Properties

Label 4002.2.a.bk
Level 4002
Weight 2
Character orbit 4002.a
Self dual yes
Analytic conductor 31.956
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 30 x^{6} + 52 x^{5} + 267 x^{4} - 352 x^{3} - 632 x^{2} + 240 x + 288\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of 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.

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} - q^{6} + ( 1 - \beta_{2} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} - q^{3} + q^{4} + \beta_{1} q^{5} - q^{6} + ( 1 - \beta_{2} ) q^{7} + q^{8} + q^{9} + \beta_{1} q^{10} -\beta_{5} q^{11} - q^{12} + ( 2 - \beta_{4} ) q^{13} + ( 1 - \beta_{2} ) q^{14} -\beta_{1} q^{15} + q^{16} + ( \beta_{1} - \beta_{5} ) q^{17} + q^{18} + ( 1 - \beta_{6} + \beta_{7} ) q^{19} + \beta_{1} q^{20} + ( -1 + \beta_{2} ) q^{21} -\beta_{5} q^{22} - q^{23} - q^{24} + ( 4 - \beta_{3} + \beta_{5} + \beta_{6} ) q^{25} + ( 2 - \beta_{4} ) q^{26} - q^{27} + ( 1 - \beta_{2} ) q^{28} - q^{29} -\beta_{1} q^{30} + ( 2 + \beta_{5} - \beta_{7} ) q^{31} + q^{32} + \beta_{5} q^{33} + ( \beta_{1} - \beta_{5} ) q^{34} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{35} + q^{36} + ( 3 - \beta_{3} + \beta_{4} + \beta_{6} ) q^{37} + ( 1 - \beta_{6} + \beta_{7} ) q^{38} + ( -2 + \beta_{4} ) q^{39} + \beta_{1} q^{40} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{41} + ( -1 + \beta_{2} ) q^{42} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{43} -\beta_{5} q^{44} + \beta_{1} q^{45} - q^{46} + ( -2 + \beta_{1} + \beta_{4} - \beta_{7} ) q^{47} - q^{48} + ( 4 + 2 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{49} + ( 4 - \beta_{3} + \beta_{5} + \beta_{6} ) q^{50} + ( -\beta_{1} + \beta_{5} ) q^{51} + ( 2 - \beta_{4} ) q^{52} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{7} ) q^{53} - q^{54} + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( 1 - \beta_{2} ) q^{56} + ( -1 + \beta_{6} - \beta_{7} ) q^{57} - q^{58} + ( -1 + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{59} -\beta_{1} q^{60} + ( 2 + \beta_{5} ) q^{61} + ( 2 + \beta_{5} - \beta_{7} ) q^{62} + ( 1 - \beta_{2} ) q^{63} + q^{64} + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{65} + \beta_{5} q^{66} + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{67} + ( \beta_{1} - \beta_{5} ) q^{68} + q^{69} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{7} ) q^{70} + ( 2 - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{71} + q^{72} + ( 3 - \beta_{1} + \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{73} + ( 3 - \beta_{3} + \beta_{4} + \beta_{6} ) q^{74} + ( -4 + \beta_{3} - \beta_{5} - \beta_{6} ) q^{75} + ( 1 - \beta_{6} + \beta_{7} ) q^{76} + ( 1 - \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{77} + ( -2 + \beta_{4} ) q^{78} + ( 1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} ) q^{79} + \beta_{1} q^{80} + q^{81} + ( -1 + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{82} + ( 5 - \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{83} + ( -1 + \beta_{2} ) q^{84} + ( 6 - \beta_{1} + \beta_{4} + \beta_{7} ) q^{85} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{86} + q^{87} -\beta_{5} q^{88} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} ) q^{89} + \beta_{1} q^{90} + ( 5 - 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{91} - q^{92} + ( -2 - \beta_{5} + \beta_{7} ) q^{93} + ( -2 + \beta_{1} + \beta_{4} - \beta_{7} ) q^{94} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{95} - q^{96} + ( \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{97} + ( 4 + 2 \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{98} -\beta_{5} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} - 8q^{3} + 8q^{4} + 2q^{5} - 8q^{6} + 5q^{7} + 8q^{8} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{2} - 8q^{3} + 8q^{4} + 2q^{5} - 8q^{6} + 5q^{7} + 8q^{8} + 8q^{9} + 2q^{10} + 3q^{11} - 8q^{12} + 13q^{13} + 5q^{14} - 2q^{15} + 8q^{16} + 5q^{17} + 8q^{18} + 7q^{19} + 2q^{20} - 5q^{21} + 3q^{22} - 8q^{23} - 8q^{24} + 24q^{25} + 13q^{26} - 8q^{27} + 5q^{28} - 8q^{29} - 2q^{30} + 15q^{31} + 8q^{32} - 3q^{33} + 5q^{34} + 9q^{35} + 8q^{36} + 22q^{37} + 7q^{38} - 13q^{39} + 2q^{40} - 8q^{41} - 5q^{42} - q^{43} + 3q^{44} + 2q^{45} - 8q^{46} - 9q^{47} - 8q^{48} + 33q^{49} + 24q^{50} - 5q^{51} + 13q^{52} + 14q^{53} - 8q^{54} - 17q^{55} + 5q^{56} - 7q^{57} - 8q^{58} - 4q^{59} - 2q^{60} + 13q^{61} + 15q^{62} + 5q^{63} + 8q^{64} + 21q^{65} - 3q^{66} - 3q^{67} + 5q^{68} + 8q^{69} + 9q^{70} + 7q^{71} + 8q^{72} + 16q^{73} + 22q^{74} - 24q^{75} + 7q^{76} - 13q^{78} + 14q^{79} + 2q^{80} + 8q^{81} - 8q^{82} + 36q^{83} - 5q^{84} + 47q^{85} - q^{86} + 8q^{87} + 3q^{88} - 12q^{89} + 2q^{90} + 20q^{91} - 8q^{92} - 15q^{93} - 9q^{94} + 7q^{95} - 8q^{96} + 10q^{97} + 33q^{98} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 30 x^{6} + 52 x^{5} + 267 x^{4} - 352 x^{3} - 632 x^{2} + 240 x + 288\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 73 \nu^{7} - 1010 \nu^{6} - 2490 \nu^{5} + 23896 \nu^{4} + 27735 \nu^{3} - 142756 \nu^{2} - 87080 \nu + 110232 \)\()/21048\)
\(\beta_{3}\)\(=\)\((\)\( 45 \nu^{7} - 94 \nu^{6} - 718 \nu^{5} + 2092 \nu^{4} + 1383 \nu^{3} - 14092 \nu^{2} + 1968 \nu + 26528 \)\()/7016\)
\(\beta_{4}\)\(=\)\((\)\( 101 \nu^{7} - 172 \nu^{6} - 2508 \nu^{5} + 1850 \nu^{4} + 17253 \nu^{3} + 9220 \nu^{2} - 28792 \nu - 30576 \)\()/10524\)
\(\beta_{5}\)\(=\)\((\)\( -68 \nu^{7} + 220 \nu^{6} + 1923 \nu^{5} - 6221 \nu^{4} - 13257 \nu^{3} + 47741 \nu^{2} - 986 \nu - 51936 \)\()/5262\)
\(\beta_{6}\)\(=\)\((\)\( 407 \nu^{7} - 1162 \nu^{6} - 9846 \nu^{5} + 31160 \nu^{4} + 57177 \nu^{3} - 212192 \nu^{2} + 9848 \nu + 97896 \)\()/21048\)
\(\beta_{7}\)\(=\)\((\)\( -269 \nu^{7} + 406 \nu^{6} + 7878 \nu^{5} - 11648 \nu^{4} - 64863 \nu^{3} + 89228 \nu^{2} + 110548 \nu - 71736 \)\()/10524\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} - \beta_{3} + 9\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + 2 \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 12 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{7} + 16 \beta_{6} + 16 \beta_{5} - 4 \beta_{4} - 14 \beta_{3} - \beta_{1} + 118\)
\(\nu^{5}\)\(=\)\(-14 \beta_{7} + 5 \beta_{6} + 37 \beta_{5} - 19 \beta_{4} + 24 \beta_{3} + 15 \beta_{2} + 155 \beta_{1} + 22\)
\(\nu^{6}\)\(=\)\(-32 \beta_{7} + 235 \beta_{6} + 245 \beta_{5} - 96 \beta_{4} - 189 \beta_{3} - 18 \beta_{2} - 12 \beta_{1} + 1637\)
\(\nu^{7}\)\(=\)\(-213 \beta_{7} + 140 \beta_{6} + 610 \beta_{5} - 287 \beta_{4} + 451 \beta_{3} + 171 \beta_{2} + 2082 \beta_{1} + 483\)

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
−3.78725
−3.42323
−1.12580
−0.652958
0.815784
2.56063
3.65713
3.95569
1.00000 −1.00000 1.00000 −3.78725 −1.00000 3.57193 1.00000 1.00000 −3.78725
1.2 1.00000 −1.00000 1.00000 −3.42323 −1.00000 −1.25167 1.00000 1.00000 −3.42323
1.3 1.00000 −1.00000 1.00000 −1.12580 −1.00000 −0.350474 1.00000 1.00000 −1.12580
