Properties

Label 2.50.a.a
Level $2$
Weight $50$
Character orbit 2.a
Self dual yes
Analytic conductor $30.413$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.4132410198\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 22129540960032\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3}\cdot 5^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 43200\sqrt{88518163840129}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -16777216 q^{2} + ( 140525537796 - \beta ) q^{3} + 281474976710656 q^{4} + ( 41587265591271750 - 349020 \beta ) q^{5} + ( -2357627301119655936 + 16777216 \beta ) q^{6} + ( -216194632642999109368 + 1176353838 \beta ) q^{7} -4722366482869645213696 q^{8} + ( -54355764372760160092467 - 281051075592 \beta ) q^{9} +O(q^{10})\) \( q -16777216 q^{2} +(140525537796 - \beta) q^{3} +281474976710656 q^{4} +(41587265591271750 - 349020 \beta) q^{5} +(-2357627301119655936 + 16777216 \beta) q^{6} +(-\)\(21\!\cdots\!68\)\( + 1176353838 \beta) q^{7} -\)\(47\!\cdots\!96\)\( q^{8} +(-\)\(54\!\cdots\!67\)\( - 281051075592 \beta) q^{9} +(-\)\(69\!\cdots\!00\)\( + 5855583928320 \beta) q^{10} +(-\)\(12\!\cdots\!68\)\( + 91137694171293 \beta) q^{11} +(\)\(39\!\cdots\!76\)\( - 281474976710656 \beta) q^{12} +(-\)\(83\!\cdots\!14\)\( + 4871108754480708 \beta) q^{13} +(\)\(36\!\cdots\!88\)\( - 19735942432555008 \beta) q^{14} +(\)\(63\!\cdots\!00\)\( - 90633488792831670 \beta) q^{15} +\)\(79\!\cdots\!36\)\( q^{16} +(\)\(62\!\cdots\!82\)\( + 1739956737232606392 \beta) q^{17} +(\)\(91\!\cdots\!72\)\( + 4715254602239311872 \beta) q^{18} +(\)\(15\!\cdots\!60\)\( - 15619643210757982437 \beta) q^{19} +(\)\(11\!\cdots\!00\)\( - 98240396371553157120 \beta) q^{20} +(-\)\(22\!\cdots\!28\)\( + \)\(38\!\cdots\!16\)\( \beta) q^{21} +(\)\(21\!\cdots\!88\)\( - \)\(15\!\cdots\!88\)\( \beta) q^{22} +(-\)\(14\!\cdots\!64\)\( + \)\(49\!\cdots\!14\)\( \beta) q^{23} +(-\)\(66\!\cdots\!16\)\( + \)\(47\!\cdots\!96\)\( \beta) q^{24} +(\)\(40\!\cdots\!75\)\( - \)\(29\!\cdots\!00\)\( \beta) q^{25} +(\)\(14\!\cdots\!24\)\( - \)\(81\!\cdots\!28\)\( \beta) q^{26} +(\)\(51\!\cdots\!00\)\( + \)\(25\!\cdots\!18\)\( \beta) q^{27} +(-\)\(60\!\cdots\!08\)\( + \)\(33\!\cdots\!28\)\( \beta) q^{28} +(-\)\(59\!\cdots\!90\)\( - \)\(12\!\cdots\!04\)\( \beta) q^{29} +(-\)\(10\!\cdots\!00\)\( + \)\(15\!