Properties

Label 3025.2.a.bm
Level $3025$
Weight $2$
Character orbit 3025.a
Self dual yes
Analytic conductor $24.155$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 18 x^{5} + 16 x^{4} - 30 x^{3} - 12 x^{2} + 12 x + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 7 x^{6} + 18 x^{5} + 16 x^{4} - 30 x^{3} - 12 x^{2} + 12 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} - 3 \nu^{4} - 5 \nu^{3} + 12 \nu^{2} + 6 \nu - 6 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 3 \nu^{5} + 5 \nu^{4} - 12 \nu^{3} - 6 \nu^{2} + 8 \nu \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 5 \nu^{4} + 12 \nu^{3} + 8 \nu^{2} - 10 \nu - 4 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{5} + 10 \nu^{4} - 13 \nu^{3} - 26 \nu^{2} + 18 \nu + 12 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} - 6 \nu^{5} + 15 \nu^{4} + 11 \nu^{3} - 18 \nu^{2} - 4 \nu + 2 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 13 \nu^{4} - 26 \nu^{3} + 18 \nu^{2} + 12 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + 3 \beta_{4} + 2 \beta_{3} - \beta_{2} + 6 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(3 \beta_{7} + 3 \beta_{6} + \beta_{5} + 13 \beta_{4} + 9 \beta_{3} - 2 \beta_{2} + 14 \beta_{1} + 11\)
\(\nu^{5}\)\(=\)\(14 \beta_{7} + 14 \beta_{6} + 3 \beta_{5} + 42 \beta_{4} + 25 \beta_{3} - 9 \beta_{2} + 54 \beta_{1} + 25\)
\(\nu^{6}\)\(=\)\(45 \beta_{7} + 45 \beta_{6} + 14 \beta_{5} + 149 \beta_{4} + 88 \beta_{3} - 25 \beta_{2} + 162 \beta_{1} + 94\)
\(\nu^{7}\)\(=\)\(163 \beta_{7} + 165 \beta_{6} + 45 \beta_{5} + 489 \beta_{4} + 275 \beta_{3} - 88 \beta_{2} + 556 \beta_{1} + 279\)

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
3.29384
1.95882
1.20828
0.860597
−0.321622
−0.672032
−1.56247
−1.76541
−2.29384 0.0424059 3.26171 0 −0.0972724 1.13968 −2.89416 −2.99820 0
1.2 −0.958815 −0.899342 −1.08067 0 0.862303 −0.761645 2.95380 −2.19118 0
1.3 −0.208285 −1.89427 −1.95662 0 0.394547 −1.28881 0.824104 0.588246 0
1.4 0.139403 2.98582 −1.98057 0 0.416233 3.56959 −0.554904 5.91512 0
1.5 1.32162 2.02642 −0.253315 0 2.67816 0.348258 −2.97803 1.10639 0
1.6 1.67203 −3.10994 0.795692 0 −5.19992 3.08998 −2.01364 6.67173 0
1.7 2.56247 −1.79260 4.56626 0 −4.59347 3.80088 6.57595 0.213399 0
1.8 2.76541 1.64150 5.64751 0 4.53942 −1.89792 10.0869 −0.305488 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
\(5\) \(1\)
\(11\) \(-1\)

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 3025.2.a.bm 8
5.b even 2 1 3025.2.a.bj 8
11.b odd 2 1 3025.2.a.bi 8
11.d odd 10 2 275.2.h.c 16
55.d odd 2 1 3025.2.a.bn 8
55.h odd 10 2 275.2.h.e yes 16
55.l even 20 4 275.2.z.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.h.c 16 11.d odd 10 2
275.2.h.e yes 16 55.h odd 10 2
275.2.z.c 32 55.l even 20 4
3025.2.a.bi 8 11.b odd 2 1
3025.2.a.bj 8 5.b even 2 1
3025.2.a.bm 8 1.a even 1 1 trivial
3025.2.a.bn 8 55.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):

