Properties

Label 2.40.a.b
Level 2
Weight 40
Character orbit 2.a
Self dual yes
Analytic conductor 19.268
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 40 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.2679102779\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 1050523661880\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 960\sqrt{4202094647521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -524288 q^{2} + ( 143709132 - \beta ) q^{3} + 274877906944 q^{4} + ( 26811369083310 - 21924 \beta ) q^{5} + ( -75344973398016 + 524288 \beta ) q^{6} + ( 3748995812923256 - 26490618 \beta ) q^{7} -144115188075855872 q^{8} + ( -159252411243429243 - 287418264 \beta ) q^{9} +O(q^{10})\) \( q -524288 q^{2} +(143709132 - \beta) q^{3} +274877906944 q^{4} +(26811369083310 - 21924 \beta) q^{5} +(-75344973398016 + 524288 \beta) q^{6} +(3748995812923256 - 26490618 \beta) q^{7} -144115188075855872 q^{8} +(-159252411243429243 - 287418264 \beta) q^{9} +(-14056879073950433280 + 11494490112 \beta) q^{10} +(-\)\(21\!\cdots\!08\)\( - 1640149371 \beta) q^{11} +(39502465412899012608 - 274877906944 \beta) q^{12} +(-\)\(55\!\cdots\!78\)\( - 1090692827172 \beta) q^{13} +(-\)\(19\!\cdots\!28\)\( + 13888713129984 \beta) q^{14} +(\)\(88\!\cdots\!20\)\( - 29962048093278 \beta) q^{15} +\)\(75\!\cdots\!36\)\( q^{16} +(-\)\(18\!\cdots\!54\)\( - 81667210689432 \beta) q^{17} +(\)\(83\!\cdots\!84\)\( + 150689946796032 \beta) q^{18} +(\)\(57\!\cdots\!80\)\( + 2573198844112971 \beta) q^{19} +(\)\(73\!\cdots\!40\)\( - 6026423231840256 \beta) q^{20} +(\)\(10\!\cdots\!92\)\( - 7555939531846832 \beta) q^{21} +(\)\(11\!\cdots\!04\)\( + 859910633422848 \beta) q^{22} +(\)\(36\!\cdots\!12\)\( + 98470758272206194 \beta) q^{23} +(-\)\(20\!\cdots\!04\)\( + 144115188075855872 \beta) q^{24} +(\)\(76\!\cdots\!75\)\( - 1175624911564976880 \beta) q^{25} +(\)\(29\!\cdots\!64\)\( + 571837160972353536 \beta) q^{26} +(\)\(50\!\cdots\!80\)\( + 4170502935022018662 \beta) q^{27} +(\)\(10\!\cdots\!64\)\( - 7281685629493051392 \beta) q^{28} +(-\)\(33\!\cdots\!30\)\( + 3076733487202628652 \beta) q^{29} +(-\)\(46\!\cdots\!60\)\( + 15708742270728536064 \beta) q^{30} +(-\)\(93\!\cdots\!28\)\( - 57411334880600430312 \beta) q^{31} -\)\(39\!\cdots\!68\)\( q^{32} +(-\)\(24\!\cdots\!56\)\( + \)\(21\!\cdots\!36\)\( \beta) q^{33} +(\)\(94\!\cdots\!52\)\( + 42817138557940924416 \beta) q^{34} +(\)\(23\!\cdots\!60\)\( - \)\(79\!\cdots\!24\)\( \beta) q^{35} +(-\)\(43\!\cdots\!92\)\( - 79004930825798025216 \beta) q^{36} +(\)\(14\!\cdots\!26\)\( + \)\(20\!\cdots\!68\)\( \beta) q^{37} +(-\)\(30\!\cdots\!40\)\( - \)\(13\!\cdots\!48\)\( \beta) q^{38} +(\)\(41\!\cdots\!04\)\( + \)\(39\!\cdots\!74\)\( \beta) q^{39} +(-\)\(38\!\cdots\!20\)\( + \)\(31\!