Properties

Label 2.44.a.b
Level 2
Weight 44
Character orbit 2.a
Self dual yes
Analytic conductor 23.422
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 44 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.4220790691\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 17280\sqrt{1589985537001}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2097152 q^{2} + ( -11170817028 - \beta ) q^{3} + 4398046511104 q^{4} + ( -23660329199010 + 19308 \beta ) q^{5} + ( -23426901271904256 - 2097152 \beta ) q^{6} + ( -111230065673519464 + 87838422 \beta ) q^{7} + 9223372036854775808 q^{8} + ( 271297323050157073557 + 22341634056 \beta ) q^{9} +O(q^{10})\) \( q +2097152 q^{2} +(-11170817028 - \beta) q^{3} +4398046511104 q^{4} +(-23660329199010 + 19308 \beta) q^{5} +(-23426901271904256 - 2097152 \beta) q^{6} +(-111230065673519464 + 87838422 \beta) q^{7} +9223372036854775808 q^{8} +(\)\(27\!\cdots\!57\)\( + 22341634056 \beta) q^{9} +(-49619306700362219520 + 40491810816 \beta) q^{10} +(-\)\(20\!\cdots\!08\)\( - 567604269771 \beta) q^{11} +(-\)\(49\!\cdots\!12\)\( - 4398046511104 \beta) q^{12} +(-\)\(75\!\cdots\!98\)\( - 1877690293332 \beta) q^{13} +(-\)\(23\!\cdots\!28\)\( + 184210522374144 \beta) q^{14} +(-\)\(89\!\cdots\!20\)\( - 192025805977614 \beta) q^{15} +\)\(19\!\cdots\!16\)\( q^{16} +(-\)\(33\!\cdots\!74\)\( - 13871340963470712 \beta) q^{17} +(\)\(56\!\cdots\!64\)\( + 46853802543808512 \beta) q^{18} +(\)\(61\!\cdots\!20\)\( + 60601238932971 \beta) q^{19} +(-\)\(10\!\cdots\!40\)\( + 84917482036396032 \beta) q^{20} +(-\)\(40\!\cdots\!08\)\( - 869996874516730352 \beta) q^{21} +(-\)\(43\!\cdots\!16\)\( - 1190352429558792192 \beta) q^{22} +(\)\(19\!\cdots\!12\)\( + 10312189640309144994 \beta) q^{23} +(-\)\(10\!\cdots\!24\)\( - 9223372036854775808 \beta) q^{24} +(-\)\(95\!\cdots\!25\)\( - 913667272348970160 \beta) q^{25} +(-\)\(15\!\cdots\!96\)\( - 3937801954041790464 \beta) q^{26} +(-\)\(99\!\cdots\!40\)\( - \)\(19\!\cdots\!98\)\( \beta) q^{27} +(-\)\(48\!\cdots\!56\)\( + \)\(38\!\cdots\!88\)\( \beta) q^{28} +(-\)\(36\!\cdots\!50\)\( + \)\(73\!\cdots\!32\)\( \beta) q^{29} +(-\)\(18\!\cdots\!40\)\( - \)\(40\!\cdots\!28\)\( \beta) q^{30} +(\)\(20\!\cdots\!12\)\( - \)\(55\!\cdots\!72\)\( \beta) q^{31} +\)\(40\!\cdots\!32\)\( q^{32} +(\)\(50\!\cdots\!24\)\( + \)\(27\!\cdots\!96\)\( \beta) q^{33} +(-\)\(70\!\cdots\!48\)\( - \)\(29\!\cdots\!24\)\( \beta) q^{34} +(\)\(80\!\cdots\!40\)\( - \)\(42\!\cdots\!32\)\( \beta) q^{35} +(\)\(11\!