Properties

Label 2.72.a.a
Level 2
Weight 72
Character orbit 2.a
Self dual yes
Analytic conductor 63.849
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 72 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.8492321122\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{6}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 1166400\sqrt{253576158163875107521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +34359738368 q^{2} +(-36645276287423412 - 3 \beta) q^{3} +\)\(11\!\cdots\!24\)\( q^{4} +(\)\(20\!\cdots\!50\)\( + 349418900 \beta) q^{5} +(-\)\(12\!\cdots\!16\)\( - 103079215104 \beta) q^{6} +(-\)\(14\!\cdots\!76\)\( - 45598388461934 \beta) q^{7} +\)\(40\!\cdots\!32\)\( q^{8} +(-\)\(30\!\cdots\!03\)\( + 219871657724540472 \beta) q^{9} +O(q^{10})\) \( q +34359738368 q^{2} +(-36645276287423412 - 3 \beta) q^{3} +\)\(11\!\cdots\!24\)\( q^{4} +(\)\(20\!\cdots\!50\)\( + 349418900 \beta) q^{5} +(-\)\(12\!\cdots\!16\)\( - 103079215104 \beta) q^{6} +(-\)\(14\!\cdots\!76\)\( - 45598388461934 \beta) q^{7} +\)\(40\!\cdots\!32\)\( q^{8} +(-\)\(30\!\cdots\!03\)\( + 219871657724540472 \beta) q^{9} +(\)\(69\!\cdots\!00\)\( + 12005941984834355200 \beta) q^{10} +(\)\(43\!\cdots\!52\)\( + \)\(15\!\cdots\!23\)\( \beta) q^{11} +(-\)\(43\!\cdots\!88\)\( - \)\(35\!\cdots\!72\)\( \beta) q^{12} +(-\)\(56\!\cdots\!82\)\( - \)\(11\!\cdots\!96\)\( \beta) q^{13} +(-\)\(50\!\cdots\!68\)\( - \)\(15\!\cdots\!12\)\( \beta) q^{14} +(-\)\(43\!\cdots\!00\)\( - \)\(18\!\cdots\!50\)\( \beta) q^{15} +\)\(13\!\cdots\!76\)\( q^{16} +(-\)\(12\!\cdots\!06\)\( + \)\(23\!\cdots\!64\)\( \beta) q^{17} +(-\)\(10\!\cdots\!04\)\( + \)\(75\!\cdots\!96\)\( \beta) q^{18} +(-\)\(75\!\cdots\!60\)\( - \)\(13\!\cdots\!03\)\( \beta) q^{19} +(\)\(23\!\cdots\!00\)\( + \)\(41\!\cdots\!00\)\( \beta) q^{20} +(\)\(52\!\cdots\!12\)\( + \)\(21\!\cdots\!36\)\( \beta) q^{21} +(\)\(14\!\cdots\!36\)\( + \)\(54\!\cdots\!64\)\( \beta) q^{22} +(\)\(11\!\cdots\!28\)\( + \)\(11\!\cdots\!22\)\( \beta) q^{23} +(-\)\(14\!\cdots\!84\)\( - \)\(12\!\cdots\!96\)\( \beta) q^{24} +(\)\(38\!\cdots\!75\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{25} +(-\)\(19\!\cdots\!76\)\( - \)\(38\!\cdots\!28\)\( \beta) q^{26} +(\)\(15\!\cdots\!00\)\( + \)\(23\!\cdots\!86\)\( \beta) q^{27} +(-\)\(17\!\cdots\!24\)\( - \)\(53\!\cdots\!16\)\( \beta) q^{28} +(-\)\(48\!\cdots\!10\)\( - \)\(41\!\cdots\!16\)\( \beta) q^{29} +(-\)\(14\!\cdots\!00\)\( - \)\(64\!\cdots\!00\)\( \beta) q^{30} +(-\)\(20\!\cdots\!88\)\( + \)\(58\!\cdots\!36\)\( \beta) q^{31} +\)\(47\!\cdots\!68\)\( q^{32} +(-\)\(32\!\cdots\!24\)\( - \)\(18\!\cdots\!32\)\( \beta) q^{33} +(-\)\(43\!\cdots\!08\)\( + \)\(80\!\cdots\!52\)\( \beta) q^{34} +(-\)\(57\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{35} +(-\)\(36\!\cdots\!72\)\( + \)\(25\!