Properties

Label 420.2.bh.a
Level $420$
Weight $2$
Character orbit 420.bh
Analytic conductor $3.354$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{6})\)
Coefficient field: 10.0.29471584693248.1
Defining polynomial: \(x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 13 x^{6} - 36 x^{5} + 39 x^{4} - 36 x^{3} + 54 x^{2} - 162 x + 243\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + ( -1 - \beta_{7} ) q^{5} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + ( \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + ( -1 - \beta_{7} ) q^{5} + ( -1 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} ) q^{7} + ( \beta_{2} + \beta_{3} - \beta_{4} ) q^{9} + ( -1 + \beta_{1} + \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{11} + ( 1 + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{13} -\beta_{8} q^{15} + ( 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} ) q^{17} + ( \beta_{6} - \beta_{8} ) q^{19} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{5} + \beta_{7} - \beta_{9} ) q^{21} + ( 3 - 2 \beta_{2} + \beta_{4} - \beta_{7} - 2 \beta_{9} ) q^{23} + \beta_{7} q^{25} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{27} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} + 2 \beta_{9} ) q^{29} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} ) q^{31} + ( -2 - \beta_{1} - 2 \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{33} + ( 1 - \beta_{2} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{35} + ( -\beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{37} + ( 1 + 2 \beta_{2} - \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{7} + 2 \beta_{8} + 2 \beta_{9} ) q^{39} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - 3 \beta_{8} ) q^{41} + ( -3 - 2 \beta_{2} - \beta_{3} + \beta_{9} ) q^{43} + ( 1 - \beta_{2} - \beta_{3} + \beta_{7} - \beta_{9} ) q^{45} + ( 2 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{9} ) q^{47} + ( -3 \beta_{1} - 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{49} + ( -5 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{51} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{53} + ( 1 - \beta_{3} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{55} + ( 4 - \beta_{2} - \beta_{3} + 4 \beta_{7} - \beta_{9} ) q^{57} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{59} + ( 3 - \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{61} + ( 1 - 5 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} ) q^{63} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{65} + ( \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} - 4 \beta_{6} - 3 \beta_{7} + \beta_{8} ) q^{67} + ( -3 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} - \beta_{8} + 3 \beta_{9} ) q^{69} + ( -1 + 3 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{71} + ( -1 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} - 4 \beta_{8} ) q^{73} + ( -\beta_{5} + \beta_{8} ) q^{75} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{6} + 4 \beta_{7} - \beta_{8} ) q^{77} + ( -4 \beta_{3} + 4 \beta_{4} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{79} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{81} + ( 1 - 4 \beta_{1} - 5 \beta_{2} + 4 \beta_{5} + 4 \beta_{6} ) q^{83} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{85} + ( -3 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} ) q^{87} + ( 6 + \beta_{1} - 3 \beta_{4} + \beta_{5} + 6 \beta_{7} ) q^{89} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 4 \beta_{7} + 3 \beta_{8} + \beta_{9} ) q^{91} + ( -2 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} + 5 \beta_{6} + 4 \beta_{7} + \beta_{8} - 4 \beta_{9} ) q^{93} + ( -\beta_{1} - \beta_{5} + \beta_{8} ) q^{95} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{97} + ( 5 - 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{3} - 5 q^{5} - 5 q^{7} - 3 q^{9} + O(q^{10}) \) \( 10 q - q^{3} - 5 q^{5} - 5 q^{7} - 3 q^{9} - 6 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 10 q^{21} + 24 q^{23} - 5 q^{25} + 8 q^{27} + 15 q^{31} - 20 q^{33} + q^{35} - q^{37} + 15 q^{39} - 8 q^{41} - 26 q^{43} + 3 q^{45} + 14 q^{47} - 13 q^{49} - 44 q^{51} - 24 q^{53} + 18 q^{57} + 42 q^{61} - q^{63} + 9 q^{65} + 7 q^{67} - 14 q^{69} - 3 q^{73} - q^{75} - 26 q^{77} + q^{79} + 41 q^{81} - 8 q^{83} - 12 q^{85} - 26 q^{87} + 28 q^{89} - 11 q^{91} - 47 q^{93} - 3 q^{95} + 36 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 2 x^{8} - 4 x^{7} + 13 x^{6} - 36 x^{5} + 39 x^{4} - 36 x^{3} + 54 x^{2} - 162 x + 243\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} - 2 \nu^{5} + 4 \nu^{4} - 4 \nu^{3} + 18 \nu^{2} - 21 \nu + 18 \)\()/18\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} - \nu^{8} - 9 \nu^{7} + 7 \nu^{6} - 18 \nu^{5} + 22 \nu^{4} - 51 \nu^{3} + 21 \nu^{2} - 171 \nu + 135 \)\()/216\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{9} - 4 \nu^{8} + 7 \nu^{7} + 28 \nu^{6} + 32 \nu^{5} - 30 \nu^{4} + 123 \nu^{3} - 216 \nu^{2} - 243 \nu - 486 \)\()/648\)
\(\beta_{5}\)\(=\)\((\)\( 3 \nu^{9} + 2 \nu^{8} - \nu^{7} + 4 \nu^{6} + 16 \nu^{5} - 58 \nu^{4} + 9 \nu^{3} + 6 \nu^{2} + 9 \nu - 486 \)\()/216\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{9} - \nu^{7} + 6 \nu^{6} - 20 \nu^{5} + 22 \nu^{4} - 15 \nu^{3} + 48 \nu^{2} - 63 \nu + 216 \)\()/72\)
\(\beta_{7}\)\(=\)\((\)\( 8 \nu^{9} - 7 \nu^{8} + 16 \nu^{7} - 23 \nu^{6} + 50 \nu^{5} - 108 \nu^{4} + 114 \nu^{3} - 153 \nu^{2} - 729 \)\()/648\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{9} - 2 \nu^{8} + 2 \nu^{7} - 4 \nu^{6} + 13 \nu^{5} - 36 \nu^{4} + 39 \nu^{3} - 36 \nu^{2} + 54 \nu - 162 \)\()/81\)
\(\beta_{9}\)\(=\)\((\)\( -13 \nu^{9} + 5 \nu^{8} + 25 \nu^{7} - 35 \nu^{6} - 94 \nu^{5} + 186 \nu^{4} + 231 \nu^{3} - 297 \nu^{2} + 243 \nu + 1701 \)\()/648\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2}\)
\(\nu^{3}\)\(=\)\(2 \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + 2 \beta_{2}\)
\(\nu^{4}\)\(=\)\(\beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{2} + 2 \beta_{1} - 3\)
\(\nu^{5}\)\(=\)\(-2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 6\)