\cdots\!28\)\( + \)\(98\!\cdots\!24\)\( \beta) q^{36} +(\)\(16\!\cdots\!26\)\( - \)\(30\!\cdots\!52\)\( \beta) q^{37} +(\)\(12\!\cdots\!40\)\( + \)\(12\!\cdots\!92\)\( \beta) q^{38} +(\)\(93\!\cdots\!44\)\( + \)\(77\!\cdots\!94\)\( \beta) q^{39} +(-\)\(21\!\cdots\!80\)\( + \)\(17\!\cdots\!64\)\( \beta) q^{40} +(-\)\(29\!\cdots\!98\)\( - \)\(10\!\cdots\!68\)\( \beta) q^{41} +(-\)\(84\!\cdots\!16\)\( - \)\(18\!\cdots\!04\)\( \beta) q^{42} +(-\)\(14\!\cdots\!28\)\( + \)\(19\!\cdots\!17\)\( \beta) q^{43} +(-\)\(92\!\cdots\!32\)\( - \)\(24\!\cdots\!84\)\( \beta) q^{44} +(\)\(19\!\cdots\!30\)\( + \)\(47\!\cdots\!96\)\( \beta) q^{45} +(\)\(40\!\cdots\!24\)\( + \)\(21\!\cdots\!88\)\( \beta) q^{46} +(\)\(91\!\cdots\!36\)\( - \)\(29\!\cdots\!68\)\( \beta) q^{47} +(-\)\(21\!\cdots\!48\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{48} +(\)\(14\!\cdots\!53\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{49} +(-\)\(20\!\cdots\!00\)\( - \)\(19\!\cdots\!20\)\( \beta) q^{50} +(\)\(69\!\cdots\!72\)\( + \)\(18\!\cdots\!10\)\( \beta) q^{51} +(-\)\(33\!\cdots\!92\)\( - \)\(82\!\cdots\!28\)\( \beta) q^{52} +(\)\(67\!\cdots\!82\)\( - \)\(35\!\cdots\!92\)\( \beta) q^{53} +(-\)\(20\!\cdots\!80\)\( - \)\(40\!\cdots\!96\)\( \beta) q^{54} +(-\)\(47\!\cdots\!20\)\( - \)\(39\!\cdots\!54\)\( \beta) q^{55} +(-\)\(10\!\cdots\!12\)\( + \)\(81\!\cdots\!76\)\( \beta) q^{56} +(-\)\(69\!\cdots\!60\)\( - \)\(61\!\cdots\!08\)\( \beta) q^{57} +(-\)\(76\!\cdots\!00\)\( + \)\(15\!\cdots\!64\)\( \beta) q^{58} +(\)\(32\!\cdots\!00\)\( + \)\(18\!\cdots\!69\)\( \beta) q^{59} +(-\)\(39\!\cdots\!80\)\( - \)\(84\!\cdots\!56\)\( \beta) q^{60} +(\)\(42\!\cdots\!22\)\( - \)\(57\!\cdots\!64\)\( \beta) q^{61} +(\)\(42\!\cdots\!24\)\( - \)\(11\!\cdots\!44\)\( \beta) q^{62} +(\)\(90\!\cdots\!52\)\( + \)\(21\!\cdots\!70\)\( \beta) q^{63} +\)\(85\!\cdots\!64\)\( q^{64} +(\)\(66\!\cdots\!80\)\( - \)\(14\!\cdots\!64\)\( \beta) q^{65} +(\)\(10\!\cdots\!48\)\( + \)\(57\!\cdots\!92\)\( \beta) q^{66} +(-\)\(19\!\cdots\!04\)\( - \)\(15\!\cdots\!09\)\( \beta) q^{67} +(-\)\(14\!\cdots\!96\)\( - \)\(61\!\cdots\!48\)\( \beta) q^{68} +(-\)\(51\!\cdots\!36\)\( - \)\(13\!\cdots\!44\)\( \beta) q^{69} +(\)\(16\!\cdots\!80\)\( - \)\(88\!\cdots\!64\)\( \beta) q^{70} +(-\)\(22\!\cdots\!48\)\( + \)\(40\!\cdots\!06\)\( \beta) q^{71} +(\)\(25\!\cdots\!56\)\( + \)\(20\!\cdots\!48\)\( \beta) q^{72} +(-\)\(51\!\cdots\!78\)\( - \)\(43\!\cdots\!52\)\( \beta) q^{73} +(\)\(35\!\cdots\!52\)\( - \)\(63\!