Properties

Label 2.46.a.a
Level 2
Weight 46
Character orbit 2.a
Self dual yes
Analytic conductor 25.651
Analytic rank 1
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5^{2}\cdot 11 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2851200\sqrt{655098313}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -4194304 q^{2} + ( -34883103276 - \beta ) q^{3} + 17592186044416 q^{4} + ( -2251248081340650 - 42476 \beta ) q^{5} + ( 146310339602939904 + 4194304 \beta ) q^{6} + ( -4782175712604552472 - 7308322 \beta ) q^{7} -73786976294838206464 q^{8} + ( 3588036050758238953533 + 69766206552 \beta ) q^{9} +O(q^{10})\) \( q -4194304 q^{2} +(-34883103276 - \beta) q^{3} +17592186044416 q^{4} +(-2251248081340650 - 42476 \beta) q^{5} +(146310339602939904 + 4194304 \beta) q^{6} +(-4782175712604552472 - 7308322 \beta) q^{7} -73786976294838206464 q^{8} +(\)\(35\!\cdots\!33\)\( + 69766206552 \beta) q^{9} +(\)\(94\!\cdots\!00\)\( + 178157256704 \beta) q^{10} +(\)\(99\!\cdots\!32\)\( + 2762949838957 \beta) q^{11} +(-\)\(61\!\cdots\!16\)\( - 17592186044416 \beta) q^{12} +(\)\(57\!\cdots\!54\)\( - 127591092331532 \beta) q^{13} +(\)\(20\!\cdots\!88\)\( + 30653324197888 \beta) q^{14} +(\)\(30\!\cdots\!00\)\( + 3732942776092026 \beta) q^{15} +\)\(30\!\cdots\!56\)\( q^{16} +(\)\(78\!\cdots\!78\)\( - 7657971672330088 \beta) q^{17} +(-\)\(15\!\cdots\!32\)\( - 292620679205879808 \beta) q^{18} +(-\)\(19\!\cdots\!00\)\( + 943670096252337787 \beta) q^{19} +(-\)\(39\!\cdots\!00\)\( - 747245694422614016 \beta) q^{20} +(\)\(20\!\cdots\!72\)\( + 5037112663704815344 \beta) q^{21} +(-\)\(41\!\cdots\!28\)\( - 11588651561336700928 \beta) q^{22} +(-\)\(15\!\cdots\!36\)\( - 44126344403460381286 \beta) q^{23} +(\)\(25\!\cdots\!64\)\( + 73786976294838206464 \beta) q^{24} +(-\)\(13\!\cdots\!25\)\( + \)\(19\!\cdots\!00\)\( \beta) q^{25} +(-\)\(24\!\cdots\!16\)\( + \)\(53\!\cdots\!28\)\( \beta) q^{26} +(-\)\(39\!\cdots\!40\)\( - \)\(30\!\cdots\!42\)\( \beta) q^{27} +(-\)\(84\!\cdots\!52\)\( - \)\(12\!\cdots\!52\)\( \beta) q^{28} +(-\)\(14\!\cdots\!10\)\( + \)\(10\!\cdots\!44\)\( \beta) q^{29} +(-\)\(12\!\cdots\!00\)\( - \)\(15\!\cdots\!04\)\( \beta) q^{30} +(-\)\(20\!\cdots\!88\)\( + \)\(35\!\cdots\!24\)\( \beta) q^{31} -\)\(12\!\cdots\!24\)\( q^{32} +(-\)\(18\!\cdots\!32\)\( - \)\(19\!