Properties

Label 16.48.a.d
Level $16$
Weight $48$
Character orbit 16.a
Self dual yes
Analytic conductor $223.852$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(223.852260248\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -9615373740 - \beta_{1} ) q^{3} + ( -7778670060568050 + 143248 \beta_{1} - 47 \beta_{2} - 3 \beta_{3} ) q^{5} + ( 9792304681472105800 + 288647031 \beta_{1} - 2171627 \beta_{2} + 4180 \beta_{3} ) q^{7} + ( -4267993104339535550043 - 69250029588 \beta_{1} + 301730130 \beta_{2} - 1508274 \beta_{3} ) q^{9} +O(q^{10})\) \( q +(-9615373740 - \beta_{1}) q^{3} +(-7778670060568050 + 143248 \beta_{1} - 47 \beta_{2} - 3 \beta_{3}) q^{5} +(9792304681472105800 + 288647031 \beta_{1} - 2171627 \beta_{2} + 4180 \beta_{3}) q^{7} +(-\)\(42\!\cdots\!43\)\( - 69250029588 \beta_{1} + 301730130 \beta_{2} - 1508274 \beta_{3}) q^{9} +(\)\(47\!\cdots\!28\)\( - 3164814174665 \beta_{1} + 43653145690 \beta_{2} + 217326120 \beta_{3}) q^{11} +(\)\(31\!\cdots\!30\)\( + 428253881669304 \beta_{1} + 636735319961 \beta_{2} - 14610507235 \beta_{3}) q^{13} +(-\)\(31\!\cdots\!00\)\( + 9683060038655529 \beta_{1} + 12264444450819 \beta_{2} + 468532892556 \beta_{3}) q^{15} +(\)\(52\!\cdots\!70\)\( - 231250822893144132 \beta_{1} - 567367762804830 \beta_{2} - 5862494129970 \beta_{3}) q^{17} +(\)\(26\!\cdots\!60\)\( - 6399332107287301719 \beta_{1} - 9378200637116970 \beta_{2} - 21154099092072 \beta_{3}) q^{19} +(-\)\(65\!\cdots\!28\)\( + 56460333251491679048 \beta_{1} - 98870608738814220 \beta_{2} + 903384551930364 \beta_{3}) q^{21} +(-\)\(34\!\cdots\!20\)\( + 38416206999361792313 \beta_{1} - 579467257376567677 \beta_{2} + 151096756188300 \beta_{3}) q^{23} +(\)\(25\!\cdots\!75\)\( + \)\(25\!\cdots\!00\)\( \beta_{1} - 11090484330999303100 \beta_{2} - 67016296066966900 \beta_{3}) q^{25} +(\)\(18\!\cdots\!80\)\( + \)\(16\!\cdots\!56\)\( \beta_{1} + 39260056568752771362 \beta_{2} - 96744589719397560 \beta_{3}) q^{27} +(-\)\(56\!\cdots\!90\)\( - \)\(46\!\cdots\!04\)\( \beta_{1} - \)\(31\!\cdots\!55\)\( \beta_{2} + 2665048030584989913 \beta_{3}) q^{29} +(-\)\(18\!\cdots\!12\)\( - \)\(11\!\cdots\!20\)\( \beta_{1} + \)\(37\!\cdots\!20\)\( \beta_{2} + 9951638514076833760 \beta_{3}) q^{31} +(\)\(65\!\cdots\!80\)\( - \)\(59\!\cdots\!48\)\( \beta_{1} - \)\(42\!\cdots\!10\)\( \beta_{2} - 37812601030217045670 \beta_{3}) q^{33} +(\)\(32\!\cdots\!00\)\( - \)\(47\!\cdots\!88\)\( \beta_{1} + \)\(55\!\cdots\!32\)\( \beta_{2} - \)\(28\!\cdots\!32\)\( \beta_{3}) q^{35} +(-\)\(28\!\cdots\!90\)\( - \)\(33\!\cdots\!04\)\( \beta_{1} + \)\(14\!\cdots\!61\)\( \beta_{2} - \)\(24\!\cdots\!55\)\( \beta_{3}) q^{37} +(-\)\(98\!\cdots\!84\)\( - \)\(55\!\cdots\!83\)\( \beta_{1} + \)\(11\!