\cdots\!28\)\( \beta) q^{40} +(\)\(83\!\cdots\!62\)\( - \)\(13\!\cdots\!08\)\( \beta) q^{41} +(-\)\(54\!\cdots\!96\)\( + \)\(39\!\cdots\!16\)\( \beta) q^{42} +(-\)\(49\!\cdots\!08\)\( + \)\(21\!\cdots\!17\)\( \beta) q^{43} +(-\)\(59\!\cdots\!52\)\( - \)\(45\!\cdots\!24\)\( \beta) q^{44} +(\)\(20\!\cdots\!70\)\( - \)\(42\!\cdots\!08\)\( \beta) q^{45} +(-\)\(19\!\cdots\!56\)\( - \)\(51\!\cdots\!72\)\( \beta) q^{46} +(\)\(33\!\cdots\!16\)\( + \)\(63\!\cdots\!92\)\( \beta) q^{47} +(\)\(10\!\cdots\!52\)\( - \)\(75\!\cdots\!36\)\( \beta) q^{48} +(\)\(18\!\cdots\!93\)\( - \)\(19\!\cdots\!16\)\( \beta) q^{49} +(-\)\(39\!\cdots\!00\)\( + \)\(61\!\cdots\!40\)\( \beta) q^{50} +(\)\(29\!\cdots\!72\)\( + \)\(16\!\cdots\!30\)\( \beta) q^{51} +(-\)\(15\!\cdots\!32\)\( - \)\(29\!\cdots\!68\)\( \beta) q^{52} +(-\)\(43\!\cdots\!18\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{53} +(-\)\(26\!\cdots\!40\)\( - \)\(21\!\cdots\!56\)\( \beta) q^{54} +(-\)\(56\!\cdots\!80\)\( + \)\(47\!\cdots\!82\)\( \beta) q^{55} +(-\)\(54\!\cdots\!32\)\( + \)\(38\!\cdots\!96\)\( \beta) q^{56} +(-\)\(91\!\cdots\!40\)\( - \)\(53\!\cdots\!08\)\( \beta) q^{57} +(\)\(17\!\cdots\!40\)\( - \)\(16\!\cdots\!76\)\( \beta) q^{58} +(-\)\(64\!\cdots\!60\)\( + \)\(12\!\cdots\!49\)\( \beta) q^{59} +(\)\(24\!\cdots\!80\)\( - \)\(82\!\cdots\!32\)\( \beta) q^{60} +(-\)\(10\!\cdots\!58\)\( - \)\(45\!\cdots\!04\)\( \beta) q^{61} +(\)\(48\!\cdots\!64\)\( + \)\(30\!\cdots\!56\)\( \beta) q^{62} +(\)\(28\!\cdots\!92\)\( + \)\(31\!\cdots\!90\)\( \beta) q^{63} +\)\(20\!\cdots\!84\)\( q^{64} +(\)\(77\!\cdots\!20\)\( - \)\(17\!\cdots\!48\)\( \beta) q^{65} +(\)\(12\!\cdots\!28\)\( - \)\(11\!\cdots\!68\)\( \beta) q^{66} +(-\)\(21\!\cdots\!04\)\( + \)\(18\!\cdots\!31\)\( \beta) q^{67} +(-\)\(49\!\cdots\!76\)\( - \)\(22\!\cdots\!08\)\( \beta) q^{68} +(-\)\(32\!\cdots\!16\)\( - \)\(35\!\cdots\!04\)\( \beta) q^{69} +(-\)\(12\!\cdots\!80\)\( + \)\(41\!\cdots\!12\)\( \beta) q^{70} +(-\)\(25\!\cdots\!68\)\( + \)\(24\!\cdots\!26\)\( \beta) q^{71} +(\)\(22\!\cdots\!96\)\( + \)\(41\!\cdots\!08\)\( \beta) q^{72} +(\)\(26\!\cdots\!62\)\( - \)\(26\!\cdots\!32\)\( \beta) q^{73} +(-\)\(78\!\cdots\!88\)\( - \)\(10\!\cdots\!84\)\( \beta) q^{74} +(\)\(46\!\cdots\!00\)\( - \)\(93\!\cdots\!35\)\( \beta) q^{75} +(\)\(15\!\cdots\!20\)\( + \)\(70\!\cdots\!24\)\( \beta) q^{76} +(-\)\(64\!\cdots\!48\)\( + \)\(57\!\cdots\!68\)\( \beta) q^{77} +(-\)\(21\!\cdots\!52\)\( - \)\(20\!\cdots\!12\)\( \beta) q^{78} +(\)\(25\!\cdots\!20\)\( - \)\(74\!\cdots\!04\)\( \beta) q^{79} +(\)\(20\!\cdots\!60\)\( - \)\(16\!\cdots\!64\)\( \beta) q^{80} +(-\)\(15\!\cdots\!59\)\( + \)\(12\!\cdots\!92\)\( \beta) q^{81} +(-\)\(43\!\cdots\!56\)\( + \)\(73\!\cdots\!04\)\( \beta) q^{82} +(-\)\(32\!