Properties

Label 144.3.q.e
Level $144$
Weight $3$
Character orbit 144.q
Analytic conductor $3.924$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.q (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.19269881856.9
Defining polynomial: \(x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} - \beta_{3} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( -3 - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} - \beta_{3} ) q^{3} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} ) q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} ) q^{7} + ( -3 - 3 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{9} + ( -5 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{11} + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{13} + ( -4 + \beta_{1} + \beta_{2} + 7 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{15} + ( 1 + 5 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{17} + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 3 \beta_{5} - 6 \beta_{7} ) q^{19} + ( -5 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{7} ) q^{21} + ( 14 + \beta_{1} - 7 \beta_{2} - 5 \beta_{3} + 6 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{23} + ( 2 - 6 \beta_{1} + 9 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} + \beta_{5} - 6 \beta_{6} + \beta_{7} ) q^{25} + ( 2 - \beta_{1} + 3 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} + \beta_{7} ) q^{27} + ( -11 + 2 \beta_{1} + \beta_{2} - 10 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + \beta_{7} ) q^{29} + ( -\beta_{1} - \beta_{2} + 13 \beta_{3} - 12 \beta_{4} - 6 \beta_{6} ) q^{31} + ( -20 - \beta_{1} - 6 \beta_{2} + 20 \beta_{3} - 7 \beta_{4} - 3 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{33} + ( 5 + 2 \beta_{2} - 12 \beta_{3} + \beta_{5} - \beta_{7} ) q^{35} + ( 16 - 6 \beta_{4} - 2 \beta_{5} + 6 \beta_{6} - 2 \beta_{7} ) q^{37} + ( -6 - 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - 12 \beta_{4} - 6 \beta_{5} - 12 \beta_{6} + 10 \beta_{7} ) q^{39} + ( 42 + 2 \beta_{1} - 6 \beta_{2} - 19 \beta_{3} + 4 \beta_{5} ) q^{41} + ( 7 - 7 \beta_{1} + 14 \beta_{2} - 14 \beta_{3} + 6 \beta_{4} + 12 \beta_{6} ) q^{43} + ( 29 - 7 \beta_{1} + 5 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} - 6 \beta_{5} - 11 \beta_{6} + 7 \beta_{7} ) q^{45} + ( -13 + 3 \beta_{1} + 5 \beta_{2} - 15 \beta_{3} - 6 \beta_{4} + \beta_{5} + 5 \beta_{7} ) q^{47} + ( 1 - 5 \beta_{1} - 5 \beta_{2} + 26 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{49} + ( -30 - 8 \beta_{1} - \beta_{2} + 19 \beta_{3} + 8 \beta_{4} - \beta_{5} - 8 \beta_{6} + 3 \beta_{7} ) q^{51} + ( -6 - 8 \beta_{1} - 12 \beta_{2} + 16 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{53} + ( 33 - 6 \beta_{1} + 12 \beta_{2} - 6 \beta_{3} + 12 \beta_{4} - 5 \beta_{5} - 12 \beta_{6} + 13 \beta_{7} ) q^{55} + ( -8 - 4 \beta_{1} - \beta_{2} - 29 \beta_{3} + 15 \beta_{4} + \beta_{5} + 18 \beta_{6} + 11 \beta_{7} ) q^{57} + ( 21 + 6 \beta_{1} - 9 \beta_{3} - 6 \beta_{5} + 14 \beta_{6} - 9 \beta_{7} ) q^{59} + ( 7 + 12 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} - 7 \beta_{5} - 18 \beta_{6} - 7 \beta_{7} ) q^{61} + ( 31 - 5 \beta_{1} - 7 \beta_{2} - 11 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} + 7 \beta_{7} ) q^{63} + ( -27 + 6 \beta_{1} - 9 \beta_{2} - 12 \beta_{3} + 7 \beta_{4} + 21 \beta_{5} - 9 \beta_{7} ) q^{65} + ( -5 + 5 \beta_{3} + 10 \beta_{5} - 5 \beta_{7} ) q^{67} + ( -25 - 3 \beta_{1} + 19 \beta_{2} + 39 \beta_{3} - 10 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} - \beta_{7} ) q^{69} + ( -30 + 12 \beta_{1} + 72 \beta_{3} - 4 \beta_{4} - 12 \beta_{5} - 4 \beta_{6} ) q^{71} + ( 3 - 7 \beta_{1} + 14 \beta_{2} - 7 \beta_{3} - 9 \beta_{4} - 25 \beta_{5} + 9 \beta_{6} - 4 \beta_{7} ) q^{73} + ( -13 + \beta_{1} - 19 \beta_{2} - 58 \beta_{3} - 18 \beta_{4} + 12 \beta_{5} - 6 \beta_{6} - 17 \beta_{7} ) q^{75} + ( 7 - 3 \beta_{1} - 5 \beta_{2} - 3 \beta_{3} + 8 \beta_{5} - 9 \beta_{6} + 7 \beta_{7} ) q^{77} + ( -7 + 5 \beta_{1} - 13 \beta_{2} + 13 \beta_{3} + 12 \beta_{4} + \beta_{5} + 24 \beta_{6} + \beta_{7} ) q^{79} + ( 24 + 15 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + 8 \beta_{4} - 11 \beta_{6} + 5 \beta_{7} ) q^{81} + ( -21 - \beta_{1} - 3 \beta_{2} - 19 \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{7} ) q^{83} + ( -6 + 4 \beta_{1} + 4 \beta_{2} - 58 \beta_{3} + 12 \beta_{5} - 6 \beta_{7} ) q^{85} + ( -29 + 7 \beta_{1} - 19 \beta_{2} + 67 \beta_{3} + 5 \beta_{5} - 6 \beta_{6} - 11 \beta_{7} ) q^{87} + ( -22 - 4 \beta_{1} + 20 \beta_{2} + 20 \beta_{3} + 14 \beta_{5} - 10 \beta_{7} ) q^{89} + ( -19 + 4 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 11 \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{91} + ( -3 + 13 \beta_{1} + 13 \beta_{2} - 21 \beta_{3} + 18 \beta_{4} - 12 \beta_{5} + 21 \beta_{6} - 23 \beta_{7} ) q^{93} + ( 82 - 4 \beta_{1} + 16 \beta_{2} - 46 \beta_{3} - 12 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{95} + ( -57 + 10 \beta_{1} - 8 \beta_{2} + 63 \beta_{3} - 4 \beta_{5} - 4 \beta_{7} ) q^{97} + ( 80 + 14 \beta_{1} + 9 \beta_{2} + 10 \beta_{3} + 22 \beta_{4} - 16 \beta_{5} + 14 \beta_{6} - 9 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 10q^{3} - 6q^{5} - 6q^{7} - 