\cdots\!04\)\( \beta) q^{74} +(\)\(11\!\cdots\!00\)\( + \)\(96\!\cdots\!05\)\( \beta) q^{75} +(\)\(27\!\cdots\!80\)\( + \)\(26\!\cdots\!84\)\( \beta) q^{76} +(-\)\(21\!\cdots\!88\)\( - \)\(17\!\cdots\!32\)\( \beta) q^{77} +(\)\(19\!\cdots\!88\)\( + \)\(16\!\cdots\!88\)\( \beta) q^{78} +(\)\(53\!\cdots\!60\)\( - \)\(42\!\cdots\!44\)\( \beta) q^{79} +(-\)\(45\!\cdots\!60\)\( + \)\(37\!\cdots\!28\)\( \beta) q^{80} +(\)\(11\!\cdots\!81\)\( + \)\(47\!\cdots\!72\)\( \beta) q^{81} +(-\)\(61\!\cdots\!96\)\( - \)\(21\!\cdots\!36\)\( \beta) q^{82} +(-\)\(49\!\cdots\!28\)\( - \)\(69\!\cdots\!45\)\( \beta) q^{83} +(-\)\(17\!\cdots\!32\)\( - \)\(38\!\cdots\!08\)\( \beta) q^{84} +(-\)\(12\!\cdots\!60\)\( - \)\(32\!\cdots\!72\)\( \beta) q^{85} +(-\)\(29\!\cdots\!56\)\( + \)\(41\!\cdots\!84\)\( \beta) q^{86} +(\)\(56\!\cdots\!00\)\( + \)\(28\!\cdots\!54\)\( \beta) q^{87} +(-\)\(19\!\cdots\!64\)\( - \)\(52\!\cdots\!68\)\( \beta) q^{88} +(\)\(87\!\cdots\!10\)\( - \)\(20\!\cdots\!20\)\( \beta) q^{89} +(\)\(41\!\cdots\!60\)\( + \)\(98\!\cdots\!92\)\( \beta) q^{90} +(\)\(57\!\cdots\!72\)\( - \)\(66\!\cdots\!08\)\( \beta) q^{91} +(\)\(83\!\cdots\!48\)\( + \)\(45\!\cdots\!76\)\( \beta) q^{92} +(\)\(24\!\cdots\!64\)\( + \)\(42\!\cdots\!04\)\( \beta) q^{93} +(\)\(19\!\cdots\!72\)\( - \)\(61\!\cdots\!36\)\( \beta) q^{94} +(-\)\(14\!\cdots\!00\)\( + \)\(11\!\cdots\!50\)\( \beta) q^{95} +(-\)\(45\!\cdots\!96\)\( - \)\(40\!\cdots\!32\)\( \beta) q^{96} +(-\)\(18\!\cdots\!94\)\( + \)\(37\!\cdots\!40\)\( \beta) q^{97} +(\)\(31\!\cdots\!56\)\( - \)\(40\!\cdots\!32\)\( \beta) q^{98} +(-\)\(11\!\cdots\!56\)\( - \)\(62\!\cdots\!95\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4194304q^{2} - 22341634056q^{3} + 8796093022208q^{4} - 47320658398020q^{5} - 46853802543808512q^{6} - 222460131347038928q^{7} + 18446744073709551616q^{8} + 542594646100314147114q^{9} + O(q^{10}) \) \( 2q + 4194304q^{2} - 22341634056q^{3} + 8796093022208q^{4} - 47320658398020q^{5} - 46853802543808512q^{6} - 222460131347038928q^{7} + 18446744073709551616q^{8} + \)\(54\!\cdots\!14\)\(q^{9} - 99238613400724439040q^{10} - \)\(41\!\cdots\!16\)\(q^{11} - \)\(98\!\cdots\!24\)\(q^{12} - \)\(15\!\cdots\!96\)\(q^{13} - \)\(46\!\cdots\!56\)\(q^{14} - \)\(17\!\cdots\!40\)\(q^{15} + \)\(38\!\cdots\!32\)\(q^{16} - \)\(67\!\cdots\!48\)\(q^{17} + \)\(11\!\cdots\!28\)\(q^{18} + \)\(12\!\cdots\!40\)\(q^{19} - \)\(20\!\cdots\!80\)\(q^{20} - \)\(80\!\cdots\!16\)\(q^{21} - \)\(87\!\cdots\!32\)\(q^{22} + \)\(38\!