\cdots\!48\)\( - \)\(21\!\cdots\!85\)\( \beta) q^{83} +(\)\(28\!\cdots\!48\)\( - \)\(20\!\cdots\!08\)\( \beta) q^{84} +(\)\(20\!\cdots\!60\)\( + \)\(17\!\cdots\!76\)\( \beta) q^{85} +(\)\(25\!\cdots\!04\)\( - \)\(11\!\cdots\!96\)\( \beta) q^{86} +(-\)\(16\!\cdots\!60\)\( + \)\(33\!\cdots\!94\)\( \beta) q^{87} +(\)\(31\!\cdots\!76\)\( + \)\(23\!\cdots\!12\)\( \beta) q^{88} +(-\)\(17\!\cdots\!90\)\( - \)\(61\!\cdots\!20\)\( \beta) q^{89} +(-\)\(10\!\cdots\!60\)\( + \)\(22\!\cdots\!04\)\( \beta) q^{90} +(\)\(10\!\cdots\!32\)\( + \)\(10\!\cdots\!72\)\( \beta) q^{91} +(\)\(10\!\cdots\!28\)\( + \)\(27\!\cdots\!36\)\( \beta) q^{92} +(\)\(20\!\cdots\!04\)\( + \)\(85\!\cdots\!44\)\( \beta) q^{93} +(-\)\(17\!\cdots\!08\)\( - \)\(33\!\cdots\!96\)\( \beta) q^{94} +(-\)\(63\!\cdots\!00\)\( - \)\(57\!\cdots\!10\)\( \beta) q^{95} +(-\)\(56\!\cdots\!76\)\( + \)\(39\!\cdots\!68\)\( \beta) q^{96} +(-\)\(72\!\cdots\!34\)\( - \)\(19\!\cdots\!00\)\( \beta) q^{97} +(-\)\(95\!\cdots\!84\)\( + \)\(10\!\cdots\!08\)\( \beta) q^{98} +(\)\(36\!\cdots\!44\)\( + \)\(62\!\cdots\!65\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 1048576q^{2} + 287418264q^{3} + 549755813888q^{4} + 53622738166620q^{5} - 150689946796032q^{6} + 7497991625846512q^{7} - 288230376151711744q^{8} - 318504822486858486q^{9} + O(q^{10}) \) \( 2q - 1048576q^{2} + 287418264q^{3} + 549755813888q^{4} + 53622738166620q^{5} - 150689946796032q^{6} + 7497991625846512q^{7} - 288230376151711744q^{8} - 318504822486858486q^{9} - 28113758147900866560q^{10} - \)\(43\!\cdots\!16\)\(q^{11} + 79004930825798025216q^{12} - \)\(11\!\cdots\!56\)\(q^{13} - \)\(39\!\cdots\!56\)\(q^{14} + \)\(17\!\cdots\!40\)\(q^{15} + \)\(15\!\cdots\!72\)\(q^{16} - \)\(36\!\cdots\!08\)\(q^{17} + \)\(16\!\cdots\!68\)\(q^{18} + \)\(11\!\cdots\!60\)\(q^{19} + \)\(14\!\cdots\!80\)\(q^{20} + \)\(20\!\cdots\!84\)\(q^{21} + \)\(22\!\cdots\!08\)\(q^{22} + \)\(73\!\cdots\!24\)\(q^{23} - \)\(41\!\cdots\!08\)\(q^{24} + \)\(15\!\cdots\!50\)\(q^{25} + \)\(58\!\cdots\!28\)\(q^{26} + \)\(10\!\cdots\!60\)\(q^{27} + \)\(20\!\cdots\!28\)\(q^{28} - \)\(66\!\cdots\!60\)\(q^{29} - \)\(93\!\cdots\!20\)\(q^{30} - \)\(18\!\cdots\!56\)\(q^{31} - \)\(79\!\cdots\!36\)\(q^{32} - \)\(49\!\cdots\!12\)\(q^{33} + \)\(18\!\cdots\!04\)\(q^{34} + \)\(46\!\cdots\!20\)\(q^{35} - \)\(87\!\cdots\!84\)\(q^{36} + \)\(29\!\cdots\!52\)\(q^{37} - \)\(60\!\cdots\!80\)\(q^{38} + \)\(82\!\cdots\!08\)\(q^{39} - \)\(77\!\cdots\!40\)\(q^{40} + \)\(16\!\cdots\!24\)\(q^{41} - \)\(10\!\cdots\!92\)\(q^{42} - \)\(98\!\cdots\!16\)\(q^{43} - \)\(11\!\cdots\!04\)\(q^{44} + \)\(40\!\cdots\!40\)\(q^{45} - \)\(38\!\cdots\!12\)\(q^{46} + \)\(67\!\cdots\!32\)\(q^{47} + \)\(21\!\cdots\!04\)\(q^{48} + \)\(36\!\cdots\!86\)\(q^{49} - \)\(79\!\cdots\!00\)\(q^{50} + \)\(58\!