Properties

Label 8016.2.a.z
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 1
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 8 x^{6} + 15 x^{5} + 19 x^{4} - 31 x^{3} - 13 x^{2} + 14 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{5} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{7} + q^{9} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{15} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{17} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{21} + ( -\beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{23} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{25} + q^{27} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{29} + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{33} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{35} + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{7} ) q^{37} + ( -\beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{39} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{41} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{43} + ( \beta_{1} + \beta_{2} - \beta_{5} ) q^{45} + ( 2 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{7} ) q^{47} + ( 2 - 2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{49} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{51} + ( -5 + 2 \beta_{1} + 3 \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{53} + ( -3 - 3 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{6} ) q^{55} + ( -5 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{57} + ( -4 - \beta_{1} + \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{59} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{7} ) q^{61} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{63} + ( 2 - 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + 8 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{65} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{5} + 3 \beta_{7} ) q^{67} + ( -\beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{69} + ( -1 - \beta_{1} + 5 \beta_{2} - \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{71} + ( -2 + 5 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{73} + ( 3 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{75} + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{77} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{79} + q^{81} + ( -1 + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( 1 - 8 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + 6 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{85} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} ) q^{87} + ( 1 + 5 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{89} + ( -4 + 2 \beta_{3} - \beta_{4} - 5 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{91} + ( -1 - 4 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} + 4 \beta_{5} + 2 \beta_{7} ) q^{93} + ( 2 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} - \beta_{4} + 8 \beta_{5} + \beta_{7} ) q^{95} + ( -6 + 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - 7 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{97} + ( 1 - 2 \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + q^{5} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + q^{5} + 8q^{9} - 5q^{11} + q^{15} - 7q^{17} - 24q^{19} - q^{23} + 3q^{25} + 8q^{27} - 11q^{29} - 30q^{31} - 5q^{33} - 26q^{35} + 11q^{37} + 10q^{41} - 24q^{43} + q^{45} + 3q^{47} + 6q^{49} - 7q^{51} - 25q^{53} - 25q^{55} - 24q^{57} - 45q^{59} + 16q^{61} - 10q^{65} - 18q^{67} - q^{69} - 21q^{71} - 8q^{73} + 3q^{75} - 18q^{77} - 10q^{79} + 8q^{81} - 7q^{83} - 11q^{85} - 11q^{87} + 26q^{89} - 15q^{91} - 30q^{93} - q^{95} - 3q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 8 x^{6} + 15 x^{5} + 19 x^{4} - 31 x^{3} - 13 x^{2} + 14 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 5 \nu^{2} + 3 \)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 7 \nu^{5} + 6 \nu^{4} + 13 \nu^{3} - 8 \nu^{2} - 5 \nu - 1 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} + \nu^{6} - 9 \nu^{5} - 8 \nu^{4} + 23 \nu^{3} + 16 \nu^{2} - 13 \nu - 3 \)\()/2\)
