Properties

Label 25.36.a.a
Level $25$
Weight $36$
Character orbit 25.a
Self dual yes
Analytic conductor $193.988$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 36 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(193.987826584\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - 12422194 x - 2645665785\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -46552 - \beta_{1} ) q^{2} + ( 34958436 - \beta_{1} + \beta_{2} ) q^{3} + ( 11613754048 + 194112 \beta_{1} + 72 \beta_{2} ) q^{4} + ( -1595510188128 - 197319012 \beta_{1} + 54528 \beta_{2} ) q^{6} + ( -292807383115352 - 723239370 \beta_{1} + 852426 \beta_{2} ) q^{7} + ( -7445336641779200 - 17597768192 \beta_{1} - 10055232 \beta_{2} ) q^{8} + ( 50030659625414517 - 228209975496 \beta_{1} - 92389176 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(-46552 - \beta_{1}) q^{2} +(34958436 - \beta_{1} + \beta_{2}) q^{3} +(11613754048 + 194112 \beta_{1} + 72 \beta_{2}) q^{4} +(-1595510188128 - 197319012 \beta_{1} + 54528 \beta_{2}) q^{6} +(-292807383115352 - 723239370 \beta_{1} + 852426 \beta_{2}) q^{7} +(-7445336641779200 - 17597768192 \beta_{1} - 10055232 \beta_{2}) q^{8} +(50030659625414517 - 228209975496 \beta_{1} - 92389176 \beta_{2}) q^{9} +(-385981809516662348 + 583640593195 \beta_{1} - 3858182955 \beta_{2}) q^{11} +(7516297175592162048 + 21885019697408 \beta_{1} - 17183392736 \beta_{2}) q^{12} +(20713203516999550186 - 54628153929972 \beta_{1} + 55424393460 \beta_{2}) q^{13} +(45303114418357146176 + 261002424214616 \beta_{1} + 98492944896 \beta_{2}) q^{14} +(\)\(71\!\cdots\!16\)\( + 5006482791469056 \beta_{1} - 1754429566464 \beta_{2}) q^{16} +(\)\(13\!\cdots\!98\)\( + 2825301075479112 \beta_{1} - 6543407393352 \beta_{2}) q^{17} +(\)\(76\!\cdots\!36\)\( - 1342002862888821 \beta_{1} + 11399973267456 \beta_{2}) q^{18} +(-\)\(10\!\cdots\!80\)\( - 84000151153806507 \beta_{1} - 9872745919317 \beta_{2}) q^{19} +(\)\(74\!\cdots\!92\)\( - 336783816338068624 \beta_{1} - 328342926322544 \beta_{2}) q^{21} +(-\)\(75\!\cdots\!04\)\( + 926845923948830028 \beta_{1} - 252123333707520 \beta_{2}) q^{22} +(\)\(17\!\cdots\!36\)\( - 2541836449593511358 \beta_{1} - 1160877406548546 \beta_{2}) q^{23} +(-\)\(12\!\cdots\!40\)\( - 1173379931106134016 \beta_{1} - 4385028066775296 \beta_{2}) q^{24} +(\)\(14\!\cdots\!12\)\( - 21659187993208072426 \beta_{1} + 6951417853215744 \beta_{2}) q^{26} +(-\)\(91\!\cdots\!00\)\( - 23978122270590027258 \beta_{1} + 34841458485708282 \beta_{2}) q^{27} +(-\)\(34\!