Properties

Label 2.52.a.a
Level 2
Weight 52
Character orbit 2.a
Self dual yes
Analytic conductor 32.946
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 52 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.9462706828\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 126606928812\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5^{2}\cdot 17 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 163200\sqrt{506427715249}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -33554432 q^{2} +(93645191388 - 17 \beta) q^{3} +1125899906842624 q^{4} +(-611368722926141250 + 3761100 \beta) q^{5} +(-3142211206555631616 + 570425344 \beta) q^{6} +(-\)\(20\!\cdots\!76\)\( + 13900943574 \beta) q^{7} -\)\(37\!\cdots\!68\)\( q^{8} +(\)\(17\!\cdots\!97\)\( - 3183936507192 \beta) q^{9} +O(q^{10})\) \( q -33554432 q^{2} +(93645191388 - 17 \beta) q^{3} +1125899906842624 q^{4} +(-611368722926141250 + 3761100 \beta) q^{5} +(-3142211206555631616 + 570425344 \beta) q^{6} +(-\)\(20\!\cdots\!76\)\( + 13900943574 \beta) q^{7} -\)\(37\!\cdots\!68\)\( q^{8} +(\)\(17\!\cdots\!97\)\( - 3183936507192 \beta) q^{9} +(\)\(20\!\cdots\!00\)\( - 126201574195200 \beta) q^{10} +(-\)\(25\!\cdots\!48\)\( - 2340849353861403 \beta) q^{11} +(\)\(10\!\cdots\!12\)\( - 19140298416324608 \beta) q^{12} +(\)\(11\!\cdots\!18\)\( - 296801475256649844 \beta) q^{13} +(\)\(68\!\cdots\!32\)\( - 466438265889619968 \beta) q^{14} +(-\)\(91\!\cdots\!00\)\( + 10745477219073808050 \beta) q^{15} +\)\(12\!\cdots\!76\)\( q^{16} +(-\)\(19\!\cdots\!06\)\( + 65544093400653633096 \beta) q^{17} +(-\)\(58\!\cdots\!04\)\( + \)\(10\!\cdots\!44\)\( \beta) q^{18} +(-\)\(68\!\cdots\!60\)\( - \)\(52\!\cdots\!17\)\( \beta) q^{19} +(-\)\(68\!\cdots\!00\)\( + \)\(42\!\cdots\!00\)\( \beta) q^{20} +(-\)\(33\!\cdots\!88\)\( + \)\(36\!\cdots\!04\)\( \beta) q^{21} +(\)\(85\!\cdots\!36\)\( + \)\(78\!\cdots\!96\)\( \beta) q^{22} +(\)\(41\!\cdots\!28\)\( - \)\(44\!\cdots\!42\)\( \beta) q^{23} +(-\)\(35\!\cdots\!84\)\( + \)\(64\!\cdots\!56\)\( \beta) q^{24} +(\)\(12\!\cdots\!75\)\( - \)\(45\!\cdots\!00\)\( \beta) q^{25} +(-\)\(40\!\cdots\!76\)\( + \)\(99\!\cdots\!08\)\( \beta) q^{26} +(\)\(69\!\cdots\!00\)\( + \)\(65\!\cdots\!54\)\( \beta) q^{27} +(-\)\(23\!\cdots\!24\)\( + \)\(15\!\cdots\!76\)\( \beta) q^{28} +(-\)\(16\!\cdots\!10\)\( - \)\(49\!\cdots\!24\)\( \beta) q^{29} +(\)\(30\!\cdots\!00\)\( - \)\(36\!\cdots\!00\)\( \beta) q^{30} +(\)\(15\!\cdots\!12\)\( + \)\(44\!\cdots\!04\)\( \beta) q^{31} -\)\(42\!\cdots\!32\)\( q^{32} +(\)\(51\!\cdots\!76\)\( + \)\(41\!