\(\beta_{7}\)\(=\)\( -\nu^{7} + \nu^{6} + 8 \nu^{5} - 7 \nu^{4} - 18 \nu^{3} + 13 \nu^{2} + 9 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4} + 5 \beta_{2} + 12\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 2 \beta_{5} + \beta_{4} + 5 \beta_{3} + 16 \beta_{1} + 2\)
\(\nu^{6}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + 8 \beta_{4} + 23 \beta_{2} + 51\)
\(\nu^{7}\)\(=\)\(8 \beta_{7} + \beta_{6} + 17 \beta_{5} + 9 \beta_{4} + 22 \beta_{3} + \beta_{2} + 65 \beta_{1} + 18\)

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
−1.63639
−0.0678707
−0.853788
0.688556
−2.05189
1.60046
2.22210
2.09883
0 1.00000 0 −3.62511 0 1.70865 0 1.00000 0
1.2 0 1.00000 0 −2.71255 0 5.14356 0 1.00000 0
1.3 0 1.00000 0 −0.621044 0 0.794861 0 1.00000 0
1.4 0 1.00000 0 0.0425261 0 −1.43576 0 1.00000 0
1.5 0 1.00000 0 0.925650 0 −2.76498 0 1.00000 0
1.6 0 1.00000 0 1.28584 0 0.120874 0 1.00000 0
1.7 0 1.00000 0 1.56175 0 −4.60397 0 1.00000 0
1.8 0 1.00000 0 4.14293 0 1.03675 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(167\) \(1\)

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 8016.2.a.z 8
4.b odd 2 1 501.2.a.d 8
12.b even 2 1 1503.2.a.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.d 8 4.b odd 2 1
1503.2.a.f 8 12.b even 2 1
8016.2.a.z 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{8} - \cdots\)
\( T_{7}^{8} - 31 T_{7}^{6} - 7 T_{7}^{5} + 175 T_{7}^{4} - 59 T_{7}^{3} - 224 T_{7}^{2} + 160 T_{7} - 16 \)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{8} - 61 T_{13}^{6} - 10 T_{13}^{5} + 1088 T_{13}^{4} - 237 T_{13}^{3} - 6922 T_{13}^{2} + 3828 T_{13} + 9224 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 - T + 19 T^{2} - 17 T^{3} + 161 T^{4} - 192 T^{5} + 927 T^{6} - 1582 T^{7} + 4718 T^{8} - 7910 T^{9} + 23175 T^{10} - 24000 T^{11} + 100625 T^{12} - 53125 T^{13} + 296875 T^{14} - 78125 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 25 T^{2} - 7 T^{3} + 245 T^{4} - 304 T^{5} + 1099 T^{6} - 4509 T^{7} + 3708 T^{8} - 31563 T^{9} + 53851 T^{10} - 104272 T^{11} + 588245 T^{12} - 117649 T^{13} + 2941225 T^{14} + 5764801 T^{16} \)
$11$ \( 1 + 5 T + 65 T^{2} + 281 T^{3} + 2021 T^{4} + 7532 T^{5} + 38975 T^{6} + 124436 T^{7} + 512616 T^{8} + 1368796 T^{9} + 4715975 T^{10} + 10025092 T^{11} + 29589461 T^{12} + 45255331 T^{13} + 115151465 T^{14} + 97435855 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 43 T^{2} - 10 T^{3} + 1062 T^{4} - 887 T^{5} + 18051 T^{6} - 22315 T^{7} + 251414 T^{8} - 290095 T^{9} + 3050619 T^{10} - 1948739 T^{11} + 30331782 T^{12} - 3712930 T^{13} + 207552787 T^{14} + 815730721 T^{16} \)
$17$ \( 1 + 7 T + 94 T^{2} + 515 T^{3} + 3983 T^{4} + 19188 T^{5} + 110490 T^{6} + 475150 T^{7} + 2210030 T^{8} + 8077550 T^{9} + 31931610 T^{10} + 94270644 T^{11} + 332664143 T^{12} + 731226355 T^{13} + 2268931486 T^{14} + 2872370711 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 24 T + 353 T^{2} + 3710 T^{3} + 31022 T^{4} + 214835 T^{5} + 1278735 T^{6} + 6653837 T^{7} + 30755986 T^{8} + 126422903 T^{9} + 461623335 T^{10} + 1473553265 T^{11} + 4042818062 T^{12} + 9186327290 T^{13} + 16607195993 T^{14} + 21452921736 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + T + 69 T^{2} - 65 T^{3} + 3165 T^{4} - 2232 T^{5} + 113855 T^{6} - 101490 T^{7} + 2806012 T^{8} - 2334270 T^{9} + 60229295 T^{10} - 27156744 T^{11} + 885696765 T^{12} - 418362295 T^{13} + 10214476341 T^{14} + 3404825447 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 + 11 T + 179 T^{2} + 1685 T^{3} + 15871 T^{4} + 120326 T^{5} + 863835 T^{6} + 5257502 T^{7} + 30730100 T^{8} + 152467558 T^{9} + 726485235 T^{10} + 2934630814 T^{11} + 11225256751 T^{12} + 34561286065 T^{13} + 106473374459 T^{14} + 189748639399 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 30 T + 494 T^{2} + 5767 T^{3} + 53036 T^{4} + 409236 T^{5} + 2761738 T^{6} + 16906143 T^{7} + 96763478 T^{8} + 524090433 T^{9} + 2654030218 T^{10} + 12191549676 T^{11} + 48979859756 T^{12} + 165104313817 T^{13} + 438426818414 T^{14} + 825378423330 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 11 T + 189 T^{2} - 1649 T^{3} + 17551 T^{4} - 135404 T^{5} + 1078209 T^{6} - 7208484 T^{7} + 46667124 T^{8} - 266713908 T^{9} + 1476068121 T^{10} - 6858618812 T^{11} + 32893399711 T^{12} - 114348185093 