\cdots\!35\)\( \beta_{2} + \)\(15\!\cdots\!96\)\( \beta_{3}) q^{39} +(\)\(32\!\cdots\!82\)\( - \)\(75\!\cdots\!80\)\( \beta_{1} + \)\(19\!\cdots\!80\)\( \beta_{2} + \)\(24\!\cdots\!40\)\( \beta_{3}) q^{41} +(\)\(11\!\cdots\!00\)\( - \)\(81\!\cdots\!63\)\( \beta_{1} - \)\(44\!\cdots\!84\)\( \beta_{2} + \)\(59\!\cdots\!00\)\( \beta_{3}) q^{43} +(\)\(21\!\cdots\!50\)\( + \)\(14\!\cdots\!16\)\( \beta_{1} - \)\(10\!\cdots\!99\)\( \beta_{2} + \)\(53\!\cdots\!49\)\( \beta_{3}) q^{45} +(-\)\(50\!\cdots\!80\)\( + \)\(55\!\cdots\!50\)\( \beta_{1} - \)\(63\!\cdots\!34\)\( \beta_{2} + \)\(22\!\cdots\!00\)\( \beta_{3}) q^{47} +(\)\(22\!\cdots\!93\)\( - \)\(34\!\cdots\!20\)\( \beta_{1} - \)\(12\!\cdots\!20\)\( \beta_{2} + \)\(27\!\cdots\!20\)\( \beta_{3}) q^{49} +(\)\(46\!\cdots\!88\)\( - \)\(82\!\cdots\!16\)\( \beta_{1} + \)\(19\!\cdots\!70\)\( \beta_{2} + \)\(31\!\cdots\!92\)\( \beta_{3}) q^{51} +(\)\(72\!\cdots\!90\)\( + \)\(15\!\cdots\!56\)\( \beta_{1} + \)\(59\!\cdots\!25\)\( \beta_{2} + \)\(25\!\cdots\!65\)\( \beta_{3}) q^{53} +(-\)\(48\!\cdots\!00\)\( - \)\(13\!\cdots\!81\)\( \beta_{1} - \)\(93\!\cdots\!91\)\( \beta_{2} + \)\(48\!\cdots\!16\)\( \beta_{3}) q^{55} +(\)\(13\!\cdots\!40\)\( - \)\(71\!\cdots\!72\)\( \beta_{1} + \)\(25\!\cdots\!86\)\( \beta_{2} - \)\(45\!\cdots\!50\)\( \beta_{3}) q^{57} +(-\)\(11\!\cdots\!20\)\( - \)\(39\!\cdots\!07\)\( \beta_{1} - \)\(41\!\cdots\!20\)\( \beta_{2} - \)\(86\!\cdots\!76\)\( \beta_{3}) q^{59} +(\)\(15\!\cdots\!22\)\( + \)\(26\!\cdots\!00\)\( \beta_{1} - \)\(49\!\cdots\!75\)\( \beta_{2} + \)\(25\!\cdots\!25\)\( \beta_{3}) q^{61} +(-\)\(14\!\cdots\!60\)\( + \)\(76\!\cdots\!35\)\( \beta_{1} + \)\(27\!\cdots\!17\)\( \beta_{2} - \)\(60\!\cdots\!80\)\( \beta_{3}) q^{63} +(\)\(32\!\cdots\!00\)\( + \)\(13\!\cdots\!76\)\( \beta_{1} - \)\(71\!\cdots\!64\)\( \beta_{2} - \)\(35\!\cdots\!36\)\( \beta_{3}) q^{65} +(-\)\(46\!\cdots\!20\)\( + \)\(28\!\cdots\!37\)\( \beta_{1} - \)\(66\!\cdots\!10\)\( \beta_{2} - \)\(10\!\cdots\!80\)\( \beta_{3}) q^{67} +(-\)\(52\!\cdots\!76\)\( + \)\(45\!\cdots\!64\)\( \beta_{1} + \)\(26\!\cdots\!60\)\( \beta_{2} + \)\(25\!\cdots\!72\)\( \beta_{3}) q^{69} +(-\)\(55\!\cdots\!92\)\( + \)\(98\!\cdots\!75\)\( \beta_{1} - \)\(31\!\cdots\!75\)\( \beta_{2} - \)\(20\!\cdots\!00\)\( \beta_{3}) q^{71} +(\)\(26\!\cdots\!70\)\( + \)\(84\!\cdots\!32\)\( \beta_{1} + \)\(14\!\cdots\!22\)\( \beta_{2} - \)\(17\!\cdots\!70\)\( \beta_{3}) q^{73} +(-\)\(80\!\cdots\!00\)\( - \)\(30\!\cdots\!75\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2} + \)\(97\!\cdots\!00\)\( \beta_{3}) q^{75} +(-\)\(65\!\cdots\!00\)\( + \)\(10\!\cdots\!08\)\( \beta_{1} - \)\(98\!\cdots\!36\)\( \beta_{2} + \)\(68\!