\cdots\!24\)\(q^{23} - \)\(20\!\cdots\!48\)\(q^{24} - \)\(19\!\cdots\!50\)\(q^{25} - \)\(31\!\cdots\!92\)\(q^{26} - \)\(19\!\cdots\!80\)\(q^{27} - \)\(97\!\cdots\!12\)\(q^{28} - \)\(72\!\cdots\!00\)\(q^{29} - \)\(37\!\cdots\!80\)\(q^{30} + \)\(40\!\cdots\!24\)\(q^{31} + \)\(81\!\cdots\!64\)\(q^{32} + \)\(10\!\cdots\!48\)\(q^{33} - \)\(14\!\cdots\!96\)\(q^{34} + \)\(16\!\cdots\!80\)\(q^{35} + \)\(23\!\cdots\!56\)\(q^{36} + \)\(33\!\cdots\!52\)\(q^{37} + \)\(25\!\cdots\!80\)\(q^{38} + \)\(18\!\cdots\!88\)\(q^{39} - \)\(43\!\cdots\!60\)\(q^{40} - \)\(58\!\cdots\!96\)\(q^{41} - \)\(16\!\cdots\!32\)\(q^{42} - \)\(28\!\cdots\!56\)\(q^{43} - \)\(18\!\cdots\!64\)\(q^{44} + \)\(39\!\cdots\!60\)\(q^{45} + \)\(80\!\cdots\!48\)\(q^{46} + \)\(18\!\cdots\!72\)\(q^{47} - \)\(43\!\cdots\!96\)\(q^{48} + \)\(29\!\cdots\!06\)\(q^{49} - \)\(40\!\cdots\!00\)\(q^{50} + \)\(13\!\cdots\!44\)\(q^{51} - \)\(66\!\cdots\!84\)\(q^{52} + \)\(13\!\cdots\!64\)\(q^{53} - \)\(41\!\cdots\!60\)\(q^{54} - \)\(94\!\cdots\!40\)\(q^{55} - \)\(20\!\cdots\!24\)\(q^{56} - \)\(13\!\cdots\!20\)\(q^{57} - \)\(15\!\cdots\!00\)\(q^{58} + \)\(65\!\cdots\!00\)\(q^{59} - \)\(78\!\cdots\!60\)\(q^{60} + \)\(84\!\cdots\!44\)\(q^{61} + \)\(84\!\cdots\!48\)\(q^{62} + \)\(18\!\cdots\!04\)\(q^{63} + \)\(17\!\cdots\!28\)\(q^{64} + \)\(13\!\cdots\!60\)\(q^{65} + \)\(21\!\cdots\!96\)\(q^{66} - \)\(38\!\cdots\!08\)\(q^{67} - \)\(29\!\cdots\!92\)\(q^{68} - \)\(10\!\cdots\!72\)\(q^{69} + \)\(33\!\cdots\!60\)\(q^{70} - \)\(45\!\cdots\!96\)\(q^{71} + \)\(50\!\cdots\!12\)\(q^{72} - \)\(10\!\cdots\!56\)\(q^{73} + \)\(70\!\cdots\!04\)\(q^{74} + \)\(22\!\cdots\!00\)\(q^{75} + \)\(54\!\cdots\!60\)\(q^{76} - \)\(42\!\cdots\!76\)\(q^{77} + \)\(39\!\cdots\!76\)\(q^{78} + \)\(10\!\cdots\!20\)\(q^{79} - \)\(91\!\cdots\!20\)\(q^{80} + \)\(22\!\cdots\!62\)\(q^{81} - \)\(12\!\cdots\!92\)\(q^{82} - \)\(99\!\cdots\!56\)\(q^{83} - \)\(35\!\cdots\!64\)\(q^{84} - \)\(25\!\cdots\!20\)\(q^{85} - \)\(59\!\cdots\!12\)\(q^{86} + \)\(11\!\cdots\!00\)\(q^{87} - \)\(38\!\cdots\!28\)\(q^{88} + \)\(17\!\cdots\!20\)\(q^{89} + \)\(83\!\cdots\!20\)\(q^{90} + \)\(11\!\cdots\!44\)\(q^{91} + \)\(16\!\cdots\!96\)\(q^{92} + \)\(48\!\cdots\!28\)\(q^{93} + \)\(38\!\cdots\!44\)\(q^{94} - \)\(28\!\cdots\!00\)\(q^{95} - \)\(90\!\cdots\!92\)\(q^{96} - \)\(36\!\cdots\!88\)\(q^{97} + \)\(62\!\cdots\!12\)\(q^{98} - \)\(23\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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
630474.
−630473.
