Properties

Label 16.50.a.b
Level 16
Weight 50
Character orbit 16.a
Self dual yes
Analytic conductor 243.306
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 50 \)
Character orbit: \([\chi]\) \(=\) 16.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(243.305928158\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 104434803447206332 x + 4289992005756109702361620\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 2)
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.

\(f(q)\) \(=\) \( q + ( 5401204796 + \beta_{1} ) q^{3} + ( -33937672133946810 - 78841 \beta_{1} - \beta_{2} ) q^{5} + ( 41826393076734599032 - 99835658 \beta_{1} + 3700 \beta_{2} ) q^{7} + ( 129810977173232307350733 - 175523252718 \beta_{1} + 1418850 \beta_{2} ) q^{9} +O(q^{10})\) \( q +(5401204796 + \beta_{1}) q^{3} +(-33937672133946810 - 78841 \beta_{1} - \beta_{2}) q^{5} +(41826393076734599032 - 99835658 \beta_{1} + 3700 \beta_{2}) q^{7} +(\)\(12\!\cdots\!33\)\( - 175523252718 \beta_{1} + 1418850 \beta_{2}) q^{9} +(\)\(86\!\cdots\!68\)\( + 76202146518283 \beta_{1} - 52537400 \beta_{2}) q^{11} +(\)\(98\!\cdots\!14\)\( - 340286117859137 \beta_{1} + 4260419575 \beta_{2}) q^{13} +(-\)\(29\!\cdots\!60\)\( - 125268300209547246 \beta_{1} - 303590419956 \beta_{2}) q^{15} +(\)\(13\!\cdots\!18\)\( - 2043132990929384798 \beta_{1} - 6325836222350 \beta_{2}) q^{17} +(-\)\(48\!\cdots\!60\)\( - 18528996595190482627 \beta_{1} - 8505702406600 \beta_{2}) q^{19} +(-\)\(36\!\cdots\!28\)\( + \)\(45\!\cdots\!44\)\( \beta_{1} + 567737584938900 \beta_{2}) q^{21} +(\)\(16\!\cdots\!36\)\( + \)\(67\!\cdots\!66\)\( \beta_{1} + 4043926350742700 \beta_{2}) q^{23} +(\)\(13\!\cdots\!75\)\( + \)\(34\!\cdots\!80\)\( \beta_{1} + 480088492044180 \beta_{2}) q^{25} +(-\)\(65\!\cdots\!00\)\( + \)\(72\!\cdots\!02\)\( \beta_{1} + 22990498274413800 \beta_{2}) q^{27} +(\)\(31\!\cdots\!10\)\( - \)\(97\!\cdots\!81\)\( \beta_{1} + 2068317919687048475 \beta_{2}) q^{29} +(-\)\(80\!\cdots\!52\)\( - \)\(21\!\cdots\!04\)\( \beta_{1} + 21160996591207173600 \beta_{2}) q^{31} +(\)\(28\!\cdots\!28\)\( - \)\(10\!\cdots\!94\)\( \beta_{1} + 98046584479571070150 \beta_{2}) q^{33} +(-\)\(10\!\cdots\!20\)\( - \)\(72\!\cdots\!32\)\( \beta_{1} + \)\(19\!\cdots\!48\)\( \beta_{2}) q^{35} +(\)\(10\!\cdots\!58\)\( + \)\(22\!\cdots\!27\)\( \beta_{1} - 1016938682892827725 \beta_{2}) q^{37} +(-\)\(12\!\cdots\!56\)\( + \)\(14\!\cdots\!82\)\( \beta_{1} + \)\(33\!\cdots\!00\)\( \beta_{2}) q^{39} +(-\)\(38\!\cdots\!18\)\( - \)\(10\!\cdots\!04\)\( \beta_{1} + \)\(88\!\cdots\!00\)\( \beta_{2}) q^{41} +(\)\(58\!\cdots\!16\)\( - \)\(10\!\cdots\!57\)\( \beta_{1} + \)\(32\!\cdots\!00\)\( \beta_{2}) q^{43} +(-\)\(38\!