Properties

Label 585.2.j.i
Level $585$
Weight $2$
Character orbit 585.j
Analytic conductor $4.671$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 2 x^{9} + 10 x^{8} - 8 x^{7} + 50 x^{6} - 42 x^{5} + 124 x^{4} - 12 x^{3} + 96 x^{2} - 36 x + 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{4} - \beta_{8} ) q^{4} - q^{5} + \beta_{7} q^{7} + ( -1 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{4} - \beta_{8} ) q^{4} - q^{5} + \beta_{7} q^{7} + ( -1 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{8} + ( \beta_{1} + \beta_{3} ) q^{10} + ( -\beta_{1} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{9} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{6} - \beta_{8} ) q^{13} + ( 1 - \beta_{2} + \beta_{6} - \beta_{9} ) q^{14} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{16} + ( -\beta_{1} + 2 \beta_{8} ) q^{17} + ( 1 - \beta_{4} - \beta_{8} ) q^{19} + ( 1 - \beta_{4} + \beta_{8} ) q^{20} + ( -2 - \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{22} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} ) q^{23} + q^{25} + ( 2 - \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{26} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{8} ) q^{28} + ( -\beta_{2} + 3 \beta_{4} + \beta_{7} + \beta_{9} ) q^{29} + ( 2 - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{9} ) q^{31} + ( 2 - 2 \beta_{4} - 2 \beta_{7} + 2 \beta_{8} ) q^{32} + ( -1 + 2 \beta_{2} - 3 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{34} -\beta_{7} q^{35} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{37} + ( -1 - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{38} + ( 1 + \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} ) q^{40} + ( -\beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{41} + ( -2 - 3 \beta_{1} + 2 \beta_{4} + \beta_{8} ) q^{43} + ( -2 - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} + 2 \beta_{6} ) q^{44} + ( 2 + \beta_{1} - 2 \beta_{4} - \beta_{6} - \beta_{7} + 3 \beta_{8} ) q^{46} + ( -3 + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} + 2 \beta_{9} ) q^{47} + ( 4 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{5} - 3 \beta_{8} ) q^{49} + ( -\beta_{1} - \beta_{3} ) q^{50} + ( -2 + 2 \beta_{1} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{52} + ( 4 - 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{9} ) q^{53} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{7} - \beta_{9} ) q^{55} + ( -1 + \beta_{1} + \beta_{4} + \beta_{7} + 2 \beta_{8} ) q^{56} + ( 2 - 4 \beta_{1} - 2 \beta_{4} + \beta_{6} + 2 \beta_{8} ) q^{58} + ( 3 - 3 \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} ) q^{59} + ( -2 - \beta_{1} + 2 \beta_{4} + \beta_{6} - \beta_{7} ) q^{61} + ( -3 \beta_{1} + \beta_{2} - 5 \beta_{4} + 3 \beta_{5} + 3 \beta_{8} ) q^{62} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{64} + ( -\beta_{1} + \beta_{2} - \beta_{6} + \beta_{8} ) q^{65} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{7} + \beta_{9} ) q^{67} + ( \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} + 2 \beta_{9} ) q^{68} + ( -1 + \beta_{2} - \beta_{6} + \beta_{9} ) q^{70} + ( 2 + \beta_{1} - 2 \beta_{4} + \beta_{6} - 