Properties

Label 2.52.a.b
Level 2
Weight 52
Character orbit 2.a
Self dual yes
Analytic conductor 32.946
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 313153613520\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{5}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 155520\sqrt{1252614454081}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 33554432 q^{2} + ( 444809887452 - \beta ) q^{3} + 1125899906842624 q^{4} + ( -252741920103024450 + 4841740 \beta ) q^{5} + ( 14925343121435787264 - 33554432 \beta ) q^{6} + ( -321677336793453348904 - 32204316298 \beta ) q^{7} + 37778931862957161709568 q^{8} + ( -1925541804684254593556043 - 889619774904 \beta ) q^{9} +O(q^{10})\) \( q +33554432 q^{2} +(444809887452 - \beta) q^{3} +1125899906842624 q^{4} +(-252741920103024450 + 4841740 \beta) q^{5} +(14925343121435787264 - 33554432 \beta) q^{6} +(-\)\(32\!\cdots\!04\)\( - 32204316298 \beta) q^{7} +\)\(37\!\cdots\!68\)\( q^{8} +(-\)\(19\!\cdots\!43\)\( - 889619774904 \beta) q^{9} +(-\)\(84\!\cdots\!00\)\( + 162461835591680 \beta) q^{10} +(\)\(43\!\cdots\!92\)\( - 1561205927883371 \beta) q^{11} +(\)\(50\!\cdots\!48\)\( - 1125899906842624 \beta) q^{12} +(-\)\(15\!\cdots\!38\)\( + 170088045644191948 \beta) q^{13} +(-\)\(10\!\cdots\!28\)\( - 1080597541327732736 \beta) q^{14} +(-\)\(25\!\cdots\!00\)\( + 2406395744574870930 \beta) q^{15} +\)\(12\!\cdots\!76\)\( q^{16} +(-\)\(14\!\cdots\!14\)\( - 9504656914909385272 \beta) q^{17} +(-\)\(64\!\cdots\!76\)\( - 29850686242871574528 \beta) q^{18} +(-\)\(36\!\cdots\!00\)\( - \)\(29\!\cdots\!29\)\( \beta) q^{19} +(-\)\(28\!\cdots\!00\)\( + \)\(54\!\cdots\!60\)\( \beta) q^{20} +(\)\(83\!\cdots\!92\)\( - \)\(14\!\cdots\!92\)\( \beta) q^{21} +(\)\(14\!\cdots\!44\)\( - \)\(52\!\cdots\!72\)\( \beta) q^{22} +(\)\(18\!\cdots\!12\)\( + \)\(28\!\cdots\!94\)\( \beta) q^{23} +(\)\(16\!\cdots\!36\)\( - \)\(37\!\cdots\!68\)\( \beta) q^{24} +(\)\(33\!\cdots\!75\)\( - \)\(24\!\cdots\!00\)\( \beta) q^{25} +(-\)\(51\!\cdots\!16\)\( + \)\(57\!\cdots\!36\)\( \beta) q^{26} +(-\)\(17\!\cdots\!80\)\( + \)\(36\!\cdots\!82\)\( \beta) q^{27} +(-\)\(36\!\cdots\!96\)\( - \)\(36\!\cdots\!52\)\( \beta) q^{28} +(-\)\(11\!\cdots\!90\)\( + \)\(31\!\cdots\!92\)\( \beta) q^{29} +(-\)\(86\!\cdots\!00\)\( + \)\(80\!\cdots\!60\)\( \beta) q^{30} +(-\)\(64\!\cdots\!08\)\( - \)\(83\!\cdots\!92\)\( \beta) q^{31} +\)\(42\!\cdots\!32\)\( q^{32} +(\)\(49\!\cdots\!84\)\( - \)\(69\!\cdots\!84\)\( \beta) q^{33} +(-\)\(48\!\cdots\!48\)\( - \)\(31\!