\cdots\!64\)\( \beta) q^{33} +(-\)\(32\!\cdots\!12\)\( + \)\(32\!\cdots\!52\)\( \beta) q^{34} +(\)\(12\!\cdots\!00\)\( + \)\(21\!\cdots\!72\)\( \beta) q^{35} +(\)\(63\!\cdots\!28\)\( + \)\(12\!\cdots\!32\)\( \beta) q^{36} +(\)\(25\!\cdots\!58\)\( - \)\(64\!\cdots\!28\)\( \beta) q^{37} +(\)\(79\!\cdots\!00\)\( - \)\(39\!\cdots\!48\)\( \beta) q^{38} +(\)\(47\!\cdots\!96\)\( - \)\(12\!\cdots\!22\)\( \beta) q^{39} +(\)\(16\!\cdots\!00\)\( + \)\(31\!\cdots\!64\)\( \beta) q^{40} +(\)\(20\!\cdots\!42\)\( + \)\(22\!\cdots\!56\)\( \beta) q^{41} +(-\)\(86\!\cdots\!88\)\( - \)\(21\!\cdots\!76\)\( \beta) q^{42} +(-\)\(17\!\cdots\!76\)\( + \)\(91\!\cdots\!37\)\( \beta) q^{43} +(\)\(17\!\cdots\!12\)\( + \)\(48\!\cdots\!12\)\( \beta) q^{44} +(-\)\(23\!\cdots\!50\)\( - \)\(30\!\cdots\!08\)\( \beta) q^{45} +(\)\(63\!\cdots\!44\)\( + \)\(18\!\cdots\!44\)\( \beta) q^{46} +(-\)\(60\!\cdots\!12\)\( + \)\(57\!\cdots\!88\)\( \beta) q^{47} +(-\)\(10\!\cdots\!56\)\( - \)\(30\!\cdots\!56\)\( \beta) q^{48} +(-\)\(83\!\cdots\!23\)\( + \)\(69\!\cdots\!68\)\( \beta) q^{49} +(\)\(57\!\cdots\!00\)\( - \)\(80\!\cdots\!00\)\( \beta) q^{50} +(\)\(13\!\cdots\!72\)\( - \)\(51\!\cdots\!90\)\( \beta) q^{51} +(\)\(10\!\cdots\!64\)\( - \)\(22\!\cdots\!12\)\( \beta) q^{52} +(\)\(70\!\cdots\!34\)\( + \)\(31\!\cdots\!88\)\( \beta) q^{53} +(\)\(16\!\cdots\!60\)\( + \)\(12\!\cdots\!68\)\( \beta) q^{54} +(-\)\(84\!\cdots\!00\)\( - \)\(10\!\cdots\!82\)\( \beta) q^{55} +(\)\(35\!\cdots\!08\)\( + \)\(53\!\cdots\!08\)\( \beta) q^{56} +(-\)\(43\!\cdots\!00\)\( - \)\(13\!\cdots\!12\)\( \beta) q^{57} +(\)\(58\!\cdots\!40\)\( - \)\(43\!\cdots\!76\)\( \beta) q^{58} +(-\)\(72\!\cdots\!20\)\( + \)\(30\!\cdots\!33\)\( \beta) q^{59} +(\)\(53\!\cdots\!00\)\( + \)\(65\!\cdots\!16\)\( \beta) q^{60} +(-\)\(10\!\cdots\!78\)\( + \)\(18\!\cdots\!28\)\( \beta) q^{61} +(\)\(87\!\cdots\!52\)\( - \)\(15\!\cdots\!96\)\( \beta) q^{62} +(-\)\(19\!\cdots\!76\)\( - \)\(35\!\cdots\!70\)\( \beta) q^{63} +\)\(54\!\cdots\!96\)\( q^{64} +(\)\(15\!\cdots\!00\)\( + \)\(44\!\cdots\!96\)\( \beta) q^{65} +(\)\(76\!\cdots\!28\)\( + \)\(82\!\cdots\!56\)\( \beta) q^{66} +(-\)\(48\!\cdots\!92\)\( - \)\(81\!\cdots\!81\)\( \beta) q^{67} +(\)\(13\!\cdots\!48\)\( - \)\(13\!\cdots\!08\)\( \beta) q^{68} +(\)\(28\!\cdots\!36\)\( + \)\(30\!