\cdots\!36\)\( - 74972221489028408832 \beta_{1} - 42717777074276544 \beta_{2}) q^{28} +(-\)\(12\!\cdots\!70\)\( - \)\(16\!\cdots\!08\)\( \beta_{1} + 30897215823693852 \beta_{2}) q^{29} +(\)\(34\!\cdots\!52\)\( + \)\(18\!\cdots\!60\)\( \beta_{1} + 125813443508636760 \beta_{2}) q^{31} +(\)\(30\!\cdots\!48\)\( - \)\(56\!\cdots\!16\)\( \beta_{1} - 110510836707594240 \beta_{2}) q^{32} +(-\)\(39\!\cdots\!28\)\( + \)\(99\!\cdots\!08\)\( \beta_{1} + 46744170434631912 \beta_{2}) q^{33} +(-\)\(18\!\cdots\!04\)\( - \)\(66\!\cdots\!46\)\( \beta_{1} - 559749470446872576 \beta_{2}) q^{34} +(-\)\(20\!\cdots\!84\)\( - \)\(14\!\cdots\!64\)\( \beta_{1} + 3891889065775683816 \beta_{2}) q^{36} +(-\)\(82\!\cdots\!02\)\( - \)\(15\!\cdots\!64\)\( \beta_{1} + 5309319421076092452 \beta_{2}) q^{37} +(\)\(41\!\cdots\!00\)\( + \)\(24\!\cdots\!12\)\( \beta_{1} + 5510380631291741952 \beta_{2}) q^{38} +(\)\(62\!\cdots\!04\)\( - \)\(23\!\cdots\!54\)\( \beta_{1} + 19171250864870765626 \beta_{2}) q^{39} +(\)\(78\!\cdots\!02\)\( - \)\(39\!\cdots\!40\)\( \beta_{1} - 24089178969404126640 \beta_{2}) q^{41} +(\)\(11\!\cdots\!96\)\( + \)\(29\!\cdots\!32\)\( \beta_{1} + 6368192380520484864 \beta_{2}) q^{42} +(\)\(15\!\cdots\!36\)\( - \)\(17\!\cdots\!35\)\( \beta_{1} - 66376501078524728013 \beta_{2}) q^{43} +(-\)\(26\!\cdots\!04\)\( - \)\(10\!\cdots\!16\)\( \beta_{1} + 52103622124984646304 \beta_{2}) q^{44} +(\)\(10\!\cdots\!52\)\( + \)\(39\!\cdots\!00\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2}) q^{46} +(-\)\(54\!\cdots\!52\)\( - \)\(86\!\cdots\!72\)\( \beta_{1} - \)\(51\!\cdots\!16\)\( \beta_{2}) q^{47} +(-\)\(14\!\cdots\!64\)\( + \)\(13\!\cdots\!12\)\( \beta_{1} + \)\(43\!\cdots\!24\)\( \beta_{2}) q^{48} +(-\)\(19\!\cdots\!07\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} - \)\(45\!\cdots\!20\)\( \beta_{2}) q^{49} +(-\)\(60\!\cdots\!68\)\( + \)\(20\!\cdots\!62\)\( \beta_{1} + \)\(18\!\cdots\!22\)\( \beta_{2}) q^{51} +(\)\(17\!\cdots\!48\)\( + \)\(25\!\cdots\!60\)\( \beta_{1} + 33640287635007496656 \beta_{2}) q^{52} +(\)\(55\!\cdots\!86\)\( + \)\(47\!\cdots\!24\)\( \beta_{1} + \)\(46\!\cdots\!76\)\( \beta_{2}) q^{53} +(\)\(14\!\cdots\!20\)\( + \)\(70\!\cdots\!28\)\( \beta_{1} + \)\(36\!\cdots\!68\)\( \beta_{2}) q^{54} +(\)\(18\!\cdots\!80\)\( + \)\(12\!\cdots\!52\)\( \beta_{1} - \)\(31\!\cdots\!88\)\( \beta_{2}) q^{56} +(-\)\(13\!\cdots\!00\)\( - \)\(14\!\cdots\!36\)\( \beta_{1} - \)\(10\!