\cdots\!80\)\( \beta_{3}) q^{77} +(\)\(33\!\cdots\!40\)\( + \)\(13\!\cdots\!14\)\( \beta_{1} + \)\(31\!\cdots\!10\)\( \beta_{2} - \)\(10\!\cdots\!28\)\( \beta_{3}) q^{79} +(-\)\(26\!\cdots\!79\)\( + \)\(42\!\cdots\!64\)\( \beta_{1} - \)\(12\!\cdots\!10\)\( \beta_{2} + \)\(57\!\cdots\!02\)\( \beta_{3}) q^{81} +(\)\(35\!\cdots\!40\)\( + \)\(56\!\cdots\!15\)\( \beta_{1} - \)\(17\!\cdots\!88\)\( \beta_{2} + \)\(10\!\cdots\!00\)\( \beta_{3}) q^{83} +(-\)\(25\!\cdots\!00\)\( + \)\(16\!\cdots\!08\)\( \beta_{1} - \)\(91\!\cdots\!62\)\( \beta_{2} - \)\(14\!\cdots\!38\)\( \beta_{3}) q^{85} +(\)\(10\!\cdots\!40\)\( + \)\(15\!\cdots\!73\)\( \beta_{1} - \)\(24\!\cdots\!69\)\( \beta_{2} - \)\(16\!\cdots\!20\)\( \beta_{3}) q^{87} +(-\)\(19\!\cdots\!70\)\( + \)\(28\!\cdots\!48\)\( \beta_{1} + \)\(19\!\cdots\!10\)\( \beta_{2} + \)\(24\!\cdots\!94\)\( \beta_{3}) q^{89} +(-\)\(13\!\cdots\!92\)\( + \)\(17\!\cdots\!04\)\( \beta_{1} + \)\(12\!\cdots\!80\)\( \beta_{2} - \)\(95\!\cdots\!88\)\( \beta_{3}) q^{91} +(\)\(26\!\cdots\!80\)\( - \)\(79\!\cdots\!48\)\( \beta_{1} - \)\(25\!\cdots\!80\)\( \beta_{2} - \)\(23\!\cdots\!60\)\( \beta_{3}) q^{93} +(-\)\(18\!\cdots\!00\)\( + \)\(13\!\cdots\!45\)\( \beta_{1} + \)\(17\!\cdots\!95\)\( \beta_{2} + \)\(11\!\cdots\!80\)\( \beta_{3}) q^{95} +(\)\(23\!\cdots\!30\)\( + \)\(11\!\cdots\!80\)\( \beta_{1} - \)\(37\!\cdots\!46\)\( \beta_{2} + \)\(23\!\cdots\!10\)\( \beta_{3}) q^{97} +(-\)\(13\!\cdots\!04\)\( - \)\(97\!\cdots\!19\)\( \beta_{1} - \)\(17\!\cdots\!80\)\( \beta_{2} - \)\(21\!\cdots\!32\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 38461494960q^{3} - 31114680242272200q^{5} + 39169218725888423200q^{7} - 17071972417358142200172q^{9} + O(q^{10}) \) \( 4q - 38461494960q^{3} - 31114680242272200q^{5} + 39169218725888423200q^{7} - \)\(17\!\cdots\!72\)\(q^{9} + \)\(19\!\cdots\!12\)\(q^{11} + \)\(12\!\cdots\!20\)\(q^{13} - \)\(12\!\cdots\!00\)\(q^{15} + \)\(21\!\cdots\!80\)\(q^{17} + \)\(10\!\cdots\!40\)\(q^{19} - \)\(26\!\cdots\!12\)\(q^{21} - \)\(13\!\cdots\!80\)\(q^{23} + \)\(10\!\cdots\!00\)\(q^{25} + \)\(73\!\cdots\!20\)\(q^{27} - \)\(22\!\cdots\!60\)\(q^{29} - \)\(75\!\cdots\!48\)\(q^{31} + \)\(26\!\cdots\!20\)\(q^{33} + \)\(13\!\cdots\!00\)\(q^{35} - \)\(11\!\cdots\!60\)\(q^{37} - \)\(39\!\cdots\!36\)\(q^{39} + \)\(13\!\cdots\!28\)\(q^{41} + \)\(44\!\cdots\!00\)\(q^{43} + \)\(86\!\cdots\!00\)\(q^{45} - \)\(20\!\cdots\!20\)\(q^{47} + \)\(91\!\cdots\!72\)\(q^{49} + \)\(18\!\cdots\!52\)\(q^{51} + \)\(29\!\cdots\!60\)\(q^{53} - \)\(19\!\cdots\!00\)\(q^{55} + \)\(55\!\cdots\!60\)\(q^{57} - \)\(47\!\cdots\!80\)\(q^{59} + \)\(62\!\cdots\!88\)\(q^{61} - \)\(58\!\cdots\!40\)\(q^{63} + \)\(12\!\cdots\!00\)\(q^{65} - \)\(18\!\cdots\!80\)\(q^{67} - \)\(20\!