\cdots\!52\)\( \beta) q^{33} +(\)\(64\!\cdots\!92\)\( - \)\(21\!\cdots\!72\)\( \beta) q^{34} +(\)\(19\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{35} +(\)\(19\!\cdots\!28\)\( - \)\(35\!\cdots\!08\)\( \beta) q^{36} +(\)\(52\!\cdots\!54\)\( - \)\(11\!\cdots\!44\)\( \beta) q^{37} +(\)\(22\!\cdots\!20\)\( + \)\(17\!\cdots\!44\)\( \beta) q^{38} +(\)\(68\!\cdots\!84\)\( - \)\(48\!\cdots\!78\)\( \beta) q^{39} +(\)\(23\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{40} +(\)\(98\!\cdots\!62\)\( - \)\(11\!\cdots\!24\)\( \beta) q^{41} +(\)\(11\!\cdots\!16\)\( - \)\(12\!\cdots\!28\)\( \beta) q^{42} +(-\)\(21\!\cdots\!12\)\( + \)\(20\!\cdots\!49\)\( \beta) q^{43} +(-\)\(28\!\cdots\!52\)\( - \)\(26\!\cdots\!72\)\( \beta) q^{44} +(-\)\(12\!\cdots\!50\)\( + \)\(85\!\cdots\!00\)\( \beta) q^{45} +(-\)\(13\!\cdots\!96\)\( + \)\(14\!\cdots\!44\)\( \beta) q^{46} +(-\)\(25\!\cdots\!56\)\( - \)\(39\!\cdots\!16\)\( \beta) q^{47} +(\)\(11\!\cdots\!88\)\( - \)\(21\!\cdots\!92\)\( \beta) q^{48} +(-\)\(57\!\cdots\!67\)\( - \)\(56\!\cdots\!48\)\( \beta) q^{49} +(-\)\(40\!\cdots\!00\)\( + \)\(15\!\cdots\!00\)\( \beta) q^{50} +(-\)\(16\!\cdots\!28\)\( + \)\(33\!\cdots\!50\)\( \beta) q^{51} +(\)\(13\!\cdots\!32\)\( - \)\(33\!\cdots\!56\)\( \beta) q^{52} +(\)\(52\!\cdots\!98\)\( - \)\(38\!\cdots\!24\)\( \beta) q^{53} +(-\)\(23\!\cdots\!00\)\( - \)\(21\!\cdots\!28\)\( \beta) q^{54} +(\)\(37\!\cdots\!00\)\( + \)\(46\!\cdots\!50\)\( \beta) q^{55} +(\)\(77\!\cdots\!68\)\( - \)\(52\!\cdots\!32\)\( \beta) q^{56} +(\)\(12\!\cdots\!20\)\( + \)\(66\!\cdots\!24\)\( \beta) q^{57} +(\)\(56\!\cdots\!20\)\( + \)\(16\!\cdots\!68\)\( \beta) q^{58} +(-\)\(55\!\cdots\!20\)\( + \)\(28\!\cdots\!77\)\( \beta) q^{59} +(-\)\(10\!\cdots\!00\)\( + \)\(12\!\cdots\!00\)\( \beta) q^{60} +(\)\(10\!\cdots\!22\)\( - \)\(40\!\cdots\!92\)\( \beta) q^{61} +(-\)\(51\!\cdots\!84\)\( - \)\(14\!\cdots\!28\)\( \beta) q^{62} +(-\)\(41\!\cdots\!72\)\( + \)\(30\!\cdots\!70\)\( \beta) q^{63} +\)\(14\!\cdots\!24\)\( q^{64} +(-\)\(15\!\cdots\!00\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{65} +(-\)\(17\!\cdots\!32\)\( - \)\(13\!\cdots\!64\)\( \beta) q^{66} +(\)\(26\!\cdots\!64\)\( - \)\(81\!\cdots\!33\)\( \beta) q^{67} +(-\)\(21\!\cdots\!44\)\( + \)\(73\!\cdots\!04\)\( \beta) q^{68} +(\)\(10\!\cdots\!64\)\( - \)\(74\!\cdots\!72\)\( \beta) q^{69} +(-\)\(65\!\cdots\!00\)\( + \)\(54\!\cdots\!00\)\( \beta) q^{70} +(\)\(35\!\cdots\!32\)\( + \)\(45\!\cdots\!18\)\( \beta) q^{71} +(-\)\(66\!\cdots\!96\)\( + \)\(12\!\cdots\!56\)\( \beta) q^{72} +(\)\(31\!