22q^{9} + O(q^{10}) \) \( 8q - 10q^{3} - 6q^{5} - 6q^{7} - 22q^{9} - 36q^{11} + 14q^{13} - 10q^{15} - 4q^{19} - 54q^{21} + 102q^{23} + 10q^{25} + 20q^{27} - 114q^{29} + 50q^{31} - 104q^{33} + 120q^{37} - 82q^{39} + 264q^{41} + 28q^{43} + 206q^{45} - 150q^{47} + 94q^{49} - 170q^{51} + 244q^{55} - 178q^{57} + 108q^{59} + 14q^{61} + 210q^{63} - 198q^{65} + 20q^{67} - 14q^{69} - 76q^{73} - 326q^{75} + 66q^{77} - 26q^{79} + 194q^{81} - 246q^{83} - 224q^{85} + 18q^{87} - 108q^{91} - 130q^{93} + 456q^{95} - 236q^{97} + 634q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 15 x^{6} - 2 x^{5} + 133 x^{4} - 84 x^{3} + 276 x^{2} + 144 x + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -6 \nu^{7} - 169 \nu^{6} + 466 \nu^{5} - 2647 \nu^{4} + 3180 \nu^{3} - 20091 \nu^{2} + 64128 \nu - 19236 \)\()/17700\)
\(\beta_{2}\)\(=\)\((\)\( 331 \nu^{7} - 756 \nu^{6} + 8709 \nu^{5} - 4178 \nu^{4} + 73845 \nu^{3} + 16116 \nu^{2} + 432972 \nu + 72936 \)\()/159300\)
\(\beta_{3}\)\(=\)\((\)\( -677 \nu^{7} + 1827 \nu^{6} - 10353 \nu^{5} + 6901 \nu^{4} - 82215 \nu^{3} + 132153 \nu^{2} - 156924 \nu + 80388 \)\()/159300\)
\(\beta_{4}\)\(=\)\((\)\( 1013 \nu^{7} - 2688 \nu^{6} + 16707 \nu^{5} - 19444 \nu^{4} + 143085 \nu^{3} - 232782 \nu^{2} + 247356 \nu - 401472 \)\()/159300\)
\(\beta_{5}\)\(=\)\((\)\( -602 \nu^{7} + 252 \nu^{6} - 5853 \nu^{5} - 12374 \nu^{4} - 64440 \nu^{3} - 39297 \nu^{2} + 41526 \nu - 95112 \)\()/79650\)
\(\beta_{6}\)\(=\)\((\)\( 644 \nu^{7} - 2019 \nu^{6} + 9966 \nu^{5} - 7447 \nu^{4} + 68730 \nu^{3} - 107691 \nu^{2} + 129078 \nu + 194364 \)\()/79650\)
\(\beta_{7}\)\(=\)\((\)\( -1004 \nu^{7} + 729 \nu^{6} - 12981 \nu^{5} - 14198 \nu^{4} - 139005 \nu^{3} - 62319 \nu^{2} - 104598 \nu - 259974 \)\()/79650\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - 2 \beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} + 1\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{7} + 6 \beta_{6} - \beta_{5} + 3 \beta_{4} + 20 \beta_{3} - \beta_{2} + 2 \beta_{1} - 19\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(-20 \beta_{7} + 3 \beta_{6} + 19 \beta_{5} - 3 \beta_{4} + 13 \beta_{3} - 26 \beta_{2} + 13 \beta_{1} - 38\)\()/3\)
\(\nu^{4}\)\(=\)\((\)\(-25 \beta_{7} - 39 \beta_{6} + 50 \beta_{5} - 78 \beta_{4} - 211 \beta_{3} - 13 \beta_{2} - 13 \beta_{1} - 25\)\()/3\)
\(\nu^{5}\)\(=\)\((\)\(79 \beta_{7} - 150 \beta_{6} + 79 \beta_{5} - 75 \beta_{4} - 626 \beta_{3} + 259 \beta_{2} - 248 \beta_{1} + 457\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(650 \beta_{7} - 507 \beta_{6} - 685 \beta_{5} + 507 \beta_{4} - 445 \beta_{3} + 890 \beta_{2} - 445 \beta_{1} + 3152\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(2293 \beta_{7} + 1335 \beta_{6} - 4586 \beta_{5} + 2670 \beta_{4} + 7609 \beta_{3} + 1039 \beta_{2} + 1039 \beta_{1} + 2293\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
