Properties

Label 2.46.a.b
Level 2
Weight 46
Character orbit 2.a
Self dual yes
Analytic conductor 25.651
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 46 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.6511452149\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 1920\sqrt{800679089088649}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4194304 q^{2} + ( 29930608596 - \beta ) q^{3} + 17592186044416 q^{4} + ( 2198556995702550 - 127980 \beta ) q^{5} + ( 125538071356637184 - 4194304 \beta ) q^{6} + ( 3951246596735649512 + 28459998 \beta ) q^{7} + 73786976294838206464 q^{8} + ( 893152018392511066173 - 59861217192 \beta ) q^{9} +O(q^{10})\) \( q +4194304 q^{2} +(29930608596 - \beta) q^{3} +17592186044416 q^{4} +(2198556995702550 - 127980 \beta) q^{5} +(125538071356637184 - 4194304 \beta) q^{6} +(3951246596735649512 + 28459998 \beta) q^{7} +73786976294838206464 q^{8} +(\)\(89\!\cdots\!73\)\( - 59861217192 \beta) q^{9} +(\)\(92\!\cdots\!00\)\( - 536787025920 \beta) q^{10} +(\)\(96\!\cdots\!72\)\( + 2197121767533 \beta) q^{11} +(\)\(52\!\cdots\!36\)\( - 17592186044416 \beta) q^{12} +(-\)\(11\!\cdots\!14\)\( + 239553473871348 \beta) q^{13} +(\)\(16\!\cdots\!48\)\( + 119369883451392 \beta) q^{14} +(\)\(44\!\cdots\!00\)\( - 6029076283818630 \beta) q^{15} +\)\(30\!\cdots\!56\)\( q^{16} +(-\)\(32\!\cdots\!18\)\( + 91984363822443672 \beta) q^{17} +(\)\(37\!\cdots\!92\)\( - 251076142713274368 \beta) q^{18} +(-\)\(45\!\cdots\!60\)\( + 815181673524211323 \beta) q^{19} +(\)\(38\!\cdots\!00\)\( - 2251447969964359680 \beta) q^{20} +(\)\(34\!\cdots\!52\)\( - 3099421535954706704 \beta) q^{21} +(\)\(40\!\cdots\!88\)\( + 9215396618050732032 \beta) q^{22} +(\)\(34\!\cdots\!56\)\( + 63330158869140020634 \beta) q^{23} +(\)\(22\!\cdots\!44\)\( - 73786976294838206464 \beta) q^{24} +(\)\(24\!\cdots\!75\)\( - \)\(56\!\cdots\!00\)\( \beta) q^{25} +(-\)\(48\!\cdots\!56\)\( + \)\(10\!\cdots\!92\)\( \beta) q^{26} +(\)\(11\!\cdots\!80\)\( + \)\(26\!\cdots\!38\)\( \beta) q^{27} +(\)\(69\!\cdots\!92\)\( + \)\(50\!\cdots\!68\)\( \beta) q^{28} +(\)\(11\!\cdots\!70\)\( + \)\(23\!\cdots\!16\)\( \beta) q^{29} +(\)\(18\!\cdots\!00\)\( - \)\(25\!\cdots\!20\)\( \beta) q^{30} +(\)\(48\!\cdots\!92\)\( + \)\(28\!\cdots\!16\)\( \beta) q^{31} +\)\(12\!\cdots\!24\)\( q^{32} +(-\)\(35\!\cdots\!88\)\( - \)\(30\!\cdots\!04\)\( \beta) q^{33} +(-\)\(13\!\cdots\!72\)\( + \)\(38\!\cdots\!88\)\( \beta) q^{34} +(-\)\(20\!\cdots\!00\)\( - \)\(44\!\cdots\!60\)\( \beta) q^{35} +(\)\(15\!\cdots\!68\)\( - \)\(10\!\cdots\!72\)\( \beta) q^{36} +(-\)\(95\!\cdots\!98\)\( - \)\(33\!\cdots\!68\)\( \beta) q^{37} +(-\)\(18\!\cdots\!40\)\( + \)\(34\!\cdots\!92\)\( \beta) q^{38} +(-\)\(74\!\cdots\!44\)\( + \)\(83\!\cdots\!22\)\( \beta) q^{39} +(\)\(16\!\cdots\!00\)\( - \)\(94\!