Properties

Label 3610.2.a.bi
Level $3610$
Weight $2$
Character orbit 3610.a
Self dual yes
Analytic conductor $28.826$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3610 = 2 \cdot 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3610.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.8259951297\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \(x^{9} - 24 x^{7} - 6 x^{6} + 183 x^{5} + 78 x^{4} - 455 x^{3} - 168 x^{2} + 228 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 24 x^{7} - 6 x^{6} + 183 x^{5} + 78 x^{4} - 455 x^{3} - 168 x^{2} + 228 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -131 \nu^{8} + 360 \nu^{7} + 2204 \nu^{6} - 5902 \nu^{5} - 8957 \nu^{4} + 24062 \nu^{3} + 1765 \nu^{2} - 15744 \nu + 6504 \)\()/3876\)
\(\beta_{3}\)\(=\)\((\)\( 70 \nu^{8} - 111 \nu^{7} - 1444 \nu^{6} + 1556 \nu^{5} + 9106 \nu^{4} - 4033 \nu^{3} - 17246 \nu^{2} - 2187 \nu + 3744 \)\()/1938\)
\(\beta_{4}\)\(=\)\((\)\( 9 \nu^{8} - 32 \nu^{7} - 140 \nu^{6} + 542 \nu^{5} + 415 \nu^{4} - 2466 \nu^{3} + 933 \nu^{2} + 2832 \nu - 1580 \)\()/204\)
\(\beta_{5}\)\(=\)\((\)\( -111 \nu^{8} + 236 \nu^{7} + 1976 \nu^{6} - 3704 \nu^{5} - 9493 \nu^{4} + 14604 \nu^{3} + 9573 \nu^{2} - 12216 \nu + 560 \)\()/1938\)
\(\beta_{6}\)\(=\)\((\)\( -134 \nu^{8} - 9 \nu^{7} + 3078 \nu^{6} + 842 \nu^{5} - 21732 \nu^{4} - 9371 \nu^{3} + 45620 \nu^{2} + 17771 \nu - 11726 \)\()/1938\)
\(\beta_{7}\)\(=\)\((\)\( 557 \nu^{8} - 1580 \nu^{7} - 10032 \nu^{6} + 26298 \nu^{5} + 49091 \nu^{4} - 113178 \nu^{3} - 49539 \nu^{2} + 106136 \nu - 10888 \)\()/3876\)
\(\beta_{8}\)\(=\)\((\)\( -291 \nu^{8} + 706 \nu^{7} + 5320 \nu^{6} - 11212 \nu^{5} - 26633 \nu^{4} + 43524 \nu^{3} + 28449 \nu^{2} - 30402 \nu + 1084 \)\()/1938\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} + \beta_{4} + 2 \beta_{3} - \beta_{2} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{5} + 2 \beta_{3} - 4 \beta_{2} + 8 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(13 \beta_{8} + \beta_{7} + \beta_{6} - 3 \beta_{5} + 9 \beta_{4} + 25 \beta_{3} - 12 \beta_{2} + 3 \beta_{1} + 44\)
\(\nu^{5}\)\(=\)\(19 \beta_{8} + 4 \beta_{7} - 2 \beta_{6} + 12 \beta_{5} + 24 \beta_{3} - 58 \beta_{2} + 74 \beta_{1} + 36\)
\(\nu^{6}\)\(=\)\(151 \beta_{8} + 13 \beta_{7} + 16 \beta_{6} - 53 \beta_{5} + 86 \beta_{4} + 280 \beta_{3} - 147 \beta_{2} + 57 \beta_{1} + 436\)
\(\nu^{7}\)\(=\)\(269 \beta_{8} + 68 \beta_{7} - 40 \beta_{6} + 113 \beta_{5} + 4 \beta_{4} + 268 \beta_{3} - 724 \beta_{2} + 743 \beta_{1} + 494\)
\(\nu^{8}\)\(=\)\(1732 \beta_{8} + 157 \beta_{7} + 181 \beta_{6} - 733 \beta_{5} + 856 \beta_{4} + 3051 \beta_{3} - 1807 \beta_{2} + 811 \beta_{1} + 4547\)

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
−3.21584
−2.37131
−2.31980
−1.10917
0.0361439
0.576411
1.79490
3.17969
3.42897
−1.00000 −3.21584 1.00000 1.00000 3.21584 −0.0233951 −1.00000 7.34161 −1.00000
1.2 −1.00000 −2.37131 1.00000 1.00000 2.37131 3.15542 −1.00000 2.62313 −1.00000
1.3 −1.00000 −2.31980 1.00000 1.00000 2.31980 −4.91674 −1.00000 2.38147 −1.00000
1.4 −1.00000 −1.10917 1.00000 1.00000 1.10917 4.92913 −1.00000 −1.76973 −1.00000
1.5 −1.00000 0.0361439 1.00000 1.00000 −0.0361439 1.83741 −1.00000 −2.99869 −1.00000
1.6 −1.00000 0.576411 1.00000 1.00000 −0.576411 −4.86419 −1.00000 −2.66775 −1.00000
1.7 −1.00000 1.79490 1.00000 1.00000 −1.79490 1.36147 −1.00000 0.221676 −1.00000
1.8 −1.00000 3.17969 1.00000 1.00000 −3.17969 −3.34610 −1.00000 7.11045 −1.00000
1.9 −1.00000 3.42897 1.00000 1.00000 −3.42897 1.86700 −1.00000 8.75785 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