\cdots\!38\)\( + \)\(11\!\cdots\!16\)\( \beta) q^{73} +(-\)\(17\!\cdots\!28\)\( + \)\(39\!\cdots\!08\)\( \beta) q^{74} +(\)\(10\!\cdots\!00\)\( - \)\(24\!\cdots\!75\)\( \beta) q^{75} +(-\)\(76\!\cdots\!40\)\( - \)\(59\!\cdots\!08\)\( \beta) q^{76} +(\)\(85\!\cdots\!48\)\( + \)\(12\!\cdots\!76\)\( \beta) q^{77} +(-\)\(22\!\cdots\!88\)\( + \)\(16\!\cdots\!96\)\( \beta) q^{78} +(\)\(99\!\cdots\!80\)\( + \)\(12\!\cdots\!88\)\( \beta) q^{79} +(-\)\(77\!\cdots\!00\)\( + \)\(47\!\cdots\!00\)\( \beta) q^{80} +(-\)\(52\!\cdots\!59\)\( - \)\(43\!\cdots\!24\)\( \beta) q^{81} +(-\)\(33\!\cdots\!84\)\( + \)\(40\!\cdots\!68\)\( \beta) q^{82} +(-\)\(36\!\cdots\!12\)\( - \)\(37\!\cdots\!25\)\( \beta) q^{83} +(-\)\(38\!\cdots\!12\)\( + \)\(40\!\cdots\!96\)\( \beta) q^{84} +(\)\(14\!\cdots\!00\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{85} +(\)\(73\!\cdots\!84\)\( - \)\(67\!\cdots\!68\)\( \beta) q^{86} +(\)\(98\!\cdots\!20\)\( + \)\(27\!\cdots\!58\)\( \beta) q^{87} +(\)\(96\!\cdots\!64\)\( + \)\(88\!\cdots\!04\)\( \beta) q^{88} +(\)\(38\!\cdots\!10\)\( + \)\(48\!\cdots\!40\)\( \beta) q^{89} +(\)\(41\!\cdots\!00\)\( - \)\(28\!\cdots\!00\)\( \beta) q^{90} +(-\)\(58\!\cdots\!68\)\( + \)\(62\!\cdots\!76\)\( \beta) q^{91} +(\)\(46\!\cdots\!72\)\( - \)\(50\!\cdots\!08\)\( \beta) q^{92} +(-\)\(87\!\cdots\!44\)\( - \)\(25\!\cdots\!52\)\( \beta) q^{93} +(\)\(85\!\cdots\!92\)\( + \)\(13\!\cdots\!12\)\( \beta) q^{94} +(-\)\(22\!\cdots\!00\)\( + \)\(29\!\cdots\!50\)\( \beta) q^{95} +(-\)\(39\!\cdots\!16\)\( + \)\(72\!\cdots\!44\)\( \beta) q^{96} +(-\)\(14\!\cdots\!86\)\( - \)\(14\!\cdots\!80\)\( \beta) q^{97} +(\)\(19\!\cdots\!44\)\( + \)\(19\!\cdots\!36\)\( \beta) q^{98} +(-\)\(34\!\cdots\!56\)\( - \)\(32\!\cdots\!75\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 67108864q^{2} + 187290382776q^{3} + 2251799813685248q^{4} - 1222737445852282500q^{5} - 6284422413111263232q^{6} - \)\(41\!\cdots\!52\)\(q^{7} - \)\(75\!\cdots\!36\)\(q^{8} + \)\(35\!\cdots\!94\)\(q^{9} + O(q^{10}) \) \( 2q - 67108864q^{2} + 187290382776q^{3} + 2251799813685248q^{4} - 1222737445852282500q^{5} - 6284422413111263232q^{6} - \)\(41\!\cdots\!52\)\(q^{7} - \)\(75\!\cdots\!36\)\(q^{8} + \)\(35\!\cdots\!94\)\(q^{9} + \)\(41\!\cdots\!00\)\(q^{10} - \)\(51\!\cdots\!96\)\(q^{11} + \)\(21\!\cdots\!24\)\(q^{12} + \)\(23\!\cdots\!36\)\(q^{13} + \)\(13\!\cdots\!64\)\(q^{14} - \)\(18\!\cdots\!00\)\(q^{15} + \)\(25\!\cdots\!52\)\(q^{16} - \)\(38\!\cdots\!12\)\(q^{17} - \)\(11\!\cdots\!08\)\(q^{18} - \)\(13\!