\cdots\!56\)\( \beta_{2}) q^{57} +(\)\(79\!\cdots\!00\)\( + \)\(32\!\cdots\!78\)\( \beta_{1} + \)\(13\!\cdots\!88\)\( \beta_{2}) q^{58} +(\)\(14\!\cdots\!60\)\( + \)\(83\!\cdots\!79\)\( \beta_{1} + \)\(13\!\cdots\!49\)\( \beta_{2}) q^{59} +(\)\(78\!\cdots\!02\)\( + \)\(49\!\cdots\!00\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{61} +(-\)\(98\!\cdots\!04\)\( - \)\(82\!\cdots\!12\)\( \beta_{1} - \)\(66\!\cdots\!60\)\( \beta_{2}) q^{62} +(-\)\(15\!\cdots\!64\)\( + \)\(44\!\cdots\!86\)\( \beta_{1} + \)\(10\!\cdots\!78\)\( \beta_{2}) q^{63} +(\)\(31\!\cdots\!28\)\( - \)\(73\!\cdots\!56\)\( \beta_{1} + \)\(95\!\cdots\!64\)\( \beta_{2}) q^{64} +(-\)\(25\!\cdots\!56\)\( + \)\(24\!\cdots\!16\)\( \beta_{1} - \)\(69\!\cdots\!04\)\( \beta_{2}) q^{66} +(\)\(62\!\cdots\!48\)\( - \)\(32\!\cdots\!53\)\( \beta_{1} + \)\(10\!\cdots\!33\)\( \beta_{2}) q^{67} +(-\)\(73\!\cdots\!36\)\( + \)\(27\!\cdots\!84\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2}) q^{68} +(-\)\(10\!\cdots\!16\)\( - \)\(23\!\cdots\!32\)\( \beta_{1} + \)\(57\!\cdots\!08\)\( \beta_{2}) q^{69} +(\)\(11\!\cdots\!52\)\( - \)\(58\!\cdots\!50\)\( \beta_{1} - \)\(52\!\cdots\!50\)\( \beta_{2}) q^{71} +(-\)\(10\!\cdots\!00\)\( + \)\(16\!\cdots\!76\)\( \beta_{1} - \)\(73\!\cdots\!04\)\( \beta_{2}) q^{72} +(\)\(95\!\cdots\!86\)\( - \)\(17\!\cdots\!28\)\( \beta_{1} + \)\(75\!\cdots\!84\)\( \beta_{2}) q^{73} +(\)\(70\!\cdots\!36\)\( + \)\(21\!\cdots\!70\)\( \beta_{1} + \)\(13\!\cdots\!20\)\( \beta_{2}) q^{74} +(-\)\(90\!\cdots\!40\)\( - \)\(58\!\cdots\!16\)\( \beta_{1} - \)\(11\!\cdots\!96\)\( \beta_{2}) q^{76} +(-\)\(23\!\cdots\!04\)\( + \)\(13\!\cdots\!40\)\( \beta_{1} + \)\(97\!\cdots\!32\)\( \beta_{2}) q^{77} +(\)\(73\!\cdots\!72\)\( - \)\(58\!\cdots\!00\)\( \beta_{1} + \)\(27\!\cdots\!44\)\( \beta_{2}) q^{78} +(-\)\(14\!\cdots\!20\)\( - \)\(26\!\cdots\!48\)\( \beta_{1} + \)\(62\!\cdots\!12\)\( \beta_{2}) q^{79} +(\)\(62\!\cdots\!21\)\( - \)\(12\!\cdots\!12\)\( \beta_{1} - \)\(65\!\cdots\!72\)\( \beta_{2}) q^{81} +(\)\(13\!\cdots\!96\)\( + \)\(18\!\cdots\!38\)\( \beta_{1} + \)\(15\!\cdots\!40\)\( \beta_{2}) q^{82} +(-\)\(49\!\cdots\!64\)\( + \)\(42\!\cdots\!51\)\( \beta_{1} + \)\(21\!\cdots\!93\)\( \beta_{2}) q^{83} +(-\)\(43\!\cdots\!84\)\( - \)\(50\!\cdots\!88\)\( \beta_{1} + \)\(95\!\cdots\!72\)\( \beta_{2}) q^{84} +(\)\(49\!\cdots\!32\)\( - \)\(24\!\cdots\!68\)\( \beta_{1} - \)\(23\!\cdots\!08\)\( \beta_{2}) q^{86} +(\)\(26\!\cdots\!00\)\( - \)\(40\!