\cdots\!28\)\( \beta) q^{36} +(-\)\(55\!\cdots\!46\)\( - \)\(69\!\cdots\!96\)\( \beta) q^{37} +(-\)\(26\!\cdots\!80\)\( - \)\(47\!\cdots\!04\)\( \beta) q^{38} +(\)\(13\!\cdots\!84\)\( + \)\(58\!\cdots\!98\)\( \beta) q^{39} +(\)\(81\!\cdots\!00\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{40} +(\)\(64\!\cdots\!62\)\( + \)\(41\!\cdots\!84\)\( \beta) q^{41} +(\)\(18\!\cdots\!16\)\( + \)\(72\!\cdots\!48\)\( \beta) q^{42} +(-\)\(78\!\cdots\!12\)\( - \)\(58\!\cdots\!09\)\( \beta) q^{43} +(\)\(51\!\cdots\!48\)\( + \)\(18\!\cdots\!52\)\( \beta) q^{44} +(\)\(20\!\cdots\!50\)\( - \)\(62\!\cdots\!00\)\( \beta) q^{45} +(\)\(37\!\cdots\!04\)\( + \)\(39\!\cdots\!96\)\( \beta) q^{46} +(-\)\(19\!\cdots\!56\)\( - \)\(36\!\cdots\!44\)\( \beta) q^{47} +(-\)\(51\!\cdots\!12\)\( - \)\(41\!\cdots\!28\)\( \beta) q^{48} +(-\)\(26\!\cdots\!67\)\( + \)\(13\!\cdots\!68\)\( \beta) q^{49} +(\)\(13\!\cdots\!00\)\( + \)\(48\!\cdots\!00\)\( \beta) q^{50} +(-\)\(19\!\cdots\!28\)\( - \)\(48\!\cdots\!50\)\( \beta) q^{51} +(-\)\(66\!\cdots\!68\)\( - \)\(13\!\cdots\!04\)\( \beta) q^{52} +(-\)\(15\!\cdots\!02\)\( - \)\(63\!\cdots\!16\)\( \beta) q^{53} +(\)\(54\!\cdots\!00\)\( + \)\(81\!\cdots\!48\)\( \beta) q^{54} +(\)\(27\!\cdots\!00\)\( + \)\(18\!\cdots\!50\)\( \beta) q^{55} +(-\)\(59\!\cdots\!32\)\( - \)\(18\!\cdots\!88\)\( \beta) q^{56} +(\)\(17\!\cdots\!20\)\( + \)\(73\!\cdots\!16\)\( \beta) q^{57} +(-\)\(16\!\cdots\!80\)\( - \)\(14\!\cdots\!88\)\( \beta) q^{58} +(-\)\(81\!\cdots\!20\)\( + \)\(66\!\cdots\!43\)\( \beta) q^{59} +(-\)\(51\!\cdots\!00\)\( - \)\(22\!\cdots\!00\)\( \beta) q^{60} +(-\)\(24\!\cdots\!78\)\( - \)\(78\!\cdots\!28\)\( \beta) q^{61} +(-\)\(68\!\cdots\!84\)\( + \)\(20\!\cdots\!48\)\( \beta) q^{62} +(-\)\(30\!\cdots\!72\)\( + \)\(10\!\cdots\!30\)\( \beta) q^{63} +\)\(16\!\cdots\!24\)\( q^{64} +(-\)\(14\!\cdots\!00\)\( - \)\(42\!\cdots\!00\)\( \beta) q^{65} +(-\)\(11\!\cdots\!32\)\( - \)\(64\!\cdots\!76\)\( \beta) q^{66} +(-\)\(67\!\cdots\!36\)\( + \)\(12\!\cdots\!53\)\( \beta) q^{67} +(-\)\(14\!\cdots\!44\)\( + \)\(27\!\cdots\!36\)\( \beta) q^{68} +(-\)\(16\!\cdots\!36\)\( - \)\(75\!\cdots\!48\)\( \beta) q^{69} +(-\)\(19\!\cdots\!00\)\( - \)\(48\!\cdots\!00\)\( \beta) q^{70} +(-\)\(73\!\cdots\!68\)\( + \)\(81\!\cdots\!62\)\( \beta) q^{71} +(-\)\(12\!\cdots\!96\)\( + \)\(89\!\cdots\!04\)\( \beta) q^{72} +(-\)\(16\!\cdots\!62\)\( + \)\(55\!\cdots\!44\)\( \beta) q^{73} +(-\)\(19\!\cdots\!28\)\( - \)\(24\!\cdots\!28\)\( \beta) q^{74} +(-\)\(15\!\cdots\!00\)\( - \)\(62\!\cdots\!25\)\( \beta) q^{75} +(-\)\(89\!\cdots\!40\)\( - \)\(16\!\cdots\!72\)\( \beta) q^{76} +(-\)\(31\!\cdots\!52\)\( - \)\(22\!\cdots\!16\)\( \beta) q^{77} +(\)\(47\!\cdots\!12\)\( + \)\(20\!\cdots\!64\)\( \beta) q^{78} +(\)\(16\!\cdots\!80\)\( + \)\(13\!\cdots\!92\)\( \beta) q^{79} +(\)\(27\!