\cdots\!44\)\(q^{51} - \)\(30\!\cdots\!64\)\(q^{52} - \)\(87\!\cdots\!36\)\(q^{53} - \)\(53\!\cdots\!80\)\(q^{54} - \)\(11\!\cdots\!60\)\(q^{55} - \)\(10\!\cdots\!64\)\(q^{56} - \)\(18\!\cdots\!80\)\(q^{57} + \)\(34\!\cdots\!80\)\(q^{58} - \)\(12\!\cdots\!20\)\(q^{59} + \)\(48\!\cdots\!60\)\(q^{60} - \)\(20\!\cdots\!16\)\(q^{61} + \)\(97\!\cdots\!28\)\(q^{62} + \)\(57\!\cdots\!84\)\(q^{63} + \)\(41\!\cdots\!68\)\(q^{64} + \)\(15\!\cdots\!40\)\(q^{65} + \)\(25\!\cdots\!56\)\(q^{66} - \)\(43\!\cdots\!08\)\(q^{67} - \)\(99\!\cdots\!52\)\(q^{68} - \)\(65\!\cdots\!32\)\(q^{69} - \)\(24\!\cdots\!60\)\(q^{70} - \)\(50\!\cdots\!36\)\(q^{71} + \)\(45\!\cdots\!92\)\(q^{72} + \)\(52\!\cdots\!24\)\(q^{73} - \)\(15\!\cdots\!76\)\(q^{74} + \)\(93\!\cdots\!00\)\(q^{75} + \)\(31\!\cdots\!40\)\(q^{76} - \)\(12\!\cdots\!96\)\(q^{77} - \)\(43\!\cdots\!04\)\(q^{78} + \)\(51\!\cdots\!40\)\(q^{79} + \)\(40\!\cdots\!20\)\(q^{80} - \)\(30\!\cdots\!18\)\(q^{81} - \)\(87\!\cdots\!12\)\(q^{82} - \)\(64\!\cdots\!96\)\(q^{83} + \)\(56\!\cdots\!96\)\(q^{84} + \)\(41\!\cdots\!20\)\(q^{85} + \)\(51\!\cdots\!08\)\(q^{86} - \)\(33\!\cdots\!20\)\(q^{87} + \)\(62\!\cdots\!52\)\(q^{88} - \)\(35\!\cdots\!80\)\(q^{89} - \)\(21\!\cdots\!20\)\(q^{90} + \)\(21\!\cdots\!64\)\(q^{91} + \)\(20\!\cdots\!56\)\(q^{92} + \)\(41\!\cdots\!08\)\(q^{93} - \)\(35\!\cdots\!16\)\(q^{94} - \)\(12\!\cdots\!00\)\(q^{95} - \)\(11\!\cdots\!52\)\(q^{96} - \)\(14\!\cdots\!68\)\(q^{97} - \)\(19\!\cdots\!68\)\(q^{98} + \)\(72\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
1.02495e6
−1.02495e6
−524288. −1.82420e9 2.74878e11 −1.63330e13 9.56404e14 −4.83820e16 −1.44115e17 −7.24864e17 8.56319e18
1.2 −524288. 2.11161e9 2.74878e11 6.99557e13 −1.10709e15 5.58800e16 −1.44115e17 4.06359e17 −3.66769e19
\(n\): 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 2.40.a.b 2
3.b odd 2 1 18.40.a.e 2
4.b odd 2 1 16.40.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.40.a.b 2 1.a even 1 1 trivial
16.40.a.b 2 4.b odd 2 1
18.40.a.e 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 \( T_{3}^{2} - 287418264 T_{3} - \)\(38\!\cdots\!76\)\( \) acting on \(S_{40}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 524288 T )^{2} \)
$3$ \( 1 - 287418264 T + 4253112193502792358 T^{2} - \)\(11\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \)
$5$ \( 1 - 53622738166620 T + \)\(24\!\cdots\!50\)\( T^{2} - \)\(97\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 7497991625846512 T - \)\(88\!\cdots\!78\)\( T^{2} - \)\(68\!\cdots\!16\)\( T^{3} + \)\(82\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + \)\(43\!\cdots\!16\)\( T + \)\(12\!\cdots\!46\)\( T^{2} + \)\(17\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \)