\cdots\!20\)\( \beta) q^{30} +(-\)\(35\!\cdots\!48\)\( - \)\(35\!\cdots\!44\)\( \beta) q^{31} -\)\(13\!\cdots\!76\)\( q^{32} +(-\)\(15\!\cdots\!28\)\( + \)\(14\!\cdots\!96\)\( \beta) q^{33} +(-\)\(10\!\cdots\!12\)\( - \)\(29\!\cdots\!72\)\( \beta) q^{34} +(-\)\(76\!\cdots\!00\)\( + \)\(12\!\cdots\!60\)\( \beta) q^{35} +(-\)\(15\!\cdots\!52\)\( - \)\(79\!\cdots\!52\)\( \beta) q^{36} +(-\)\(22\!\cdots\!58\)\( - \)\(60\!\cdots\!08\)\( \beta) q^{37} +(-\)\(26\!\cdots\!60\)\( + \)\(26\!\cdots\!92\)\( \beta) q^{38} +(-\)\(92\!\cdots\!44\)\( + \)\(15\!\cdots\!82\)\( \beta) q^{39} +(-\)\(19\!\cdots\!00\)\( + \)\(16\!\cdots\!20\)\( \beta) q^{40} +(\)\(34\!\cdots\!82\)\( - \)\(38\!\cdots\!76\)\( \beta) q^{41} +(\)\(37\!\cdots\!48\)\( - \)\(64\!\cdots\!56\)\( \beta) q^{42} +(\)\(11\!\cdots\!16\)\( - \)\(24\!\cdots\!63\)\( \beta) q^{43} +(-\)\(35\!\cdots\!08\)\( + \)\(25\!\cdots\!08\)\( \beta) q^{44} +(\)\(13\!\cdots\!50\)\( + \)\(72\!\cdots\!40\)\( \beta) q^{45} +(\)\(24\!\cdots\!24\)\( - \)\(82\!\cdots\!24\)\( \beta) q^{46} +(-\)\(59\!\cdots\!48\)\( + \)\(19\!\cdots\!48\)\( \beta) q^{47} +(\)\(11\!\cdots\!56\)\( - \)\(79\!\cdots\!36\)\( \beta) q^{48} +(\)\(18\!\cdots\!17\)\( - \)\(50\!\cdots\!68\)\( \beta) q^{49} +(-\)\(68\!\cdots\!00\)\( + \)\(48\!\cdots\!00\)\( \beta) q^{50} +(-\)\(20\!\cdots\!28\)\( - \)\(37\!\cdots\!50\)\( \beta) q^{51} +(-\)\(23\!\cdots\!84\)\( + \)\(13\!\cdots\!48\)\( \beta) q^{52} +(-\)\(19\!\cdots\!34\)\( + \)\(10\!\cdots\!28\)\( \beta) q^{53} +(-\)\(86\!\cdots\!00\)\( - \)\(42\!\cdots\!88\)\( \beta) q^{54} +(-\)\(53\!\cdots\!00\)\( + \)\(42\!\cdots\!10\)\( \beta) q^{55} +(\)\(10\!\cdots\!28\)\( - \)\(55\!\cdots\!48\)\( \beta) q^{56} +(\)\(48\!\cdots\!60\)\( - \)\(18\!\cdots\!12\)\( \beta) q^{57} +(\)\(99\!\cdots\!40\)\( + \)\(21\!\cdots\!64\)\( \beta) q^{58} +(\)\(10\!\cdots\!20\)\( + \)\(71\!\cdots\!77\)\( \beta) q^{59} +(\)\(17\!\cdots\!00\)\( - \)\(25\!\cdots\!20\)\( \beta) q^{60} +(\)\(28\!\cdots\!02\)\( - \)\(15\!\cdots\!48\)\( \beta) q^{61} +(\)\(59\!\cdots\!68\)\( + \)\(59\!\cdots\!04\)\( \beta) q^{62} +(-\)\(42\!\cdots\!44\)\( - \)\(31\!\cdots\!90\)\( \beta) q^{63} +\)\(22\!\cdots\!16\)\( q^{64} +(-\)\(31\!\cdots\!00\)\( + \)\(49\!\cdots\!80\)\( \beta) q^{65} +(\)\(25\!\cdots\!48\)\( - \)\(23\!\cdots\!36\)\( \beta) q^{66} +(-\)\(56\!\cdots\!08\)\( - \)\(89\!