\cdots\!04\)\( \beta) q^{34} +(-\)\(46\!\cdots\!00\)\( + \)\(65\!\cdots\!40\)\( \beta) q^{35} +(-\)\(21\!\cdots\!32\)\( - \)\(10\!\cdots\!96\)\( \beta) q^{36} +(-\)\(12\!\cdots\!74\)\( - \)\(24\!\cdots\!92\)\( \beta) q^{37} +(-\)\(12\!\cdots\!00\)\( - \)\(99\!\cdots\!28\)\( \beta) q^{38} +(-\)\(12\!\cdots\!76\)\( + \)\(91\!\cdots\!34\)\( \beta) q^{39} +(-\)\(95\!\cdots\!00\)\( + \)\(18\!\cdots\!20\)\( \beta) q^{40} +(-\)\(10\!\cdots\!18\)\( - \)\(72\!\cdots\!88\)\( \beta) q^{41} +(\)\(27\!\cdots\!44\)\( - \)\(46\!\cdots\!44\)\( \beta) q^{42} +(-\)\(16\!\cdots\!68\)\( + \)\(38\!\cdots\!17\)\( \beta) q^{43} +(\)\(48\!\cdots\!08\)\( - \)\(17\!\cdots\!04\)\( \beta) q^{44} +(\)\(35\!\cdots\!50\)\( - \)\(90\!\cdots\!20\)\( \beta) q^{45} +(\)\(62\!\cdots\!84\)\( + \)\(95\!\cdots\!08\)\( \beta) q^{46} +(\)\(44\!\cdots\!76\)\( + \)\(36\!\cdots\!12\)\( \beta) q^{47} +(\)\(56\!\cdots\!52\)\( - \)\(12\!\cdots\!76\)\( \beta) q^{48} +(\)\(18\!\cdots\!73\)\( + \)\(20\!\cdots\!84\)\( \beta) q^{49} +(\)\(11\!\cdots\!00\)\( - \)\(82\!\cdots\!00\)\( \beta) q^{50} +(-\)\(61\!\cdots\!28\)\( + \)\(10\!\cdots\!70\)\( \beta) q^{51} +(-\)\(17\!\cdots\!12\)\( + \)\(19\!\cdots\!52\)\( \beta) q^{52} +(-\)\(49\!\cdots\!18\)\( - \)\(24\!\cdots\!12\)\( \beta) q^{53} +(-\)\(59\!\cdots\!60\)\( + \)\(12\!\cdots\!24\)\( \beta) q^{54} +(-\)\(23\!\cdots\!00\)\( + \)\(41\!\cdots\!30\)\( \beta) q^{55} +(-\)\(12\!\cdots\!72\)\( - \)\(12\!\cdots\!64\)\( \beta) q^{56} +(-\)\(15\!\cdots\!00\)\( + \)\(23\!\cdots\!92\)\( \beta) q^{57} +(-\)\(38\!\cdots\!80\)\( + \)\(10\!\cdots\!44\)\( \beta) q^{58} +(\)\(35\!\cdots\!20\)\( + \)\(16\!\cdots\!09\)\( \beta) q^{59} +(-\)\(29\!\cdots\!00\)\( + \)\(27\!\cdots\!20\)\( \beta) q^{60} +(\)\(12\!\cdots\!82\)\( - \)\(19\!\cdots\!84\)\( \beta) q^{61} +(-\)\(21\!\cdots\!56\)\( - \)\(27\!\cdots\!44\)\( \beta) q^{62} +(\)\(14\!\cdots\!72\)\( + \)\(62\!\cdots\!30\)\( \beta) q^{63} +\)\(14\!\cdots\!24\)\( q^{64} +(\)\(28\!\cdots\!00\)\( - \)\(11\!\cdots\!20\)\( \beta) q^{65} +(\)\(16\!\cdots\!88\)\( - \)\(23\!\cdots\!88\)\( \beta) q^{66} +(\)\(29\!\cdots\!96\)\( + \)\(15\!\cdots\!11\)\( \beta) q^{67} +(-\)\(16\!\cdots\!36\)\( - \)\(10\!\cdots\!28\)\( \beta) q^{68} +(-\)\(33\!\cdots\!76\)\( + \)\(10\!\cdots\!76\)\( \beta) q^{69} +(-\)\(15\!\cdots\!00\)\( + \)\(22\!\cdots\!80\)\( \beta) q^{70} +(\)\(23\!\cdots\!92\)\( - \)\(12\!\cdots\!34\)\( \beta) q^{71} +(-\)\(72\!\cdots\!24\)\( - \)\(33\!\cdots\!72\)\( \beta) q^{72} +(-\)\(18\!\cdots\!58\)\( + \)\(24\!