\( T_{2}^{8} - 5 T_{2}^{7} + 31 T_{2}^{5} - 34 T_{2}^{4} - 25 T_{2}^{3} + 34 T_{2}^{2} + 3 T_{2} - 1 \)
\(T_{3}^{8} + \cdots\)
\(T_{19}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + 3 T + 34 T^{2} - 25 T^{3} - 34 T^{4} + 31 T^{5} - 5 T^{7} + T^{8} \)
$3$ \( 4 - 90 T - 105 T^{2} + 67 T^{3} + 74 T^{4} - 15 T^{5} - 16 T^{6} + T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( -31 + 62 T + 122 T^{2} - 104 T^{3} - 86 T^{4} + 50 T^{5} + 10 T^{6} - 8 T^{7} + T^{8} \)
$11$ \( T^{8} \)
$13$ \( 3475 - 7525 T + 5340 T^{2} - 835 T^{3} - 604 T^{4} + 241 T^{5} - 4 T^{6} - 9 T^{7} + T^{8} \)
$17$ \( 439 - 4151 T - 1263 T^{2} + 4710 T^{3} - 1814 T^{4} + 6 T^{5} + 107 T^{6} - 19 T^{7} + T^{8} \)
$19$ \( 10480 - 5520 T - 5555 T^{2} + 1735 T^{3} + 1016 T^{4} - 119 T^{5} - 64 T^{6} + T^{7} + T^{8} \)
$23$ \( -121 - 1078 T + 778 T^{2} + 1430 T^{3} + 131 T^{4} - 252 T^{5} - 62 T^{6} + 2 T^{7} + T^{8} \)
$29$ \( -35495 - 34565 T + 5630 T^{2} + 8565 T^{3} + 566 T^{4} - 503 T^{5} - 62 T^{6} + 7 T^{7} + T^{8} \)
$31$ \( 33569 + 14183 T - 40606 T^{2} + 3465 T^{3} + 3456 T^{4} - 289 T^{5} - 100 T^{6} + 5 T^{7} + T^{8} \)
$37$ \( -4451 - 7920 T + 14810 T^{2} - 4876 T^{3} - 1021 T^{4} + 640 T^{5} - 54 T^{6} - 8 T^{7} + T^{8} \)
$41$ \( 442576 + 368384 T - 156464 T^{2} - 25956 T^{3} + 10064 T^{4} + 396 T^{5} - 197 T^{6} + T^{8} \)
$43$ \( -101 - 9376 T - 20940 T^{2} - 10950 T^{3} + 676 T^{4} + 732 T^{5} - 28 T^{6} - 14 T^{7} + T^{8} \)
$47$ \( -459469 - 371481 T + 52702 T^{2} + 49507 T^{3} + 1850 T^{4} - 1417 T^{5} - 108 T^{6} + 11 T^{7} + T^{8} \)
$53$ \( -48896 + 307600 T - 504485 T^{2} + 62287 T^{3} + 16944 T^{4} - 1745 T^{5} - 216 T^{6} + 11 T^{7} + T^{8} \)
$59$ \( -78605 + 340405 T + 285410 T^{2} - 70965 T^{3} - 13486 T^{4} + 3841 T^{5} - 140 T^{6} - 17 T^{7} + T^{8} \)
$61$ \( 3319739 - 770562 T - 679310 T^{2} + 22610 T^{3} + 23756 T^{4} - 46 T^{5} - 282 T^{6} - 2 T^{7} + T^{8} \)
$67$ \( 23994961 - 3514469 T - 1516857 T^{2} + 150560 T^{3} + 34476 T^{4} - 1940 T^{5} - 323 T^{6} + 7 T^{7} + T^{8} \)
$71$ \( 960859 + 887655 T + 112325 T^{2} - 71914 T^{3} - 6916 T^{4} + 2890 T^{5} - 99 T^{6} - 17 T^{7} + T^{8} \)
$73$ \( -466721 - 193916 T + 325508 T^{2} + 35058 T^{3} - 24991 T^{4} + 1350 T^{5} + 290 T^{6} - 34 T^{7} + T^{8} \)
$79$ \( -524695 - 1204485 T + 375760 T^{2} + 130915 T^{3} - 9364 T^{4} - 3517 T^{5} - 42 T^{6} + 23 T^{7} + T^{8} \)
$83$ \( -8329 + 75983 T - 134518 T^{2} + 72175 T^{3} - 6974 T^{4} - 2485 T^{5} + 568 T^{6} - 41 T^{7} + T^{8} \)
$89$ \( -278125 - 1576875 T + 211750 T^{2} + 195325 T^{3} + 11416 T^{4} - 3101 T^{5} - 258 T^{6} + 11 T^{7} + T^{8} \)
$97$ \( 85616 - 377392 T + 347248 T^{2} - 113740 T^{3} + 11136 T^{4} + 1230 T^{5} - 257 T^{6} + 2 T^{7} + T^{8} \)
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