\cdots\!34\)\( \beta_{1} + \)\(13\!\cdots\!86\)\( \beta_{2}) q^{87} +(\)\(62\!\cdots\!00\)\( + \)\(26\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!36\)\( \beta_{2}) q^{88} +(\)\(10\!\cdots\!90\)\( - \)\(67\!\cdots\!44\)\( \beta_{1} + \)\(91\!\cdots\!36\)\( \beta_{2}) q^{89} +(\)\(33\!\cdots\!32\)\( - \)\(19\!\cdots\!48\)\( \beta_{1} + \)\(53\!\cdots\!12\)\( \beta_{2}) q^{91} +(-\)\(27\!\cdots\!52\)\( - \)\(93\!\cdots\!08\)\( \beta_{1} + \)\(18\!\cdots\!28\)\( \beta_{2}) q^{92} +(\)\(13\!\cdots\!72\)\( + \)\(81\!\cdots\!28\)\( \beta_{1} - \)\(35\!\cdots\!68\)\( \beta_{2}) q^{93} +(\)\(63\!\cdots\!56\)\( + \)\(15\!\cdots\!48\)\( \beta_{1} - \)\(21\!\cdots\!12\)\( \beta_{2}) q^{94} +(-\)\(10\!\cdots\!28\)\( - \)\(86\!\cdots\!32\)\( \beta_{1} + \)\(74\!\cdots\!08\)\( \beta_{2}) q^{96} +(\)\(35\!\cdots\!98\)\( - \)\(99\!\cdots\!92\)\( \beta_{1} - \)\(89\!\cdots\!36\)\( \beta_{2}) q^{97} +(\)\(44\!\cdots\!64\)\( + \)\(25\!\cdots\!27\)\( \beta_{1} - \)\(32\!\cdots\!80\)\( \beta_{2}) q^{98} +(\)\(10\!\cdots\!84\)\( + \)\(15\!\cdots\!23\)\( \beta_{1} - \)\(30\!\cdots\!87\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 139656q^{2} + 104875308q^{3} + 34841262144q^{4} - 4786530564384q^{6} - 878422149346056q^{7} - 22336009925337600q^{8} + 150091978876243551q^{9} + O(q^{10}) \) \( 3q - 139656q^{2} + 104875308q^{3} + 34841262144q^{4} - 4786530564384q^{6} - 878422149346056q^{7} - 22336009925337600q^{8} + 150091978876243551q^{9} - 1157945428549987044q^{11} + 22548891526776486144q^{12} + 62139610550998650558q^{13} + \)\(13\!\cdots\!28\)\(q^{14} + \)\(21\!\cdots\!48\)\(q^{16} + \)\(39\!\cdots\!94\)\(q^{17} + \)\(23\!\cdots\!08\)\(q^{18} - \)\(32\!\cdots\!40\)\(q^{19} + \)\(22\!\cdots\!76\)\(q^{21} - \)\(22\!\cdots\!12\)\(q^{22} + \)\(51\!\cdots\!08\)\(q^{23} - \)\(37\!\cdots\!20\)\(q^{24} + \)\(42\!\cdots\!36\)\(q^{26} - \)\(27\!\cdots\!00\)\(q^{27} - \)\(10\!\cdots\!08\)\(q^{28} - \)\(38\!\cdots\!10\)\(q^{29} + \)\(10\!\cdots\!56\)\(q^{31} + \)\(92\!\cdots\!44\)\(q^{32} - \)\(11\!\cdots\!84\)\(q^{33} - \)\(55\!\cdots\!12\)\(q^{34} - \)\(60\!\cdots\!52\)\(q^{36} - \)\(24\!\cdots\!06\)\(q^{37} + \)\(12\!\cdots\!00\)\(q^{38} + \)\(18\!\cdots\!12\)\(q^{39} + \)\(23\!\cdots\!06\)\(q^{41} + \)\(33\!\cdots\!88\)\(q^{42} + \)\(47\!\cdots\!08\)\(q^{43} - \)\(80\!\cdots\!12\)\(q^{44} + \)\(31\!\cdots\!56\)\(q^{46} - \)\(16\!\cdots\!56\)\(q^{47} - \)\(44\!\cdots\!92\)\(q^{48} - \)\(59\!\cdots\!21\)\(q^{49} - \)\(18\!