta_{4} - 8257716 \beta_{5} ) q^{57} + ( 224206542 \beta_{1} + 175785984 \beta_{2} + 8785152 \beta_{4} - 8785152 \beta_{5} ) q^{58} + ( -21581469008 - 3957488 \beta_{1} - 14383516 \beta_{3} + 13127990 \beta_{4} + 11149246 \beta_{5} ) q^{59} + ( -54702882816 + 1085009920 \beta_{1} - 86036480 \beta_{2} + 4947968 \beta_{3} - 1503232 \beta_{4} - 36864 \beta_{5} ) q^{60} + ( 368061078560 + 5167104 \beta_{1} - 20768574 \beta_{3} + 2633631 \beta_{4} + 5217183 \beta_{5} ) q^{61} + ( -3717244256 \beta_{1} + 96946176 \beta_{2} + 10812672 \beta_{4} - 10812672 \beta_{5} ) q^{62} + ( 1452549907 \beta_{1} - 45884245 \beta_{2} + 23459631 \beta_{4} - 23459631 \beta_{5} ) q^{63} -68719476736 q^{64} + ( -79193229804 + 8457388980 \beta_{1} - 174278370 \beta_{2} + 25991292 \beta_{3} + 28821117 \beta_{4} - 24990741 \beta_{5} ) q^{65} + ( 238177259304 + 7713416 \beta_{1} - 26292088 \beta_{3} + 1575920 \beta_{4} + 5432628 \beta_{5} ) q^{66} + ( -15734278419 \beta_{1} + 317678671 \beta_{2} - 13917202 \beta_{4} + 13917202 \beta_{5} ) q^{67} + ( 1794129920 \beta_{1} - 45907968 \beta_{2} - 26468352 \beta_{4} + 26468352 \beta_{5} ) q^{68} + ( -411762656778 - 22480192 \beta_{1} + 678266 \beta_{3} + 33381155 \beta_{4} + 22141059 \beta_{5} ) q^{69} + ( -220754863354 + 6823111530 \beta_{1} + 388602880 \beta_{2} - 26087058 \beta_{3} - 35705308 \beta_{4} + 807259 \beta_{5} ) q^{70} + ( 169439525640 - 11978880 \beta_{1} + 40721376 \beta_{3} - 2392368 \beta_{4} - 8381808 \beta_{5} ) q^{71} + ( -2901766144 \beta_{1} - 127926272 \beta_{2} - 8650752 \beta_{4} + 8650752 \beta_{5} ) q^{72} + ( 13785698244 \beta_{1} - 92547572 \beta_{2} - 20801872 \beta_{4} + 20801872 \beta_{5} ) q^{73} + ( 258044982940 + 37056172 \beta_{1} + 6642476 \beta_{3} - 58905496 \beta_{4} - 40377410 \beta_{5} ) q^{74} + ( -249281379120 + 12821459525 \beta_{1} - 302281725 \beta_{2} - 5131240 \beta_{3} - 23578240 \beta_{4} + 54644520 \beta_{5} ) q^{75} + ( -97362444288 + 34406400 \beta_{1} + 29835264 \beta_{3} - 66527232 \beta_{4} - 49324032 \beta_{5} ) q^{76} + ( -44120806252 \beta_{1} - 2140835412 \beta_{2} - 1875888 \beta_{4} + 1875888 \beta_{5} ) q^{77} + ( 13518621624 \beta_{1} - 374310912 \beta_{2} - 18720000 \beta_{4} + 18720000 \beta_{5} ) q^{78} + ( -3923935992 + 25923744 \beta_{1} + 119991096 \beta_{3} - 98881164 \beta_{4} - 85919292 \beta_{5} ) q^{79} + ( -6912212992 + 1509949440 \beta_{1} + 251658240 \beta_{2} + 16777216 \beta_{3} + 16777216 \beta_{4} + 33554432 \beta_{5} ) q^{80} + ( -430933536153 - 123145016 \beta_{1} + 103905082 \beta_{3} + 132764983 \beta_{4} + 71192475 \beta_{5} ) q^{81} + ( -26629770294 \beta_{1} + 768008704 \beta_{2} - 48838336 \beta_{4} + 48838336 \beta_{5} ) q^{82} + ( 25902259349 \beta_{1} + 180468135 \beta_{2} + 8585838 \beta_{4} - 