\cdots\!30\)\( - \)\(19\!\cdots\!73\)\( \beta_{1} + \)\(33\!\cdots\!47\)\( \beta_{2}) q^{45} +(\)\(72\!\cdots\!52\)\( - \)\(72\!\cdots\!88\)\( \beta_{1} - \)\(31\!\cdots\!00\)\( \beta_{2}) q^{47} +(\)\(12\!\cdots\!17\)\( + \)\(10\!\cdots\!48\)\( \beta_{1} - \)\(12\!\cdots\!00\)\( \beta_{2}) q^{49} +(-\)\(74\!\cdots\!72\)\( + \)\(10\!\cdots\!90\)\( \beta_{1} - \)\(41\!\cdots\!00\)\( \beta_{2}) q^{51} +(\)\(24\!\cdots\!34\)\( - \)\(10\!\cdots\!37\)\( \beta_{1} - \)\(14\!\cdots\!25\)\( \beta_{2}) q^{53} +(-\)\(10\!\cdots\!80\)\( - \)\(90\!\cdots\!18\)\( \beta_{1} - \)\(34\!\cdots\!48\)\( \beta_{2}) q^{55} +(-\)\(68\!\cdots\!60\)\( - \)\(24\!\cdots\!82\)\( \beta_{1} - \)\(27\!\cdots\!50\)\( \beta_{2}) q^{57} +(\)\(17\!\cdots\!80\)\( + \)\(19\!\cdots\!47\)\( \beta_{1} + \)\(81\!\cdots\!00\)\( \beta_{2}) q^{59} +(-\)\(41\!\cdots\!98\)\( + \)\(48\!\cdots\!43\)\( \beta_{1} + \)\(18\!\cdots\!75\)\( \beta_{2}) q^{61} +(\)\(15\!\cdots\!56\)\( - \)\(34\!\cdots\!30\)\( \beta_{1} - \)\(13\!\cdots\!00\)\( \beta_{2}) q^{63} +(-\)\(14\!\cdots\!40\)\( - \)\(12\!\cdots\!04\)\( \beta_{1} - \)\(63\!\cdots\!44\)\( \beta_{2}) q^{65} +(-\)\(37\!\cdots\!08\)\( + \)\(51\!\cdots\!01\)\( \beta_{1} + \)\(27\!\cdots\!00\)\( \beta_{2}) q^{67} +(\)\(25\!\cdots\!56\)\( + \)\(19\!\cdots\!12\)\( \beta_{1} + \)\(17\!\cdots\!00\)\( \beta_{2}) q^{69} +(\)\(13\!\cdots\!08\)\( + \)\(18\!\cdots\!62\)\( \beta_{1} - \)\(31\!\cdots\!00\)\( \beta_{2}) q^{71} +(-\)\(33\!\cdots\!06\)\( - \)\(40\!\cdots\!62\)\( \beta_{1} + \)\(60\!\cdots\!50\)\( \beta_{2}) q^{73} +(\)\(12\!\cdots\!00\)\( + \)\(69\!\cdots\!55\)\( \beta_{1} + \)\(49\!\cdots\!80\)\( \beta_{2}) q^{75} +(-\)\(77\!\cdots\!24\)\( + \)\(31\!\cdots\!52\)\( \beta_{1} + \)\(92\!\cdots\!00\)\( \beta_{2}) q^{77} +(-\)\(12\!\cdots\!20\)\( - \)\(32\!\cdots\!32\)\( \beta_{1} + \)\(91\!\cdots\!00\)\( \beta_{2}) q^{79} +(-\)\(48\!\cdots\!39\)\( - \)\(33\!\cdots\!34\)\( \beta_{1} - \)\(23\!\cdots\!50\)\( \beta_{2}) q^{81} +(-\)\(50\!\cdots\!64\)\( + \)\(72\!\cdots\!65\)\( \beta_{1} - \)\(57\!\cdots\!00\)\( \beta_{2}) q^{83} +(\)\(18\!\cdots\!20\)\( + \)\(29\!\cdots\!42\)\( \beta_{1} - \)\(10\!\cdots\!38\)\( \beta_{2}) q^{85} +(-\)\(35\!\cdots\!40\)\( + \)\(70\!\cdots\!94\)\( \beta_{1} - \)\(98\!\cdots\!00\)\( \beta_{2}) q^{87} +(\)\(18\!\cdots\!90\)\( + \)\(60\!\cdots\!10\)\( \beta_{1} + \)\(75\!\cdots\!50\)\( \beta_{2}) q^{89} +(\)\(48\!\cdots\!48\)\( - \)\(77\!\cdots\!56\)\( \beta_{1} + \)\(18\!\cdots\!00\)\( \beta_{2}) q^{91} +(-\)\(78\!\cdots\!92\)\( + \)\(18\!\cdots\!04\)\( \beta_{1} + \)\(10\!\cdots\!00\)\( \beta_{2}) q^{93} +(\)\(93\!\cdots\!00\)\( + \)\(28\!\cdots\!30\)\( \beta_{1} + \)\(99\!