\cdots\!00\)\( + \)\(48\!\cdots\!00\)\( \beta) q^{80} +(-\)\(73\!\cdots\!59\)\( - \)\(29\!\cdots\!16\)\( \beta) q^{81} +(\)\(22\!\cdots\!16\)\( + \)\(14\!\cdots\!12\)\( \beta) q^{82} +(\)\(14\!\cdots\!88\)\( - \)\(15\!\cdots\!75\)\( \beta) q^{83} +(\)\(62\!\cdots\!88\)\( + \)\(24\!\cdots\!64\)\( \beta) q^{84} +(\)\(25\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( \beta) q^{85} +(-\)\(26\!\cdots\!16\)\( - \)\(20\!\cdots\!12\)\( \beta) q^{86} +(\)\(61\!\cdots\!20\)\( + \)\(29\!\cdots\!22\)\( \beta) q^{87} +(\)\(17\!\cdots\!64\)\( + \)\(64\!\cdots\!36\)\( \beta) q^{88} +(\)\(19\!\cdots\!10\)\( - \)\(17\!\cdots\!40\)\( \beta) q^{89} +(\)\(69\!\cdots\!00\)\( - \)\(21\!\cdots\!00\)\( \beta) q^{90} +(\)\(18\!\cdots\!32\)\( + \)\(42\!\cdots\!84\)\( \beta) q^{91} +(\)\(13\!\cdots\!72\)\( + \)\(13\!\cdots\!28\)\( \beta) q^{92} +(-\)\(53\!\cdots\!44\)\( - \)\(15\!\cdots\!68\)\( \beta) q^{93} +(-\)\(66\!\cdots\!08\)\( - \)\(12\!\cdots\!92\)\( \beta) q^{94} +(-\)\(18\!\cdots\!00\)\( - \)\(54\!\cdots\!50\)\( \beta) q^{95} +(-\)\(17\!\cdots\!16\)\( - \)\(14\!\cdots\!04\)\( \beta) q^{96} +(-\)\(28\!\cdots\!86\)\( + \)\(17\!\cdots\!80\)\( \beta) q^{97} +(-\)\(91\!\cdots\!56\)\( + \)\(45\!\cdots\!24\)\( \beta) q^{98} +(-\)\(12\!\cdots\!56\)\( + \)\(46\!\cdots\!75\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 68719476736q^{2} - 73290552574846824q^{3} + \)\(23\!\cdots\!48\)\(q^{4} + \)\(40\!\cdots\!00\)\(q^{5} - \)\(25\!\cdots\!32\)\(q^{6} - \)\(29\!\cdots\!52\)\(q^{7} + \)\(81\!\cdots\!64\)\(q^{8} - \)\(61\!\cdots\!06\)\(q^{9} + O(q^{10}) \) \( 2q + 68719476736q^{2} - 73290552574846824q^{3} + \)\(23\!\cdots\!48\)\(q^{4} + \)\(40\!\cdots\!00\)\(q^{5} - \)\(25\!\cdots\!32\)\(q^{6} - \)\(29\!\cdots\!52\)\(q^{7} + \)\(81\!\cdots\!64\)\(q^{8} - \)\(61\!\cdots\!06\)\(q^{9} + \)\(13\!\cdots\!00\)\(q^{10} + \)\(86\!\cdots\!04\)\(q^{11} - \)\(86\!\cdots\!76\)\(q^{12} - \)\(11\!\cdots\!64\)\(q^{13} - \)\(10\!\cdots\!36\)\(q^{14} - \)\(87\!\cdots\!00\)\(q^{15} + \)\(27\!\cdots\!52\)\(q^{16} - \)\(25\!\cdots\!12\)\(q^{17} - \)\(21\!\cdots\!08\)\(q^{18} - \)\(15\!\cdots\!20\)\(q^{19} + \)\(47\!\cdots\!00\)\(q^{20} + \)\(10\!\cdots\!24\)\(q^{21} + \)\(29\!\cdots\!72\)\(q^{22} + \)\(22\!\cdots\!56\)\(q^{23} - \)\(29\!\cdots\!68\)\(q^{24} + \)\(76\!\cdots\!50\)\(q^{25} - \)\(38\!\cdots\!52\)\(q^{26} + \)\(31\!\cdots\!00\)\(q^{27} - \)\(34\!\cdots\!48\)\(q^{28} - \)\(96\!\cdots\!20\)\(q^{29} - \)\(29\!\cdots\!00\)\(q^{30} - \)\(40\!\cdots\!76\)\(q^{31} + \)\(95\!\cdots\!36\)\(q^{32} - \)\(64\!\cdots\!48\)\(q^{33} - \)\(86\!\cdots\!16\)\(q^{34} - \)\(11\!\cdots\!00\)\(q^{35} - \)\(72\!\cdots\!44\)\(q^{36} - \)\(11\!\cdots\!92\)\(q^{37} - \)\(52\!\cdots\!60\)\(q^{38} + \)\(27\!\cdots\!68\)\(q^{39} + \)\(16\!\cdots\!00\)\(q^{40} + \)\(12\!\cdots\!24\)\(q^{41} + \)\(36\!