\cdots\!20\)\(q^{19} - \)\(13\!\cdots\!00\)\(q^{20} - \)\(67\!\cdots\!76\)\(q^{21} + \)\(17\!\cdots\!72\)\(q^{22} + \)\(82\!\cdots\!56\)\(q^{23} - \)\(70\!\cdots\!68\)\(q^{24} + \)\(24\!\cdots\!50\)\(q^{25} - \)\(80\!\cdots\!52\)\(q^{26} + \)\(13\!\cdots\!00\)\(q^{27} - \)\(46\!\cdots\!48\)\(q^{28} - \)\(33\!\cdots\!20\)\(q^{29} + \)\(61\!\cdots\!00\)\(q^{30} + \)\(30\!\cdots\!24\)\(q^{31} - \)\(85\!\cdots\!64\)\(q^{32} + \)\(10\!\cdots\!52\)\(q^{33} + \)\(12\!\cdots\!84\)\(q^{34} + \)\(39\!\cdots\!00\)\(q^{35} + \)\(39\!\cdots\!56\)\(q^{36} + \)\(10\!\cdots\!08\)\(q^{37} + \)\(45\!\cdots\!40\)\(q^{38} + \)\(13\!\cdots\!68\)\(q^{39} + \)\(46\!\cdots\!00\)\(q^{40} + \)\(19\!\cdots\!24\)\(q^{41} + \)\(22\!\cdots\!32\)\(q^{42} - \)\(43\!\cdots\!24\)\(q^{43} - \)\(57\!\cdots\!04\)\(q^{44} - \)\(24\!\cdots\!00\)\(q^{45} - \)\(27\!\cdots\!92\)\(q^{46} - \)\(50\!\cdots\!12\)\(q^{47} + \)\(23\!\cdots\!76\)\(q^{48} - \)\(11\!\cdots\!34\)\(q^{49} - \)\(80\!\cdots\!00\)\(q^{50} - \)\(33\!\cdots\!56\)\(q^{51} + \)\(26\!\cdots\!64\)\(q^{52} + \)\(10\!\cdots\!96\)\(q^{53} - \)\(46\!\cdots\!00\)\(q^{54} + \)\(75\!\cdots\!00\)\(q^{55} + \)\(15\!\cdots\!36\)\(q^{56} + \)\(24\!\cdots\!40\)\(q^{57} + \)\(11\!\cdots\!40\)\(q^{58} - \)\(11\!\cdots\!40\)\(q^{59} - \)\(20\!\cdots\!00\)\(q^{60} + \)\(21\!\cdots\!44\)\(q^{61} - \)\(10\!\cdots\!68\)\(q^{62} - \)\(83\!\cdots\!44\)\(q^{63} + \)\(28\!\cdots\!48\)\(q^{64} - \)\(31\!\cdots\!00\)\(q^{65} - \)\(34\!\cdots\!64\)\(q^{66} + \)\(52\!\cdots\!28\)\(q^{67} - \)\(42\!\cdots\!88\)\(q^{68} + \)\(21\!\cdots\!28\)\(q^{69} - \)\(13\!\cdots\!00\)\(q^{70} + \)\(71\!\cdots\!64\)\(q^{71} - \)\(13\!\cdots\!92\)\(q^{72} + \)\(63\!\cdots\!76\)\(q^{73} - \)\(35\!\cdots\!56\)\(q^{74} + \)\(21\!\cdots\!00\)\(q^{75} - \)\(15\!\cdots\!80\)\(q^{76} + \)\(17\!\cdots\!96\)\(q^{77} - \)\(45\!\cdots\!76\)\(q^{78} + \)\(19\!\cdots\!60\)\(q^{79} - \)\(15\!\cdots\!00\)\(q^{80} - \)\(10\!\cdots\!18\)\(q^{81} - \)\(66\!\cdots\!68\)\(q^{82} - \)\(72\!\cdots\!24\)\(q^{83} - \)\(76\!\cdots\!24\)\(q^{84} + \)\(29\!\cdots\!00\)\(q^{85} + \)\(14\!\cdots\!68\)\(q^{86} + \)\(19\!\cdots\!40\)\(q^{87} + \)\(19\!\cdots\!28\)\(q^{88} + \)\(76\!\cdots\!20\)\(q^{89} + \)\(82\!\cdots\!00\)\(q^{90} - \)\(11\!\cdots\!36\)\(q^{91} + \)\(92\!\cdots\!44\)\(q^{92} - \)\(17\!\cdots\!88\)\(q^{93} + \)\(17\!\cdots\!84\)\(q^{94} - \)\(45\!\cdots\!00\)\(q^{95} - \)\(79\!\cdots\!32\)\(q^{96} - \)\(29\!\cdots\!72\)\(q^{97} + \)\(38\!\cdots\!88\)\(q^{98} - \)\(69\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
355819.