2 \beta_{8} ) q^{71} + ( -2 - \beta_{2} - 6 \beta_{3} - \beta_{5} + \beta_{6} ) q^{73} + ( -4 + 4 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{8} ) q^{74} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{76} + ( -7 + \beta_{2} + \beta_{3} - 3 \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{77} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{79} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{80} + ( -2 - 4 \beta_{1} + 2 \beta_{4} + \beta_{6} - 2 \beta_{7} ) q^{82} + ( -3 + 3 \beta_{2} + \beta_{3} + \beta_{5} - 3 \beta_{6} ) q^{83} + ( \beta_{1} - 2 \beta_{8} ) q^{85} + ( -8 + \beta_{2} + 3 \beta_{3} + 2 \beta_{5} - \beta_{6} ) q^{86} + ( 2 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} + 6 \beta_{8} ) q^{88} + ( 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + 2 \beta_{4} ) q^{89} + ( -1 + \beta_{1} + 2 \beta_{5} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{91} + ( 4 + 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} + 2 \beta_{9} ) q^{92} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + 3 \beta_{7} - 4 \beta_{8} + 3 \beta_{9} ) q^{94} + ( -1 + \beta_{4} + \beta_{8} ) q^{95} + ( 2 - \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{97} + ( 7 - 7 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} - 3 \beta_{8} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{2} - 6 q^{4} - 10 q^{5} - q^{7} - 12 q^{8} + O(q^{10}) \) \( 10 q + 2 q^{2} - 6 q^{4} - 10 q^{5} - q^{7} - 12 q^{8} - 2 q^{10} + 8 q^{11} + q^{13} + 8 q^{14} - 4 q^{16} + 4 q^{19} + 6 q^{20} - 14 q^{22} - 6 q^{23} + 10 q^{25} + 10 q^{26} + 2 q^{28} + 16 q^{29} + 18 q^{31} + 14 q^{32} + q^{35} + 4 q^{37} - 12 q^{38} + 12 q^{40} + 6 q^{41} - 15 q^{43} - 28 q^{44} + 16 q^{46} - 20 q^{47} - 10 q^{49} + 2 q^{50} - 22 q^{52} + 40 q^{53} - 8 q^{55} - 2 q^{56} + 4 q^{58} + 12 q^{59} - 11 q^{61} - 22 q^{62} + 8 q^{64} - q^{65} - 5 q^{67} + 50 q^{68} - 8 q^{70} + 10 q^{71} + 2 q^{73} - 26 q^{74} - 24 q^{76} - 84 q^{77} - 34 q^{79} + 4 q^{80} - 16 q^{82} - 32 q^{83} - 88 q^{86} - 20 q^{88} + 4 q^{89} - q^{91} + 68 q^{92} + 16 q^{94} - 4 q^{95} + 11 q^{97} + 30 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 10 x^{8} - 8 x^{7} + 50 x^{6} - 42 x^{5} + 124 x^{4} - 12 x^{3} + 96 x^{2} - 36 x + 36\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 355 \nu^{9} - 2199 \nu^{8} + 7398 \nu^{7} - 18528 \nu^{6} + 39184 \nu^{5} - 87134 \nu^{4} + 143476 \nu^{3} - 156238 \nu^{2} + 124656 \nu - 55404 \)\()/40746\)
\(\beta_{3}\)\(=\)\((\)\( 4282 \nu^{9} - 12311 \nu^{8} + 50191 \nu^{7} - 64307 \nu^{6} + 231068 \nu^{5} - 318192 \nu^{4} + 721480 \nu^{3} - 289782 \nu^{2} + 136368 \nu - 235800 \)\()/448206\)
\(\beta_{4}\)\(=\)\((\)\( -6550 \nu^{9} + 8818 \nu^{8} - 53189 \nu^{7} + 2209 \nu^{6} - 263193 \nu^{5} + 44032 \nu^{4} - 494008 \nu^{3} - 642880 \nu^{2} - 339018 \nu + 99432 \)\()/448206\)
\(\beta_{5}\)\(=\)\((\)\( -8029 \nu^{9} + 19682 \nu^{8} - 80242 \nu^{7} + 81275 \nu^{6} - 369416 \nu^{5} + 508704 \nu^{4} - 959878 \nu^{3} + 463284 \nu^{2} - 218016 \nu + 1426266 \)\()/448206\)
\(\beta_{6}\)\(=\)\((\)\( -9476 \nu^{9} + 17684 \nu^{8} - 89335 \nu^{7} + 36452 \nu^{6} - 354900 \nu^{5} + 123782 \nu^{4} - 621080 \nu^{3} - 732992 \nu^{2} + 907392 \nu - 408504 \)\()/448206\)