\cdots\!08\)\( \beta) q^{73} +(-\)\(43\!\cdots\!68\)\( - \)\(82\!\cdots\!44\)\( \beta) q^{74} +(\)\(22\!\cdots\!00\)\( - \)\(14\!\cdots\!75\)\( \beta) q^{75} +(-\)\(41\!\cdots\!00\)\( - \)\(33\!\cdots\!96\)\( \beta) q^{76} +(\)\(15\!\cdots\!32\)\( + \)\(36\!\cdots\!68\)\( \beta) q^{77} +(-\)\(40\!\cdots\!32\)\( + \)\(30\!\cdots\!88\)\( \beta) q^{78} +(\)\(22\!\cdots\!40\)\( - \)\(59\!\cdots\!24\)\( \beta) q^{79} +(-\)\(32\!\cdots\!00\)\( + \)\(61\!\cdots\!40\)\( \beta) q^{80} +(\)\(32\!\cdots\!61\)\( + \)\(53\!\cdots\!32\)\( \beta) q^{81} +(-\)\(34\!\cdots\!76\)\( - \)\(24\!\cdots\!16\)\( \beta) q^{82} +(\)\(32\!\cdots\!12\)\( + \)\(31\!\cdots\!35\)\( \beta) q^{83} +(\)\(93\!\cdots\!08\)\( - \)\(15\!\cdots\!08\)\( \beta) q^{84} +(\)\(22\!\cdots\!00\)\( - \)\(67\!\cdots\!60\)\( \beta) q^{85} +(-\)\(56\!\cdots\!76\)\( + \)\(12\!\cdots\!44\)\( \beta) q^{86} +(-\)\(61\!\cdots\!80\)\( + \)\(25\!\cdots\!74\)\( \beta) q^{87} +(\)\(16\!\cdots\!56\)\( - \)\(58\!\cdots\!28\)\( \beta) q^{88} +(-\)\(37\!\cdots\!90\)\( - \)\(10\!\cdots\!20\)\( \beta) q^{89} +(\)\(11\!\cdots\!00\)\( - \)\(30\!\cdots\!40\)\( \beta) q^{90} +(-\)\(16\!\cdots\!48\)\( + \)\(44\!\cdots\!32\)\( \beta) q^{91} +(\)\(20\!\cdots\!88\)\( + \)\(32\!\cdots\!56\)\( \beta) q^{92} +(-\)\(26\!\cdots\!16\)\( + \)\(27\!\cdots\!24\)\( \beta) q^{93} +(\)\(14\!\cdots\!32\)\( + \)\(12\!\cdots\!84\)\( \beta) q^{94} +(\)\(49\!\cdots\!00\)\( - \)\(17\!\cdots\!50\)\( \beta) q^{95} +(\)\(18\!\cdots\!64\)\( - \)\(42\!\cdots\!32\)\( \beta) q^{96} +(\)\(39\!\cdots\!86\)\( + \)\(11\!\cdots\!20\)\( \beta) q^{97} +(\)\(63\!\cdots\!36\)\( + \)\(69\!\cdots\!88\)\( \beta) q^{98} +(\)\(33\!\cdots\!44\)\( + \)\(30\!\cdots\!85\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 67108864q^{2} + 889619774904q^{3} + 2251799813685248q^{4} - 505483840206048900q^{5} + 29850686242871574528q^{6} - 643354673586906697808q^{7} + 75557863725914323419136q^{8} - 3851083609368509187112086q^{9} + O(q^{10}) \) \( 2q + 67108864q^{2} + 889619774904q^{3} + 2251799813685248q^{4} - 505483840206048900q^{5} + 29850686242871574528q^{6} - \)\(64\!\cdots\!08\)\(q^{7} + \)\(75\!\cdots\!36\)\(q^{8} - \)\(38\!\cdots\!86\)\(q^{9} - \)\(16\!\cdots\!00\)\(q^{10} + \)\(86\!\cdots\!84\)\(q^{11} + \)\(10\!\cdots\!96\)\(q^{12} - \)\(30\!\cdots\!76\)\(q^{13} - \)\(21\!\cdots\!56\)\(q^{14} - \)\(51\!\cdots\!00\)\(q^{15} + \)\(25\!\cdots\!52\)\(q^{16} - \)\(28\!\cdots\!28\)\(q^{17} - \)\(12\!\cdots\!52\)\(q^{18} - \)\(73\!\cdots\!00\)\(q^{19} - \)\(56\!\cdots\!00\)\(q^{20} + \)\(16\!