−355818.
−3.35544e7 −1.88072e12 1.12590e15 −1.74558e17 6.31065e19 −4.35744e20 −3.77789e22 1.38342e24 5.85718e24
1.2 −3.35544e7 2.06801e12 1.12590e15 −1.04818e18 −6.93910e19 −3.66463e21 −3.77789e22 2.12298e24 3.51711e25
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2.52.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.52.a.a 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 187290382776 T_{3} - \)\(38\!\cdots\!56\)\( \) acting on \(S_{52}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 33554432 T )^{2} \)
$3$ \( 1 - 187290382776 T + \)\(41\!\cdots\!38\)\( T^{2} - \)\(40\!\cdots\!72\)\( T^{3} + \)\(46\!\cdots\!09\)\( T^{4} \)
$5$ \( 1 + 1222737445852282500 T + \)\(10\!\cdots\!50\)\( T^{2} + \)\(54\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + \)\(41\!\cdots\!52\)\( T + \)\(26\!\cdots\!62\)\( T^{2} + \)\(51\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + \)\(51\!\cdots\!96\)\( T + \)\(24\!\cdots\!26\)\( T^{2} + \)\(66\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 - \)\(23\!\cdots\!36\)\( T + \)\(10\!\cdots\!98\)\( T^{2} - \)\(15\!\cdots\!32\)\( T^{3} + \)\(41\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 + \)\(38\!\cdots\!12\)\( T + \)\(14\!\cdots\!02\)\( T^{2} + \)\(21\!\cdots\!96\)\( T^{3} + \)\(32\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 + \)\(13\!\cdots\!20\)\( T - \)\(41\!\cdots\!62\)\( T^{2} + \)\(22\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 - \)\(82\!\cdots\!56\)\( T + \)\(46\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!12\)\( T^{3} + \)\(78\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 + \)\(33\!\cdots\!20\)\( T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(12\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 - \)\(30\!\cdots\!24\)\( T + \)\(44\!\cdots\!06\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 - \)\(10\!\cdots\!08\)\( T + \)\(21\!\cdots\!42\)\( T^{2} - \)\(10\!\cdots\!04\)\( T^{3} + \)\(90\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 - \)\(19\!\cdots\!24\)\( T + \)\(45\!\cdots\!26\)\( T^{2} - \)\(35\!\cdots\!84\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 + \)\(43\!\cdots\!24\)\( T + \)\(39\!\cdots\!58\)\( T^{2} + \)\(88\!\cdots\!68\)\( T^{3} + \)\(41\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(50\!\cdots\!12\)\( T + \)\(22\!\cdots\!42\)\( T^{2} + \)\(95\!\cdots\!36\)\( T^{3} + \)\(35\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 - \)\(10\!\cdots\!96\)\( T + \)\(18\!\cdots\!98\)\( T^{2} - \)\(91\!\cdots\!12\)\( T^{3} + \)\(75\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 + \)\(11\!\cdots\!40\)\( T + \)\(43\!\cdots\!18\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 - \)\(21\!\cdots\!44\)\( T + \)\(17\!\cdots\!06\)\( T^{2} - \)\(24\!\cdots\!84\)\( T^{3} + \)\(12\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 - \)\(52\!\cdots\!28\)\( T + \)\(33\!\cdots\!62\)\( T^{2} - \)\(71\!\cdots\!24\)\( T^{3} + \)\(18\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 - \)\(71\!\cdots\!64\)\( T + \)\(50\!\cdots\!66\)\( T^{2} - \)\(18\!\cdots\!44\)\( T^{3} + \)\(67\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 - \)\(63\!\cdots\!76\)\( T + \)\(29\!\cdots\!98\)\( T^{2} - \)\(67\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(10\!\cdots\!58\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(36\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 + \)\(72\!\cdots\!24\)\( T + \)\(14\!\cdots\!78\)\( T^{2} + \)\(54\!\cdots\!08\)\( T^{3} + \)\(55\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 - \)\(76\!\cdots\!20\)\( T + \)\(67\!\cdots\!78\)\( T^{2} - \)\(20\!\cdots\!80\)\( T^{3} + \)\(68\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 + \)\(29\!\cdots\!72\)\( T + \)\(41\!\cdots\!02\)\( T^{2} + \)\(63\!\cdots\!16\)\( T^{3} + \)\(44\!\cdots\!09\)\( T^{4} \)
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