\cdots\!20\)\( \beta) q^{40} +(-\)\(92\!\cdots\!38\)\( - \)\(26\!\cdots\!76\)\( \beta) q^{41} +(\)\(14\!\cdots\!08\)\( - \)\(12\!\cdots\!16\)\( \beta) q^{42} +(-\)\(40\!\cdots\!64\)\( + \)\(98\!\cdots\!57\)\( \beta) q^{43} +(\)\(17\!\cdots\!52\)\( + \)\(38\!\cdots\!28\)\( \beta) q^{44} +(\)\(24\!\cdots\!50\)\( - \)\(24\!\cdots\!40\)\( \beta) q^{45} +(\)\(14\!\cdots\!24\)\( + \)\(26\!\cdots\!36\)\( \beta) q^{46} +(\)\(18\!\cdots\!52\)\( - \)\(84\!\cdots\!72\)\( \beta) q^{47} +(\)\(92\!\cdots\!76\)\( - \)\(30\!\cdots\!56\)\( \beta) q^{48} +(-\)\(89\!\cdots\!63\)\( + \)\(22\!\cdots\!52\)\( \beta) q^{49} +(\)\(10\!\cdots\!00\)\( - \)\(23\!\cdots\!00\)\( \beta) q^{50} +(-\)\(36\!\cdots\!28\)\( + \)\(60\!\cdots\!30\)\( \beta) q^{51} +(-\)\(20\!\cdots\!24\)\( + \)\(42\!\cdots\!68\)\( \beta) q^{52} +(-\)\(33\!\cdots\!94\)\( - \)\(15\!\cdots\!52\)\( \beta) q^{53} +(\)\(48\!\cdots\!20\)\( + \)\(11\!\cdots\!52\)\( \beta) q^{54} +(-\)\(61\!\cdots\!00\)\( - \)\(75\!\cdots\!10\)\( \beta) q^{55} +(\)\(29\!\cdots\!68\)\( + \)\(20\!\cdots\!72\)\( \beta) q^{56} +(-\)\(37\!\cdots\!60\)\( + \)\(69\!\cdots\!68\)\( \beta) q^{57} +(\)\(46\!\cdots\!80\)\( + \)\(97\!\cdots\!64\)\( \beta) q^{58} +(-\)\(73\!\cdots\!60\)\( - \)\(12\!\cdots\!63\)\( \beta) q^{59} +(\)\(78\!\cdots\!00\)\( - \)\(10\!\cdots\!80\)\( \beta) q^{60} +(\)\(84\!\cdots\!82\)\( + \)\(17\!\cdots\!52\)\( \beta) q^{61} +(\)\(20\!\cdots\!68\)\( + \)\(11\!\cdots\!64\)\( \beta) q^{62} +(-\)\(14\!\cdots\!24\)\( - \)\(21\!\cdots\!50\)\( \beta) q^{63} +\)\(54\!\cdots\!96\)\( q^{64} +(-\)\(93\!\cdots\!00\)\( + \)\(67\!\cdots\!20\)\( \beta) q^{65} +(-\)\(15\!\cdots\!52\)\( - \)\(12\!\cdots\!16\)\( \beta) q^{66} +(-\)\(41\!\cdots\!88\)\( - \)\(17\!\cdots\!41\)\( \beta) q^{67} +(-\)\(57\!\cdots\!88\)\( + \)\(16\!\cdots\!52\)\( \beta) q^{68} +(-\)\(82\!\cdots\!24\)\( - \)\(16\!\cdots\!92\)\( \beta) q^{69} +(-\)\(86\!\cdots\!00\)\( - \)\(18\!\cdots\!40\)\( \beta) q^{70} +(\)\(55\!\cdots\!32\)\( + \)\(77\!\cdots\!02\)\( \beta) q^{71} +(\)\(65\!\cdots\!72\)\( - \)\(44\!\cdots\!88\)\( \beta) q^{72} +(\)\(68\!\cdots\!66\)\( + \)\(10\!\cdots\!08\)\( \beta) q^{73} +(-\)\(40\!\cdots\!92\)\( - \)\(13\!\cdots\!72\)\( \beta) q^{74} +(\)\(24\!\cdots\!00\)\( - \)\(41\!\cdots\!75\)\( \beta) q^{75} +(-\)\(79\!\cdots\!60\)\( + \)\(14\!\cdots\!68\)\( \beta) q^{76} +(\)\(56\!\cdots\!64\)\( + \)\(11\!\cdots\!52\)\( \beta) q^{77} +(-\)\(31\!\cdots\!76\)\( + \)\(34\!\cdots\!88\)\( \beta) q^{78} +(\)\(16\!\cdots\!00\)\( + \)\(19\!\cdots\!88\)\( \beta) q^{79} +(\)\(68\!\cdots\!00\)\( - \)\(39\!\cdots\!80\)\( \beta) q^{80} +(\)\(78\!\cdots\!41\)\( + \)\(69\!\cdots\!24\)\( \beta) q^{81} +(-\)\(38\!\cdots\!52\)\( - \)\(11\!\cdots\!04\)\( \beta) q^{82} +(-\)\(47\!\cdots\!24\)\( - \)\(22\!