Properties

Label 2.38.a.a
Level $2$
Weight $38$
Character orbit 2.a
Self dual yes
Analytic conductor $17.343$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.3428076249\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 756643680\)
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 \(\beta = 17280\sqrt{3026574721}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -262144 q^{2} + ( 211535604 - \beta ) q^{3} + 68719476736 q^{4} + ( -6753765277770 + 3444 \beta ) q^{5} + ( -55452789374976 + 262144 \beta ) q^{6} + ( 1553087081481128 + 5126718 \beta ) q^{7} -18014398509481984 q^{8} + ( 498193775039693853 - 423071208 \beta ) q^{9} +O(q^{10})\) \( q -262144 q^{2} +(211535604 - \beta) q^{3} +68719476736 q^{4} +(-6753765277770 + 3444 \beta) q^{5} +(-55452789374976 + 262144 \beta) q^{6} +(1553087081481128 + 5126718 \beta) q^{7} -18014398509481984 q^{8} +(498193775039693853 - 423071208 \beta) q^{9} +(1770459044975738880 - 902823936 \beta) q^{10} +(2001439330770516252 + 25266324237 \beta) q^{11} +(14536616017913708544 - 68719476736 \beta) q^{12} +(-\)\(20\!\cdots\!86\)\( + 40178550228 \beta) q^{13} +(-\)\(40\!\cdots\!32\)\( - 1343938363392 \beta) q^{14} +(-\)\(45\!\cdots\!80\)\( + 7482293897946 \beta) q^{15} +\)\(47\!\cdots\!96\)\( q^{16} +(\)\(46\!\cdots\!18\)\( - 73067502067368 \beta) q^{17} +(-\)\(13\!\cdots\!32\)\( + 110905578749952 \beta) q^{18} +(-\)\(75\!\cdots\!00\)\( + 83542782562107 \beta) q^{19} +(-\)\(46\!\cdots\!20\)\( + 236669877878784 \beta) q^{20} +(-\)\(43\!\cdots\!88\)\( - 468603692813456 \beta) q^{21} +(-\)\(52\!\cdots\!88\)\( - 6623415300784128 \beta) q^{22} +(\)\(24\!\cdots\!04\)\( + 13722480149762874 \beta) q^{23} +(-\)\(38\!\cdots\!36\)\( + 18014398509481984 \beta) q^{24} +(-\)\(16\!\cdots\!25\)\( - 46519935233279760 \beta) q^{25} +(\)\(54\!\cdots\!84\)\( - 10532565870968832 \beta) q^{26} +(\)\(39\!\cdots\!60\)\( - 137404492667986122 \beta) q^{27} +(\)\(10\!\cdots\!08\)\( + 352305378333032448 \beta) q^{28} +(\)\(58\!\cdots\!90\)\( + 872730741021530244 \beta) q^{29} +(\)\(11\!\cdots\!20\)\( - 1961438451583156224 \beta) q^{30} +(-\)\(27\!\cdots\!28\)\( - 872718796832206536 \beta) q^{31} -\)\(12\!\cdots\!24\)\( q^{32} +(-\)\(22\!\cdots\!92\)\( + 3343287827563117896 \beta) q^{33} +(-\)\(12\!\cdots\!92\)\( + 19154207261948116992 \beta) q^{34} +(\)\(54\!\cdots\!40\)\( - 29275818108697454028 \beta) q^{35} +(\)\(34\!\cdots\!08\)\( - 29073232035827417088 \beta) q^{36} +(-\)\(57\!\cdots\!22\)\( + 28974975230710445412 \beta) q^{37} +(\)\(19\!\cdots\!00\)\( - 21900239191960977408 \beta) q^{38} +(-\)\(80\!\cdots\!44\)\( + \)\(21\!\cdots\!98\)\( \beta) q^{39} +(\)\(12\!