\(\beta_{7}\)\(=\)\((\)\( -10145 \nu^{9} - 7555 \nu^{8} - 72631 \nu^{7} - 139765 \nu^{6} - 519015 \nu^{5} - 700936 \nu^{4} - 1168190 \nu^{3} - 1844240 \nu^{2} - 2260776 \nu - 953850 \)\()/448206\)
\(\beta_{8}\)\(=\)\((\)\( -5301 \nu^{9} + 6361 \nu^{8} - 43172 \nu^{7} - 3447 \nu^{6} - 217077 \nu^{5} - 19472 \nu^{4} - 414542 \nu^{3} - 551312 \nu^{2} - 311802 \nu - 297390 \)\()/149402\)
\(\beta_{9}\)\(=\)\((\)\( -9385 \nu^{9} + 23127 \nu^{8} - 94287 \nu^{7} + 111339 \nu^{6} - 434076 \nu^{5} + 597744 \nu^{4} - 956678 \nu^{3} + 544374 \nu^{2} - 256176 \nu + 675982 \)\()/149402\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{5} - 3 \beta_{4} + \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{5} + 5 \beta_{3} - \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(-7 \beta_{8} + \beta_{7} + \beta_{6} + 14 \beta_{4} - \beta_{1} - 14\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 10 \beta_{8} + 2 \beta_{7} - 10 \beta_{5} + 10 \beta_{4} - 30 \beta_{3} + 8 \beta_{2} - 20 \beta_{1}\)
\(\nu^{6}\)\(=\)\(10 \beta_{9} - 12 \beta_{6} - 46 \beta_{5} - 58 \beta_{3} + 12 \beta_{2} + 76\)
\(\nu^{7}\)\(=\)\(82 \beta_{8} - 22 \beta_{7} - 56 \beta_{6} - 84 \beta_{4} + 110 \beta_{1} + 84\)
\(\nu^{8}\)\(=\)\(-78 \beta_{9} + 304 \beta_{8} - 78 \beta_{7} + 304 \beta_{5} - 446 \beta_{4} + 414 \beta_{3} - 104 \beta_{2} + 110 \beta_{1}\)
\(\nu^{9}\)\(=\)\(-182 \beta_{9} + 382 \beta_{6} + 622 \beta_{5} + 1268 \beta_{3} - 382 \beta_{2} - 660\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
−1.06141 + 1.83842i
−0.473183 + 0.819577i
0.313396 0.542817i
0.905157 1.56778i
1.31604 2.27945i
−1.06141 1.83842i
−0.473183 0.819577i
0.313396 + 0.542817i
0.905157 + 1.56778i
1.31604 + 2.27945i
−1.06141 1.83842i 0 −1.25320 + 2.17061i −1.00000 0 −0.733534 + 1.27052i 1.07500 0 1.06141 + 1.83842i
406.2 −0.473183 0.819577i 0 0.552196 0.956432i −1.00000 0 0.781529 1.35365i −2.93789 0 0.473183 + 0.819577i
406.3 0.313396 + 0.542817i 0 0.803566 1.39182i −1.00000 0 −2.21563 + 3.83759i 2.26092 0 −0.313396 0.542817i
406.4 0.905157 + 1.56778i 0 −0.638619 + 1.10612i −1.00000 0 2.21251 3.83219i 1.30843 0 −0.905157 1.56778i
406.5 1.31604 + 2.27945i 0 −2.46394 + 4.26767i −1.00000 0 −0.544875 + 0.943751i −7.70645 0 −1.31604 2.27945i
451.1 −1.06141 + 1.83842i 0 −1.25320 2.17061i −1.00000 0 −0.733534 1.27052i 1.07500 0 1.06141 1.83842i
451.2 −0.473183 + 0.819577i 0 0.552196 + 0.956432i −1.00000 0 0.781529 + 1.35365i −2.93789 0 0.473183 0.819577i
451.3 0.313396 0.542817i 0 0.803566 + 1.39182i −1.00000 0 −2.21563 3.83759i 2.26092 0 −0.313396 + 0.542817i
451.4 0.905157 1.56778i 0 −0.638619 1.10612i −1.00000 0 2.21251 + 3.83219i 1.30843 0 −0.905157 + 1.56778i
451.5 1.31604 2.27945i 0 −2.46394 4.26767i −1.00000 0 −0.544875 0.943751i −7.70645 0 −1.31604 + 2.27945i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.i yes 10
3.b odd 2 1 585.2.j.h 10
13.c even 3 1 inner 585.2.j.i yes 10
13.c even 3 1 7605.2.a.cm 5
13.e even 6 1 7605.2.a.cn 5
39.h odd 6 1 7605.2.a.cl 5
39.i odd 6 1 585.2.j.h 10
39.i odd 6 1 7605.2.a.co 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.j.h 10 3.b odd 2 1
