Properties

Label 688.5.b.d
Level 688
Weight 5
Character orbit 688.b
Analytic conductor 71.119
Analytic rank 0
Dimension 12
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 688.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(71.1185346017\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{3} + \beta_{7} q^{5} + ( \beta_{1} - 2 \beta_{3} + \beta_{10} ) q^{7} + ( -39 + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{3} + \beta_{7} q^{5} + ( \beta_{1} - 2 \beta_{3} + \beta_{10} ) q^{7} + ( -39 + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{8} ) q^{9} + ( 14 + \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{11} + ( -19 - \beta_{2} - 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 4 \beta_{8} ) q^{13} + ( 10 + 5 \beta_{2} + \beta_{4} - 5 \beta_{5} - 9 \beta_{6} + 2 \beta_{8} ) q^{15} + ( 55 + 10 \beta_{2} + 3 \beta_{4} + \beta_{5} - 7 \beta_{6} + 6 \beta_{8} ) q^{17} + ( \beta_{1} - 4 \beta_{3} + \beta_{7} - 2 \beta_{10} - \beta_{11} ) q^{19} + ( -208 + 21 \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - 23 \beta_{6} + 25 \beta_{8} ) q^{21} + ( -120 + \beta_{2} - 2 \beta_{4} - 14 \beta_{5} - 15 \beta_{6} - 8 \beta_{8} ) q^{23} + ( -15 - 14 \beta_{2} - 15 \beta_{5} + \beta_{6} + 10 \beta_{8} ) q^{25} + ( -19 \beta_{1} + 12 \beta_{3} - 7 \beta_{7} + 3 \beta_{9} - 7 \beta_{10} - \beta_{11} ) q^{27} + ( 6 \beta_{1} - 14 \beta_{3} - 2 \beta_{7} - \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{29} + ( -492 - 4 \beta_{2} + 3 \beta_{4} + 28 \beta_{5} + 2 \beta_{6} + 25 \beta_{8} ) q^{31} + ( -50 \beta_{1} - 2 \beta_{3} + 6 \beta_{7} + 4 \beta_{9} + 4 \beta_{10} + 3 \beta_{11} ) q^{33} + ( -72 + 3 \beta_{2} + 14 \beta_{5} - 49 \beta_{6} + 15 \beta_{8} ) q^{35} + ( -2 \beta_{1} - 29 \beta_{3} - 30 \beta_{7} + 4 \beta_{9} + 15 \beta_{10} - 4 \beta_{11} ) q^{37} + ( -101 \beta_{1} + 37 \beta_{3} + 22 \beta_{7} + 9 \beta_{9} + 19 \beta_{10} + 4 \beta_{11} ) q^{39} + ( 415 - 42 \beta_{2} - 16 \beta_{4} + 15 \beta_{5} + 23 \beta_{6} - 34 \beta_{8} ) q^{41} + ( 74 - 9 \beta_{1} + 4 \beta_{2} + 65 \beta_{3} - 15 \beta_{4} - 16 \beta_{5} + 7 \beta_{6} + 17 \beta_{7} + 26 \beta_{8} + 4 \beta_{9} - 13 \beta_{10} - 3 \beta_{11} ) q^{43} + ( \beta_{1} + 44 \beta_{3} + 11 \beta_{7} - 7 \beta_{9} - 29 \beta_{10} - 4 \beta_{11} ) q^{45} + ( 492 - 12 \beta_{2} + 18 \beta_{4} + 9 \beta_{5} - 39 \beta_{6} - 24 \beta_{8} ) q^{47} + ( -693 - 24 \beta_{4} - 33 \beta_{5} - 96 \beta_{6} + 21 \beta_{8} ) q^{49} + ( 16 \beta_{1} - \beta_{3} - 57 \beta_{7} - 5 \beta_{9} - 41 \beta_{10} + 3 \beta_{11} ) q^{51} + ( 