Properties

Label 5054.2.a.bf
Level $5054$
Weight $2$
Character orbit 5054.a
Self dual yes
Analytic conductor $40.356$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.19520000000.1
Defining polynomial: \(x^{8} - 2 x^{7} - 12 x^{6} + 16 x^{5} + 50 x^{4} - 24 x^{3} - 72 x^{2} - 32 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 12 x^{6} + 16 x^{5} + 50 x^{4} - 24 x^{3} - 72 x^{2} - 32 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{4} - 2 \nu^{3} - 6 \nu^{2} + 8 \nu + 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{4} + 4 \nu^{3} + 4 \nu^{2} - 18 \nu - 4 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 10 \nu^{2} + 16 \nu - 2 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{5} - 2 \nu^{4} - 8 \nu^{3} + 8 \nu^{2} + 18 \nu + 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 2 \nu^{6} - 12 \nu^{5} + 17 \nu^{4} + 48 \nu^{3} - 32 \nu^{2} - 64 \nu - 16 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -2 \nu^{7} + 5 \nu^{6} + 22 \nu^{5} - 43 \nu^{4} - 84 \nu^{3} + 86 \nu^{2} + 116 \nu + 24 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-8 \beta_{5} + 8 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 10 \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(-14 \beta_{5} + 16 \beta_{4} + 12 \beta_{3} + 16 \beta_{2} + 42 \beta_{1} + 20\)
\(\nu^{6}\)\(=\)\(2 \beta_{7} + 4 \beta_{6} - 66 \beta_{5} + 70 \beta_{4} + 30 \beta_{3} + 56 \beta_{2} + 92 \beta_{1} + 102\)
\(\nu^{7}\)\(=\)\(4 \beta_{7} + 10 \beta_{6} - 148 \beta_{5} + 180 \beta_{4} + 122 \beta_{3} + 188 \beta_{2} + 326 \beta_{1} + 188\)

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
−1.91631
2.04552
−0.368905
2.35877
−0.236286
−2.15124
3.02989
−0.761439
−1.00000 −3.35992 1.00000 −1.35877 3.35992 −1.00000 −1.00000 8.28904 1.35877
1.2 −1.00000 −2.40760 1.00000 1.76144 2.40760 −1.00000 −1.00000 2.79656 −1.76144
1.3 −1.00000 −2.30521 1.00000 3.15124 2.30521 −1.00000 −1.00000 2.31400 −3.15124
1.4 −1.00000 −0.717767 1.00000 2.91631 0.717767 −1.00000 −1.00000 −2.48481 −2.91631
1.5 −1.00000 0.0295377 1.00000 −2.02989 −0.0295377 −1.00000 −1.00000 −2.99913 2.02989
1.6 −1.00000 0.578669 1.00000 1.36891 −0.578669 −1.00000 −1.00000 −2.66514 −1.36891
1.7 −1.00000 2.04815 1.00000 1.23629 −2.04815 −1.00000 −1.00000 1.19490 −1.23629
1.8 −1.00000 2.13415 1.00000 −1.04552 −2.13415 −1.00000 −1.00000 1.55458 1.04552
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(19\) \(1\)

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 5054.2.a.bf 8
19.b odd 2 1 5054.2.a.bi yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.bf 8 1.a even 1 1 trivial
5054.2.a.bi yes 8 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5054))\):