\cdots\!41\)\( \beta) q^{67} +(\)\(17\!\cdots\!92\)\( + \)\(48\!\cdots\!52\)\( \beta) q^{68} +(-\)\(10\!\cdots\!44\)\( + \)\(21\!\cdots\!08\)\( \beta) q^{69} +(\)\(12\!\cdots\!00\)\( - \)\(20\!\cdots\!60\)\( \beta) q^{70} +(-\)\(12\!\cdots\!08\)\( - \)\(42\!\cdots\!78\)\( \beta) q^{71} +(\)\(25\!\cdots\!32\)\( + \)\(13\!\cdots\!32\)\( \beta) q^{72} +(-\)\(48\!\cdots\!94\)\( + \)\(61\!\cdots\!28\)\( \beta) q^{73} +(\)\(37\!\cdots\!28\)\( + \)\(10\!\cdots\!28\)\( \beta) q^{74} +(\)\(53\!\cdots\!00\)\( - \)\(81\!\cdots\!75\)\( \beta) q^{75} +(\)\(44\!\cdots\!60\)\( - \)\(43\!\cdots\!72\)\( \beta) q^{76} +(\)\(17\!\cdots\!24\)\( - \)\(21\!\cdots\!08\)\( \beta) q^{77} +(\)\(15\!\cdots\!04\)\( - \)\(25\!\cdots\!12\)\( \beta) q^{78} +(-\)\(84\!\cdots\!80\)\( + \)\(65\!\cdots\!48\)\( \beta) q^{79} +(\)\(32\!\cdots\!00\)\( - \)\(27\!\cdots\!20\)\( \beta) q^{80} +(-\)\(28\!\cdots\!39\)\( + \)\(97\!\cdots\!64\)\( \beta) q^{81} +(-\)\(58\!\cdots\!12\)\( + \)\(63\!\cdots\!16\)\( \beta) q^{82} +(-\)\(12\!\cdots\!64\)\( - \)\(30\!\cdots\!25\)\( \beta) q^{83} +(-\)\(63\!\cdots\!68\)\( + \)\(10\!\cdots\!96\)\( \beta) q^{84} +(-\)\(74\!\cdots\!00\)\( - \)\(14\!\cdots\!40\)\( \beta) q^{85} +(-\)\(19\!\cdots\!56\)\( + \)\(40\!\cdots\!08\)\( \beta) q^{86} +(\)\(12\!\cdots\!60\)\( + \)\(41\!\cdots\!06\)\( \beta) q^{87} +(\)\(59\!\cdots\!28\)\( - \)\(43\!\cdots\!28\)\( \beta) q^{88} +(\)\(50\!\cdots\!90\)\( + \)\(84\!\cdots\!40\)\( \beta) q^{89} +(-\)\(23\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( \beta) q^{90} +(\)\(11\!\cdots\!52\)\( - \)\(20\!\cdots\!76\)\( \beta) q^{91} +(-\)\(40\!\cdots\!84\)\( + \)\(13\!\cdots\!84\)\( \beta) q^{92} +(\)\(90\!\cdots\!92\)\( + \)\(30\!\cdots\!24\)\( \beta) q^{93} +(\)\(99\!\cdots\!68\)\( - \)\(32\!\cdots\!68\)\( \beta) q^{94} +(\)\(15\!\cdots\!00\)\( - \)\(61\!\cdots\!50\)\( \beta) q^{95} +(-\)\(18\!\cdots\!96\)\( + \)\(13\!\cdots\!76\)\( \beta) q^{96} +(\)\(43\!\cdots\!62\)\( + \)\(97\!\cdots\!40\)\( \beta) q^{97} +(-\)\(30\!\cdots\!72\)\( + \)\(85\!\cdots\!88\)\( \beta) q^{98} +(-\)\(41\!\cdots\!44\)\( - \)\(46\!\cdots\!75\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 33554432q^{2} + 281051075592q^{3} + 562949953421312q^{4} + 83174531182543500q^{5} - 4715254602239311872q^{6} - 432389265285998218736q^{7} - 9444732965739290427392q^{8} - 108711528745520320184934q^{9} + O(q^{10}) \) \( 2q - 33554432q^{2} + 281051075592q^{3} + 562949953421312q^{4} + 83174531182543500q^{5} - 4715254602239311872q^{6} - \)\(43\!