\cdots\!04\)\(q^{51} + \)\(51\!\cdots\!44\)\(q^{52} + \)\(16\!\cdots\!58\)\(q^{53} + \)\(44\!\cdots\!60\)\(q^{54} + \)\(56\!\cdots\!40\)\(q^{56} - \)\(40\!\cdots\!00\)\(q^{57} + \)\(23\!\cdots\!00\)\(q^{58} + \)\(43\!\cdots\!80\)\(q^{59} + \)\(23\!\cdots\!06\)\(q^{61} - \)\(29\!\cdots\!12\)\(q^{62} - \)\(45\!\cdots\!92\)\(q^{63} + \)\(93\!\cdots\!84\)\(q^{64} - \)\(75\!\cdots\!68\)\(q^{66} + \)\(18\!\cdots\!44\)\(q^{67} - \)\(21\!\cdots\!08\)\(q^{68} - \)\(32\!\cdots\!48\)\(q^{69} + \)\(34\!\cdots\!56\)\(q^{71} - \)\(31\!\cdots\!00\)\(q^{72} + \)\(28\!\cdots\!58\)\(q^{73} + \)\(21\!\cdots\!08\)\(q^{74} - \)\(27\!\cdots\!20\)\(q^{76} - \)\(69\!\cdots\!12\)\(q^{77} + \)\(22\!\cdots\!16\)\(q^{78} - \)\(42\!\cdots\!60\)\(q^{79} + \)\(18\!\cdots\!63\)\(q^{81} + \)\(40\!\cdots\!88\)\(q^{82} - \)\(14\!\cdots\!92\)\(q^{83} - \)\(13\!\cdots\!52\)\(q^{84} + \)\(14\!\cdots\!96\)\(q^{86} + \)\(78\!\cdots\!00\)\(q^{87} + \)\(18\!\cdots\!00\)\(q^{88} + \)\(30\!\cdots\!70\)\(q^{89} + \)\(10\!\cdots\!96\)\(q^{91} - \)\(83\!\cdots\!56\)\(q^{92} + \)\(40\!\cdots\!16\)\(q^{93} + \)\(19\!\cdots\!68\)\(q^{94} - \)\(32\!\cdots\!84\)\(q^{96} + \)\(10\!\cdots\!94\)\(q^{97} + \)\(13\!\cdots\!92\)\(q^{98} + \)\(30\!\cdots\!52\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 12422194 x - 2645665785\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 24 \nu^{2} + 44712 \nu - 198755104 \)\()/979\)
\(\beta_{2}\)\(=\)\((\)\( -39144 \nu^{2} + 84967848 \nu + 324169574624 \)\()/979\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 1631 \beta_{1}\)\()/161280\)
\(\nu^{2}\)\(=\)\((\)\(-621 \beta_{2} + 1180109 \beta_{1} + 445211432960\)\()/53760\)

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
3626.53
−3412.77
−213.765
−331573. 1.54691e8 7.55810e10 0 −5.12913e13 −3.96640e14 −1.36679e16 −2.61023e16 0
1.2 26808.0 −3.95729e8 −3.36411e10 0 −1.06087e13 −6.06942e14 −1.82297e15 1.06570e17 0
1.3 165109. 3.45913e8 −7.09870e9 0 5.71135e13 1.25160e14 −6.84517e15 6.96245e16 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 25.36.a.a 3
5.b even 2 1 1.36.a.a 3
5.c odd 4 2 25.36.b.a 6
15.d odd 2 1 9.36.a.b 3
20.d odd 2 1 16.36.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.36.a.a 3 5.b even 2 1
9.36.a.b 3 15.d odd 2 1
16.36.a.d 3 20.d odd 2 1
25.36.a.a 3 1.a even 1 1 trivial
25.36.b.a 6 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 139656 T_{2}^{2} - 59208339456 T_{2} + \)\(14\!\cdots\!64\)\( \) acting on \(S_{36}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1467625047588864 - 59208339456 T + 139656 T^{2} + T^{3} \)