95 - 19 \beta_{2} + 15 \beta_{4} + 22 \beta_{5} - 81 \beta_{6} + 75 \beta_{8} ) q^{53} + ( -6 \beta_{1} - 71 \beta_{3} + 76 \beta_{7} - 7 \beta_{9} + \beta_{10} ) q^{55} + ( -670 + 32 \beta_{2} - 3 \beta_{4} + 57 \beta_{5} + 110 \beta_{6} - 47 \beta_{8} ) q^{57} + ( -1182 + 2 \beta_{2} + 2 \beta_{4} + 79 \beta_{5} + 50 \beta_{6} - 31 \beta_{8} ) q^{59} + ( -187 \beta_{1} - 35 \beta_{3} - 88 \beta_{7} + 19 \beta_{9} - 44 \beta_{10} + 3 \beta_{11} ) q^{61} + ( -204 \beta_{1} + 322 \beta_{3} - 128 \beta_{7} + 2 \beta_{9} - 63 \beta_{10} - 2 \beta_{11} ) q^{63} + ( -77 \beta_{1} - 213 \beta_{3} + 74 \beta_{7} - 15 \beta_{9} + 3 \beta_{11} ) q^{65} + ( 52 - 36 \beta_{2} + 45 \beta_{4} + 157 \beta_{5} + 93 \beta_{6} + 5 \beta_{8} ) q^{67} + ( -15 \beta_{1} + 231 \beta_{3} - 131 \beta_{7} - 10 \beta_{9} - 9 \beta_{10} - 14 \beta_{11} ) q^{69} + ( 251 \beta_{1} - 84 \beta_{3} + 162 \beta_{7} + 6 \beta_{9} + 30 \beta_{10} - 12 \beta_{11} ) q^{71} + ( -176 \beta_{1} - 237 \beta_{3} + 122 \beta_{7} - 7 \beta_{9} - 46 \beta_{10} - \beta_{11} ) q^{73} + ( -31 \beta_{1} - 210 \beta_{3} + 59 \beta_{7} - 15 \beta_{9} - 16 \beta_{10} - 13 \beta_{11} ) q^{75} + ( -203 \beta_{1} - 82 \beta_{3} + 4 \beta_{7} + 16 \beta_{9} + 42 \beta_{10} + 15 \beta_{11} ) q^{77} + ( -2042 + 142 \beta_{2} - 3 \beta_{4} - 142 \beta_{5} + 49 \beta_{6} - \beta_{8} ) q^{79} + ( -1939 - 38 \beta_{2} + 23 \beta_{4} + 80 \beta_{5} + 60 \beta_{6} - 104 \beta_{8} ) q^{81} + ( 628 - 3 \beta_{2} - 121 \beta_{4} + 39 \beta_{5} + 38 \beta_{6} - 202 \beta_{8} ) q^{83} + ( 136 \beta_{1} + 370 \beta_{3} + 177 \beta_{7} - 7 \beta_{9} - 13 \beta_{10} - 14 \beta_{11} ) q^{85} + ( -1420 - 115 \beta_{2} - 63 \beta_{4} - 111 \beta_{5} - 298 \beta_{6} + 97 \beta_{8} ) q^{87} + ( 37 \beta_{1} - 196 \beta_{3} + 190 \beta_{7} + 10 \beta_{9} - 30 \beta_{10} + 15 \beta_{11} ) q^{89} + ( -576 \beta_{1} - 350 \beta_{3} + 4 \beta_{7} + 28 \beta_{9} + 5 \beta_{10} + 34 \beta_{11} ) q^{91} + ( 365 \beta_{1} + 678 \beta_{3} + 187 \beta_{7} + 22 \beta_{9} - 34 \beta_{10} - 2 \beta_{11} ) q^{93} + ( -16 - 95 \beta_{2} + 20 \beta_{4} - 49 \beta_{5} + 170 \beta_{6} - 121 \beta_{8} ) q^{95} + ( -489 + 85 \beta_{2} + 48 \beta_{4} + 67 \beta_{5} + 226 \beta_{6} - 165 \beta_{8} ) q^{97} + ( 2320 - 19 \beta_{2} + 105 \beta_{4} - 143 \beta_{5} - 380 \beta_{6} + 58 \beta_{8} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 462q^{9} + O(q^{10}) \) \( 12q - 462q^{9} + 180q^{11} - 216q^{13} + 92q^{15} + 678q^{17} - 2392q^{21} - 1566q^{23} - 174q^{25} - 5710q^{31} - 936q^{35} + 4878q^{41} + 1108q^{43} + 