\cdots\!84\)\(q^{21} + \)\(28\!\cdots\!88\)\(q^{22} + \)\(37\!\cdots\!24\)\(q^{23} + \)\(33\!\cdots\!72\)\(q^{24} + \)\(66\!\cdots\!50\)\(q^{25} - \)\(10\!\cdots\!32\)\(q^{26} - \)\(35\!\cdots\!60\)\(q^{27} - \)\(72\!\cdots\!92\)\(q^{28} - \)\(23\!\cdots\!80\)\(q^{29} - \)\(17\!\cdots\!00\)\(q^{30} - \)\(12\!\cdots\!16\)\(q^{31} + \)\(85\!\cdots\!64\)\(q^{32} + \)\(98\!\cdots\!68\)\(q^{33} - \)\(96\!\cdots\!96\)\(q^{34} - \)\(92\!\cdots\!00\)\(q^{35} - \)\(43\!\cdots\!64\)\(q^{36} - \)\(25\!\cdots\!48\)\(q^{37} - \)\(24\!\cdots\!00\)\(q^{38} - \)\(24\!\cdots\!52\)\(q^{39} - \)\(19\!\cdots\!00\)\(q^{40} - \)\(20\!\cdots\!36\)\(q^{41} + \)\(55\!\cdots\!88\)\(q^{42} - \)\(33\!\cdots\!36\)\(q^{43} + \)\(97\!\cdots\!16\)\(q^{44} + \)\(71\!\cdots\!00\)\(q^{45} + \)\(12\!\cdots\!68\)\(q^{46} + \)\(88\!\cdots\!52\)\(q^{47} + \)\(11\!\cdots\!04\)\(q^{48} + \)\(37\!\cdots\!46\)\(q^{49} + \)\(22\!\cdots\!00\)\(q^{50} - \)\(12\!\cdots\!56\)\(q^{51} - \)\(34\!\cdots\!24\)\(q^{52} - \)\(99\!\cdots\!36\)\(q^{53} - \)\(11\!\cdots\!20\)\(q^{54} - \)\(46\!\cdots\!00\)\(q^{55} - \)\(24\!\cdots\!44\)\(q^{56} - \)\(30\!\cdots\!00\)\(q^{57} - \)\(77\!\cdots\!60\)\(q^{58} + \)\(71\!\cdots\!40\)\(q^{59} - \)\(58\!\cdots\!00\)\(q^{60} + \)\(24\!\cdots\!64\)\(q^{61} - \)\(43\!\cdots\!12\)\(q^{62} + \)\(29\!\cdots\!44\)\(q^{63} + \)\(28\!\cdots\!48\)\(q^{64} + \)\(57\!\cdots\!00\)\(q^{65} + \)\(33\!\cdots\!76\)\(q^{66} + \)\(58\!\cdots\!92\)\(q^{67} - \)\(32\!\cdots\!72\)\(q^{68} - \)\(67\!\cdots\!52\)\(q^{69} - \)\(31\!\cdots\!00\)\(q^{70} + \)\(46\!\cdots\!84\)\(q^{71} - \)\(14\!\cdots\!48\)\(q^{72} - \)\(37\!\cdots\!16\)\(q^{73} - \)\(86\!\cdots\!36\)\(q^{74} + \)\(44\!\cdots\!00\)\(q^{75} - \)\(82\!\cdots\!00\)\(q^{76} + \)\(30\!\cdots\!64\)\(q^{77} - \)\(80\!\cdots\!64\)\(q^{78} + \)\(45\!\cdots\!80\)\(q^{79} - \)\(64\!\cdots\!00\)\(q^{80} + \)\(64\!\cdots\!22\)\(q^{81} - \)\(68\!\cdots\!52\)\(q^{82} + \)\(65\!\cdots\!24\)\(q^{83} + \)\(18\!\cdots\!16\)\(q^{84} + \)\(44\!\cdots\!00\)\(q^{85} - \)\(11\!\cdots\!52\)\(q^{86} - \)\(12\!\cdots\!60\)\(q^{87} + \)\(32\!\cdots\!12\)\(q^{88} - \)\(74\!\cdots\!80\)\(q^{89} + \)\(23\!\cdots\!00\)\(q^{90} - \)\(32\!\cdots\!96\)\(q^{91} + \)\(41\!\cdots\!76\)\(q^{92} - \)\(52\!\cdots\!32\)\(q^{93} + \)\(29\!\cdots\!64\)\(q^{94} + \)\(98\!\cdots\!00\)\(q^{95} + \)\(37\!\cdots\!28\)\(q^{96} + \)\(78\!\cdots\!72\)\(q^{97} + \)\(12\!\cdots\!72\)\(q^{98} + \)\(67\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
559602.