\cdots\!05\)\( \beta) q^{83} +(\)\(60\!\cdots\!32\)\( - \)\(54\!\cdots\!64\)\( \beta) q^{84} +(-\)\(41\!\cdots\!00\)\( + \)\(62\!\cdots\!40\)\( \beta) q^{85} +(-\)\(16\!\cdots\!56\)\( + \)\(41\!\cdots\!28\)\( \beta) q^{86} +(\)\(26\!\cdots\!20\)\( - \)\(10\!\cdots\!34\)\( \beta) q^{87} +(\)\(71\!\cdots\!08\)\( + \)\(16\!\cdots\!12\)\( \beta) q^{88} +(\)\(33\!\cdots\!90\)\( + \)\(45\!\cdots\!40\)\( \beta) q^{89} +(\)\(10\!\cdots\!00\)\( - \)\(10\!\cdots\!60\)\( \beta) q^{90} +(\)\(15\!\cdots\!32\)\( + \)\(91\!\cdots\!04\)\( \beta) q^{91} +(\)\(61\!\cdots\!96\)\( + \)\(11\!\cdots\!44\)\( \beta) q^{92} +(\)\(61\!\cdots\!32\)\( - \)\(39\!\cdots\!56\)\( \beta) q^{93} +(\)\(75\!\cdots\!08\)\( - \)\(35\!\cdots\!88\)\( \beta) q^{94} +(-\)\(40\!\cdots\!00\)\( + \)\(75\!\cdots\!50\)\( \beta) q^{95} +(\)\(38\!\cdots\!04\)\( - \)\(12\!\cdots\!24\)\( \beta) q^{96} +(-\)\(41\!\cdots\!18\)\( - \)\(11\!\cdots\!80\)\( \beta) q^{97} +(-\)\(37\!\cdots\!52\)\( + \)\(94\!\cdots\!08\)\( \beta) q^{98} +(-\)\(30\!\cdots\!44\)\( - \)\(38\!\cdots\!15\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8388608q^{2} + 59861217192q^{3} + 35184372088832q^{4} + 4397113991405100q^{5} + 251076142713274368q^{6} + 7902493193471299024q^{7} + 147573952589676412928q^{8} + 1786304036785022132346q^{9} + O(q^{10}) \) \( 2q + 8388608q^{2} + 59861217192q^{3} + 35184372088832q^{4} + 4397113991405100q^{5} + 251076142713274368q^{6} + 7902493193471299024q^{7} + \)\(14\!\cdots\!28\)\(q^{8} + \)\(17\!\cdots\!46\)\(q^{9} + \)\(18\!\cdots\!00\)\(q^{10} + \)\(19\!\cdots\!44\)\(q^{11} + \)\(10\!\cdots\!72\)\(q^{12} - \)\(23\!\cdots\!28\)\(q^{13} + \)\(33\!\cdots\!96\)\(q^{14} + \)\(88\!\cdots\!00\)\(q^{15} + \)\(61\!\cdots\!12\)\(q^{16} - \)\(65\!\cdots\!36\)\(q^{17} + \)\(74\!\cdots\!84\)\(q^{18} - \)\(90\!\cdots\!20\)\(q^{19} + \)\(77\!\cdots\!00\)\(q^{20} + \)\(68\!\cdots\!04\)\(q^{21} + \)\(81\!\cdots\!76\)\(q^{22} + \)\(69\!\cdots\!12\)\(q^{23} + \)\(44\!\cdots\!88\)\(q^{24} + \)\(49\!\cdots\!50\)\(q^{25} - \)\(96\!\cdots\!12\)\(q^{26} + \)\(22\!\cdots\!60\)\(q^{27} + \)\(13\!\cdots\!84\)\(q^{28} + \)\(22\!\cdots\!40\)\(q^{29} + \)\(37\!\cdots\!00\)\(q^{30} + \)\(96\!\cdots\!84\)\(q^{31} + \)\(25\!\cdots\!48\)\(q^{32} - \)\(71\!\cdots\!76\)\(q^{33} - \)\(27\!\cdots\!44\)\(q^{34} - \)\(41\!\cdots\!00\)\(q^{35} + \)\(31\!\cdots\!36\)\(q^{36} - \)\(19\!\cdots\!96\)\(q^{37} - \)\(37\!\cdots\!80\)\(q^{38} - \)\(14\!\cdots\!88\)\(q^{39} + \)\(32\!\cdots\!00\)\(q^{40} - \)\(18\!\cdots\!76\)\(q^{41} + \)\(28\!\cdots\!16\)\(q^{42} - \)\(80\!\cdots\!28\)\(q^{43} + \)\(34\!\cdots\!04\)\(q^{44} + \)\(49\!\cdots\!00\)\(q^{45} + \)\(29\!\cdots\!48\)\(q^{46} + \)\(36\!\cdots\!04\)\(q^{47} + \)\(18\!\cdots\!52\)\(q^{48} - \)\(17\!\cdots\!26\)\(q^{49} + \)\(20\!\cdots\!00\)\(q^{50} - \)\(73\!