2.09715e6 −3.29600e10 4.39805e12 3.97045e14 −6.91221e16 1.80269e18 9.22337e18 7.58103e20 8.32663e20
1.2 2.09715e6 1.06183e10 4.39805e12 −4.44365e14 2.22683e16 −2.02515e18 9.22337e18 −2.15508e20 −9.31902e20
\(n\): 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 2.44.a.b 2
3.b odd 2 1 18.44.a.c 2
4.b odd 2 1 16.44.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.44.a.b 2 1.a even 1 1 trivial
16.44.a.b 2 4.b odd 2 1
18.44.a.c 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 \( T_{3}^{2} + 22341634056 T_{3} - \)\(34\!\cdots\!16\)\( \) acting on \(S_{44}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2097152 T )^{2} \)
$3$ \( 1 + 22341634056 T + \)\(30\!\cdots\!38\)\( T^{2} + \)\(73\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \)
$5$ \( 1 + 47320658398020 T + \)\(20\!\cdots\!50\)\( T^{2} + \)\(53\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 222460131347038928 T + \)\(71\!\cdots\!82\)\( T^{2} + \)\(48\!\cdots\!04\)\( T^{3} + \)\(47\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + \)\(41\!\cdots\!16\)\( T + \)\(14\!\cdots\!26\)\( T^{2} + \)\(25\!\cdots\!96\)\( T^{3} + \)\(36\!\cdots\!61\)\( T^{4} \)
$13$ \( 1 + \)\(15\!\cdots\!96\)\( T + \)\(21\!\cdots\!98\)\( T^{2} + \)\(11\!\cdots\!12\)\( T^{3} + \)\(62\!\cdots\!09\)\( T^{4} \)
$17$ \( 1 + \)\(67\!\cdots\!48\)\( T + \)\(72\!\cdots\!02\)\( T^{2} + \)\(54\!\cdots\!24\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \)
$19$ \( 1 - \)\(12\!\cdots\!40\)\( T + \)\(19\!\cdots\!18\)\( T^{2} - \)\(11\!\cdots\!60\)\( T^{3} + \)\(93\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 - \)\(38\!\cdots\!24\)\( T + \)\(21\!\cdots\!78\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!89\)\( T^{4} \)
$29$ \( 1 + \)\(72\!\cdots\!00\)\( T + \)\(25\!\cdots\!78\)\( T^{2} + \)\(55\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!21\)\( T^{4} \)
$31$ \( 1 - \)\(40\!\cdots\!24\)\( T + \)\(12\!\cdots\!26\)\( T^{2} - \)\(54\!\cdots\!84\)\( T^{3} + \)\(18\!\cdots\!81\)\( T^{4} \)
$37$ \( 1 - \)\(33\!\cdots\!52\)\( T + \)\(13\!\cdots\!82\)\( T^{2} - \)\(91\!\cdots\!56\)\( T^{3} + \)\(73\!\cdots\!09\)\( T^{4} \)
$41$ \( 1 + \)\(58\!\cdots\!96\)\( T + \)\(48\!\cdots\!46\)\( T^{2} + \)\(13\!\cdots\!16\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$43$ \( 1 + \)\(28\!\cdots\!56\)\( T + \)\(53\!\cdots\!98\)\( T^{2} + \)\(49\!\cdots\!92\)\( T^{3} + \)\(30\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(18\!\cdots\!72\)\( T + \)\(20\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \)
$53$ \( 1 - \)\(13\!\cdots\!64\)\( T + \)\(32\!\cdots\!78\)\( T^{2} - \)\(18\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!29\)\( T^{4} \)
$59$ \( 1 - \)\(65\!\cdots\!00\)\( T + \)\(26\!\cdots\!58\)\( T^{2} - \)\(91\!\cdots\!00\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \)
$61$ \( 1 - \)\(84\!\cdots\!44\)\( T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(49\!\cdots\!64\)\( T^{3} + \)\(34\!\cdots\!61\)\( T^{4} \)
$67$ \( 1 + \)\(38\!\cdots\!08\)\( T + \)\(10\!\cdots\!42\)\( T^{2} + \)\(12\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!69\)\( T^{4} \)
$71$ \( 1 + \)\(45\!\cdots\!96\)\( T + \)\(76\!\cdots\!26\)\( T^{2} + \)\(18\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!21\)\( T^{4} \)
$73$ \( 1 + \)\(10\!\cdots\!56\)\( T + \)\(20\!\cdots\!18\)\( T^{2} + \)\(13\!\cdots\!52\)\( T^{3} + \)\(17\!\cdots\!89\)\( T^{4} \)
$79$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(42\!\cdots\!80\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + \)\(99\!\cdots\!56\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(32\!\cdots\!72\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \)
$89$ \( 1 - \)\(17\!\cdots\!20\)\( T + \)\(11\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!61\)\( T^{4} \)
$97$ \( 1 + \)\(36\!\cdots\!88\)\( T - \)\(10\!\cdots\!18\)\( T^{2} + \)\(98\!\cdots\!24\)\( T^{3} + \)\(72\!\cdots\!29\)\( T^{4} \)
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