585.2.j.h 10 39.i odd 6 1
585.2.j.i yes 10 1.a even 1 1 trivial
585.2.j.i yes 10 13.c even 3 1 inner
7605.2.a.cl 5 39.h odd 6 1
7605.2.a.cm 5 13.c even 3 1
7605.2.a.cn 5 13.e even 6 1
7605.2.a.co 5 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 36 - 36 T + 96 T^{2} - 12 T^{3} + 124 T^{4} - 42 T^{5} + 50 T^{6} - 8 T^{7} + 10 T^{8} - 2 T^{9} + T^{10} \)
$3$ \( T^{10} \)
$5$ \( ( 1 + T )^{10} \)
$7$ \( 2401 + 2303 T + 3287 T^{2} + 1122 T^{3} + 1469 T^{4} + 439 T^{5} + 459 T^{6} + 22 T^{7} + 23 T^{8} + T^{9} + T^{10} \)
$11$ \( 576 - 1152 T + 3840 T^{2} + 3072 T^{3} + 3904 T^{4} + 744 T^{5} + 464 T^{6} - 128 T^{7} + 64 T^{8} - 8 T^{9} + T^{10} \)
$13$ \( 371293 - 28561 T - 46137 T^{2} + 17914 T^{3} + 2405 T^{4} - 2013 T^{5} + 185 T^{6} + 106 T^{7} - 21 T^{8} - T^{9} + T^{10} \)
$17$ \( 412164 + 365940 T + 333888 T^{2} + 56220 T^{3} + 28696 T^{4} + 1342 T^{5} + 1930 T^{6} + 28 T^{7} + 50 T^{8} + T^{10} \)
$19$ \( 144 + 48 T + 448 T^{2} - 336 T^{3} + 1216 T^{4} - 332 T^{5} + 212 T^{6} - 40 T^{7} + 24 T^{8} - 4 T^{9} + T^{10} \)
$23$ \( 5184 - 8640 T + 11808 T^{2} - 7200 T^{3} + 4128 T^{4} - 792 T^{5} + 496 T^{6} - 48 T^{7} + 56 T^{8} + 6 T^{9} + T^{10} \)
$29$ \( 66564 - 99072 T + 123720 T^{2} - 65256 T^{3} + 34864 T^{4} - 6694 T^{5} + 5220 T^{6} - 1112 T^{7} + 198 T^{8} - 16 T^{9} + T^{10} \)
$31$ \( ( 603 + 693 T + 178 T^{2} - 30 T^{3} - 9 T^{4} + T^{5} )^{2} \)
$37$ \( 20736 + 34560 T + 65664 T^{2} + 4992 T^{3} + 19072 T^{4} + 5648 T^{5} + 3632 T^{6} + 368 T^{7} + 80 T^{8} - 4 T^{9} + T^{10} \)
$41$ \( 141562404 - 25414128 T + 11272968 T^{2} - 1127304 T^{3} + 456036 T^{4} - 41538 T^{5} + 10852 T^{6} - 540 T^{7} + 134 T^{8} - 6 T^{9} + T^{10} \)
$43$ \( 4239481 + 4585393 T + 4016507 T^{2} + 1143506 T^{3} + 307459 T^{4} + 51011 T^{5} + 9997 T^{6} + 1366 T^{7} + 195 T^{8} + 15 T^{9} + T^{10} \)
$47$ \( ( 7434 + 618 T - 752 T^{2} - 90 T^{3} + 10 T^{4} + T^{5} )^{2} \)
$53$ \( ( 15552 - 10368 T + 1872 T^{2} - 20 T^{4} + T^{5} )^{2} \)
$59$ \( 7584516 - 892296 T + 1195560 T^{2} - 103032 T^{3} + 137376 T^{4} - 11610 T^{5} + 6192 T^{6} - 288 T^{7} + 186 T^{8} - 12 T^{9} + T^{10} \)
$61$ \( 9138529 - 1048981 T + 1347747 T^{2} + 225526 T^{3} + 126725 T^{4} + 16341 T^{5} + 5009 T^{6} + 658 T^{7} + 135 T^{8} + 11 T^{9} + T^{10} \)
$67$ \( 289 + 1343 T + 6309 T^{2} + 1180 T^{3} + 3407 T^{4} - 597 T^{5} + 1877 T^{6} - 212 T^{7} + 69 T^{8} + 5 T^{9} + T^{10} \)
$71$ \( 26244 - 83592 T + 208908 T^{2} - 175536 T^{3} + 115584 T^{4} - 17946 T^{5} + 4540 T^{6} - 488 T^{7} + 122 T^{8} - 10 T^{9} + T^{10} \)
$73$ \( ( 9693 + 12105 T + 144 T^{2} - 236 T^{3} - T^{4} + T^{5} )^{2} \)
$79$ \( ( 2349 - 1899 T - 1086 T^{2} - 26 T^{3} + 17 T^{4} + T^{5} )^{2} \)
$83$ \( ( 6264 - 1368 T - 2060 T^{2} - 116 T^{3} + 16 T^{4} + T^{5} )^{2} \)
$89$ \( 2916 - 3888 T + 9396 T^{2} - 5400 T^{3} + 13212 T^{4} - 7434 T^{5} + 10644 T^{6} + 252 T^{7} + 118 T^{8} - 4 T^{9} + T^{10} \)
$97$ \( 58967041 - 62591529 T + 50266827 T^{2} - 15169466 T^{3} + 3460075 T^{4} - 445423 T^{5} + 48217 T^{6} - 2782 T^{7} + 251 T^{8} - 11 T^{9} + T^{10} \)
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