\cdots\!72\)\( \beta) q^{69} +(-\)\(52\!\cdots\!00\)\( - \)\(92\!\cdots\!88\)\( \beta) q^{70} +(\)\(22\!\cdots\!72\)\( - \)\(43\!\cdots\!62\)\( \beta) q^{71} +(-\)\(26\!\cdots\!12\)\( - \)\(51\!\cdots\!28\)\( \beta) q^{72} +(-\)\(17\!\cdots\!26\)\( + \)\(56\!\cdots\!48\)\( \beta) q^{73} +(-\)\(10\!\cdots\!32\)\( + \)\(27\!\cdots\!12\)\( \beta) q^{74} +(-\)\(53\!\cdots\!00\)\( + \)\(70\!\cdots\!25\)\( \beta) q^{75} +(-\)\(33\!\cdots\!00\)\( + \)\(16\!\cdots\!92\)\( \beta) q^{76} +(-\)\(58\!\cdots\!04\)\( - \)\(13\!\cdots\!08\)\( \beta) q^{77} +(-\)\(20\!\cdots\!84\)\( + \)\(53\!\cdots\!88\)\( \beta) q^{78} +(-\)\(31\!\cdots\!40\)\( - \)\(12\!\cdots\!68\)\( \beta) q^{79} +(-\)\(69\!\cdots\!00\)\( - \)\(13\!\cdots\!56\)\( \beta) q^{80} +(\)\(19\!\cdots\!21\)\( + \)\(29\!\cdots\!96\)\( \beta) q^{81} +(-\)\(87\!\cdots\!68\)\( - \)\(96\!\cdots\!24\)\( \beta) q^{82} +(\)\(48\!\cdots\!24\)\( - \)\(11\!\cdots\!65\)\( \beta) q^{83} +(\)\(36\!\cdots\!52\)\( + \)\(88\!\cdots\!04\)\( \beta) q^{84} +(-\)\(27\!\cdots\!00\)\( - \)\(15\!\cdots\!28\)\( \beta) q^{85} +(\)\(74\!\cdots\!04\)\( - \)\(38\!\cdots\!48\)\( \beta) q^{86} +(-\)\(50\!\cdots\!40\)\( - \)\(22\!\cdots\!34\)\( \beta) q^{87} +(-\)\(73\!\cdots\!48\)\( - \)\(20\!\cdots\!48\)\( \beta) q^{88} +(-\)\(62\!\cdots\!10\)\( - \)\(26\!\cdots\!40\)\( \beta) q^{89} +(\)\(10\!\cdots\!00\)\( + \)\(12\!\cdots\!32\)\( \beta) q^{90} +(-\)\(22\!\cdots\!88\)\( + \)\(56\!\cdots\!16\)\( \beta) q^{91} +(-\)\(26\!\cdots\!76\)\( - \)\(77\!\cdots\!76\)\( \beta) q^{92} +(-\)\(11\!\cdots\!12\)\( + \)\(84\!\cdots\!64\)\( \beta) q^{93} +(\)\(25\!\cdots\!48\)\( - \)\(24\!\cdots\!52\)\( \beta) q^{94} +(-\)\(17\!\cdots\!00\)\( - \)\(13\!\cdots\!50\)\( \beta) q^{95} +(\)\(45\!\cdots\!24\)\( + \)\(12\!\cdots\!24\)\( \beta) q^{96} +(\)\(23\!\cdots\!18\)\( - \)\(26\!\cdots\!20\)\( \beta) q^{97} +(\)\(35\!\cdots\!92\)\( - \)\(29\!\cdots\!72\)\( \beta) q^{98} +(\)\(13\!\cdots\!56\)\( + \)\(16\!\cdots\!45\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 8388608q^{2} - 69766206552q^{3} + 35184372088832q^{4} - 4502496162681300q^{5} + 292620679205879808q^{6} - 9564351425209104944q^{7} - 147573952589676412928q^{8} + 7176072101516477907066q^{9} + O(q^{10}) \) \( 2q - 8388608q^{2} - 69766206552q^{3} + 35184372088832q^{4} - 4502496162681300q^{5} + 292620679205879808q^{6} - 9564351425209104944q^{7} - \)\(14\!