\cdots\!80\)\( - 62041588466655952896 \beta) q^{40} +(\)\(23\!\cdots\!22\)\( - \)\(33\!\cdots\!24\)\( \beta) q^{41} +(\)\(11\!\cdots\!72\)\( + \)\(12\!\cdots\!64\)\( \beta) q^{42} +(-\)\(15\!\cdots\!76\)\( - \)\(13\!\cdots\!03\)\( \beta) q^{43} +(\)\(13\!\cdots\!72\)\( + \)\(17\!\cdots\!32\)\( \beta) q^{44} +(-\)\(46\!\cdots\!10\)\( + \)\(45\!\cdots\!92\)\( \beta) q^{45} +(-\)\(65\!\cdots\!76\)\( - \)\(35\!\cdots\!56\)\( \beta) q^{46} +(-\)\(56\!\cdots\!52\)\( - \)\(75\!\cdots\!12\)\( \beta) q^{47} +(\)\(99\!\cdots\!84\)\( - \)\(47\!\cdots\!96\)\( \beta) q^{48} +(\)\(76\!\cdots\!77\)\( + \)\(15\!\cdots\!08\)\( \beta) q^{49} +(\)\(43\!\cdots\!00\)\( + \)\(12\!\cdots\!40\)\( \beta) q^{50} +(\)\(66\!\cdots\!72\)\( - \)\(15\!\cdots\!90\)\( \beta) q^{51} +(-\)\(14\!\cdots\!96\)\( + \)\(27\!\cdots\!08\)\( \beta) q^{52} +(\)\(26\!\cdots\!14\)\( - \)\(10\!\cdots\!12\)\( \beta) q^{53} +(-\)\(10\!\cdots\!40\)\( + \)\(36\!\cdots\!68\)\( \beta) q^{54} +(\)\(65\!\cdots\!60\)\( - \)\(16\!\cdots\!02\)\( \beta) q^{55} +(-\)\(27\!\cdots\!52\)\( - \)\(92\!\cdots\!12\)\( \beta) q^{56} +(-\)\(23\!\cdots\!00\)\( + \)\(76\!\cdots\!28\)\( \beta) q^{57} +(-\)\(15\!\cdots\!60\)\( - \)\(22\!\cdots\!36\)\( \beta) q^{58} +(\)\(29\!\cdots\!80\)\( - \)\(81\!\cdots\!07\)\( \beta) q^{59} +(-\)\(31\!\cdots\!80\)\( + \)\(51\!\cdots\!56\)\( \beta) q^{60} +(\)\(47\!\cdots\!42\)\( - \)\(77\!\cdots\!52\)\( \beta) q^{61} +(\)\(72\!\cdots\!32\)\( + \)\(22\!\cdots\!84\)\( \beta) q^{62} +(-\)\(11\!\cdots\!16\)\( + \)\(18\!\cdots\!30\)\( \beta) q^{63} +\)\(32\!\cdots\!56\)\( q^{64} +(\)\(15\!\cdots\!20\)\( - \)\(99\!\cdots\!44\)\( \beta) q^{65} +(\)\(58\!\cdots\!48\)\( - \)\(87\!\cdots\!24\)\( \beta) q^{66} +(\)\(16\!\cdots\!48\)\( + \)\(19\!\cdots\!59\)\( \beta) q^{67} +(\)\(31\!\cdots\!48\)\( - \)\(50\!\cdots\!48\)\( \beta) q^{68} +(-\)\(11\!\cdots\!84\)\( + \)\(41\!\cdots\!92\)\( \beta) q^{69} +(-\)\(14\!\cdots\!60\)\( + \)\(76\!\cdots\!32\)\( \beta) q^{70} +(-\)\(18\!\cdots\!88\)\( + \)\(11\!\cdots\!38\)\( \beta) q^{71} +(-\)\(89\!\cdots\!52\)\( + \)\(76\!\cdots\!72\)\( \beta) q^{72} +(-\)\(53\!\cdots\!86\)\( - \)\(38\!\cdots\!32\)\( \beta) q^{73} +(\)\(15\!\cdots\!68\)\( - \)\(75\!\cdots\!28\)\( \beta) q^{74} +(\)\(38\!\cdots\!00\)\( + \)\(65\!\cdots\!85\)\( \beta) q^{75} +(-\)\(51\!\cdots\!00\)\( + \)\(57\!\cdots\!52\)\( \beta) q^{76} +(\)\(12\!\cdots\!56\)\( + \)\(49\!\cdots\!72\)\( \beta) q^{77} +(\)\(21\!\cdots\!36\)\( - \)\(56\!\cdots\!12\)\( \beta) q^{78} +(\)\(20\!\cdots\!60\)\( + \)\(10\!\cdots\!32\)\( \beta) q^{79} +(-\)\(31\!\cdots\!20\)\( + \)\(16\!\cdots\!24\)\( \beta) q^{80} +(-\)\(17\!\cdots\!99\)\( - \)\(23\!\cdots\!44\)\( \beta) q^{81} +(-\)\(62\!