8585838 \beta_{5} ) q^{83} + ( 632418607104 + 21168128 \beta_{1} - 70524928 \beta_{3} + 3510272 \beta_{4} + 14094336 \beta_{5} ) q^{84} + ( 1576794903608 + 36108111540 \beta_{1} - 970950760 \beta_{2} - 104644584 \beta_{3} + 186117766 \beta_{4} + 1275482 \beta_{5} ) q^{85} + ( -1818023954194 - 46840138 \beta_{1} - 36260426 \beta_{3} + 88390420 \beta_{4} + 64970351 \beta_{5} ) q^{86} + ( -36477321570 \beta_{1} + 2212026354 \beta_{2} + 48635640 \beta_{4} - 48635640 \beta_{5} ) q^{87} + ( -410763264 \beta_{1} - 1035993088 \beta_{2} + 115867648 \beta_{4} - 115867648 \beta_{5} ) q^{88} + ( -183808237150 - 5054320 \beta_{1} - 45884120 \beta_{3} + 30523540 \beta_{4} + 27996380 \beta_{5} ) q^{89} + ( -185403725838 + 18669012935 \beta_{1} + 228911360 \beta_{2} + 88776874 \beta_{3} + 96428524 \beta_{4} - 106210527 \beta_{5} ) q^{90} + ( -1606536208152 + 344920224 \beta_{1} - 590523648 \beta_{3} - 222118512 \beta_{4} - 49658400 \beta_{5} ) q^{91} + ( -8159547392 \beta_{1} + 1912713216 \beta_{2} + 51277824 \beta_{4} - 51277824 \beta_{5} ) q^{92} + ( -23182618544 \beta_{1} + 5924646944 \beta_{2} - 1710216 \beta_{4} + 1710216 \beta_{5} ) q^{93} + ( 3492611731114 - 158015742 \beta_{1} + 167208834 \beta_{3} + 153419196 \beta_{4} + 74411325 \beta_{5} ) q^{94} + ( 4483315842900 + 54839034000 \beta_{1} + 2453509500 \beta_{2} + 136945800 \beta_{3} - 520308450 \beta_{4} + 14285850 \beta_{5} ) q^{95} + ( -65129152512 - 33554432 \beta_{1} - 33554432 \beta_{3} + 67108864 \beta_{4} + 50331648 \beta_{5} ) q^{96} + ( -37797913764 \beta_{1} + 1313113120 \beta_{2} + 252891458 \beta_{4} - 252891458 \beta_{5} ) q^{97} + ( -23463331893 \beta_{1} - 5756616192 \beta_{2} - 20304576 \beta_{4} + 20304576 \beta_{5} ) q^{98} + ( -1962591349536 + 367877872 \beta_{1} - 712639460 \beta_{3} - 195497078 \beta_{4} - 11558142 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 24576q^{4} - 2470q^{5} - 23296q^{6} + 4260922q^{9} + O(q^{10}) \) \( 6q - 24576q^{4} - 2470q^{5} - 23296q^{6} + 4260922q^{9} - 2269440q^{10} + 673672q^{11} - 1211648q^{14} + 80128760q^{15} + 100663296q^{16} + 142606200q^{19} + 10117120q^{20} - 926360008q^{21} + 95420416q^{24} + 1820907150q^{25} - 3369156096q^{26} + 1402368660q^{29} + 6491083520q^{30} - 22270466688q^{31} + 10743816192q^{34} + 40910703880q^{35} - 17452736512q^{36} + 80990077584q^{39} + 9295626240q^{40} - 159550828628q^{41} - 2759360512q^{44} + 112298555110q^{45} - 48346742016q^{46} - 142584010062q^{49} + 33045516800q^{50} + 47596879232q^{51} + 104187005440q^{54} - 465712133640q^{55} + 4962910208q^{56} - 129517581080q^{59} - 328207400960q^{60} + 2208324934212q^{61} - 412316860416q^{64} - 475107396240q^{65} + 1429010971648q^{66} - 2470574584136q^{69} - 1324581354240q^{70} + 1016718596592q^{71} + 1548283182592q^{74} - 