\cdots\!28\)\(q^{8} + \)\(71\!\cdots\!66\)\(q^{9} + \)\(18\!\cdots\!00\)\(q^{10} + \)\(19\!\cdots\!64\)\(q^{11} - \)\(12\!\cdots\!32\)\(q^{12} + \)\(11\!\cdots\!08\)\(q^{13} + \)\(40\!\cdots\!76\)\(q^{14} + \)\(60\!\cdots\!00\)\(q^{15} + \)\(61\!\cdots\!12\)\(q^{16} + \)\(15\!\cdots\!56\)\(q^{17} - \)\(30\!\cdots\!64\)\(q^{18} - \)\(38\!\cdots\!00\)\(q^{19} - \)\(79\!\cdots\!00\)\(q^{20} + \)\(41\!\cdots\!44\)\(q^{21} - \)\(83\!\cdots\!56\)\(q^{22} - \)\(30\!\cdots\!72\)\(q^{23} + \)\(51\!\cdots\!28\)\(q^{24} - \)\(27\!\cdots\!50\)\(q^{25} - \)\(48\!\cdots\!32\)\(q^{26} - \)\(78\!\cdots\!80\)\(q^{27} - \)\(16\!\cdots\!04\)\(q^{28} - \)\(28\!\cdots\!20\)\(q^{29} - \)\(25\!\cdots\!00\)\(q^{30} - \)\(41\!\cdots\!76\)\(q^{31} - \)\(25\!\cdots\!48\)\(q^{32} - \)\(36\!\cdots\!64\)\(q^{33} - \)\(65\!\cdots\!24\)\(q^{34} + \)\(24\!\cdots\!00\)\(q^{35} + \)\(12\!\cdots\!56\)\(q^{36} + \)\(50\!\cdots\!16\)\(q^{37} + \)\(15\!\cdots\!00\)\(q^{38} + \)\(95\!\cdots\!92\)\(q^{39} + \)\(33\!\cdots\!00\)\(q^{40} + \)\(41\!\cdots\!84\)\(q^{41} - \)\(17\!\cdots\!76\)\(q^{42} - \)\(35\!\cdots\!52\)\(q^{43} + \)\(34\!\cdots\!24\)\(q^{44} - \)\(47\!\cdots\!00\)\(q^{45} + \)\(12\!\cdots\!88\)\(q^{46} - \)\(12\!\cdots\!24\)\(q^{47} - \)\(21\!\cdots\!12\)\(q^{48} - \)\(16\!\cdots\!46\)\(q^{49} + \)\(11\!\cdots\!00\)\(q^{50} + \)\(27\!\cdots\!44\)\(q^{51} + \)\(20\!\cdots\!28\)\(q^{52} + \)\(14\!\cdots\!68\)\(q^{53} + \)\(33\!\cdots\!20\)\(q^{54} - \)\(16\!\cdots\!00\)\(q^{55} + \)\(70\!\cdots\!16\)\(q^{56} - \)\(87\!\cdots\!00\)\(q^{57} + \)\(11\!\cdots\!80\)\(q^{58} - \)\(14\!\cdots\!40\)\(q^{59} + \)\(10\!\cdots\!00\)\(q^{60} - \)\(21\!\cdots\!56\)\(q^{61} + \)\(17\!\cdots\!04\)\(q^{62} - \)\(39\!\cdots\!52\)\(q^{63} + \)\(10\!\cdots\!92\)\(q^{64} + \)\(31\!\cdots\!00\)\(q^{65} + \)\(15\!\cdots\!56\)\(q^{66} - \)\(96\!\cdots\!84\)\(q^{67} + \)\(27\!\cdots\!96\)\(q^{68} + \)\(57\!\cdots\!72\)\(q^{69} - \)\(10\!\cdots\!00\)\(q^{70} + \)\(44\!\cdots\!44\)\(q^{71} - \)\(52\!\cdots\!24\)\(q^{72} - \)\(34\!\cdots\!52\)\(q^{73} - \)\(21\!\cdots\!64\)\(q^{74} - \)\(10\!\cdots\!00\)\(q^{75} - \)\(66\!\cdots\!00\)\(q^{76} - \)\(11\!\cdots\!08\)\(q^{77} - \)\(40\!\cdots\!68\)\(q^{78} - \)\(62\!\cdots\!80\)\(q^{79} - \)\(13\!\cdots\!00\)\(q^{80} + \)\(38\!