5526q^{47} - 8544q^{49} + 1212q^{53} - 7692q^{57} - 14016q^{59} + 1088q^{67} - 24302q^{79} - 23660q^{81} + 7032q^{83} - 17850q^{87} - 606q^{95} - 5842q^{97} + 25924q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} + 142 x^{10} + 7173 x^{8} + 157368 x^{6} + 1510016 x^{4} + 5098688 x^{2} + 90352\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 1013 \nu^{10} + 51724 \nu^{8} - 3751835 \nu^{6} - 249791190 \nu^{4} - 2971827868 \nu^{2} + 390161384 \)\()/ 218680368 \)
\(\beta_{3}\)\(=\)\((\)\( 7373 \nu^{11} + 1060068 \nu^{9} + 54814785 \nu^{7} + 1256312182 \nu^{5} + 12979904372 \nu^{3} + 48750171768 \nu \)\()/ 437360736 \)
\(\beta_{4}\)\(=\)\((\)\( -8029 \nu^{10} - 1003616 \nu^{8} - 42059193 \nu^{6} - 669406178 \nu^{4} - 3108074228 \nu^{2} + 2072740888 \)\()/ 218680368 \)
\(\beta_{5}\)\(=\)\((\)\( 3645 \nu^{10} + 491936 \nu^{8} + 22519538 \nu^{6} + 398171095 \nu^{4} + 2226381792 \nu^{2} + 109255312 \)\()/54670092\)
\(\beta_{6}\)\(=\)\((\)\( -15593 \nu^{10} - 2019468 \nu^{8} - 86326317 \nu^{6} - 1342893190 \nu^{4} - 5496338564 \nu^{2} + 9669475032 \)\()/ 218680368 \)
\(\beta_{7}\)\(=\)\((\)\( -10113 \nu^{11} - 1379868 \nu^{9} - 65318629 \nu^{7} - 1285172518 \nu^{5} - 10485382436 \nu^{3} - 28620915064 \nu \)\()/ 145786912 \)
\(\beta_{8}\)\(=\)\((\)\( -28089 \nu^{10} - 3610960 \nu^{8} - 153145033 \nu^{6} - 2392704686 \nu^{4} - 11543951796 \nu^{2} - 967119080 \)\()/ 218680368 \)
\(\beta_{9}\)\(=\)\((\)\( -36865 \nu^{11} - 5300340 \nu^{9} - 274073925 \nu^{7} - 6281560910 \nu^{5} - 64024800388 \nu^{3} - 208761999960 \nu \)\()/ 437360736 \)
\(\beta_{10}\)\(=\)\((\)\( -43059 \nu^{11} - 6156292 \nu^{9} - 313089127 \nu^{7} - 6867724298 \nu^{5} - 63796972524 \nu^{3} - 193079423624 \nu \)\()/ 437360736 \)
\(\beta_{11}\)\(=\)\((\)\( -130223 \nu^{11} - 18990012 \nu^{9} - 996377731 \nu^{7} - 23117282490 \nu^{5} - 239736846012 \nu^{3} - 898096461064 \nu \)\()/ 437360736 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{2} - 48\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{9} + 5 \beta_{3} - 40 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(-7 \beta_{8} - 21 \beta_{6} - 32 \beta_{5} + 3 \beta_{4} - 33 \beta_{2} + 992\)
\(\nu^{5}\)\(=\)\(-3 \beta_{11} + 9 \beta_{10} - 34 \beta_{9} - 5 \beta_{7} - 191 \beta_{3} + 946 \beta_{1}\)
\(\nu^{6}\)\(=\)\(491 \beta_{8} + 1051 \beta_{6} + 1766 \beta_{5} - 327 \beta_{4} + 1783 \beta_{2} - 47912\)
\(\nu^{7}\)\(=\)\(327 \beta_{11} - 801 \beta_{10} + 1938 \beta_{9} + 517 \beta_{7} + 12915 \beta_{3} - 47576 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-28143 \beta_{8} - 55849 \beta_{6} - 93232 \beta_{5} + 25791 \beta_{4} - 93745 \beta_{2} + 2454152\)