1495698537200q^{75} - 584114995200q^{76} - 23303633760q^{79} - 41439723520q^{80} - 2585393406754q^{81} + 3794370592768q^{84} + 9460560132480q^{85} - 10908216246016q^{86} - 1102941191140q^{89} - 1112244801280q^{90} - 9640398296208q^{91} + 20956004804352q^{94} + 26900168949000q^{95} - 390842023936q^{96} - 11776973376136q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} + 2 x^{4} + 160950 x^{3} + 43599609 x^{2} + 975553632 x + 10914144768\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 2902156 \nu^{5} - 38332616 \nu^{4} + 467184200 \nu^{3} + 252252560280 \nu^{2} + 123915151592604 \nu + 1444881606555456 \)\()/ 21586271220765 \)
\(\beta_{2}\)\(=\)\((\)\( 8857615 \nu^{5} - 124928126 \nu^{4} + 53829158030 \nu^{3} + 717491956170 \nu^{2} + 722940932399175 \nu + 8279858082468816 \)\()/ 34538033953224 \)
\(\beta_{3}\)\(=\)\((\)\(-8853546833 \nu^{5} + 730507083298 \nu^{4} - 24595132422850 \nu^{3} - 1322005625015190 \nu^{2} - 203160410986305897 \nu + 11635391903364851652\)\()/ 949795933713660 \)
\(\beta_{4}\)\(=\)\((\)\( 117783373 \nu^{5} - 586906082 \nu^{4} + 15870764330 \nu^{3} + 23032569783630 \nu^{2} + 4563380738283957 \nu + 86914657131672672 \)\()/ 5276644076187 \)
\(\beta_{5}\)\(=\)\((\)\(-14657213123 \nu^{5} + 279534354118 \nu^{4} + 5659609021850 \nu^{3} - 3008911425433890 \nu^{2} - 613155973504765707 \nu - 4139867393445834288\)\()/ 474897966856830 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-3 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 64 \beta_{2} - 8 \beta_{1} + 426\)\()/1280\)
\(\nu^{2}\)\(=\)\((\)\(-66 \beta_{5} + 66 \beta_{4} + 760 \beta_{2} - 27559 \beta_{1}\)\()/400\)
\(\nu^{3}\)\(=\)\((\)\(98691 \beta_{5} + 134404 \beta_{4} - 65090 \beta_{3} + 2125120 \beta_{2} - 8225496 \beta_{1} - 515022570\)\()/6400\)
\(\nu^{4}\)\(=\)\((\)\(88137 \beta_{5} + 146578 \beta_{4} + 57490 \beta_{3} - 116882 \beta_{1} - 2342499750\)\()/80\)
\(\nu^{5}\)\(=\)\((\)\(161899005 \beta_{5} + 179082748 \beta_{4} - 71150078 \beta_{3} - 3012454336 \beta_{2} + 17767644824 \beta_{1} - 1133922685014\)\()/1280\)

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
62.8643 + 62.8643i
−11.7164 11.7164i
−50.1479 50.1479i
−50.1479 + 50.1479i
−11.7164 + 11.7164i
62.8643 62.8643i
64.0000i 1311.29i −4096.00 −34287.6 + 6712.97i −83922.3 77461.9i 262144.i −125146. 429630. + 2.19441e6i
9.2 64.0000i 180.328i −4096.00 33325.2 + 10494.5i 11541.0 386506.i 262144.i 1.56180e6 671650. 2.13281e6i
9.3 64.0000i 948.958i −4096.00 −272.592 34937.5i 60733.3 454502.i 262144.i 693802. −2.23600e6 + 17445.9i
9.4 64.0000i 948.958i −4096.00 −272.592 + 34937.5i 60733.3 454502.i 262144.i 693802. −2.23600e6 17445.9i
9.5 64.0000i 180.328i −4096.00 33325.2 10494.5i 11541.0 386506.i 262144.i 1.56180e6 671650. + 2.13281e6i