\cdots\!56\)\(q^{51} - \)\(40\!\cdots\!48\)\(q^{52} - \)\(66\!\cdots\!88\)\(q^{53} + \)\(96\!\cdots\!40\)\(q^{54} - \)\(12\!\cdots\!00\)\(q^{55} + \)\(58\!\cdots\!36\)\(q^{56} - \)\(75\!\cdots\!20\)\(q^{57} + \)\(92\!\cdots\!60\)\(q^{58} - \)\(14\!\cdots\!20\)\(q^{59} + \)\(15\!\cdots\!00\)\(q^{60} + \)\(16\!\cdots\!64\)\(q^{61} + \)\(40\!\cdots\!36\)\(q^{62} - \)\(29\!\cdots\!48\)\(q^{63} + \)\(10\!\cdots\!92\)\(q^{64} - \)\(18\!\cdots\!00\)\(q^{65} - \)\(30\!\cdots\!04\)\(q^{66} - \)\(82\!\cdots\!76\)\(q^{67} - \)\(11\!\cdots\!76\)\(q^{68} - \)\(16\!\cdots\!48\)\(q^{69} - \)\(17\!\cdots\!00\)\(q^{70} + \)\(11\!\cdots\!64\)\(q^{71} + \)\(13\!\cdots\!44\)\(q^{72} + \)\(13\!\cdots\!32\)\(q^{73} - \)\(80\!\cdots\!84\)\(q^{74} + \)\(48\!\cdots\!00\)\(q^{75} - \)\(15\!\cdots\!20\)\(q^{76} + \)\(11\!\cdots\!28\)\(q^{77} - \)\(62\!\cdots\!52\)\(q^{78} + \)\(33\!\cdots\!00\)\(q^{79} + \)\(13\!\cdots\!00\)\(q^{80} + \)\(15\!\cdots\!82\)\(q^{81} - \)\(77\!\cdots\!04\)\(q^{82} - \)\(95\!\cdots\!48\)\(q^{83} + \)\(12\!\cdots\!64\)\(q^{84} - \)\(83\!\cdots\!00\)\(q^{85} - \)\(33\!\cdots\!12\)\(q^{86} + \)\(52\!\cdots\!40\)\(q^{87} + \)\(14\!\cdots\!16\)\(q^{88} + \)\(67\!\cdots\!80\)\(q^{89} + \)\(20\!\cdots\!00\)\(q^{90} + \)\(31\!\cdots\!64\)\(q^{91} + \)\(12\!\cdots\!92\)\(q^{92} + \)\(12\!\cdots\!64\)\(q^{93} + \)\(15\!\cdots\!16\)\(q^{94} - \)\(81\!\cdots\!00\)\(q^{95} + \)\(77\!\cdots\!08\)\(q^{96} - \)\(82\!\cdots\!36\)\(q^{97} - \)\(74\!\cdots\!04\)\(q^{98} - \)\(60\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
1.41481e7
−1.41481e7
4.19430e6 −2.43982e10 1.75922e13 −4.75445e15 −1.02334e17 5.49745e18 7.37870e19 −2.35904e21 −1.99416e22
1.2 4.19430e6 8.42595e10 1.75922e13 9.15156e15 3.53410e17 2.40505e18 7.37870e19 4.14534e21 3.83844e22
\(n\): 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 2.46.a.b 2
3.b odd 2 1 18.46.a.b 2
4.b odd 2 1 16.46.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.46.a.b 2 1.a even 1 1 trivial
16.46.a.a 2 4.b odd 2 1
18.46.a.b 2 3.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 59861217192 T_{3} - \)\(20\!\cdots\!84\)\( \) acting on \(S_{46}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 4194304 T )^{2} \)
$3$ \( 1 - 59861217192 T + \)\(38\!\cdots\!02\)\( T^{2} - \)\(17\!\cdots\!56\)\( T^{3} + \)\(87\!\cdots\!49\)\( T^{4} \)
$5$ \( 1 - 4397113991405100 T + \)\(13\!\cdots\!50\)\( T^{2} - \)\(12\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 7902493193471299024 T + \)\(22\!\cdots\!58\)\( T^{2} - \)\(84\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - \)\(19\!\cdots\!44\)\( T + \)\(14\!\cdots\!86\)\( T^{2} - \)\(14\!\cdots\!44\)\( T^{3} + \)\(53\!\cdots\!01\)\( T^{4} \)
$13$ \( 1 + \)\(23\!\cdots\!28\)\( T + \)\(10\!\cdots\!82\)\( T^{2} + \)\(31\!\cdots\!04\)\( T^{3} + \)\(17\!\cdots\!49\)\( T^{4} \)