\(\nu^{9}\)\(=\)\(-25791 \beta_{11} + 53421 \beta_{10} - 106384 \beta_{9} - 37293 \beta_{7} - 828917 \beta_{3} + 2458520 \beta_{1}\)
\(\nu^{10}\)\(=\)\(1529399 \beta_{8} + 3032785 \beta_{6} + 4877264 \beta_{5} - 1788243 \beta_{4} + 4935709 \beta_{2} - 128941408\)
\(\nu^{11}\)\(=\)\(1788243 \beta_{11} - 3259197 \beta_{10} + 5800608 \beta_{9} + 2370193 \beta_{7} + 51365613 \beta_{3} - 129062592 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
0.133471i
7.49282i
2.75662i
6.72223i
4.43775i
3.65497i
3.65497i
4.43775i
6.72223i
2.75662i
7.49282i
0.133471i
0 14.8068i 0 26.0324i 0 87.9232i 0 −138.241 0
257.2 0 14.5182i 0 9.40180i 0 53.5326i 0 −129.777 0
257.3 0 12.3317i 0 21.9831i 0 63.3272i 0 −71.0696 0
257.4 0 8.31879i 0 1.48242i 0 13.7963i 0 11.7977 0
257.5 0 6.93950i 0 22.3554i 0 51.9837i 0 32.8434 0
257.6 0 4.18965i 0 45.6695i 0 34.3337i 0 63.4469 0
257.7 0 4.18965i 0 45.6695i 0 34.3337i 0 63.4469 0
257.8 0 6.93950i 0 22.3554i 0 51.9837i 0 32.8434 0
257.9 0 8.31879i 0 1.48242i 0 13.7963i 0 11.7977 0
257.10 0 12.3317i 0 21.9831i 0 63.3272i 0 −71.0696 0
257.11 0 14.5182i 0 9.40180i 0 53.5326i 0 −129.777 0
257.12 0 14.8068i 0 26.0324i 0 87.9232i 0 −138.241 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.5.b.d 12
4.b odd 2 1 43.5.b.b 12
12.b even 2 1 387.5.b.c 12
43.b odd 2 1 inner 688.5.b.d 12
172.d even 2 1 43.5.b.b 12
516.h odd 2 1 387.5.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.b.b 12 4.b odd 2 1
43.5.b.b 12 172.d even 2 1
387.5.b.c 12 12.b even 2 1
387.5.b.c 12 516.h odd 2 1
688.5.b.d 12 1.a even 1 1 trivial
688.5.b.d 12 43.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{12} + 717 T_{3}^{10} + 195527 T_{3}^{8} + 25281430 T_{3}^{6} + 1583947512 T_{3}^{4} + 44423599176 T_{3}^{2} + 411073500528 \)
\( T_{11}^{6} - 90 T_{11}^{5} - 18220 T_{11}^{4} + 2312718 T_{11}^{3} - 63695785 T_{11}^{2} + 475717044 T_{11} - 473184052 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 255 T^{2} + 47783 T^{4} - 6646421 T^{6} + 799989903 T^{8} - 79711532676 T^{10} + 6990833564658 T^{12} - 522987365887236 T^{14} + 34436942157258063 T^{16} - 1877145602287584501 T^{18} + 88542863683907518503 T^{20} - \)\(31\!\cdots\!55\)\( T^{22} + \)\(79\!\cdots\!61\)\( T^{24} \)
$5$ \( 1 - 3663 T^{2} + 6511015 T^{4} - 7365160765 T^{6} + 5973224140031 T^{8} - 3902236317085676 T^{10} + 2399998678381934162 T^{12} - \)\(15\!\cdots\!00\)\( T^{14} + \)\(91\!\cdots\!75\)\( T^{16} - \)\(43\!\cdots\!25\)\( T^{18} + \)\(15\!\cdots\!75\)\( T^{20} - \)\(33\!\cdots\!75\)\( T^{22} + \)\(35\!\cdots\!25\)\( T^{24} \)