T^{13} + 484922291301 T^{14} - 1044250648463 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 10 T + 232 T^{2} - 1869 T^{3} + 25858 T^{4} - 173932 T^{5} + 1795868 T^{6} - 10291247 T^{7} + 87043884 T^{8} - 421941127 T^{9} + 3018854108 T^{10} - 11987567372 T^{11} + 73068527938 T^{12} - 216535239669 T^{13} + 1102024183912 T^{14} - 1947542738810 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 24 T + 480 T^{2} + 6573 T^{3} + 79642 T^{4} + 786764 T^{5} + 7047644 T^{6} + 54159389 T^{7} + 381070708 T^{8} + 2328853727 T^{9} + 13031093756 T^{10} + 62553245348 T^{11} + 272280149242 T^{12} + 966286495839 T^{13} + 3034254263520 T^{14} + 6523646666568 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 3 T + 73 T^{2} - 628 T^{3} + 6578 T^{4} - 33029 T^{5} + 475607 T^{6} - 2491768 T^{7} + 21399806 T^{8} - 117113096 T^{9} + 1050615863 T^{10} - 3429169867 T^{11} + 32098541618 T^{12} - 144028664396 T^{13} + 786882719017 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 25 T + 579 T^{2} + 8774 T^{3} + 120610 T^{4} + 1329957 T^{5} + 13375065 T^{6} + 114505786 T^{7} + 896980008 T^{8} + 6068806658 T^{9} + 37570557585 T^{10} + 198000008289 T^{11} + 951670913410 T^{12} + 3669247255582 T^{13} + 12833165093691 T^{14} + 29367778495925 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 45 T + 1223 T^{2} + 23998 T^{3} + 374940 T^{4} + 4846851 T^{5} + 53304093 T^{6} + 504932174 T^{7} + 4157961070 T^{8} + 29790998266 T^{9} + 185551547733 T^{10} + 995441411529 T^{11} + 4543283333340 T^{12} + 17156753327402 T^{13} + 51586792642943 T^{14} + 111989316816855 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 16 T + 449 T^{2} - 5808 T^{3} + 91382 T^{4} - 952529 T^{5} + 10887925 T^{6} - 91603311 T^{7} + 824334746 T^{8} - 5587801971 T^{9} + 40513968925 T^{10} - 216205984949 T^{11} + 1265260642262 T^{12} - 4905415316208 T^{13} + 23132648088089 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 18 T + 475 T^{2} + 6870 T^{3} + 105772 T^{4} + 1204725 T^{5} + 13774321 T^{6} + 126193127 T^{7} + 1141351048 T^{8} + 8454939509 T^{9} + 61832926969 T^{10} + 362336705175 T^{11} + 2131424370412 T^{12} + 9275359485090 T^{13} + 42967731530275 T^{14} + 109092808895814 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + 21 T + 408 T^{2} + 4842 T^{3} + 53937 T^{4} + 469505 T^{5} + 3864512 T^{6} + 29321392 T^{7} + 233979356 T^{8} + 2081818832 T^{9} + 19481004992 T^{10} + 168041004055 T^{11} + 1370629838097 T^{12} + 8736078517542 T^{13} + 52264915839768 T^{14} + 190997523326211 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 8 T + 361 T^{2} + 2398 T^{3} + 62862 T^{4} + 365975 T^{5} + 7401469 T^{6} + 38062427 T^{7} + 634690566 T^{8} + 2778557171 T^{9} + 39442428301 T^{10} + 142370496575 T^{11} + 1785170225742 T^{12} + 4971225680014 T^{13} + 54631655690329 T^{14} + 88379188152776 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 10 T + 266 T^{2} + 1825 T^{3} + 37702 T^{4} + 229058 T^{5} + 4155060 T^{6} + 23409469 T^{7} + 372961876 T^{8} + 1849348051 T^{9} + 25931729460 T^{10} + 112934527262 T^{11} + 1468495953862 T^{12} + 5615627928175 T^{13} + 64661263168586 T^{14} + 192039089861590 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 7 T + 397 T^{2} + 3500 T^{3} + 78910 T^{4} + 711159 T^{5} + 10765451 T^{6} + 84640014 T^{7} + 1058869618 T^{8} + 7025121162 T^{9} + 74163191939 T^{10} + 406631471133 T^{11} + 3744936110110 T^{12} + 13786642250500 T^{13} + 129795328227493 T^{14} + 189952356927389 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 26 T + 761 T^{2} - 12182 T^{3} + 203638 T^{4} - 2354455 T^{5} + 29224825 T^{6} - 274376689 T^{7} + 2923563118 T^{8} - 24419525321 T^{9} + 231489838825 T^{10} - 1659817786895 T^{11} + 12776704472758 T^{12} - 68025012207718 T^{13} + 378202762421321 T^{14} - 1150014707283754 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 3 T + 402 T^{2} + 2024 T^{3} + 73933 T^{4} + 595959 T^{5} + 8542284 T^{6} + 96913774 T^{7} + 827247384 T^{8} + 9400636078 T^{9} + 80374350156 T^{10} + 543915688407 T^{11} + 6545235332173 T^{12} + 17380776680168 T^{13} + 334854745981458 T^{14} + 242394853434339 T^{15} + 7837433594376961 T^{16} \)
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