−1.41950 2.45865i
−0.331167 0.573598i
0.831167 + 1.43962i
1.91950 + 3.32468i
−1.41950 + 2.45865i
−0.331167 + 0.573598i
0.831167 1.43962i
1.91950 3.32468i
0 −2.91950 + 0.690286i 0 1.80902 1.04444i 0 0.781452 1.35351i 0 8.04701 4.03058i 0
65.2 0 −1.83117 2.37631i 0 3.44299 1.98781i 0 1.80469 3.12582i 0 −2.29365 + 8.70282i 0
65.3 0 −0.668833 + 2.92449i 0 −0.0440114 + 0.0254100i 0 −4.52944 + 7.84521i 0 −8.10532 3.91200i 0
65.4 0 0.419504 2.97052i 0 −8.20800 + 4.73889i 0 −1.05671 + 1.83027i 0 −8.64803 2.49230i 0
113.1 0 −2.91950 0.690286i 0 1.80902 + 1.04444i 0 0.781452 + 1.35351i 0 8.04701 + 4.03058i 0
113.2 0 −1.83117 + 2.37631i 0 3.44299 + 1.98781i 0 1.80469 + 3.12582i 0 −2.29365 8.70282i 0
113.3 0 −0.668833 2.92449i 0 −0.0440114 0.0254100i 0 −4.52944 7.84521i 0 −8.10532 + 3.91200i 0
113.4 0 0.419504 + 2.97052i 0 −8.20800 4.73889i 0 −1.05671 1.83027i 0 −8.64803 + 2.49230i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 113.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.q.e 8
3.b odd 2 1 432.3.q.e 8
4.b odd 2 1 72.3.m.b 8
8.b even 2 1 576.3.q.j 8
8.d odd 2 1 576.3.q.i 8
9.c even 3 1 432.3.q.e 8
9.c even 3 1 1296.3.e.i 8
9.d odd 6 1 inner 144.3.q.e 8
9.d odd 6 1 1296.3.e.i 8
12.b even 2 1 216.3.m.b 8
24.f even 2 1 1728.3.q.j 8
24.h odd 2 1 1728.3.q.i 8
36.f odd 6 1 216.3.m.b 8
36.f odd 6 1 648.3.e.c 8
36.h even 6 1 72.3.m.b 8
36.h even 6 1 648.3.e.c 8
72.j odd 6 1 576.3.q.j 8
72.l even 6 1 576.3.q.i 8
72.n even 6 1 1728.3.q.i 8
72.p odd 6 1 1728.3.q.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.m.b 8 4.b odd 2 1
72.3.m.b 8 36.h even 6 1
144.3.q.e 8 1.a even 1 1 trivial
144.3.q.e 8 9.d odd 6 1 inner
216.3.m.b 8 12.b even 2 1
216.3.m.b 8 36.f odd 6 1
432.3.q.e 8 3.b odd 2 1
432.3.q.e 8 9.c even 3 1
576.3.q.i 8 8.d odd 2 1
576.3.q.i 8 72.l even 6 1
576.3.q.j 8 8.b even 2 1
576.3.q.j 8 72.j odd 6 1
648.3.e.c 8 36.f odd 6 1
648.3.e.c 8 36.h even 6 1
1296.3.e.i 8 9.c even 3 1
1296.3.e.i 8 9.d odd 6 1
1728.3.q.i 8 24.h odd 2 1
1728.3.q.i 8 72.n even 6 1
1728.3.q.j 8 24.f even 2 1
1728.3.q.j 8 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{8} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 6561 + 7290 T + 4941 T^{2} + 2430 T^{3} + 912 T^{4} + 270 T^{5} + 61 T^{6} + 10 T^{7} + T^{8} \)
$5$ \( 16 + 528 T + 5612 T^{2} - 6468 T^{3} + 2661 T^{4} - 294 T^{5} - 37 T^{6} + 6 T^{7} + T^{8} \)