$7$ \( 1 - 10134 T^{2} + 60060006 T^{4} - 249160434566 T^{6} + 812915964019071 T^{8} - 2246049549704995236 T^{10} + \)\(56\!\cdots\!48\)\( T^{12} - \)\(12\!\cdots\!36\)\( T^{14} + \)\(27\!\cdots\!71\)\( T^{16} - \)\(47\!\cdots\!66\)\( T^{18} + \)\(66\!\cdots\!06\)\( T^{20} - \)\(64\!\cdots\!34\)\( T^{22} + \)\(36\!\cdots\!01\)\( T^{24} \)
$11$ \( ( 1 - 90 T + 69626 T^{2} - 4275732 T^{3} + 2084651350 T^{4} - 90865763142 T^{5} + 37469241503078 T^{6} - 1330365638162022 T^{7} + 446863530661139350 T^{8} - 13419078640054034772 T^{9} + \)\(31\!\cdots\!86\)\( T^{10} - \)\(60\!\cdots\!90\)\( T^{11} + \)\(98\!\cdots\!41\)\( T^{12} )^{2} \)
$13$ \( ( 1 + 108 T + 78130 T^{2} + 14124934 T^{3} + 3639591614 T^{4} + 705010114760 T^{5} + 121575713991590 T^{6} + 20135793887660360 T^{7} + 2968926691433773694 T^{8} + \)\(32\!\cdots\!54\)\( T^{9} + \)\(51\!\cdots\!30\)\( T^{10} + \)\(20\!\cdots\!08\)\( T^{11} + \)\(54\!\cdots\!61\)\( T^{12} )^{2} \)
$17$ \( ( 1 - 339 T + 345843 T^{2} - 103717833 T^{3} + 57770879546 T^{4} - 15179725016055 T^{5} + 6016834256618483 T^{6} - 1267825813065929655 T^{7} + \)\(40\!\cdots\!86\)\( T^{8} - \)\(60\!\cdots\!13\)\( T^{9} + \)\(16\!\cdots\!83\)\( T^{10} - \)\(13\!\cdots\!39\)\( T^{11} + \)\(33\!\cdots\!21\)\( T^{12} )^{2} \)
$19$ \( 1 - 1175657 T^{2} + 671010023409 T^{4} - 244979577528319381 T^{6} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(12\!\cdots\!10\)\( T^{10} + \)\(18\!\cdots\!98\)\( T^{12} - \)\(20\!\cdots\!10\)\( T^{14} + \)\(18\!\cdots\!03\)\( T^{16} - \)\(12\!\cdots\!01\)\( T^{18} + \)\(55\!\cdots\!49\)\( T^{20} - \)\(16\!\cdots\!57\)\( T^{22} + \)\(23\!\cdots\!41\)\( T^{24} \)
$23$ \( ( 1 + 783 T + 1163077 T^{2} + 843582057 T^{3} + 713091466102 T^{4} + 397049506101027 T^{5} + 259574077696956889 T^{6} + \)\(11\!\cdots\!07\)\( T^{7} + \)\(55\!\cdots\!62\)\( T^{8} + \)\(18\!\cdots\!97\)\( T^{9} + \)\(71\!\cdots\!97\)\( T^{10} + \)\(13\!\cdots\!83\)\( T^{11} + \)\(48\!\cdots\!41\)\( T^{12} )^{2} \)
$29$ \( 1 - 5231219 T^{2} + 13592412986871 T^{4} - 23367533349246758977 T^{6} + \)\(29\!\cdots\!35\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{10} + \)\(23\!\cdots\!82\)\( T^{12} - \)\(14\!\cdots\!68\)\( T^{14} + \)\(74\!\cdots\!35\)\( T^{16} - \)\(29\!\cdots\!37\)\( T^{18} + \)\(85\!\cdots\!11\)\( T^{20} - \)\(16\!\cdots\!19\)\( T^{22} + \)\(15\!\cdots\!61\)\( T^{24} \)
$31$ \( ( 1 + 2855 T + 6518969 T^{2} + 8600011177 T^{3} + 10228398490358 T^{4} + 8682474860253963 T^{5} + 8850492205571119245 T^{6} + \)\(80\!\cdots\!23\)\( T^{7} + \)\(87\!\cdots\!78\)\( T^{8} + \)\(67\!\cdots\!97\)\( T^{9} + \)\(47\!\cdots\!89\)\( T^{10} + \)\(19\!\cdots\!55\)\( T^{11} + \)\(62\!\cdots\!21\)\( T^{12} )^{2} \)