\cdots\!68\)\( + \)\(88\!\cdots\!56\)\( \beta) q^{82} +(\)\(86\!\cdots\!44\)\( - \)\(17\!\cdots\!65\)\( \beta) q^{83} +(-\)\(29\!\cdots\!68\)\( - \)\(32\!\cdots\!16\)\( \beta) q^{84} +(-\)\(23\!\cdots\!60\)\( + \)\(49\!\cdots\!52\)\( \beta) q^{85} +(\)\(41\!\cdots\!44\)\( + \)\(35\!\cdots\!32\)\( \beta) q^{86} +(-\)\(66\!\cdots\!40\)\( - \)\(40\!\cdots\!14\)\( \beta) q^{87} +(-\)\(36\!\cdots\!68\)\( - \)\(45\!\cdots\!08\)\( \beta) q^{88} +(-\)\(38\!\cdots\!10\)\( + \)\(65\!\cdots\!60\)\( \beta) q^{89} +(\)\(12\!\cdots\!40\)\( - \)\(11\!\cdots\!48\)\( \beta) q^{90} +(-\)\(13\!\cdots\!08\)\( - \)\(10\!\cdots\!64\)\( \beta) q^{91} +(\)\(17\!\cdots\!44\)\( + \)\(94\!\cdots\!64\)\( \beta) q^{92} +(\)\(20\!\cdots\!88\)\( + \)\(25\!\cdots\!84\)\( \beta) q^{93} +(\)\(14\!\cdots\!88\)\( + \)\(19\!\cdots\!28\)\( \beta) q^{94} +(\)\(53\!\cdots\!00\)\( - \)\(31\!\cdots\!90\)\( \beta) q^{95} +(-\)\(26\!\cdots\!96\)\( + \)\(12\!\cdots\!24\)\( \beta) q^{96} +(-\)\(32\!\cdots\!22\)\( - \)\(57\!\cdots\!20\)\( \beta) q^{97} +(-\)\(19\!\cdots\!88\)\( - \)\(41\!\cdots\!52\)\( \beta) q^{98} +(-\)\(86\!\cdots\!44\)\( + \)\(11\!\cdots\!45\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 524288q^{2} + 423071208q^{3} + 137438953472q^{4} - 13507530555540q^{5} - 110905578749952q^{6} + 3106174162962256q^{7} - 36028797018963968q^{8} + 996387550079387706q^{9} + O(q^{10}) \) \( 2q - 524288q^{2} + 423071208q^{3} + 137438953472q^{4} - 13507530555540q^{5} - 110905578749952q^{6} + 3106174162962256q^{7} - 36028797018963968q^{8} + 996387550079387706q^{9} + 3540918089951477760q^{10} + 4002878661541032504q^{11} + 29073232035827417088q^{12} - \)\(41\!\cdots\!72\)\(q^{13} - \)\(81\!\cdots\!64\)\(q^{14} - \)\(90\!\cdots\!60\)\(q^{15} + \)\(94\!\cdots\!92\)\(q^{16} + \)\(92\!\cdots\!36\)\(q^{17} - \)\(26\!\cdots\!64\)\(q^{18} - \)\(15\!\cdots\!00\)\(q^{19} - \)\(92\!\cdots\!40\)\(q^{20} - \)\(86\!\cdots\!76\)\(q^{21} - \)\(10\!\cdots\!76\)\(q^{22} + \)\(49\!\cdots\!08\)\(q^{23} - \)\(76\!\cdots\!72\)\(q^{24} - \)\(32\!\cdots\!50\)\(q^{25} + \)\(10\!\cdots\!68\)\(q^{26} + \)\(78\!\cdots\!20\)\(q^{27} + \)\(21\!\cdots\!16\)\(q^{28} + \)\(11\!\cdots\!80\)\(q^{29} + \)\(23\!\cdots\!40\)\(q^{30} - \)\(55\!\cdots\!56\)\(q^{31} - \)\(24\!\cdots\!48\)\(q^{32} - \)\(44\!\cdots\!84\)\(q^{33} - \)\(24\!\cdots\!84\)\(q^{34} + \)\(10\!\cdots\!80\)\(q^{35} + \)\(68\!\cdots\!16\)\(q^{36} - \)\(11\!\cdots\!44\)\(q^{37} + \)\(39\!\cdots\!00\)\(q^{38} - \)\(16\!\cdots\!88\)\(q^{39} + \)\(24\!\cdots\!60\)\(q^{40} + \)\(47\!\cdots\!44\)\(q^{41} + \)\(22\!\cdots\!44\)\(q^{42} - \)\(31\!\cdots\!52\)\(q^{43} + \)\(27\!