9.6 64.0000i 1311.29i −4096.00 −34287.6 6712.97i −83922.3 77461.9i 262144.i −125146. 429630. 2.19441e6i
\(n\): 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 10.14.b.a 6
3.b odd 2 1 90.14.c.b 6
4.b odd 2 1 80.14.c.b 6
5.b even 2 1 inner 10.14.b.a 6
5.c odd 4 1 50.14.a.i 3
5.c odd 4 1 50.14.a.j 3
15.d odd 2 1 90.14.c.b 6
20.d odd 2 1 80.14.c.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.14.b.a 6 1.a even 1 1 trivial
10.14.b.a 6 5.b even 2 1 inner
50.14.a.i 3 5.c odd 4 1
50.14.a.j 3 5.c odd 4 1
80.14.c.b 6 4.b odd 2 1
80.14.c.b 6 20.d odd 2 1
90.14.c.b 6 3.b odd 2 1
90.14.c.b 6 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(10, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 4096 T^{2} )^{3} \)
$3$ \( 1 - 6913430 T^{2} + 22845784047687 T^{4} - 45755853786256361940 T^{6} + \)\(58\!\cdots\!23\)\( T^{8} - \)\(44\!\cdots\!30\)\( T^{10} + \)\(16\!\cdots\!89\)\( T^{12} \)
$5$ \( 1 + 2470 T - 907403125 T^{2} + 3538476562500 T^{3} - 1107669830322265625 T^{4} + \)\(36\!\cdots\!50\)\( T^{5} + \)\(18\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - 219375026190 T^{2} + \)\(33\!\cdots\!47\)\( T^{4} - \)\(40\!\cdots\!20\)\( T^{6} + \)\(31\!\cdots\!03\)\( T^{8} - \)\(19\!\cdots\!90\)\( T^{10} + \)\(82\!\cdots\!49\)\( T^{12} \)
$11$ \( ( 1 - 336836 T + 3915679625825 T^{2} - \)\(19\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!75\)\( T^{4} - \)\(40\!\cdots\!96\)\( T^{5} + \)\(41\!\cdots\!91\)\( T^{6} )^{2} \)
$13$ \( 1 - 393621383117310 T^{2} + \)\(14\!\cdots\!27\)\( T^{4} - \)\(24\!\cdots\!80\)\( T^{6} + \)\(13\!\cdots\!43\)\( T^{8} - \)\(33\!\cdots\!10\)\( T^{10} + \)\(77\!\cdots\!29\)\( T^{12} \)
$17$ \( 1 - 24453031619484390 T^{2} + \)\(47\!\cdots\!07\)\( T^{4} - \)\(52\!\cdots\!20\)\( T^{6} + \)\(47\!\cdots\!83\)\( T^{8} - \)\(23\!\cdots\!90\)\( T^{10} + \)\(94\!\cdots\!09\)\( T^{12} \)
$19$ \( ( 1 - 71303100 T + 41817006257447577 T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!43\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{5} + \)\(74\!\cdots\!79\)\( T^{6} )^{2} \)
$23$ \( 1 - 2298256175304463470 T^{2} + \)\(24\!\cdots\!67\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(61\!\cdots\!63\)\( T^{8} - \)\(14\!\cdots\!70\)\( T^{10} + \)\(16\!\cdots\!69\)\( T^{12} \)
$29$ \( ( 1 - 701184330 T + 14276632748319196467 T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!63\)\( T^{4} - \)\(73\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!69\)\( T^{6} )^{2} \)
$31$ \( ( 1 + 11135233344 T + \)\(10\!\cdots\!85\)\( T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!35\)\( T^{4} + \)\(66\!\cdots\!64\)\( T^{5} + \)\(14\!\cdots\!71\)\( T^{6} )^{2} \)
$37$ \( 1 - \)\(94\!\cdots\!50\)\( T^{2} + \)\(40\!\cdots\!27\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(24\!\cdots\!43\)\( T^{8} - \)\(33\!\cdots\!50\)\( T^{10} + \)\(20\!\cdots\!29\)\( T^{12} \)