$13$ \( 1 + \)\(11\!\cdots\!56\)\( T + \)\(51\!\cdots\!38\)\( T^{2} + \)\(30\!\cdots\!12\)\( T^{3} + \)\(77\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 + \)\(36\!\cdots\!08\)\( T + \)\(19\!\cdots\!22\)\( T^{2} + \)\(35\!\cdots\!24\)\( T^{3} + \)\(94\!\cdots\!09\)\( T^{4} \)
$19$ \( 1 - \)\(11\!\cdots\!60\)\( T + \)\(15\!\cdots\!58\)\( T^{2} - \)\(85\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \)
$23$ \( 1 - \)\(73\!\cdots\!24\)\( T + \)\(35\!\cdots\!18\)\( T^{2} - \)\(93\!\cdots\!88\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \)
$29$ \( 1 + \)\(66\!\cdots\!60\)\( T + \)\(32\!\cdots\!38\)\( T^{2} + \)\(71\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$31$ \( 1 + \)\(18\!\cdots\!56\)\( T + \)\(25\!\cdots\!26\)\( T^{2} + \)\(27\!\cdots\!76\)\( T^{3} + \)\(21\!\cdots\!41\)\( T^{4} \)
$37$ \( 1 - \)\(29\!\cdots\!52\)\( T + \)\(15\!\cdots\!22\)\( T^{2} - \)\(42\!\cdots\!96\)\( T^{3} + \)\(20\!\cdots\!29\)\( T^{4} \)
$41$ \( 1 - \)\(16\!\cdots\!24\)\( T + \)\(89\!\cdots\!66\)\( T^{2} - \)\(13\!\cdots\!64\)\( T^{3} + \)\(62\!\cdots\!21\)\( T^{4} \)
$43$ \( 1 + \)\(98\!\cdots\!16\)\( T + \)\(83\!\cdots\!78\)\( T^{2} + \)\(49\!\cdots\!12\)\( T^{3} + \)\(25\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - \)\(67\!\cdots\!32\)\( T + \)\(42\!\cdots\!22\)\( T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 + \)\(87\!\cdots\!36\)\( T + \)\(49\!\cdots\!58\)\( T^{2} + \)\(15\!\cdots\!12\)\( T^{3} + \)\(31\!\cdots\!89\)\( T^{4} \)
$59$ \( 1 + \)\(12\!\cdots\!20\)\( T + \)\(23\!\cdots\!78\)\( T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!21\)\( T^{4} \)
$61$ \( 1 + \)\(20\!\cdots\!16\)\( T + \)\(85\!\cdots\!46\)\( T^{2} + \)\(88\!\cdots\!56\)\( T^{3} + \)\(18\!\cdots\!81\)\( T^{4} \)
$67$ \( 1 + \)\(43\!\cdots\!08\)\( T + \)\(24\!\cdots\!22\)\( T^{2} + \)\(70\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!09\)\( T^{4} \)
$71$ \( 1 + \)\(50\!\cdots\!36\)\( T + \)\(29\!\cdots\!86\)\( T^{2} + \)\(79\!\cdots\!16\)\( T^{3} + \)\(25\!\cdots\!61\)\( T^{4} \)
$73$ \( 1 - \)\(52\!\cdots\!24\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!88\)\( T^{3} + \)\(21\!\cdots\!69\)\( T^{4} \)
$79$ \( 1 - \)\(51\!\cdots\!40\)\( T - \)\(73\!\cdots\!62\)\( T^{2} - \)\(52\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!61\)\( T^{4} \)
$83$ \( 1 + \)\(64\!\cdots\!96\)\( T + \)\(24\!\cdots\!98\)\( T^{2} + \)\(45\!\cdots\!12\)\( T^{3} + \)\(48\!\cdots\!09\)\( T^{4} \)
$89$ \( 1 + \)\(35\!\cdots\!80\)\( T + \)\(69\!\cdots\!18\)\( T^{2} + \)\(38\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \)
$97$ \( 1 + \)\(14\!\cdots\!68\)\( T + \)\(46\!\cdots\!22\)\( T^{2} + \)\(44\!\cdots\!44\)\( T^{3} + \)\(92\!\cdots\!89\)\( T^{4} \)
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