$37$ \( 1 - 6978249 T^{2} + 25643197308697 T^{4} - 70260487375765593853 T^{6} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(39\!\cdots\!70\)\( T^{10} + \)\(80\!\cdots\!70\)\( T^{12} - \)\(13\!\cdots\!70\)\( T^{14} + \)\(21\!\cdots\!19\)\( T^{16} - \)\(30\!\cdots\!33\)\( T^{18} + \)\(39\!\cdots\!57\)\( T^{20} - \)\(37\!\cdots\!49\)\( T^{22} + \)\(18\!\cdots\!21\)\( T^{24} \)
$41$ \( ( 1 - 2439 T + 13822603 T^{2} - 19943096547 T^{3} + 69051390319336 T^{4} - 65595635499999207 T^{5} + \)\(21\!\cdots\!21\)\( T^{6} - \)\(18\!\cdots\!27\)\( T^{7} + \)\(55\!\cdots\!56\)\( T^{8} - \)\(44\!\cdots\!07\)\( T^{9} + \)\(88\!\cdots\!23\)\( T^{10} - \)\(43\!\cdots\!39\)\( T^{11} + \)\(50\!\cdots\!61\)\( T^{12} )^{2} \)
$43$ \( 1 - 1108 T + 1897246 T^{2} + 4681128092 T^{3} + 681017924063 T^{4} + 846563366867960 T^{5} + 64607141521498794692 T^{6} + \)\(28\!\cdots\!60\)\( T^{7} + \)\(79\!\cdots\!63\)\( T^{8} + \)\(18\!\cdots\!92\)\( T^{9} + \)\(25\!\cdots\!46\)\( T^{10} - \)\(51\!\cdots\!08\)\( T^{11} + \)\(15\!\cdots\!01\)\( T^{12} \)
$47$ \( ( 1 - 2763 T + 21754149 T^{2} - 55851382377 T^{3} + 234662825346711 T^{4} - 481604898621884868 T^{5} + \)\(14\!\cdots\!34\)\( T^{6} - \)\(23\!\cdots\!08\)\( T^{7} + \)\(55\!\cdots\!71\)\( T^{8} - \)\(64\!\cdots\!57\)\( T^{9} + \)\(12\!\cdots\!29\)\( T^{10} - \)\(76\!\cdots\!63\)\( T^{11} + \)\(13\!\cdots\!81\)\( T^{12} )^{2} \)
$53$ \( ( 1 - 606 T + 25718024 T^{2} - 24674633094 T^{3} + 346581142619392 T^{4} - 336085601563787574 T^{5} + \)\(32\!\cdots\!82\)\( T^{6} - \)\(26\!\cdots\!94\)\( T^{7} + \)\(21\!\cdots\!12\)\( T^{8} - \)\(12\!\cdots\!54\)\( T^{9} + \)\(99\!\cdots\!04\)\( T^{10} - \)\(18\!\cdots\!06\)\( T^{11} + \)\(24\!\cdots\!81\)\( T^{12} )^{2} \)
$59$ \( ( 1 + 7008 T + 80137258 T^{2} + 394451068512 T^{3} + 2532059909380847 T^{4} + 9247600606403540352 T^{5} + \)\(41\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!72\)\( T^{7} + \)\(37\!\cdots\!87\)\( T^{8} + \)\(70\!\cdots\!72\)\( T^{9} + \)\(17\!\cdots\!78\)\( T^{10} + \)\(18\!\cdots\!08\)\( T^{11} + \)\(31\!\cdots\!61\)\( T^{12} )^{2} \)
$61$ \( 1 - 32665486 T^{2} + 1118892201421590 T^{4} - \)\(24\!\cdots\!14\)\( T^{6} + \)\(52\!\cdots\!07\)\( T^{8} - \)\(83\!\cdots\!88\)\( T^{10} + \)\(13\!\cdots\!08\)\( T^{12} - \)\(16\!\cdots\!28\)\( T^{14} + \)\(19\!\cdots\!27\)\( T^{16} - \)\(17\!\cdots\!74\)\( T^{18} + \)\(15\!\cdots\!90\)\( T^{20} - \)\(84\!\cdots\!86\)\( T^{22} + \)\(49\!\cdots\!81\)\( T^{24} \)