$3$ \( \)\(21\!\cdots\!32\)\( - 144593891972573904 T - 104875308 T^{2} + T^{3} \)
$5$ \( T^{3} \)
$7$ \( -\)\(30\!\cdots\!16\)\( + \)\(11\!\cdots\!64\)\( T + 878422149346056 T^{2} + T^{3} \)
$11$ \( -\)\(14\!\cdots\!08\)\( - \)\(17\!\cdots\!88\)\( T + 1157945428549987044 T^{2} + T^{3} \)
$13$ \( -\)\(49\!\cdots\!48\)\( + \)\(63\!\cdots\!56\)\( T - 62139610550998650558 T^{2} + T^{3} \)
$17$ \( \)\(68\!\cdots\!24\)\( - \)\(17\!\cdots\!96\)\( T - \)\(39\!\cdots\!94\)\( T^{2} + T^{3} \)
$19$ \( -\)\(49\!\cdots\!00\)\( - \)\(13\!\cdots\!00\)\( T + \)\(32\!\cdots\!40\)\( T^{2} + T^{3} \)
$23$ \( \)\(20\!\cdots\!72\)\( - \)\(53\!\cdots\!84\)\( T - \)\(51\!\cdots\!08\)\( T^{2} + T^{3} \)
$29$ \( -\)\(25\!\cdots\!00\)\( - \)\(14\!\cdots\!00\)\( T + \)\(38\!\cdots\!10\)\( T^{2} + T^{3} \)
$31$ \( \)\(11\!\cdots\!92\)\( - \)\(10\!\cdots\!88\)\( T - \)\(10\!\cdots\!56\)\( T^{2} + T^{3} \)
$37$ \( -\)\(36\!\cdots\!96\)\( - \)\(17\!\cdots\!16\)\( T + \)\(24\!\cdots\!06\)\( T^{2} + T^{3} \)
$41$ \( \)\(11\!\cdots\!92\)\( - \)\(27\!\cdots\!88\)\( T - \)\(23\!\cdots\!06\)\( T^{2} + T^{3} \)
$43$ \( \)\(12\!\cdots\!12\)\( + \)\(78\!\cdots\!36\)\( T - \)\(47\!\cdots\!08\)\( T^{2} + T^{3} \)
$47$ \( -\)\(47\!\cdots\!56\)\( - \)\(30\!\cdots\!76\)\( T + \)\(16\!\cdots\!56\)\( T^{2} + T^{3} \)
$53$ \( \)\(44\!\cdots\!32\)\( - \)\(38\!\cdots\!04\)\( T - \)\(16\!\cdots\!58\)\( T^{2} + T^{3} \)
$59$ \( \)\(10\!\cdots\!00\)\( - \)\(25\!\cdots\!00\)\( T - \)\(43\!\cdots\!80\)\( T^{2} + T^{3} \)
$61$ \( \)\(57\!\cdots\!92\)\( + \)\(98\!\cdots\!12\)\( T - \)\(23\!\cdots\!06\)\( T^{2} + T^{3} \)
$67$ \( \)\(10\!\cdots\!24\)\( + \)\(34\!\cdots\!04\)\( T - \)\(18\!\cdots\!44\)\( T^{2} + T^{3} \)
$71$ \( \)\(33\!\cdots\!92\)\( - \)\(22\!\cdots\!88\)\( T - \)\(34\!\cdots\!56\)\( T^{2} + T^{3} \)
$73$ \( -\)\(22\!\cdots\!28\)\( - \)\(26\!\cdots\!84\)\( T - \)\(28\!\cdots\!58\)\( T^{2} + T^{3} \)
$79$ \( -\)\(88\!\cdots\!00\)\( - \)\(61\!\cdots\!00\)\( T + \)\(42\!\cdots\!60\)\( T^{2} + T^{3} \)
$83$ \( \)\(11\!\cdots\!92\)\( + \)\(71\!\cdots\!76\)\( T + \)\(14\!\cdots\!92\)\( T^{2} + T^{3} \)
$89$ \( -\)\(91\!\cdots\!00\)\( + \)\(29\!\cdots\!00\)\( T - \)\(30\!\cdots\!70\)\( T^{2} + T^{3} \)
$97$ \( \)\(83\!\cdots\!44\)\( + \)\(19\!\cdots\!24\)\( T - \)\(10\!\cdots\!94\)\( T^{2} + T^{3} \)
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