−559601.
3.35544e7 2.70751e11 1.12590e15 5.90004e17 9.08491e18 −5.92711e21 3.77789e22 −2.08039e24 1.97972e25
1.2 3.35544e7 6.18868e11 1.12590e15 −1.09549e18 2.07658e19 5.28375e21 3.77789e22 −1.77070e24 −3.67585e25
\(n\): 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.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.52.a.b 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 889619774904 T_{3} + \)\(16\!\cdots\!04\)\( \) acting on \(S_{52}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 33554432 T )^{2} \)
$3$ \( 1 - 889619774904 T + \)\(44\!\cdots\!98\)\( T^{2} - \)\(19\!\cdots\!88\)\( T^{3} + \)\(46\!\cdots\!09\)\( T^{4} \)
$5$ \( 1 + 505483840206048900 T + \)\(24\!\cdots\!50\)\( T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + \)\(64\!\cdots\!08\)\( T - \)\(61\!\cdots\!98\)\( T^{2} + \)\(80\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - \)\(86\!\cdots\!84\)\( T + \)\(18\!\cdots\!86\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 + \)\(30\!\cdots\!76\)\( T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(19\!\cdots\!12\)\( T^{3} + \)\(41\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 + \)\(28\!\cdots\!28\)\( T + \)\(13\!\cdots\!62\)\( T^{2} + \)\(16\!\cdots\!24\)\( T^{3} + \)\(32\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 + \)\(73\!\cdots\!00\)\( T + \)\(46\!\cdots\!38\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 - \)\(37\!\cdots\!24\)\( T + \)\(34\!\cdots\!98\)\( T^{2} - \)\(10\!\cdots\!48\)\( T^{3} + \)\(78\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 + \)\(23\!\cdots\!80\)\( T + \)\(86\!\cdots\!58\)\( T^{2} + \)\(88\!\cdots\!20\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 + \)\(12\!\cdots\!16\)\( T + \)\(26\!\cdots\!26\)\( T^{2} + \)\(14\!\cdots\!96\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 + \)\(25\!\cdots\!48\)\( T + \)\(33\!\cdots\!02\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(90\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 + \)\(20\!\cdots\!36\)\( T + \)\(30\!\cdots\!06\)\( T^{2} + \)\(36\!\cdots\!76\)\( T^{3} + \)\(31\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 + \)\(33\!\cdots\!36\)\( T - \)\(40\!\cdots\!62\)\( T^{2} + \)\(67\!\cdots\!52\)\( T^{3} + \)\(41\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(88\!\cdots\!52\)\( T + \)\(56\!\cdots\!82\)\( T^{2} - \)\(16\!\cdots\!56\)\( T^{3} + \)\(35\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 + \)\(99\!\cdots\!36\)\( T + \)\(18\!\cdots\!18\)\( T^{2} + \)\(85\!\cdots\!92\)\( T^{3} + \)\(75\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 - \)\(71\!\cdots\!40\)\( T + \)\(41\!\cdots\!18\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(42\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 - \)\(24\!\cdots\!64\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(27\!\cdots\!04\)\( T^{3} + \)\(12\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(28\!\cdots\!82\)\( T^{2} - \)\(79\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 - \)\(46\!\cdots\!84\)\( T + \)\(64\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!64\)\( T^{3} + \)\(67\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 + \)\(37\!\cdots\!16\)\( T + \)\(65\!\cdots\!18\)\( T^{2} + \)\(40\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 - \)\(45\!\cdots\!80\)\( T + \)\(16\!\cdots\!58\)\( T^{2} - \)\(27\!\cdots\!20\)\( T^{3} + \)\(36\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 - \)\(65\!\cdots\!24\)\( T + \)\(12\!\cdots\!78\)\( T^{2} - \)\(49\!\cdots\!08\)\( T^{3} + \)\(55\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 + \)\(74\!\cdots\!80\)\( T + \)\(62\!\cdots\!78\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 - \)\(78\!\cdots\!72\)\( T + \)\(53\!\cdots\!02\)\( T^{2} - \)\(16\!\cdots\!16\)\( T^{3} + \)\(44\!\cdots\!09\)\( T^{4} \)
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