\cdots\!32\)\(q^{42} - \)\(15\!\cdots\!24\)\(q^{43} + \)\(10\!\cdots\!96\)\(q^{44} + \)\(40\!\cdots\!00\)\(q^{45} + \)\(75\!\cdots\!08\)\(q^{46} - \)\(38\!\cdots\!12\)\(q^{47} - \)\(10\!\cdots\!24\)\(q^{48} - \)\(53\!\cdots\!34\)\(q^{49} + \)\(26\!\cdots\!00\)\(q^{50} - \)\(39\!\cdots\!56\)\(q^{51} - \)\(13\!\cdots\!36\)\(q^{52} - \)\(30\!\cdots\!04\)\(q^{53} + \)\(10\!\cdots\!00\)\(q^{54} + \)\(55\!\cdots\!00\)\(q^{55} - \)\(11\!\cdots\!64\)\(q^{56} + \)\(34\!\cdots\!40\)\(q^{57} - \)\(33\!\cdots\!60\)\(q^{58} - \)\(16\!\cdots\!40\)\(q^{59} - \)\(10\!\cdots\!00\)\(q^{60} - \)\(48\!\cdots\!56\)\(q^{61} - \)\(13\!\cdots\!68\)\(q^{62} - \)\(60\!\cdots\!44\)\(q^{63} + \)\(32\!\cdots\!48\)\(q^{64} - \)\(29\!\cdots\!00\)\(q^{65} - \)\(22\!\cdots\!64\)\(q^{66} - \)\(13\!\cdots\!72\)\(q^{67} - \)\(29\!\cdots\!88\)\(q^{68} - \)\(32\!\cdots\!72\)\(q^{69} - \)\(39\!\cdots\!00\)\(q^{70} - \)\(14\!\cdots\!36\)\(q^{71} - \)\(24\!\cdots\!92\)\(q^{72} - \)\(32\!\cdots\!24\)\(q^{73} - \)\(38\!\cdots\!56\)\(q^{74} - \)\(31\!\cdots\!00\)\(q^{75} - \)\(17\!\cdots\!80\)\(q^{76} - \)\(62\!\cdots\!04\)\(q^{77} + \)\(94\!\cdots\!24\)\(q^{78} + \)\(32\!\cdots\!60\)\(q^{79} + \)\(55\!\cdots\!00\)\(q^{80} - \)\(14\!\cdots\!18\)\(q^{81} + \)\(44\!\cdots\!32\)\(q^{82} + \)\(29\!\cdots\!76\)\(q^{83} + \)\(12\!\cdots\!76\)\(q^{84} + \)\(51\!\cdots\!00\)\(q^{85} - \)\(53\!\cdots\!32\)\(q^{86} + \)\(12\!\cdots\!40\)\(q^{87} + \)\(35\!\cdots\!28\)\(q^{88} + \)\(38\!\cdots\!20\)\(q^{89} + \)\(13\!\cdots\!00\)\(q^{90} + \)\(37\!\cdots\!64\)\(q^{91} + \)\(26\!\cdots\!44\)\(q^{92} - \)\(10\!\cdots\!88\)\(q^{93} - \)\(13\!\cdots\!16\)\(q^{94} - \)\(36\!\cdots\!00\)\(q^{95} - \)\(35\!\cdots\!32\)\(q^{96} - \)\(57\!\cdots\!72\)\(q^{97} - \)\(18\!\cdots\!12\)\(q^{98} - \)\(25\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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
7.96204e9
−7.96204e9
3.43597e10 −9.23668e16 1.18059e21 8.49883e24 −3.17370e27 −9.92557e29 4.05648e31 1.02216e33 2.92018e35
1.2 3.43597e10 1.90762e16 1.18059e21 −4.48127e24 6.55455e26 7.01318e29 4.05648e31 −7.14556e33 −1.53975e35
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + \)\(73\!\cdots\!24\)\( T_{3} - \)\(17\!\cdots\!56\)\( \) acting on \(S_{72}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 34359738368 T )^{2} \)
$3$ \( 1 + 73290552574846824 T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(55\!\cdots\!28\)\( T^{3} + \)\(56\!\cdots\!09\)\( T^{4} \)