\cdots\!80\)\( \beta_{2}) q^{95} +(\)\(14\!\cdots\!38\)\( + \)\(28\!\cdots\!30\)\( \beta_{1} + \)\(28\!\cdots\!50\)\( \beta_{2}) q^{97} +(-\)\(58\!\cdots\!56\)\( + \)\(22\!\cdots\!55\)\( \beta_{1} + \)\(16\!\cdots\!00\)\( \beta_{2}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 16203614388q^{3} - 101813016401840430q^{5} + 125479179230203797096q^{7} + 389432931519696922052199q^{9} + O(q^{10}) \) \( 3q + 16203614388q^{3} - 101813016401840430q^{5} + \)\(12\!\cdots\!96\)\(q^{7} + \)\(38\!\cdots\!99\)\(q^{9} + \)\(25\!\cdots\!04\)\(q^{11} + \)\(29\!\cdots\!42\)\(q^{13} - \)\(87\!\cdots\!80\)\(q^{15} + \)\(39\!\cdots\!54\)\(q^{17} - \)\(14\!\cdots\!80\)\(q^{19} - \)\(10\!\cdots\!84\)\(q^{21} + \)\(49\!\cdots\!08\)\(q^{23} + \)\(39\!\cdots\!25\)\(q^{25} - \)\(19\!\cdots\!00\)\(q^{27} + \)\(93\!\cdots\!30\)\(q^{29} - \)\(24\!\cdots\!56\)\(q^{31} + \)\(84\!\cdots\!84\)\(q^{33} - \)\(30\!\cdots\!60\)\(q^{35} + \)\(30\!\cdots\!74\)\(q^{37} - \)\(36\!\cdots\!68\)\(q^{39} - \)\(11\!\cdots\!54\)\(q^{41} + \)\(17\!\cdots\!48\)\(q^{43} - \)\(11\!\cdots\!90\)\(q^{45} + \)\(21\!\cdots\!56\)\(q^{47} + \)\(37\!\cdots\!51\)\(q^{49} - \)\(22\!\cdots\!16\)\(q^{51} + \)\(74\!\cdots\!02\)\(q^{53} - \)\(32\!\cdots\!40\)\(q^{55} - \)\(20\!\cdots\!80\)\(q^{57} + \)\(53\!\cdots\!40\)\(q^{59} - \)\(12\!\cdots\!94\)\(q^{61} + \)\(46\!\cdots\!68\)\(q^{63} - \)\(42\!\cdots\!20\)\(q^{65} - \)\(11\!\cdots\!24\)\(q^{67} + \)\(77\!\cdots\!68\)\(q^{69} + \)\(40\!\cdots\!24\)\(q^{71} - \)\(10\!\cdots\!18\)\(q^{73} + \)\(38\!\cdots\!00\)\(q^{75} - \)\(23\!\cdots\!72\)\(q^{77} - \)\(36\!\cdots\!60\)\(q^{79} - \)\(14\!\cdots\!17\)\(q^{81} - \)\(15\!\cdots\!92\)\(q^{83} + \)\(56\!\cdots\!60\)\(q^{85} - \)\(10\!\cdots\!20\)\(q^{87} + \)\(54\!\cdots\!70\)\(q^{89} + \)\(14\!\cdots\!44\)\(q^{91} - \)\(23\!\cdots\!76\)\(q^{93} + \)\(28\!\cdots\!00\)\(q^{95} + \)\(43\!\cdots\!14\)\(q^{97} - \)\(17\!\cdots\!68\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 104434803447206332 x + 4289992005756109702361620\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 160 \nu^{2} + 17410920480 \nu - 11139712373505648960 \)\()/12670249\)
\(\beta_{2}\)\(=\)\((\)\( -11698720 \nu^{2} + 6966187410422880 \nu + 814502346867205307288640 \)\()/12670249\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 73117 \beta_{1} + 216760320\)\()/ 650280960 \)
\(\nu^{2}\)\(=\)\((\)\(-12090917 \beta_{2} + 4837630146127 \beta_{1} + 5030515869856343941939200\)\()/72253440\)

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
4.17763e7
−3.42020e8
3.00244e8
0 −7.94355e11 0 −5.65262e16 0 4.38546e20 0 3.91700e23 0
1.2 0 1.33407e11 0 1.87739e17 0 −8.28497e20 0 −2.21502e23 0