\cdots\!36\)\(q^{7} - \)\(94\!\cdots\!92\)\(q^{8} - \)\(10\!\cdots\!34\)\(q^{9} - \)\(13\!\cdots\!00\)\(q^{10} - \)\(25\!\cdots\!36\)\(q^{11} + \)\(79\!\cdots\!52\)\(q^{12} - \)\(16\!\cdots\!28\)\(q^{13} + \)\(72\!\cdots\!76\)\(q^{14} + \)\(12\!\cdots\!00\)\(q^{15} + \)\(15\!\cdots\!72\)\(q^{16} + \)\(12\!\cdots\!64\)\(q^{17} + \)\(18\!\cdots\!44\)\(q^{18} + \)\(31\!\cdots\!20\)\(q^{19} + \)\(23\!\cdots\!00\)\(q^{20} - \)\(44\!\cdots\!56\)\(q^{21} + \)\(42\!\cdots\!76\)\(q^{22} - \)\(28\!\cdots\!28\)\(q^{23} - \)\(13\!\cdots\!32\)\(q^{24} + \)\(81\!\cdots\!50\)\(q^{25} + \)\(28\!\cdots\!48\)\(q^{26} + \)\(10\!\cdots\!00\)\(q^{27} - \)\(12\!\cdots\!16\)\(q^{28} - \)\(11\!\cdots\!80\)\(q^{29} - \)\(21\!\cdots\!00\)\(q^{30} - \)\(71\!\cdots\!96\)\(q^{31} - \)\(26\!\cdots\!52\)\(q^{32} - \)\(30\!\cdots\!56\)\(q^{33} - \)\(20\!\cdots\!24\)\(q^{34} - \)\(15\!\cdots\!00\)\(q^{35} - \)\(30\!\cdots\!04\)\(q^{36} - \)\(45\!\cdots\!16\)\(q^{37} - \)\(53\!\cdots\!20\)\(q^{38} - \)\(18\!\cdots\!88\)\(q^{39} - \)\(39\!\cdots\!00\)\(q^{40} + \)\(69\!\cdots\!64\)\(q^{41} + \)\(75\!\cdots\!96\)\(q^{42} + \)\(23\!\cdots\!32\)\(q^{43} - \)\(70\!\cdots\!16\)\(q^{44} + \)\(27\!\cdots\!00\)\(q^{45} + \)\(48\!\cdots\!48\)\(q^{46} - \)\(11\!\cdots\!96\)\(q^{47} + \)\(22\!\cdots\!12\)\(q^{48} + \)\(36\!\cdots\!34\)\(q^{49} - \)\(13\!\cdots\!00\)\(q^{50} - \)\(40\!\cdots\!56\)\(q^{51} - \)\(47\!\cdots\!68\)\(q^{52} - \)\(39\!\cdots\!68\)\(q^{53} - \)\(17\!\cdots\!00\)\(q^{54} - \)\(10\!\cdots\!00\)\(q^{55} + \)\(20\!\cdots\!56\)\(q^{56} + \)\(96\!\cdots\!20\)\(q^{57} + \)\(19\!\cdots\!80\)\(q^{58} + \)\(21\!\cdots\!40\)\(q^{59} + \)\(35\!\cdots\!00\)\(q^{60} + \)\(57\!\cdots\!04\)\(q^{61} + \)\(11\!\cdots\!36\)\(q^{62} - \)\(85\!\cdots\!88\)\(q^{63} + \)\(44\!\cdots\!32\)\(q^{64} - \)\(63\!\cdots\!00\)\(q^{65} + \)\(51\!\cdots\!96\)\(q^{66} - \)\(11\!\cdots\!16\)\(q^{67} + \)\(34\!\cdots\!84\)\(q^{68} - \)\(20\!\cdots\!88\)\(q^{69} + \)\(25\!\cdots\!00\)\(q^{70} - \)\(25\!\cdots\!16\)\(q^{71} + \)\(51\!\cdots\!64\)\(q^{72} - \)\(97\!\cdots\!88\)\(q^{73} + \)\(75\!\cdots\!56\)\(q^{74} + \)\(10\!\cdots\!00\)\(q^{75} + \)\(89\!\cdots\!20\)\(q^{76} + \)\(35\!