$17$ \( 1 + \)\(65\!\cdots\!36\)\( T + \)\(32\!\cdots\!38\)\( T^{2} + \)\(15\!\cdots\!52\)\( T^{3} + \)\(55\!\cdots\!49\)\( T^{4} \)
$19$ \( 1 + \)\(90\!\cdots\!20\)\( T + \)\(70\!\cdots\!98\)\( T^{2} + \)\(31\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!01\)\( T^{4} \)
$23$ \( 1 - \)\(69\!\cdots\!12\)\( T + \)\(38\!\cdots\!22\)\( T^{2} - \)\(13\!\cdots\!16\)\( T^{3} + \)\(35\!\cdots\!49\)\( T^{4} \)
$29$ \( 1 - \)\(22\!\cdots\!40\)\( T + \)\(24\!\cdots\!98\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + \)\(41\!\cdots\!01\)\( T^{4} \)
$31$ \( 1 - \)\(96\!\cdots\!84\)\( T + \)\(46\!\cdots\!66\)\( T^{2} - \)\(12\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!01\)\( T^{4} \)
$37$ \( 1 + \)\(19\!\cdots\!96\)\( T + \)\(82\!\cdots\!18\)\( T^{2} + \)\(70\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!49\)\( T^{4} \)
$41$ \( 1 + \)\(18\!\cdots\!76\)\( T + \)\(63\!\cdots\!46\)\( T^{2} + \)\(69\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!01\)\( T^{4} \)
$43$ \( 1 + \)\(80\!\cdots\!28\)\( T + \)\(35\!\cdots\!82\)\( T^{2} + \)\(25\!\cdots\!04\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(36\!\cdots\!04\)\( T + \)\(38\!\cdots\!18\)\( T^{2} - \)\(63\!\cdots\!28\)\( T^{3} + \)\(30\!\cdots\!49\)\( T^{4} \)
$53$ \( 1 + \)\(66\!\cdots\!88\)\( T + \)\(20\!\cdots\!22\)\( T^{2} + \)\(26\!\cdots\!84\)\( T^{3} + \)\(15\!\cdots\!49\)\( T^{4} \)
$59$ \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(53\!\cdots\!98\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} \)
$61$ \( 1 - \)\(16\!\cdots\!64\)\( T + \)\(42\!\cdots\!26\)\( T^{2} - \)\(36\!\cdots\!64\)\( T^{3} + \)\(47\!\cdots\!01\)\( T^{4} \)
$67$ \( 1 + \)\(82\!\cdots\!76\)\( T + \)\(22\!\cdots\!58\)\( T^{2} + \)\(12\!\cdots\!32\)\( T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \)
$71$ \( 1 - \)\(11\!\cdots\!64\)\( T + \)\(23\!\cdots\!26\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(41\!\cdots\!01\)\( T^{4} \)
$73$ \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(15\!\cdots\!42\)\( T^{2} - \)\(96\!\cdots\!76\)\( T^{3} + \)\(50\!\cdots\!49\)\( T^{4} \)
$79$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(51\!\cdots\!98\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!01\)\( T^{4} \)
$83$ \( 1 + \)\(95\!\cdots\!48\)\( T + \)\(32\!\cdots\!62\)\( T^{2} + \)\(21\!\cdots\!64\)\( T^{3} + \)\(52\!\cdots\!49\)\( T^{4} \)
$89$ \( 1 - \)\(67\!\cdots\!80\)\( T + \)\(11\!\cdots\!98\)\( T^{2} - \)\(35\!\cdots\!20\)\( T^{3} + \)\(27\!\cdots\!01\)\( T^{4} \)
$97$ \( 1 + \)\(82\!\cdots\!36\)\( T + \)\(67\!\cdots\!38\)\( T^{2} + \)\(21\!\cdots\!52\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \)
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