\(\nu^{6}\)\(=\)\(-4 \beta_{9} - 8 \beta_{7} + 4 \beta_{6} + 12 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 3\)
\(\nu^{7}\)\(=\)\(-8 \beta_{9} - 10 \beta_{8} + 18 \beta_{7} + 12 \beta_{6} - 4 \beta_{5} + 6 \beta_{4} - 18 \beta_{3} - 8 \beta_{2} - \beta_{1}\)
\(\nu^{8}\)\(=\)\(16 \beta_{9} - 16 \beta_{8} - 7 \beta_{7} - 10 \beta_{6} + 38 \beta_{5} - 5 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} - 8 \beta_{1} + 30\)
\(\nu^{9}\)\(=\)\(16 \beta_{9} - 29 \beta_{8} + 88 \beta_{7} - 9 \beta_{6} + 25 \beta_{5} + 26 \beta_{3} + 18 \beta_{2} + 46 \beta_{1} + 48\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(1 + \beta_{7}\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.527154 1.64988i
1.72689 + 0.133595i
−1.31611 1.12599i
1.15038 + 1.29484i
−1.08831 + 1.34743i
0.527154 + 1.64988i
1.72689 0.133595i
−1.31611 + 1.12599i
1.15038 1.29484i
−1.08831 1.34743i
0 −1.69242 0.368412i 0 −0.500000 0.866025i 0 −1.80025 + 1.93884i 0 2.72854 + 1.24701i 0
101.2 0 −0.747749 + 1.56233i 0 −0.500000 0.866025i 0 −0.456468 2.60608i 0 −1.88174 2.33646i 0
101.3 0 −0.317079 1.70278i 0 −0.500000 0.866025i 0 −1.73439 1.99797i 0 −2.79892 + 1.07983i 0
101.4 0 0.546177 + 1.64368i 0 −0.500000 0.866025i 0 −1.08214 + 2.41433i 0 −2.40338 + 1.79548i 0
101.5 0 1.71107 0.268793i 0 −0.500000 0.866025i 0 2.57325 0.615143i 0 2.85550 0.919845i 0
341.1 0 −1.69242 + 0.368412i 0 −0.500000 + 0.866025i 0 −1.80025 1.93884i 0 2.72854 1.24701i 0
341.2 0 −0.747749 1.56233i 0 −0.500000 + 0.866025i 0 −0.456468 + 2.60608i 0 −1.88174 + 2.33646i 0
341.3 0 −0.317079 + 1.70278i 0 −0.500000 + 0.866025i 0 −1.73439 + 1.99797i 0 −2.79892 1.07983i 0
341.4 0 0.546177 1.64368i 0 −0.500000 + 0.866025i 0 −1.08214 2.41433i 0 −2.40338 1.79548i 0
341.5 0 1.71107 + 0.268793i 0 −0.500000 + 0.866025i 0 2.57325 + 0.615143i 0 2.85550 + 0.919845i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 341.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 420.2.bh.a 10
3.b odd 2 1 420.2.bh.b yes 10
5.b even 2 1 2100.2.bi.k 10
5.c odd 4 2 2100.2.bo.h 20
7.c even 3 1 2940.2.d.b 10
7.d odd 6 1 420.2.bh.b yes 10
7.d odd 6 1 2940.2.d.a 10
15.d odd 2 1 2100.2.bi.j 10
15.e even 4 2 2100.2.bo.g 20
21.g even 6 1 inner 420.2.bh.a 10
21.g even 6 1 2940.2.d.b 10
21.h odd 6 1 2940.2.d.a 10
35.i odd 6 1 2100.2.bi.j 10
35.k even 12 2 2100.2.bo.g 20
105.p even 6 1 2100.2.bi.k 10
105.w odd 12 2 2100.2.bo.h 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.bh.a 10 1.a even 1 1 trivial
420.2.bh.a 10 21.g even 6 1 inner
420.2.bh.b yes 10 3.b odd 2 1
420.2.bh.b yes 10 7.d odd 6 1
2100.2.bi.j 10 15.d odd 2 1
2100.2.bi.j 10 35.i odd 6 1
2100.2.bi.k 10 5.b even 2 1
2100.2.bi.k 10 105.p even 6 1
2100.2.bo.g 20 15.e even 4 2
2100.2.bo.g 20 35.k even 12 2
2100.2.bo.h 20 5.c odd 4 2
2100.2.bo.h 20 105.w odd 12 2
2940.2.d.a 10 7.d odd 6 1
2940.2.d.a 10 21.h odd 6 1
2940.2.d.b 10 7.c even 3 1
2940.2.d.b 10 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{11}^{10} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(420, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \)