\(T_{3}^{8} + \cdots\)
\(T_{5}^{8} - \cdots\)
\(T_{13}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{8} \)
$3$ \( 1 - 34 T + 2 T^{2} + 98 T^{3} + 15 T^{4} - 38 T^{5} - 8 T^{6} + 4 T^{7} + T^{8} \)
$5$ \( -79 + 46 T + 122 T^{2} - 82 T^{3} - 50 T^{4} + 42 T^{5} + 2 T^{6} - 6 T^{7} + T^{8} \)
$7$ \( ( 1 + T )^{8} \)
$11$ \( -419 + 1004 T - 388 T^{2} - 428 T^{3} + 210 T^{4} + 68 T^{5} - 28 T^{6} - 4 T^{7} + T^{8} \)
$13$ \( 181 + 144 T - 888 T^{2} + 422 T^{3} + 355 T^{4} - 182 T^{5} - 38 T^{6} + 6 T^{7} + T^{8} \)
$17$ \( -779 + 3528 T - 1732 T^{2} - 1054 T^{3} + 515 T^{4} + 86 T^{5} - 42 T^{6} - 2 T^{7} + T^{8} \)
$19$ \( T^{8} \)
$23$ \( 1525 + 8100 T + 8200 T^{2} - 4700 T^{3} - 2190 T^{4} + 460 T^{5} + 80 T^{6} - 20 T^{7} + T^{8} \)
$29$ \( -7619 - 11776 T + 22292 T^{2} + 8192 T^{3} - 2030 T^{4} - 752 T^{5} + 12 T^{6} + 16 T^{7} + T^{8} \)
$31$ \( -919 + 2782 T + 3438 T^{2} - 3006 T^{3} - 1170 T^{4} + 274 T^{5} + 158 T^{6} + 22 T^{7} + T^{8} \)
$37$ \( -410699 - 33492 T + 216968 T^{2} - 64804 T^{3} - 270 T^{4} + 1796 T^{5} - 112 T^{6} - 12 T^{7} + T^{8} \)
$41$ \( 515341 + 223094 T - 148318 T^{2} - 80998 T^{3} - 7865 T^{4} + 1778 T^{5} + 452 T^{6} + 36 T^{7} + T^{8} \)
$43$ \( 3305 - 36080 T - 56550 T^{2} + 2560 T^{3} + 7215 T^{4} - 80 T^{5} - 190 T^{6} + T^{8} \)
$47$ \( 121 + 1166 T + 3562 T^{2} + 3718 T^{3} + 1350 T^{4} + 2 T^{5} - 78 T^{6} - 6 T^{7} + T^{8} \)
$53$ \( -24979 - 102536 T + 87272 T^{2} + 36992 T^{3} - 1910 T^{4} - 1512 T^{5} - 48 T^{6} + 16 T^{7} + T^{8} \)
$59$ \( -3744859 - 3276814 T - 363538 T^{2} + 189738 T^{3} + 15690 T^{4} - 2978 T^{5} - 218 T^{6} + 14 T^{7} + T^{8} \)
$61$ \( -411095 + 258950 T + 82470 T^{2} - 71270 T^{3} + 6270 T^{4} + 1650 T^{5} - 170 T^{6} - 10 T^{7} + T^{8} \)
$67$ \( -4692995 + 1048580 T + 666950 T^{2} - 193520 T^{3} - 4185 T^{4} + 4000 T^{5} - 130 T^{6} - 20 T^{7} + T^{8} \)
$71$ \( 14480 - 24320 T - 177920 T^{2} + 12480 T^{3} + 13400 T^{4} - 320 T^{5} - 240 T^{6} + T^{8} \)
$73$ \( -58519 - 37032 T + 33828 T^{2} + 14426 T^{3} - 2005 T^{4} - 1014 T^{5} - 2 T^{6} + 18 T^{7} + T^{8} \)
$79$ \( 1204961 - 628544 T - 540258 T^{2} + 61028 T^{3} + 29555 T^{4} - 908 T^{5} - 338 T^{6} + 4 T^{7} + T^{8} \)
$83$ \( 24520336 - 20068064 T + 4609152 T^{2} - 5312 T^{3} - 92180 T^{4} + 7312 T^{5} + 172 T^{6} - 36 T^{7} + T^{8} \)
$89$ \( -122399 - 150332 T + 25948 T^{2} + 53666 T^{3} + 915 T^{4} - 2194 T^{5} - 102 T^{6} + 18 T^{7} + T^{8} \)
$97$ \( -7416859 - 3247266 T + 712262 T^{2} + 273502 T^{3} - 13390 T^{4} - 5182 T^{5} - 58 T^{6} + 26 T^{7} + T^{8} \)
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