$41$ \( ( 1 + 79775414314 T + \)\(44\!\cdots\!95\)\( T^{2} + \)\(15\!\cdots\!60\)\( T^{3} + \)\(41\!\cdots\!95\)\( T^{4} + \)\(68\!\cdots\!74\)\( T^{5} + \)\(79\!\cdots\!61\)\( T^{6} )^{2} \)
$43$ \( 1 - \)\(66\!\cdots\!30\)\( T^{2} + \)\(20\!\cdots\!47\)\( T^{4} - \)\(42\!\cdots\!40\)\( T^{6} + \)\(61\!\cdots\!03\)\( T^{8} - \)\(58\!\cdots\!30\)\( T^{10} + \)\(25\!\cdots\!49\)\( T^{12} \)
$47$ \( 1 - \)\(18\!\cdots\!10\)\( T^{2} + \)\(14\!\cdots\!87\)\( T^{4} - \)\(82\!\cdots\!80\)\( T^{6} + \)\(43\!\cdots\!23\)\( T^{8} - \)\(16\!\cdots\!10\)\( T^{10} + \)\(26\!\cdots\!89\)\( T^{12} \)
$53$ \( 1 - \)\(10\!\cdots\!90\)\( T^{2} + \)\(55\!\cdots\!87\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(37\!\cdots\!23\)\( T^{8} - \)\(47\!\cdots\!90\)\( T^{10} + \)\(31\!\cdots\!89\)\( T^{12} \)
$59$ \( ( 1 + 64758790540 T + \)\(21\!\cdots\!37\)\( T^{2} + \)\(63\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!23\)\( T^{4} + \)\(71\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!39\)\( T^{6} )^{2} \)
$61$ \( ( 1 - 1104162467106 T + \)\(79\!\cdots\!55\)\( T^{2} - \)\(38\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!55\)\( T^{4} - \)\(28\!\cdots\!66\)\( T^{5} + \)\(42\!\cdots\!41\)\( T^{6} )^{2} \)
$67$ \( 1 + \)\(17\!\cdots\!50\)\( T^{2} + \)\(24\!\cdots\!07\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(74\!\cdots\!83\)\( T^{8} + \)\(16\!\cdots\!50\)\( T^{10} + \)\(27\!\cdots\!09\)\( T^{12} \)
$71$ \( ( 1 - 508359298296 T + \)\(32\!\cdots\!05\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!55\)\( T^{4} - \)\(69\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!31\)\( T^{6} )^{2} \)
$73$ \( 1 - \)\(73\!\cdots\!30\)\( T^{2} + \)\(25\!\cdots\!67\)\( T^{4} - \)\(54\!\cdots\!40\)\( T^{6} + \)\(72\!\cdots\!63\)\( T^{8} - \)\(57\!\cdots\!30\)\( T^{10} + \)\(21\!\cdots\!69\)\( T^{12} \)
$79$ \( ( 1 + 11651816880 T + \)\(79\!\cdots\!17\)\( T^{2} + \)\(33\!\cdots\!40\)\( T^{3} + \)\(36\!\cdots\!63\)\( T^{4} + \)\(25\!\cdots\!80\)\( T^{5} + \)\(10\!\cdots\!19\)\( T^{6} )^{2} \)
$83$ \( 1 - \)\(44\!\cdots\!30\)\( T^{2} + \)\(90\!\cdots\!07\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!83\)\( T^{8} - \)\(27\!\cdots\!30\)\( T^{10} + \)\(48\!\cdots\!09\)\( T^{12} \)
$89$ \( ( 1 + 551470595570 T + \)\(65\!\cdots\!07\)\( T^{2} + \)\(23\!\cdots\!60\)\( T^{3} + \)\(14\!\cdots\!83\)\( T^{4} + \)\(26\!\cdots\!70\)\( T^{5} + \)\(10\!\cdots\!09\)\( T^{6} )^{2} \)
$97$ \( 1 - \)\(33\!\cdots\!70\)\( T^{2} + \)\(50\!\cdots\!87\)\( T^{4} - \)\(43\!\cdots\!60\)\( T^{6} + \)\(22\!\cdots\!23\)\( T^{8} - \)\(68\!\cdots\!70\)\( T^{10} + \)\(92\!\cdots\!89\)\( T^{12} \)
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