1.4 1.00000 −1.00000 1.00000 −0.652958 −1.00000 −3.89658 1.00000 1.00000 −0.652958
1.5 1.00000 −1.00000 1.00000 0.815784 −1.00000 2.48945 1.00000 1.00000 0.815784
1.6 1.00000 −1.00000 1.00000 2.56063 −1.00000 3.94138 1.00000 1.00000 2.56063
1.7 1.00000 −1.00000 1.00000 3.65713 −1.00000 −4.08293 1.00000 1.00000 3.65713
1.8 1.00000 −1.00000 1.00000 3.95569 −1.00000 4.57890 1.00000 1.00000 3.95569
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 4002.2.a.bk 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4002.2.a.bk 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4002))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( 1 - 2 T + 10 T^{2} - 18 T^{3} + 67 T^{4} - 102 T^{5} + 458 T^{6} - 790 T^{7} + 2768 T^{8} - 3950 T^{9} + 11450 T^{10} - 12750 T^{11} + 41875 T^{12} - 56250 T^{13} + 156250 T^{14} - 156250 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 5 T + 24 T^{2} - 73 T^{3} + 288 T^{4} - 805 T^{5} + 2824 T^{6} - 7665 T^{7} + 24094 T^{8} - 53655 T^{9} + 138376 T^{10} - 276115 T^{11} + 691488 T^{12} - 1226911 T^{13} + 2823576 T^{14} - 4117715 T^{15} + 5764801 T^{16} \)
$11$ \( 1 - 3 T + 39 T^{2} - 154 T^{3} + 974 T^{4} - 3596 T^{5} + 17345 T^{6} - 54121 T^{7} + 226690 T^{8} - 595331 T^{9} + 2098745 T^{10} - 4786276 T^{11} + 14260334 T^{12} - 24801854 T^{13} + 69090879 T^{14} - 58461513 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 13 T + 105 T^{2} - 522 T^{3} + 2068 T^{4} - 7084 T^{5} + 31911 T^{6} - 147353 T^{7} + 619590 T^{8} - 1915589 T^{9} + 5392959 T^{10} - 15563548 T^{11} + 59064148 T^{12} - 193814946 T^{13} + 506814945 T^{14} - 815730721 T^{15} + 815730721 T^{16} \)
$17$ \( 1 - 5 T + 80 T^{2} - 331 T^{3} + 3160 T^{4} - 11297 T^{5} + 83360 T^{6} - 263535 T^{7} + 1628590 T^{8} - 4480095 T^{9} + 24091040 T^{10} - 55502161 T^{11} + 263926360 T^{12} - 469972667 T^{13} + 1931005520 T^{14} - 2051693365 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 - 7 T + 62 T^{2} - 259 T^{3} + 1524 T^{4} - 6555 T^{5} + 37058 T^{6} - 190391 T^{7} + 895158 T^{8} - 3617429 T^{9} + 13377938 T^{10} - 44960745 T^{11} + 198609204 T^{12} - 641309641 T^{13} + 2916844622 T^{14} - 6257102173 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 + T )^{8} \)
$29$ \( ( 1 + T )^{8} \)
$31$ \( 1 - 15 T + 273 T^{2} - 2596 T^{3} + 27084 T^{4} - 189138 T^{5} + 1460487 T^{6} - 8237389 T^{7} + 52912694 T^{8} - 255359059 T^{9} + 1403528007 T^{10} - 5634610158 T^{11} + 25012642764 T^{12} - 74321275996 T^{13} + 242288504913 T^{14} - 412689211665 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 22 T + 352 T^{2} - 4098 T^{3} + 40849 T^{4} - 345810 T^{5} + 2646836 T^{6} - 18170950 T^{7} + 115687748 T^{8} - 672325150 T^{9} + 3623518484 T^{10} - 17516313930 T^{11} + 76557602689 T^{12} - 284171535786 T^{13} + 903135695968 T^{14} - 2088501296926 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 8 T + 194 T^{2} + 1052 T^{3} + 17737 T^{4} + 77524 T^{5} + 1130186 T^{6} + 4344968 T^{7} + 54209140 T^{8} + 178143688 T^{9} + 1899842666 T^{10} + 5343031604 T^{11} + 50120522857 T^{12} + 121880723452 T^{13} + 921520222754 T^{14} + 1558034191048 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + T + 216 T^{2} - 27 T^{3} + 23428 T^{4} - 14131 T^{5} + 1664632 T^{6} - 1274687 T^{7} + 83748086 T^{8} - 54811541 T^{9} + 3077904568 T^{10} - 1123513417 T^{11} + 80095669828 T^{12} - 3969227961 T^{13} + 1365414418584 T^{14} + 271818611107 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 9 T + 304 T^{2} + 2189 T^{3} + 