1.3 0 6.77152e11 0 −2.33026e17 0 5.15430e20 0 2.19235e23 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 16.50.a.b 3
4.b odd 2 1 2.50.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.50.a.b 3 4.b odd 2 1
16.50.a.b 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 16203614388 T_{3}^{2} - \)\(55\!\cdots\!52\)\( T_{3} + \)\(71\!\cdots\!64\)\( \) acting on \(S_{50}^{\mathrm{new}}(\Gamma_0(16))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 16203614388 T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(64\!\cdots\!56\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} - \)\(92\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!87\)\( T^{6} \)
$5$ \( 1 + 101813016401840430 T + \)\(12\!\cdots\!75\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(21\!\cdots\!75\)\( T^{4} + \)\(32\!\cdots\!50\)\( T^{5} + \)\(56\!\cdots\!25\)\( T^{6} \)
$7$ \( 1 - \)\(12\!\cdots\!96\)\( T + \)\(20\!\cdots\!93\)\( T^{2} + \)\(12\!\cdots\!88\)\( T^{3} + \)\(53\!\cdots\!51\)\( T^{4} - \)\(82\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!43\)\( T^{6} \)
$11$ \( 1 - \)\(25\!\cdots\!04\)\( T + \)\(96\!\cdots\!45\)\( T^{2} + \)\(35\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(29\!\cdots\!24\)\( T^{5} + \)\(12\!\cdots\!71\)\( T^{6} \)
$13$ \( 1 - \)\(29\!\cdots\!42\)\( T + \)\(13\!\cdots\!07\)\( T^{2} - \)\(22\!\cdots\!76\)\( T^{3} + \)\(51\!\cdots\!11\)\( T^{4} - \)\(43\!\cdots\!18\)\( T^{5} + \)\(56\!\cdots\!17\)\( T^{6} \)
$17$ \( 1 - \)\(39\!\cdots\!54\)\( T + \)\(70\!\cdots\!63\)\( T^{2} - \)\(93\!\cdots\!08\)\( T^{3} + \)\(13\!\cdots\!11\)\( T^{4} - \)\(15\!\cdots\!86\)\( T^{5} + \)\(75\!\cdots\!73\)\( T^{6} \)
$19$ \( 1 + \)\(14\!\cdots\!80\)\( T + \)\(12\!\cdots\!37\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(56\!\cdots\!23\)\( T^{4} + \)\(30\!\cdots\!80\)\( T^{5} + \)\(94\!\cdots\!39\)\( T^{6} \)
$23$ \( 1 - \)\(49\!\cdots\!08\)\( T + \)\(23\!\cdots\!77\)\( T^{2} - \)\(55\!\cdots\!64\)\( T^{3} + \)\(12\!\cdots\!51\)\( T^{4} - \)\(13\!\cdots\!52\)\( T^{5} + \)\(14\!\cdots\!47\)\( T^{6} \)
$29$ \( 1 - \)\(93\!\cdots\!30\)\( T + \)\(95\!\cdots\!07\)\( T^{2} - \)\(86\!\cdots\!40\)\( T^{3} + \)\(43\!\cdots\!83\)\( T^{4} - \)\(19\!\cdots\!30\)\( T^{5} + \)\(93\!\cdots\!09\)\( T^{6} \)
$31$ \( 1 + \)\(24\!\cdots\!56\)\( T + \)\(16\!\cdots\!25\)\( T^{2} + \)\(71\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!75\)\( T^{4} + \)\(34\!\cdots\!96\)\( T^{5} + \)\(16\!\cdots\!11\)\( T^{6} \)
$37$ \( 1 - \)\(30\!\cdots\!74\)\( T + \)\(21\!\cdots\!23\)\( T^{2} - \)\(40\!\cdots\!08\)\( T^{3} + \)\(14\!\cdots\!71\)\( T^{4} - \)\(14\!\cdots\!46\)\( T^{5} + \)\(33\!\cdots\!33\)\( T^{6} \)