\cdots\!04\)\(q^{69} - \)\(22\!\cdots\!68\)\(q^{71} + \)\(10\!\cdots\!80\)\(q^{73} - \)\(32\!\cdots\!00\)\(q^{75} - \)\(26\!\cdots\!00\)\(q^{77} + \)\(13\!\cdots\!60\)\(q^{79} - \)\(10\!\cdots\!16\)\(q^{81} + \)\(14\!\cdots\!60\)\(q^{83} - \)\(10\!\cdots\!00\)\(q^{85} + \)\(43\!\cdots\!60\)\(q^{87} - \)\(79\!\cdots\!80\)\(q^{89} - \)\(53\!\cdots\!68\)\(q^{91} + \)\(10\!\cdots\!20\)\(q^{93} - \)\(75\!\cdots\!00\)\(q^{95} + \)\(95\!\cdots\!20\)\(q^{97} - \)\(53\!\cdots\!16\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 832803191366 x^{2} + 3710135215485780 x + 13175318942671469337000\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 60053 \nu^{2} - 774048298232 \nu - 22223994704270844 \)\()/514752\)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{3} - 420371 \nu^{2} + 9119526154952 \nu + 155567037632879076 \)\()/73536\)
\(\beta_{3}\)\(=\)\((\)\( -15527 \nu^{3} + 8555465933 \nu^{2} + 12502154924993992 \nu - 3605708545770388002972 \)\()/514752\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 49 \beta_{1} + 12582912\)\()/50331648\)
\(\nu^{2}\)\(=\)\((\)\(8192 \beta_{3} - 152881 \beta_{2} + 119706015 \beta_{1} + 62874535621702975488\)\()/ 150994944 \)
\(\nu^{3}\)\(=\)\((\)\(-491954176 \beta_{3} + 2331325857389 \beta_{2} + 184321343935197 \beta_{1} - 420064432517594927333376\)\()/ 150994944 \)

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
−124721.
901372.
−906006.
129356.
0 −1.52034e11 0 4.23962e16 0 3.90714e19 0 −3.47448e21 0
1.2 0 −1.28510e11 0 −1.15086e16 0 −1.54211e19 0 −1.00741e22 0
1.3 0 2.01642e10 0 −3.11682e16 0 1.26592e20 0 −2.61822e22 0
1.4 0 2.21918e11 0 −3.08341e16 0 −1.11073e20 0 2.26588e22 0
\(n\): 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 16.48.a.d 4
4.b odd 2 1 1.48.a.a 4
12.b even 2 1 9.48.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.48.a.a 4 4.b odd 2 1
9.48.a.c 4 12.b even 2 1
16.48.a.d 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 38461494960 T_{3}^{3} - \)\(43\!\cdots\!88\)\( T_{3}^{2} - \)\(34\!\cdots\!40\)\( T_{3} + \)\(87\!\cdots\!36\)\( \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( \)\(87\!\cdots\!36\)\( - \)\(34\!\cdots\!40\)\( T - \)\(43\!\cdots\!88\)\( T^{2} + 38461494960 T^{3} + T^{4} \)
$5$ \( -\)\(46\!\cdots\!00\)\( - \)\(59\!\cdots\!00\)\( T - \)\(14\!\cdots\!00\)\( T^{2} + 31114680242272200 T^{3} + T^{4} \)
$7$ \( \)\(84\!\cdots\!96\)\( + \)\(34\!\cdots\!00\)\( T - \)\(14\!\cdots\!72\)\( T^{2} - 39169218725888423200 T^{3} + T^{4} \)
$11$ \( -\)\(18\!\cdots\!44\)\( + \)\(11\!\cdots\!92\)\( T - \)\(75\!\cdots\!96\)\( T^{2} - \)\(19\!\cdots\!12\)\( T^{3} + T^{4} \)
$13$ \( \)\(10\!\cdots\!76\)\( + \)\(33\!\cdots\!80\)\( T - \)\(23\!\cdots\!48\)\( T^{2} - \)\(12\!\cdots\!20\)\( T^{3} + T^{4} \)