\cdots\!44\)\(q^{44} - \)\(93\!\cdots\!20\)\(q^{45} - \)\(13\!\cdots\!52\)\(q^{46} - \)\(11\!\cdots\!04\)\(q^{47} + \)\(19\!\cdots\!68\)\(q^{48} + \)\(15\!\cdots\!54\)\(q^{49} + \)\(86\!\cdots\!00\)\(q^{50} + \)\(13\!\cdots\!44\)\(q^{51} - \)\(28\!\cdots\!92\)\(q^{52} + \)\(53\!\cdots\!28\)\(q^{53} - \)\(20\!\cdots\!80\)\(q^{54} + \)\(13\!\cdots\!20\)\(q^{55} - \)\(55\!\cdots\!04\)\(q^{56} - \)\(46\!\cdots\!00\)\(q^{57} - \)\(30\!\cdots\!20\)\(q^{58} + \)\(58\!\cdots\!60\)\(q^{59} - \)\(62\!\cdots\!60\)\(q^{60} + \)\(94\!\cdots\!84\)\(q^{61} + \)\(14\!\cdots\!64\)\(q^{62} - \)\(23\!\cdots\!32\)\(q^{63} + \)\(64\!\cdots\!12\)\(q^{64} + \)\(30\!\cdots\!40\)\(q^{65} + \)\(11\!\cdots\!96\)\(q^{66} + \)\(32\!\cdots\!96\)\(q^{67} + \)\(63\!\cdots\!96\)\(q^{68} - \)\(23\!\cdots\!68\)\(q^{69} - \)\(28\!\cdots\!20\)\(q^{70} - \)\(36\!\cdots\!76\)\(q^{71} - \)\(17\!\cdots\!04\)\(q^{72} - \)\(10\!\cdots\!72\)\(q^{73} + \)\(30\!\cdots\!36\)\(q^{74} + \)\(77\!\cdots\!00\)\(q^{75} - \)\(10\!\cdots\!00\)\(q^{76} + \)\(24\!\cdots\!12\)\(q^{77} + \)\(42\!\cdots\!72\)\(q^{78} + \)\(41\!\cdots\!20\)\(q^{79} - \)\(63\!\cdots\!40\)\(q^{80} - \)\(34\!\cdots\!98\)\(q^{81} - \)\(12\!\cdots\!36\)\(q^{82} + \)\(17\!\cdots\!88\)\(q^{83} - \)\(59\!\cdots\!36\)\(q^{84} - \)\(46\!\cdots\!20\)\(q^{85} + \)\(83\!\cdots\!88\)\(q^{86} - \)\(13\!\cdots\!80\)\(q^{87} - \)\(72\!\cdots\!36\)\(q^{88} - \)\(77\!\cdots\!20\)\(q^{89} + \)\(24\!\cdots\!80\)\(q^{90} - \)\(27\!\cdots\!16\)\(q^{91} + \)\(34\!\cdots\!88\)\(q^{92} + \)\(40\!\cdots\!76\)\(q^{93} + \)\(29\!\cdots\!76\)\(q^{94} + \)\(10\!\cdots\!00\)\(q^{95} - \)\(52\!\cdots\!92\)\(q^{96} - \)\(65\!\cdots\!44\)\(q^{97} - \)\(39\!\cdots\!76\)\(q^{98} - \)\(17\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
27507.7
−27506.7
−262144. −7.39112e8 6.87195e10 −3.47974e12 1.93754e14 6.42679e15 −1.80144e16 9.60023e16 9.12192e17
1.2 −262144. 1.16218e9 6.87195e10 −1.00278e13 −3.04659e14 −3.32061e15 −1.80144e16 9.00385e17 2.62873e18
\(n\): 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 2.38.a.a 2
3.b odd 2 1 18.38.a.f 2
4.b odd 2 1 16.38.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.38.a.a 2 1.a even 1 1 trivial
16.38.a.a 2 4.b odd 2 1
18.38.a.f 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 423071208 T_{3} - \)858983057411401584

'>\(85\!\cdots\!84\)\( \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 262144 T )^{2} \)
$3$ \( 1 - 423071208 T + 41584754370593142 T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!69\)\( T^{4} \)
