Properties

Label 1008.2.r.m
Level $1008$
Weight $2$
Character orbit 1008.r
Analytic conductor $8.049$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.04892052375\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.508277025.1
Defining polynomial: \(x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( 1 - \beta_{2} ) q^{3} + ( \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{5} + ( 1 - \beta_{6} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{2} ) q^{3} + ( \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{5} + ( 1 - \beta_{6} ) q^{7} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} ) q^{9} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{11} + ( -2 \beta_{1} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{13} + ( -1 - \beta_{1} + \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 3 \beta_{7} ) q^{15} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{17} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{19} + ( \beta_{1} - \beta_{6} ) q^{21} + ( \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} ) q^{23} + ( -5 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{25} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{27} + ( 1 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{7} ) q^{29} + ( \beta_{2} - 3 \beta_{6} + \beta_{7} ) q^{31} + ( -3 + 2 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{33} + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{35} + ( 6 + \beta_{1} + \beta_{4} - \beta_{7} ) q^{37} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + 3 \beta_{4} + 4 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{39} + ( 4 \beta_{1} - 3 \beta_{2} + \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{41} + ( 2 - \beta_{1} + \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} ) q^{43} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{4} + 3 \beta_{5} + 5 \beta_{6} ) q^{45} + ( -1 + 4 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{47} -\beta_{6} q^{49} + ( -9 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} ) q^{51} + ( -1 + 3 \beta_{2} - 3 \beta_{4} - 3 \beta_{6} ) q^{53} + ( -3 + \beta_{1} + 4 \beta_{2} - \beta_{3} - 3 \beta_{4} - \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{55} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{57} + ( -\beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{7} ) q^{59} + ( -4 + 3 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{63} + ( -6 - 5 \beta_{1} + 5 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} + 2 \beta_{5} + \beta_{6} - 8 \beta_{7} ) q^{65} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{67} + ( 5 - 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} ) q^{69} + ( 6 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 6 \beta_{7} ) q^{71} + ( 1 + 5 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - 5 \beta_{7} ) q^{73} + ( -1 - 4 \beta_{1} + 5 \beta_{2} - 5 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - \beta_{6} - 7 \beta_{7} ) q^{75} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{77} + ( 10 - 4 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 7 \beta_{4} - \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{79} + ( 4 + 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 7 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} ) q^{81} + ( 3 - 6 \beta_{1} - 2 \beta_{2} - 6 \beta_{3} + \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{83} + ( 4 \beta_{1} + 5 \beta_{3} - 9 \beta_{4} - 10 \beta_{5} - 2 \beta_{6} + 10 \beta_{7} ) q^{85} + ( -4 - \beta_{1} - 4 \beta_{2} + \beta_{3} - 4 \beta_{4} + 8 \beta_{6} + 5 \beta_{7} ) q^{87} + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{5} - \beta_{6} - \beta_{7} ) q^{89} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{91} + ( -1 + \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} - 5 \beta_{6} + \beta_{7} ) q^{93} + ( \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{95} + ( 5 + \beta_{1} - 7 \beta_{2} + 2 \beta_{3} - 8 \beta_{4} - \beta_{5} + 2 \beta_{6} + 7 \beta_{7} ) q^{97} + ( 5 \beta_{2} - 4 \beta_{3} + 8 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 8 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{3} + 4q^{5} + 4q^{7} + 10q^{9} + O(q^{10}) \) \( 8q + 