41456 T^{4} + 247729 T^{5} + 3441408 T^{6} + 17318733 T^{7} + 193709886 T^{8} + 813980451 T^{9} + 7602070272 T^{10} + 25719967967 T^{11} + 202292055536 T^{12} + 502036220323 T^{13} + 3276881460016 T^{14} + 4559608084167 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 14 T + 300 T^{2} - 2958 T^{3} + 38988 T^{4} - 313626 T^{5} + 3319444 T^{6} - 23179914 T^{7} + 206455430 T^{8} - 1228535442 T^{9} + 9324318196 T^{10} - 46691698002 T^{11} + 307634073228 T^{12} - 1237022268294 T^{13} + 6649308338700 T^{14} - 16445955957718 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 4 T + 222 T^{2} + 896 T^{3} + 27357 T^{4} + 102548 T^{5} + 2428622 T^{6} + 8399528 T^{7} + 163415196 T^{8} + 495572152 T^{9} + 8454033182 T^{10} + 21061205692 T^{11} + 331494644877 T^{12} + 640572171904 T^{13} + 9364078468302 T^{14} + 9954605939276 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - 13 T + 509 T^{2} - 5236 T^{3} + 110804 T^{4} - 921462 T^{5} + 13630243 T^{6} - 91869197 T^{7} + 1041659494 T^{8} - 5604021017 T^{9} + 50718134203 T^{10} - 209154366222 T^{11} + 1534174566164 T^{12} - 4422306232036 T^{13} + 26223870549749 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 3 T + 281 T^{2} - 216 T^{3} + 37366 T^{4} - 107138 T^{5} + 3826511 T^{6} - 10693043 T^{7} + 304683698 T^{8} - 716433881 T^{9} + 17177207879 T^{10} - 32223146294 T^{11} + 752966787286 T^{12} - 291627023112 T^{13} + 25418805389489 T^{14} + 18182134815969 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 7 T + 317 T^{2} - 2068 T^{3} + 46432 T^{4} - 298298 T^{5} + 4360091 T^{6} - 28261253 T^{7} + 327322606 T^{8} - 2006548963 T^{9} + 21979218731 T^{10} - 106764135478 T^{11} + 1179915172192 T^{12} - 3731146297868 T^{13} + 40607790002957 T^{14} - 63665841108737 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 - 16 T + 356 T^{2} - 4196 T^{3} + 53148 T^{4} - 507284 T^{5} + 4711356 T^{6} - 40976280 T^{7} + 340756294 T^{8} - 2991268440 T^{9} + 25106816124 T^{10} - 197342099828 T^{11} + 1509309712668 T^{12} - 8698608404228 T^{13} + 53874984558884 T^{14} - 176758376305552 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 14 T + 480 T^{2} - 5706 T^{3} + 111164 T^{4} - 1111882 T^{5} + 15864416 T^{6} - 132890126 T^{7} + 1515327366 T^{8} - 10498319954 T^{9} + 99009820256 T^{10} - 548201189398 T^{11} + 4329846804284 T^{12} - 17557683812694 T^{13} + 116681978650080 T^{14} - 268854725806226 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 36 T + 1068 T^{2} - 21540 T^{3} + 379932 T^{4} - 5414068 T^{5} + 69163188 T^{6} - 750478036 T^{7} + 7372769638 T^{8} - 62289676988 T^{9} + 476465202132 T^{10} - 3095693699516 T^{11} + 18030934814172 T^{12} - 84846935450220 T^{13} + 349172318758092 T^{14} - 976897835626572 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 12 T + 480 T^{2} + 5664 T^{3} + 113508 T^{4} + 1285880 T^{5} + 17169344 T^{6} + 176415740 T^{7} + 1808026422 T^{8} + 15701000860 T^{9} + 135998373824 T^{10} + 906505537720 T^{11} + 7121746291428 T^{12} + 31628112719136 T^{13} + 238551019661280 T^{14} + 530776018746348 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - 10 T + 524 T^{2} - 6122 T^{3} + 138388 T^{4} - 1555974 T^{5} + 24094964 T^{6} - 228532070 T^{7} + 2849699030 T^{8} - 22167610790 T^{9} + 226709516276 T^{10} - 1420095458502 T^{11} + 12251390139028 T^{12} - 52571697053354 T^{13} + 436477330582796 T^{14} - 807982844781130 T^{15} + 7837433594376961 T^{16} \)
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