$41$ \( 1 + \)\(11\!\cdots\!54\)\( T + \)\(28\!\cdots\!55\)\( T^{2} + \)\(24\!\cdots\!20\)\( T^{3} + \)\(30\!\cdots\!55\)\( T^{4} + \)\(12\!\cdots\!34\)\( T^{5} + \)\(12\!\cdots\!81\)\( T^{6} \)
$43$ \( 1 - \)\(17\!\cdots\!48\)\( T + \)\(32\!\cdots\!97\)\( T^{2} - \)\(36\!\cdots\!24\)\( T^{3} + \)\(35\!\cdots\!71\)\( T^{4} - \)\(21\!\cdots\!52\)\( T^{5} + \)\(13\!\cdots\!07\)\( T^{6} \)
$47$ \( 1 - \)\(21\!\cdots\!56\)\( T + \)\(34\!\cdots\!13\)\( T^{2} - \)\(34\!\cdots\!12\)\( T^{3} + \)\(29\!\cdots\!71\)\( T^{4} - \)\(16\!\cdots\!84\)\( T^{5} + \)\(62\!\cdots\!63\)\( T^{6} \)
$53$ \( 1 - \)\(74\!\cdots\!02\)\( T + \)\(61\!\cdots\!67\)\( T^{2} - \)\(55\!\cdots\!36\)\( T^{3} + \)\(19\!\cdots\!11\)\( T^{4} - \)\(70\!\cdots\!78\)\( T^{5} + \)\(29\!\cdots\!37\)\( T^{6} \)
$59$ \( 1 - \)\(53\!\cdots\!40\)\( T + \)\(22\!\cdots\!17\)\( T^{2} - \)\(63\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!63\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(20\!\cdots\!19\)\( T^{6} \)
$61$ \( 1 + \)\(12\!\cdots\!94\)\( T + \)\(12\!\cdots\!35\)\( T^{2} + \)\(78\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!14\)\( T^{5} + \)\(27\!\cdots\!21\)\( T^{6} \)
$67$ \( 1 + \)\(11\!\cdots\!24\)\( T + \)\(13\!\cdots\!33\)\( T^{2} + \)\(71\!\cdots\!68\)\( T^{3} + \)\(39\!\cdots\!51\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(27\!\cdots\!23\)\( T^{6} \)
$71$ \( 1 - \)\(40\!\cdots\!24\)\( T + \)\(18\!\cdots\!85\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(96\!\cdots\!35\)\( T^{4} - \)\(10\!\cdots\!64\)\( T^{5} + \)\(13\!\cdots\!91\)\( T^{6} \)
$73$ \( 1 + \)\(10\!\cdots\!18\)\( T + \)\(83\!\cdots\!47\)\( T^{2} + \)\(39\!\cdots\!84\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} + \)\(40\!\cdots\!42\)\( T^{5} + \)\(80\!\cdots\!97\)\( T^{6} \)
$79$ \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(33\!\cdots\!57\)\( T^{2} + \)\(71\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!83\)\( T^{4} + \)\(33\!\cdots\!60\)\( T^{5} + \)\(89\!\cdots\!59\)\( T^{6} \)
$83$ \( 1 + \)\(15\!\cdots\!92\)\( T + \)\(23\!\cdots\!97\)\( T^{2} + \)\(20\!\cdots\!96\)\( T^{3} + \)\(25\!\cdots\!91\)\( T^{4} + \)\(17\!\cdots\!28\)\( T^{5} + \)\(12\!\cdots\!27\)\( T^{6} \)
$89$ \( 1 - \)\(54\!\cdots\!70\)\( T + \)\(86\!\cdots\!27\)\( T^{2} - \)\(34\!\cdots\!60\)\( T^{3} + \)\(28\!\cdots\!43\)\( T^{4} - \)\(59\!\cdots\!70\)\( T^{5} + \)\(36\!\cdots\!29\)\( T^{6} \)
$97$ \( 1 - \)\(43\!\cdots\!14\)\( T + \)\(35\!\cdots\!83\)\( T^{2} - \)\(13\!\cdots\!48\)\( T^{3} + \)\(80\!\cdots\!11\)\( T^{4} - \)\(21\!\cdots\!46\)\( T^{5} + \)\(11\!\cdots\!13\)\( T^{6} \)
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