\cdots\!48\)\(q^{77} + \)\(30\!\cdots\!08\)\(q^{78} - \)\(16\!\cdots\!60\)\(q^{79} + \)\(65\!\cdots\!00\)\(q^{80} - \)\(56\!\cdots\!78\)\(q^{81} - \)\(11\!\cdots\!24\)\(q^{82} - \)\(24\!\cdots\!28\)\(q^{83} - \)\(12\!\cdots\!36\)\(q^{84} - \)\(14\!\cdots\!00\)\(q^{85} - \)\(38\!\cdots\!12\)\(q^{86} + \)\(24\!\cdots\!20\)\(q^{87} + \)\(11\!\cdots\!56\)\(q^{88} + \)\(10\!\cdots\!80\)\(q^{89} - \)\(46\!\cdots\!00\)\(q^{90} + \)\(22\!\cdots\!04\)\(q^{91} - \)\(81\!\cdots\!68\)\(q^{92} + \)\(18\!\cdots\!84\)\(q^{93} + \)\(19\!\cdots\!36\)\(q^{94} + \)\(31\!\cdots\!00\)\(q^{95} - \)\(37\!\cdots\!92\)\(q^{96} + \)\(86\!\cdots\!24\)\(q^{97} - \)\(61\!\cdots\!44\)\(q^{98} - \)\(83\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
4.70421e6
−4.70420e6
−1.67772e7 −2.65918e11 2.81475e14 −1.00270e17 4.46136e18 2.61926e20 −4.72237e21 −1.68587e23 1.68224e24
1.2 −1.67772e7 5.46969e11 2.81475e14 1.83444e17 −9.17661e18 −6.94316e20 −4.72237e21 5.98756e22 −3.07768e24
\(n\): 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 2.50.a.a 2
4.b odd 2 1 16.50.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.50.a.a 2 1.a even 1 1 trivial
16.50.a.a 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 281051075592 T_{3} - \)145448711312147320422384

'>\(14\!\cdots\!84\)\( \) acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + 16777216 T )^{2} \)
$3$ \( 1 - 281051075592 T + \)\(33\!\cdots\!82\)\( T^{2} - \)\(67\!\cdots\!36\)\( T^{3} + \)\(57\!\cdots\!89\)\( T^{4} \)
$5$ \( 1 - 83174531182543500 T + \)\(17\!\cdots\!50\)\( T^{2} - \)\(14\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 + \)\(43\!\cdots\!36\)\( T + \)\(33\!\cdots\!38\)\( T^{2} + \)\(11\!\cdots\!52\)\( T^{3} + \)\(66\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + \)\(25\!\cdots\!36\)\( T + \)\(76\!\cdots\!06\)\( T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!81\)\( T^{4} \)
$13$ \( 1 + \)\(16\!\cdots\!28\)\( T + \)\(44\!\cdots\!42\)\( T^{2} + \)\(64\!\cdots\!44\)\( T^{3} + \)\(14\!\cdots\!29\)\( T^{4} \)
$17$ \( 1 - \)\(12\!\cdots\!64\)\( T + \)\(38\!\cdots\!18\)\( T^{2} - \)\(24\!\cdots\!08\)\( T^{3} + \)\(38\!\cdots\!09\)\( T^{4} \)
$19$ \( 1 - \)\(31\!\cdots\!20\)\( T + \)\(11\!\cdots\!58\)\( T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!41\)\( T^{4} \)