\cdots\!42\)\(q^{81} - \)\(17\!\cdots\!36\)\(q^{82} + \)\(96\!\cdots\!48\)\(q^{83} + \)\(72\!\cdots\!04\)\(q^{84} - \)\(54\!\cdots\!00\)\(q^{85} + \)\(14\!\cdots\!08\)\(q^{86} - \)\(10\!\cdots\!80\)\(q^{87} - \)\(14\!\cdots\!96\)\(q^{88} - \)\(12\!\cdots\!20\)\(q^{89} + \)\(20\!\cdots\!00\)\(q^{90} - \)\(44\!\cdots\!76\)\(q^{91} - \)\(53\!\cdots\!52\)\(q^{92} - \)\(23\!\cdots\!24\)\(q^{93} + \)\(50\!\cdots\!96\)\(q^{94} - \)\(34\!\cdots\!00\)\(q^{95} + \)\(90\!\cdots\!48\)\(q^{96} + \)\(47\!\cdots\!36\)\(q^{97} + \)\(70\!\cdots\!84\)\(q^{98} + \)\(27\!\cdots\!12\)\(q^{99} + O(q^{100}) \)

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
12797.9
−12796.9
−4.19430e6 −1.07859e11 1.75922e13 −5.35098e15 4.52394e17 −5.31551e18 −7.37870e19 8.67930e21 2.24436e22
1.2 −4.19430e6 3.80930e10 1.75922e13 8.48487e14 −1.59774e17 −4.24884e18 −7.37870e19 −1.50323e21 −3.55881e21
\(n\): 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.a 2
3.b odd 2 1 18.46.a.f 2
4.b odd 2 1 16.46.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.46.a.a 2 1.a even 1 1 trivial
16.46.a.b 2 4.b odd 2 1
18.46.a.f 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 \( T_{3}^{2} + 69766206552 T_{3} - \)\(41\!\cdots\!24\)\( \) acting on \(S_{46}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 4194304 T )^{2} \)
$3$ \( 1 + 69766206552 T + \)\(17\!\cdots\!62\)\( T^{2} + \)\(20\!\cdots\!36\)\( T^{3} + \)\(87\!\cdots\!49\)\( T^{4} \)
$5$ \( 1 + 4502496162681300 T + \)\(52\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(80\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + 9564351425209104944 T + \)\(23\!\cdots\!98\)\( T^{2} + \)\(10\!\cdots\!08\)\( T^{3} + \)\(11\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - \)\(19\!\cdots\!64\)\( T + \)\(11\!\cdots\!26\)\( T^{2} - \)\(14\!\cdots\!64\)\( T^{3} + \)\(53\!\cdots\!01\)\( T^{4} \)
$13$ \( 1 - \)\(11\!\cdots\!08\)\( T + \)\(21\!\cdots\!02\)\( T^{2} - \)\(15\!\cdots\!44\)\( T^{3} + \)\(17\!\cdots\!49\)\( T^{4} \)
$17$ \( 1 - \)\(15\!\cdots\!56\)\( T + \)\(47\!\cdots\!98\)\( T^{2} - \)\(36\!\cdots\!92\)\( T^{3} + \)\(55\!\cdots\!49\)\( T^{4} \)
$19$ \( 1 + \)\(38\!\cdots\!00\)\( T + \)\(26\!\cdots\!98\)\( T^{2} + \)\(13\!\cdots\!00\)\( T^{3} + \)\(12\!\cdots\!01\)\( T^{4} \)