$3$ \( 243 + 81 T + 54 T^{2} - 9 T^{3} - 33 T^{4} - 12 T^{5} - 11 T^{6} - T^{7} + 2 T^{8} + T^{9} + T^{10} \)
$5$ \( ( 1 + T + T^{2} )^{5} \)
$7$ \( 16807 + 12005 T + 6517 T^{2} + 294 T^{3} - 637 T^{4} - 533 T^{5} - 91 T^{6} + 6 T^{7} + 19 T^{8} + 5 T^{9} + T^{10} \)
$11$ \( 1728 + 7488 T + 14128 T^{2} + 14352 T^{3} + 8156 T^{4} + 2280 T^{5} + 84 T^{6} - 96 T^{7} - 4 T^{8} + 6 T^{9} + T^{10} \)
$13$ \( 3888 + 65704 T^{2} + 22251 T^{4} + 2183 T^{6} + 81 T^{8} + T^{10} \)
$17$ \( 69696 + 21120 T + 55504 T^{2} - 32832 T^{3} + 30292 T^{4} - 7548 T^{5} + 2352 T^{6} - 168 T^{7} + 70 T^{8} - 6 T^{9} + T^{10} \)
$19$ \( 192 + 960 T + 1864 T^{2} + 1320 T^{3} - 213 T^{4} - 519 T^{5} + 152 T^{6} + 45 T^{7} - 12 T^{8} - 3 T^{9} + T^{10} \)
$23$ \( 2462508 - 3527964 T + 2097940 T^{2} - 591888 T^{3} + 50818 T^{4} + 11418 T^{5} - 1389 T^{6} - 744 T^{7} + 223 T^{8} - 24 T^{9} + T^{10} \)
$29$ \( 8748 + 129448 T^{2} + 100708 T^{4} + 6537 T^{6} + 142 T^{8} + T^{10} \)
$31$ \( 1978032 - 2143680 T + 211684 T^{2} + 609840 T^{3} + 92463 T^{4} - 41223 T^{5} - 376 T^{6} + 945 T^{7} + 12 T^{8} - 15 T^{9} + T^{10} \)
$37$ \( 12544 + 95872 T + 732848 T^{2} + 14152 T^{3} + 57241 T^{4} - 1533 T^{5} + 3634 T^{6} - 65 T^{7} + 68 T^{8} + T^{9} + T^{10} \)
$41$ \( ( 1338 + 1084 T - 64 T^{2} - 115 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$43$ \( ( 1559 + 1473 T - 568 T^{2} - 42 T^{3} + 13 T^{4} + T^{5} )^{2} \)
$47$ \( 295289856 + 26119680 T + 29254912 T^{2} - 5407744 T^{3} + 2084288 T^{4} - 197728 T^{5} + 31216 T^{6} - 1904 T^{7} + 284 T^{8} - 14 T^{9} + T^{10} \)
$53$ \( 113246208 + 103809024 T + 17563648 T^{2} - 12976128 T^{3} + 946176 T^{4} + 258048 T^{5} - 9664 T^{6} - 2880 T^{7} + 72 T^{8} + 24 T^{9} + T^{10} \)
$59$ \( 125081856 - 49030656 T + 23447008 T^{2} - 1921728 T^{3} + 844324 T^{4} - 71664 T^{5} + 21216 T^{6} - 756 T^{7} + 160 T^{8} + T^{10} \)
$61$ \( 1338672 - 3398784 T + 3926512 T^{2} - 2666112 T^{3} + 1167096 T^{4} - 342264 T^{5} + 68273 T^{6} - 9198 T^{7} + 807 T^{8} - 42 T^{9} + T^{10} \)
$67$ \( 45684081 - 30841317 T + 24268059 T^{2} - 268326 T^{3} + 1088883 T^{4} - 40797 T^{5} + 35871 T^{6} + 324 T^{7} + 241 T^{8} - 7 T^{9} + T^{10} \)
$71$ \( 88259328 + 17753152 T^{2} + 1171116 T^{4} + 28832 T^{6} + 288 T^{8} + T^{10} \)
$73$ \( 1572528 + 10668864 T + 22887532 T^{2} - 8414256 T^{3} - 266785 T^{4} + 431217 T^{5} + 57384 T^{6} - 759 T^{7} - 250 T^{8} + 3 T^{9} + T^{10} \)
$79$ \( 138485824 + 65147648 T + 32471336 T^{2} + 3684368 T^{3} + 1104241 T^{4} + 52755 T^{5} + 31558 T^{6} + 503 T^{7} + 194 T^{8} - T^{9} + T^{10} \)
$83$ \( ( -28794 + 28510 T - 1024 T^{2} - 379 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$89$ \( 272484 - 48024 T + 256936 T^{2} - 186932 T^{3} + 261524 T^{4} - 109826 T^{5} + 35605 T^{6} - 5236 T^{7} + 563 T^{8} - 28 T^{9} + T^{10} \)
$97$ \( 1051392 + 1222528 T^{2} + 205424 T^{4} + 11040 T^{6} + 212 T^{8} + T^{10} \)
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