$5$ \( 1 + 13507530555540 T + \)\(18\!\cdots\!50\)\( T^{2} + \)\(98\!\cdots\!00\)\( T^{3} + \)\(52\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 3106174162962256 T + \)\(15\!\cdots\!98\)\( T^{2} - \)\(57\!\cdots\!92\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 4002878661541032504 T + \)\(10\!\cdots\!46\)\( T^{2} - \)\(13\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 + \)\(41\!\cdots\!72\)\( T + \)\(37\!\cdots\!62\)\( T^{2} + \)\(68\!\cdots\!76\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 - \)\(92\!\cdots\!36\)\( T + \)\(18\!\cdots\!78\)\( T^{2} - \)\(31\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(97\!\cdots\!78\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - \)\(49\!\cdots\!08\)\( T + \)\(32\!\cdots\!22\)\( T^{2} - \)\(12\!\cdots\!24\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 - \)\(11\!\cdots\!80\)\( T + \)\(22\!\cdots\!18\)\( T^{2} - \)\(15\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 + \)\(55\!\cdots\!56\)\( T + \)\(37\!\cdots\!06\)\( T^{2} + \)\(83\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 + \)\(11\!\cdots\!44\)\( T + \)\(23\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 - \)\(47\!\cdots\!44\)\( T + \)\(89\!\cdots\!46\)\( T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 + \)\(31\!\cdots\!52\)\( T + \)\(63\!\cdots\!62\)\( T^{2} + \)\(87\!\cdots\!36\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(11\!\cdots\!04\)\( T + \)\(12\!\cdots\!78\)\( T^{2} + \)\(82\!\cdots\!48\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 - \)\(53\!\cdots\!28\)\( T + \)\(13\!\cdots\!22\)\( T^{2} - \)\(33\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 - \)\(58\!\cdots\!60\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - \)\(94\!\cdots\!84\)\( T + \)\(19\!\cdots\!06\)\( T^{2} - \)\(10\!\cdots\!64\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 - \)\(32\!\cdots\!96\)\( T + \)\(76\!\cdots\!58\)\( T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + \)\(36\!\cdots\!76\)\( T + \)\(84\!\cdots\!26\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + \)\(10\!\cdots\!72\)\( T + \)\(44\!\cdots\!02\)\( T^{2} + \)\(94\!\cdots\!16\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - \)\(41\!\cdots\!20\)\( T + \)\(22\!\cdots\!18\)\( T^{2} - \)\(68\!\cdots\!80\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - \)\(17\!\cdots\!88\)\( T + \)\(18\!\cdots\!82\)\( T^{2} - \)\(17\!\cdots\!24\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + \)\(77\!\cdots\!20\)\( T + \)\(24\!\cdots\!58\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 + \)\(65\!\cdots\!44\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(21\!\cdots\!28\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \)
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