$23$ \( 1 + \)\(28\!\cdots\!28\)\( T + \)\(86\!\cdots\!22\)\( T^{2} + \)\(15\!\cdots\!64\)\( T^{3} + \)\(28\!\cdots\!69\)\( T^{4} \)
$29$ \( 1 + \)\(11\!\cdots\!80\)\( T + \)\(99\!\cdots\!38\)\( T^{2} + \)\(53\!\cdots\!20\)\( T^{3} + \)\(20\!\cdots\!61\)\( T^{4} \)
$31$ \( 1 + \)\(71\!\cdots\!96\)\( T + \)\(34\!\cdots\!46\)\( T^{2} + \)\(84\!\cdots\!16\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \)
$37$ \( 1 + \)\(45\!\cdots\!16\)\( T + \)\(12\!\cdots\!18\)\( T^{2} + \)\(31\!\cdots\!32\)\( T^{3} + \)\(48\!\cdots\!29\)\( T^{4} \)
$41$ \( 1 - \)\(69\!\cdots\!64\)\( T + \)\(31\!\cdots\!46\)\( T^{2} - \)\(74\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!21\)\( T^{4} \)
$43$ \( 1 - \)\(23\!\cdots\!32\)\( T + \)\(35\!\cdots\!42\)\( T^{2} - \)\(25\!\cdots\!76\)\( T^{3} + \)\(12\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(11\!\cdots\!96\)\( T + \)\(10\!\cdots\!38\)\( T^{2} + \)\(10\!\cdots\!32\)\( T^{3} + \)\(73\!\cdots\!89\)\( T^{4} \)
$53$ \( 1 + \)\(39\!\cdots\!68\)\( T + \)\(99\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!44\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \)
$59$ \( 1 - \)\(21\!\cdots\!40\)\( T + \)\(46\!\cdots\!78\)\( T^{2} - \)\(12\!\cdots\!60\)\( T^{3} + \)\(34\!\cdots\!21\)\( T^{4} \)
$61$ \( 1 - \)\(57\!\cdots\!04\)\( T + \)\(27\!\cdots\!86\)\( T^{2} - \)\(17\!\cdots\!64\)\( T^{3} + \)\(91\!\cdots\!81\)\( T^{4} \)
$67$ \( 1 + \)\(11\!\cdots\!16\)\( T + \)\(79\!\cdots\!58\)\( T^{2} + \)\(34\!\cdots\!52\)\( T^{3} + \)\(90\!\cdots\!09\)\( T^{4} \)
$71$ \( 1 + \)\(25\!\cdots\!16\)\( T + \)\(89\!\cdots\!26\)\( T^{2} + \)\(13\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!61\)\( T^{4} \)
$73$ \( 1 + \)\(97\!\cdots\!88\)\( T + \)\(57\!\cdots\!62\)\( T^{2} + \)\(19\!\cdots\!44\)\( T^{3} + \)\(40\!\cdots\!69\)\( T^{4} \)
$79$ \( 1 + \)\(16\!\cdots\!60\)\( T + \)\(12\!\cdots\!38\)\( T^{2} + \)\(16\!\cdots\!40\)\( T^{3} + \)\(92\!\cdots\!61\)\( T^{4} \)
$83$ \( 1 + \)\(24\!\cdots\!28\)\( T + \)\(64\!\cdots\!02\)\( T^{2} + \)\(26\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!09\)\( T^{4} \)
$89$ \( 1 - \)\(10\!\cdots\!80\)\( T + \)\(80\!\cdots\!18\)\( T^{2} - \)\(33\!\cdots\!20\)\( T^{3} + \)\(10\!\cdots\!81\)\( T^{4} \)
$97$ \( 1 - \)\(86\!\cdots\!24\)\( T + \)\(29\!\cdots\!78\)\( T^{2} - \)\(19\!\cdots\!08\)\( T^{3} + \)\(50\!\cdots\!89\)\( T^{4} \)
show more
show less