$17$ \( -\)\(14\!\cdots\!24\)\( + \)\(21\!\cdots\!80\)\( T + \)\(10\!\cdots\!48\)\( T^{2} - \)\(21\!\cdots\!80\)\( T^{3} + T^{4} \)
$19$ \( \)\(37\!\cdots\!00\)\( + \)\(81\!\cdots\!00\)\( T - \)\(16\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!40\)\( T^{3} + T^{4} \)
$23$ \( \)\(56\!\cdots\!16\)\( + \)\(10\!\cdots\!80\)\( T + \)\(63\!\cdots\!92\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + T^{4} \)
$29$ \( -\)\(69\!\cdots\!00\)\( - \)\(16\!\cdots\!00\)\( T - \)\(80\!\cdots\!00\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + T^{4} \)
$31$ \( \)\(13\!\cdots\!36\)\( - \)\(24\!\cdots\!88\)\( T - \)\(36\!\cdots\!36\)\( T^{2} + \)\(75\!\cdots\!48\)\( T^{3} + T^{4} \)
$37$ \( -\)\(15\!\cdots\!64\)\( - \)\(78\!\cdots\!60\)\( T - \)\(49\!\cdots\!12\)\( T^{2} + \)\(11\!\cdots\!60\)\( T^{3} + T^{4} \)
$41$ \( \)\(89\!\cdots\!76\)\( + \)\(37\!\cdots\!28\)\( T - \)\(23\!\cdots\!56\)\( T^{2} - \)\(13\!\cdots\!28\)\( T^{3} + T^{4} \)
$43$ \( -\)\(57\!\cdots\!04\)\( + \)\(12\!\cdots\!00\)\( T + \)\(50\!\cdots\!72\)\( T^{2} - \)\(44\!\cdots\!00\)\( T^{3} + T^{4} \)
$47$ \( \)\(84\!\cdots\!16\)\( - \)\(55\!\cdots\!20\)\( T - \)\(50\!\cdots\!92\)\( T^{2} + \)\(20\!\cdots\!20\)\( T^{3} + T^{4} \)
$53$ \( \)\(10\!\cdots\!36\)\( + \)\(86\!\cdots\!40\)\( T - \)\(10\!\cdots\!88\)\( T^{2} - \)\(29\!\cdots\!60\)\( T^{3} + T^{4} \)
$59$ \( -\)\(14\!\cdots\!00\)\( + \)\(36\!\cdots\!00\)\( T + \)\(36\!\cdots\!00\)\( T^{2} + \)\(47\!\cdots\!80\)\( T^{3} + T^{4} \)
$61$ \( -\)\(47\!\cdots\!44\)\( + \)\(26\!\cdots\!08\)\( T - \)\(28\!\cdots\!96\)\( T^{2} - \)\(62\!\cdots\!88\)\( T^{3} + T^{4} \)
$67$ \( -\)\(23\!\cdots\!24\)\( + \)\(46\!\cdots\!20\)\( T + \)\(86\!\cdots\!48\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + T^{4} \)
$71$ \( -\)\(15\!\cdots\!04\)\( - \)\(25\!\cdots\!48\)\( T - \)\(86\!\cdots\!16\)\( T^{2} + \)\(22\!\cdots\!68\)\( T^{3} + T^{4} \)
$73$ \( -\)\(60\!\cdots\!84\)\( + \)\(28\!\cdots\!20\)\( T - \)\(66\!\cdots\!08\)\( T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + T^{4} \)
$79$ \( \)\(29\!\cdots\!00\)\( - \)\(74\!\cdots\!00\)\( T + \)\(55\!\cdots\!00\)\( T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + T^{4} \)
$83$ \( -\)\(10\!\cdots\!44\)\( + \)\(38\!\cdots\!40\)\( T - \)\(23\!\cdots\!68\)\( T^{2} - \)\(14\!\cdots\!60\)\( T^{3} + T^{4} \)
$89$ \( -\)\(18\!\cdots\!00\)\( - \)\(94\!\cdots\!00\)\( T - \)\(94\!\cdots\!00\)\( T^{2} + \)\(79\!\cdots\!80\)\( T^{3} + T^{4} \)
$97$ \( -\)\(75\!\cdots\!84\)\( + \)\(26\!\cdots\!20\)\( T + \)\(19\!\cdots\!08\)\( T^{2} - \)\(95\!\cdots\!20\)\( T^{3} + T^{4} \)
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