$23$ \( 1 + \)\(30\!\cdots\!72\)\( T + \)\(29\!\cdots\!82\)\( T^{2} + \)\(57\!\cdots\!96\)\( T^{3} + \)\(35\!\cdots\!49\)\( T^{4} \)
$29$ \( 1 + \)\(28\!\cdots\!20\)\( T + \)\(73\!\cdots\!98\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(41\!\cdots\!01\)\( T^{4} \)
$31$ \( 1 + \)\(41\!\cdots\!76\)\( T + \)\(23\!\cdots\!46\)\( T^{2} + \)\(54\!\cdots\!76\)\( T^{3} + \)\(16\!\cdots\!01\)\( T^{4} \)
$37$ \( 1 - \)\(50\!\cdots\!16\)\( T + \)\(13\!\cdots\!78\)\( T^{2} - \)\(18\!\cdots\!12\)\( T^{3} + \)\(13\!\cdots\!49\)\( T^{4} \)
$41$ \( 1 - \)\(41\!\cdots\!84\)\( T + \)\(47\!\cdots\!66\)\( T^{2} - \)\(15\!\cdots\!84\)\( T^{3} + \)\(14\!\cdots\!01\)\( T^{4} \)
$43$ \( 1 + \)\(35\!\cdots\!52\)\( T + \)\(67\!\cdots\!62\)\( T^{2} + \)\(11\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(12\!\cdots\!24\)\( T + \)\(17\!\cdots\!58\)\( T^{2} + \)\(21\!\cdots\!68\)\( T^{3} + \)\(30\!\cdots\!49\)\( T^{4} \)
$53$ \( 1 - \)\(14\!\cdots\!68\)\( T + \)\(12\!\cdots\!42\)\( T^{2} - \)\(55\!\cdots\!24\)\( T^{3} + \)\(15\!\cdots\!49\)\( T^{4} \)
$59$ \( 1 + \)\(14\!\cdots\!40\)\( T + \)\(14\!\cdots\!98\)\( T^{2} + \)\(70\!\cdots\!60\)\( T^{3} + \)\(23\!\cdots\!01\)\( T^{4} \)
$61$ \( 1 + \)\(21\!\cdots\!56\)\( T + \)\(37\!\cdots\!86\)\( T^{2} + \)\(47\!\cdots\!56\)\( T^{3} + \)\(47\!\cdots\!01\)\( T^{4} \)
$67$ \( 1 + \)\(96\!\cdots\!84\)\( T + \)\(28\!\cdots\!78\)\( T^{2} + \)\(14\!\cdots\!88\)\( T^{3} + \)\(22\!\cdots\!49\)\( T^{4} \)
$71$ \( 1 - \)\(44\!\cdots\!44\)\( T + \)\(35\!\cdots\!86\)\( T^{2} - \)\(89\!\cdots\!44\)\( T^{3} + \)\(41\!\cdots\!01\)\( T^{4} \)
$73$ \( 1 + \)\(34\!\cdots\!52\)\( T + \)\(12\!\cdots\!62\)\( T^{2} + \)\(24\!\cdots\!36\)\( T^{3} + \)\(50\!\cdots\!49\)\( T^{4} \)
$79$ \( 1 + \)\(62\!\cdots\!80\)\( T - \)\(27\!\cdots\!02\)\( T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(61\!\cdots\!01\)\( T^{4} \)
$83$ \( 1 - \)\(96\!\cdots\!48\)\( T + \)\(40\!\cdots\!62\)\( T^{2} - \)\(21\!\cdots\!64\)\( T^{3} + \)\(52\!\cdots\!49\)\( T^{4} \)
$89$ \( 1 + \)\(12\!\cdots\!20\)\( T + \)\(14\!\cdots\!98\)\( T^{2} + \)\(66\!\cdots\!80\)\( T^{3} + \)\(27\!\cdots\!01\)\( T^{4} \)
$97$ \( 1 - \)\(47\!\cdots\!36\)\( T + \)\(47\!\cdots\!38\)\( T^{2} - \)\(12\!\cdots\!52\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \)
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