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 3610.2.a.bi 9
19.b odd 2 1 3610.2.a.bj 9
19.e even 9 2 190.2.k.d 18
95.p even 18 2 950.2.l.i 18
95.q odd 36 4 950.2.u.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.d 18 19.e even 9 2
950.2.l.i 18 95.p even 18 2
950.2.u.g 36 95.q odd 36 4
3610.2.a.bi 9 1.a even 1 1 trivial
3610.2.a.bj 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3610))\):

\(T_{3}^{9} - \cdots\)
\(T_{7}^{9} - \cdots\)
\(T_{13}^{9} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{9} \)
$3$ \( -8 + 228 T - 168 T^{2} - 455 T^{3} + 78 T^{4} + 183 T^{5} - 6 T^{6} - 24 T^{7} + T^{9} \)
$5$ \( ( -1 + T )^{9} \)
$7$ \( -136 - 5592 T + 9372 T^{2} - 3540 T^{3} - 1236 T^{4} + 804 T^{5} + 35 T^{6} - 51 T^{7} + T^{9} \)
$11$ \( 71271 - 86517 T + 8253 T^{2} + 21790 T^{3} - 6156 T^{4} - 1221 T^{5} + 547 T^{6} - 9 T^{7} - 12 T^{8} + T^{9} \)
$13$ \( -13544 - 27240 T - 8136 T^{2} + 8288 T^{3} + 3894 T^{4} - 486 T^{5} - 415 T^{6} - 27 T^{7} + 9 T^{8} + T^{9} \)
$17$ \( -14328 + 9108 T + 17856 T^{2} - 6409 T^{3} - 6102 T^{4} + 903 T^{5} + 484 T^{6} - 75 T^{7} - 6 T^{8} + T^{9} \)
$19$ \( T^{9} \)
$23$ \( -11016 - 14904 T + 4212 T^{2} + 9324 T^{3} - 432 T^{4} - 1548 T^{5} + 171 T^{6} + 81 T^{7} - 18 T^{8} + T^{9} \)
$29$ \( -9792 - 61920 T + 97920 T^{2} - 27080 T^{3} - 9024 T^{4} + 3744 T^{5} + 128 T^{6} - 114 T^{7} + T^{9} \)
$31$ \( -23104 + 57792 T + 35280 T^{2} - 26024 T^{3} - 7080 T^{4} + 2460 T^{5} + 384 T^{6} - 84 T^{7} - 6 T^{8} + T^{9} \)
$37$ \( 25992 + 53352 T - 44388 T^{2} - 21012 T^{3} + 12096 T^{4} + 2544 T^{5} - 743 T^{6} - 129 T^{7} + 6 T^{8} + T^{9} \)
$41$ \( 3749517 + 1387935 T - 1005381 T^{2} - 354448 T^{3} + 37338 T^{4} + 15975 T^{5} - 293 T^{6} - 225 T^{7} + T^{9} \)
$43$ \( 21176 - 90612 T + 153588 T^{2} - 134207 T^{3} + 64734 T^{4} - 16737 T^{5} + 1920 T^{6} + 3 T^{7} - 18 T^{8} + T^{9} \)
$47$ \( -1368 + 21960 T - 52560 T^{2} - 9136 T^{3} + 15594 T^{4} + 3102 T^{5} - 773 T^{6} - 159 T^{7} + 3 T^{8} + T^{9} \)
$53$ \( 155592 + 118728 T - 31428 T^{2} - 33220 T^{3} + 1356 T^{4} + 2820 T^{5} - 7 T^{6} - 93 T^{7} + T^{9} \)
$59$ \( 3345957 + 6059619 T + 3263418 T^{2} + 567728 T^{3} - 61869 T^{4} - 34671 T^{5} - 3979 T^{6} - 33 T^{7} + 21 T^{8} + T^{9} \)
$61$ \( 420444224 - 188973984 T + 16265424 T^{2} + 4135416 T^{3} - 758424 T^{4} - 1428 T^{5} + 6992 T^{6} - 276 T^{7} - 18 T^{8} + T^{9} \)
$67$ \( -511552 - 48720 T + 388128 T^{2} - 88059 T^{3} - 33294 T^{4} + 9237 T^{5} + 566 T^{6} - 204 T^{7} + T^{9} \)
$71$ \( 13042368 + 375552 T - 4282128 T^{2} - 122968 T^{3} + 225816 T^{4} + 8796 T^{5} - 3904 T^{6} - 192 T^{7} + 18 T^{8} + T^{9} \)
$73$ \( -187272 - 732564 T - 975240 T^{2} - 588609 T^{3} - 169794 T^{4} - 19539 T^{5} + 838 T^{6} + 420 T^{7} + 36 T^{8} + T^{9} \)
$79$ \( -5796352 - 15510528 T - 11510400 T^{2} - 2957888 T^{3} - 45600 T^{4} + 64512 T^{5} + 2016 T^{6} - 468 T^{7} - 6 T^{8} + T^{9} \)
$83$ \( -7573752 - 12492324 T - 6242850 T^{2} - 587683 T^{3} + 252720 T^{4} + 35145 T^{5} - 2354 T^{6} - 360 T^{7} + 6 T^{8} + T^{9} \)
$89$ \( 12500559 - 14292513 T - 1453905 T^{2} + 3811960 T^{3} - 689901 T^{4} - 4860 T^{5} + 8129 T^{6} - 348 T^{7} - 18 T^{8} + T^{9} \)
$97$ \( -58248 - 769824 T - 2339550 T^{2} + 644007 T^{3} + 334728 T^{4} + 9915 T^{5} - 4838 T^{6} - 240 T^{7} + 18 T^{8} + T^{9} \)
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