$67$ \( ( 1 - 544 T + 66684038 T^{2} - 18315837698 T^{3} + 2452178722840730 T^{4} - 587634295199545788 T^{5} + \)\(60\!\cdots\!70\)\( T^{6} - \)\(11\!\cdots\!48\)\( T^{7} + \)\(99\!\cdots\!30\)\( T^{8} - \)\(14\!\cdots\!78\)\( T^{9} + \)\(10\!\cdots\!78\)\( T^{10} - \)\(18\!\cdots\!44\)\( T^{11} + \)\(66\!\cdots\!21\)\( T^{12} )^{2} \)
$71$ \( 1 - 101562020 T^{2} + 5784818387848562 T^{4} - \)\(20\!\cdots\!64\)\( T^{6} + \)\(47\!\cdots\!67\)\( T^{8} - \)\(75\!\cdots\!84\)\( T^{10} + \)\(13\!\cdots\!88\)\( T^{12} - \)\(48\!\cdots\!24\)\( T^{14} + \)\(20\!\cdots\!07\)\( T^{16} - \)\(54\!\cdots\!84\)\( T^{18} + \)\(10\!\cdots\!42\)\( T^{20} - \)\(11\!\cdots\!20\)\( T^{22} + \)\(72\!\cdots\!61\)\( T^{24} \)
$73$ \( 1 - 193491698 T^{2} + 19000747654703030 T^{4} - \)\(12\!\cdots\!18\)\( T^{6} + \)\(60\!\cdots\!75\)\( T^{8} - \)\(23\!\cdots\!56\)\( T^{10} + \)\(72\!\cdots\!44\)\( T^{12} - \)\(18\!\cdots\!36\)\( T^{14} + \)\(39\!\cdots\!75\)\( T^{16} - \)\(65\!\cdots\!38\)\( T^{18} + \)\(80\!\cdots\!30\)\( T^{20} - \)\(66\!\cdots\!98\)\( T^{22} + \)\(27\!\cdots\!81\)\( T^{24} \)
$79$ \( ( 1 + 12151 T + 155715789 T^{2} + 800821687889 T^{3} + 4128429862975343 T^{4} - 5685482156720652576 T^{5} - \)\(30\!\cdots\!42\)\( T^{6} - \)\(22\!\cdots\!56\)\( T^{7} + \)\(62\!\cdots\!23\)\( T^{8} + \)\(47\!\cdots\!49\)\( T^{9} + \)\(35\!\cdots\!69\)\( T^{10} + \)\(10\!\cdots\!51\)\( T^{11} + \)\(34\!\cdots\!81\)\( T^{12} )^{2} \)
$83$ \( ( 1 - 3516 T + 117162268 T^{2} - 863865543048 T^{3} + 7914887456622244 T^{4} - 75598926847021341108 T^{5} + \)\(40\!\cdots\!22\)\( T^{6} - \)\(35\!\cdots\!68\)\( T^{7} + \)\(17\!\cdots\!04\)\( T^{8} - \)\(92\!\cdots\!28\)\( T^{9} + \)\(59\!\cdots\!08\)\( T^{10} - \)\(84\!\cdots\!16\)\( T^{11} + \)\(11\!\cdots\!21\)\( T^{12} )^{2} \)
$89$ \( 1 - 474457834 T^{2} + 111655192487117862 T^{4} - \)\(17\!\cdots\!66\)\( T^{6} + \)\(19\!\cdots\!11\)\( T^{8} - \)\(17\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!28\)\( T^{12} - \)\(68\!\cdots\!52\)\( T^{14} + \)\(30\!\cdots\!71\)\( T^{16} - \)\(10\!\cdots\!06\)\( T^{18} + \)\(26\!\cdots\!02\)\( T^{20} - \)\(44\!\cdots\!34\)\( T^{22} + \)\(37\!\cdots\!81\)\( T^{24} \)
$97$ \( ( 1 + 2921 T + 384124299 T^{2} + 837647682043 T^{3} + 69794294829141482 T^{4} + \)\(11\!\cdots\!33\)\( T^{5} + \)\(76\!\cdots\!95\)\( T^{6} + \)\(10\!\cdots\!73\)\( T^{7} + \)\(54\!\cdots\!02\)\( T^{8} + \)\(58\!\cdots\!63\)\( T^{9} + \)\(23\!\cdots\!79\)\( T^{10} + \)\(15\!\cdots\!21\)\( T^{11} + \)\(48\!\cdots\!81\)\( T^{12} )^{2} \)
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