$7$ \( 11664 - 3888 T + 4860 T^{2} - 108 T^{3} + 1197 T^{4} - 126 T^{5} + 69 T^{6} + 6 T^{7} + T^{8} \)
$11$ \( 105616729 + 58455576 T + 11688824 T^{2} + 500544 T^{3} - 50235 T^{4} - 3168 T^{5} + 344 T^{6} + 36 T^{7} + T^{8} \)
$13$ \( 2611456 - 6270080 T + 14487184 T^{2} - 1407128 T^{3} + 179137 T^{4} - 2846 T^{5} + 547 T^{6} - 14 T^{7} + T^{8} \)
$17$ \( 7020428944 + 120824648 T^{2} + 687753 T^{4} + 1454 T^{6} + T^{8} \)
$19$ \( ( 226348 + 4004 T - 1179 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$23$ \( 11198718976 - 3042651648 T + 228044192 T^{2} + 12909648 T^{3} - 670143 T^{4} - 45798 T^{5} + 3917 T^{6} - 102 T^{7} + T^{8} \)
$29$ \( 106450807824 + 9055894608 T - 37821492 T^{2} - 25063668 T^{3} + 86949 T^{4} + 102942 T^{5} + 5235 T^{6} + 114 T^{7} + T^{8} \)
$31$ \( 152712134656 + 12067409920 T + 1224387712 T^{2} + 17678560 T^{3} + 1633465 T^{4} - 27110 T^{5} + 3193 T^{6} - 50 T^{7} + T^{8} \)
$37$ \( ( 206496 + 27360 T - 276 T^{2} - 60 T^{3} + T^{4} )^{2} \)
$41$ \( 1919025613521 - 435967071768 T + 44376688026 T^{2} - 2581267824 T^{3} + 93582171 T^{4} - 2165328 T^{5} + 31434 T^{6} - 264 T^{7} + T^{8} \)
$43$ \( 1352729498761 - 211873953592 T + 28244463112 T^{2} - 838981528 T^{3} + 24309277 T^{4} - 245392 T^{5} + 5032 T^{6} - 28 T^{7} + T^{8} \)
$47$ \( 4615347568896 + 475641590400 T + 16526225232 T^{2} + 19261800 T^{3} - 8914095 T^{4} - 13050 T^{5} + 7413 T^{6} + 150 T^{7} + T^{8} \)
$53$ \( 78435844096 + 2590797824 T^{2} + 12396816 T^{4} + 7016 T^{6} + T^{8} \)
$59$ \( 127589696809 - 43823785536 T + 6086181872 T^{2} - 367082496 T^{3} + 4892493 T^{4} + 323136 T^{5} + 896 T^{6} - 108 T^{7} + T^{8} \)
$61$ \( 133593174016 + 46766967808 T + 13427579584 T^{2} + 1020419248 T^{3} + 63457201 T^{4} + 368674 T^{5} + 8251 T^{6} - 14 T^{7} + T^{8} \)
$67$ \( 17391015625 + 4879375000 T + 1606375000 T^{2} - 61325000 T^{3} + 3848125 T^{4} - 38000 T^{5} + 2200 T^{6} - 20 T^{7} + T^{8} \)
$71$ \( 114698616545536 + 406952754944 T^{2} + 206774880 T^{4} + 26864 T^{6} + T^{8} \)
$73$ \( ( 2961976 - 487192 T - 9831 T^{2} + 38 T^{3} + T^{4} )^{2} \)
$79$ \( 103529078405776 + 749891898800 T + 123470983324 T^{2} - 1384089748 T^{3} + 122492077 T^{4} - 449026 T^{5} + 12277 T^{6} + 26 T^{7} + T^{8} \)
$83$ \( 1085363908864 + 263110694016 T + 28654548944 T^{2} + 1792361544 T^{3} + 70034865 T^{4} + 1745862 T^{5} + 27269 T^{6} + 246 T^{7} + T^{8} \)
$89$ \( 309931236458496 + 1618052474880 T^{2} + 407712528 T^{4} + 34920 T^{6} + T^{8} \)
$97$ \( 3435006304129 + 837036947756 T + 171307640890 T^{2} + 7083794672 T^{3} + 202097299 T^{4} + 3255536 T^{5} + 38074 T^{6} + 236 T^{7} + T^{8} \)
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