$5$ \( 1 - \)\(40\!\cdots\!00\)\( T + \)\(46\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + \)\(29\!\cdots\!52\)\( T + \)\(13\!\cdots\!62\)\( T^{2} + \)\(29\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - \)\(86\!\cdots\!04\)\( T + \)\(18\!\cdots\!26\)\( T^{2} - \)\(75\!\cdots\!44\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 + \)\(11\!\cdots\!64\)\( T + \)\(20\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!68\)\( T^{3} + \)\(15\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 + \)\(25\!\cdots\!12\)\( T + \)\(28\!\cdots\!02\)\( T^{2} + \)\(58\!\cdots\!96\)\( T^{3} + \)\(52\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(64\!\cdots\!38\)\( T^{2} + \)\(93\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 - \)\(22\!\cdots\!56\)\( T + \)\(61\!\cdots\!38\)\( T^{2} - \)\(10\!\cdots\!12\)\( T^{3} + \)\(23\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 + \)\(96\!\cdots\!20\)\( T + \)\(98\!\cdots\!58\)\( T^{2} + \)\(65\!\cdots\!80\)\( T^{3} + \)\(45\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 + \)\(40\!\cdots\!76\)\( T + \)\(39\!\cdots\!06\)\( T^{2} + \)\(30\!\cdots\!56\)\( T^{3} + \)\(59\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 + \)\(11\!\cdots\!92\)\( T + \)\(73\!\cdots\!42\)\( T^{2} + \)\(24\!\cdots\!96\)\( T^{3} + \)\(48\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 - \)\(12\!\cdots\!24\)\( T + \)\(62\!\cdots\!26\)\( T^{2} - \)\(41\!\cdots\!84\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 + \)\(15\!\cdots\!24\)\( T + \)\(71\!\cdots\!58\)\( T^{2} + \)\(14\!\cdots\!68\)\( T^{3} + \)\(89\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(38\!\cdots\!12\)\( T + \)\(13\!\cdots\!42\)\( T^{2} + \)\(20\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 + \)\(30\!\cdots\!04\)\( T + \)\(39\!\cdots\!98\)\( T^{2} + \)\(82\!\cdots\!88\)\( T^{3} + \)\(70\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 + \)\(16\!\cdots\!40\)\( T + \)\(17\!\cdots\!18\)\( T^{2} + \)\(87\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 + \)\(48\!\cdots\!56\)\( T + \)\(15\!\cdots\!06\)\( T^{2} + \)\(27\!\cdots\!16\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 + \)\(13\!\cdots\!72\)\( T + \)\(12\!\cdots\!62\)\( T^{2} + \)\(60\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 + \)\(14\!\cdots\!36\)\( T + \)\(10\!\cdots\!66\)\( T^{2} + \)\(40\!\cdots\!56\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 + \)\(32\!\cdots\!24\)\( T + \)\(56\!\cdots\!98\)\( T^{2} + \)\(65\!\cdots\!48\)\( T^{3} + \)\(39\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 - \)\(32\!\cdots\!60\)\( T + \)\(74\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 - \)\(29\!\cdots\!76\)\( T + \)\(35\!\cdots\!78\)\( T^{2} - \)\(52\!\cdots\!92\)\( T^{3} + \)\(32\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 - \)\(38\!\cdots\!20\)\( T + \)\(87\!\cdots\!78\)\( T^{2} - \)\(99\!\cdots\!80\)\( T^{3} + \)\(65\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 + \)\(57\!\cdots\!72\)\( T + \)\(21\!\cdots\!02\)\( T^{2} + \)\(66\!\cdots\!16\)\( T^{3} + \)\(13\!\cdots\!09\)\( T^{4} \)
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