4q^{3} + 4q^{5} + 4q^{7} + 10q^{9} + 6q^{11} - 3q^{13} - 4q^{15} - 16q^{17} + 4q^{19} - q^{21} + 5q^{23} - 14q^{25} - 5q^{27} + q^{29} - 11q^{31} + 8q^{35} + 54q^{37} + 12q^{39} + 2q^{41} + 11q^{43} + 26q^{45} - 7q^{47} - 4q^{49} - 17q^{51} - 8q^{53} - 12q^{55} - 13q^{57} - 9q^{59} - 7q^{61} + 5q^{63} - 9q^{65} + 12q^{67} + 4q^{69} + 24q^{71} + 26q^{73} + 23q^{75} - 6q^{77} + 22q^{79} + 34q^{81} + 6q^{83} - 11q^{85} - 37q^{87} - 28q^{89} - 6q^{91} - 13q^{93} + 23q^{95} - q^{97} + 42q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 5 x^{6} - 15 x^{5} + 21 x^{4} + 3 x^{3} - 22 x^{2} + 3 x + 19\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -4 \nu^{7} + 79 \nu^{6} - 177 \nu^{5} + 459 \nu^{4} - 1008 \nu^{3} + 1011 \nu^{2} - 752 \nu - 478 \)\()/933\)
\(\beta_{3}\)\(=\)\((\)\( 35 \nu^{7} + 164 \nu^{6} - 395 \nu^{5} + 260 \nu^{4} - 2687 \nu^{3} + 2894 \nu^{2} + 1604 \nu - 2193 \)\()/1866\)
\(\beta_{4}\)\(=\)\((\)\( 217 \nu^{7} + 146 \nu^{6} + 39 \nu^{5} - 876 \nu^{4} - 4095 \nu^{3} + 6498 \nu^{2} + 1610 \nu - 2525 \)\()/5598\)
\(\beta_{5}\)\(=\)\((\)\( 241 \nu^{7} - 328 \nu^{6} + 1101 \nu^{5} - 3630 \nu^{4} + 1953 \nu^{3} - 5166 \nu^{2} + 6122 \nu + 343 \)\()/5598\)
\(\beta_{6}\)\(=\)\((\)\( -145 \nu^{7} + 298 \nu^{6} - 585 \nu^{5} + 1944 \nu^{4} - 2019 \nu^{3} - 438 \nu^{2} + 730 \nu + 1799 \)\()/1866\)
\(\beta_{7}\)\(=\)\((\)\( 701 \nu^{7} - 1016 \nu^{6} + 1863 \nu^{5} - 6966 \nu^{4} + 2181 \nu^{3} + 10122 \nu^{2} - 6296 \nu - 6265 \)\()/5598\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{5} + \beta_{4} - \beta_{2}\)
\(\nu^{3}\)\(=\)\(-3 \beta_{7} - 5 \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-\beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3} + 6 \beta_{2} + 7 \beta_{1}\)
\(\nu^{5}\)\(=\)\(5 \beta_{7} + 12 \beta_{6} - \beta_{5} + 12 \beta_{4} - 8 \beta_{3} - 8 \beta_{2} - \beta_{1} - 14\)
\(\nu^{6}\)\(=\)\(-29 \beta_{7} - 35 \beta_{6} - 7 \beta_{5} + 32 \beta_{4} - \beta_{3} + 2 \beta_{2} - 18 \beta_{1} + 16\)
\(\nu^{7}\)\(=\)\(-38 \beta_{7} - 77 \beta_{6} + 20 \beta_{5} - 13 \beta_{4} - 10 \beta_{3} + 92 \beta_{2} + 52 \beta_{1} + 60\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−0.577806 2.22188i
−0.734668 + 0.348716i
1.86526 + 0.199842i
0.947217 + 0.807294i
−0.577806 + 2.22188i
−0.734668 0.348716i
1.86526 0.199842i
0.947217 0.807294i
0 −1.71311 + 0.255482i 0 1.81197 3.13842i 0 0.500000 + 0.866025i 0 2.86946 0.875335i 0
337.2 0 0.434663 + 1.67662i 0 −1.21814 + 2.10988i 0 0.500000 + 0.866025i 0 −2.62214 + 1.45753i 0
337.3 0 1.60570 0.649414i 0 −0.468293 + 0.811107i 0 0.500000 + 0.866025i 0 2.15652 2.08552i 0
337.4 0 1.67275 + 0.449358i 0 1.87447 3.24667i 0 0.500000 + 0.866025i 0 2.59615 + 1.50332i 0
673.1 0 −1.71311 0.255482i 0 1.81197 + 3.13842i 0 0.500000 0.866025i 0 2.86946 + 0.875335i 0
673.2 0 0.434663 1.67662i 0 −1.21814 2.10988i 0 0.500000 0.866025i 0 −2.62214 1.45753i 0
673.3 0 1.60570 + 0.649414i 0 −0.468293 0.811107i 0 0.500000 0.866025i 0 2.15652 + 2.08552i 0
673.4 0 1.67275 0.449358i 0 1.87447 + 3.24667i 0 0.500000 0.866025i 0 2.59615 1.50332i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 673.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.2.r.m 8
3.b odd 2 1 3024.2.r.l 8
4.b odd 2 1 504.2.r.d 8
9.c even 3 1 inner 1008.2.r.m 8
9.c even 3 1 9072.2.a.ce 4
9.d odd 6 1 3024.2.r.l 8
9.d odd 6 1 9072.2.a.cl 4
12.b even 2 1 1512.2.r.d 8
36.f odd 6 1 504.2.r.d 8
36.f odd 6 1 4536.2.a.x 4
36.h even 6 1 1512.2.r.d 8
36.h even 6 1 4536.2.a.ba 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.r.d 8 4.b odd 2 1
504.2.r.d 8 36.f odd 6 1
1008.2.r.m 8 1.a even 1 1 trivial
1008.2.r.m 8 9.c even 3 1 inner
1512.2.r.d 8 12.b even 2 1
1512.2.r.d 8 36.h even 6 1
3024.2.r.l 8 3.b odd 2 1
3024.2.r.l 8 9.d odd 6 1
4536.2.a.x 4 36.f odd 6 1
4536.2.a.ba 4 36.h even 6 1
9072.2.a.ce 4 9.c even 3 1
9072.2.a.cl 4 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1008, [\chi])\):

\(T_{5}^{8} - \cdots\)
\(T_{11}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 - 4 T + 3 T^{2} + 11 T^{3} - 32 T^{4} + 33 T^{5} + 27 T^{6} - 108 T^{7} + 81 T^{8} \)
$5$ \( 1 - 4 T + 5 T^{2} + 18 T^{3} - 94 T^{4} + 232 T^{5} - 110 T^{6} - 1011 T^{7} + 3826 T^{8} - 5055 T^{9} - 2750 T^{10} + 29000 T^{11} - 58750 T^{12} + 56250 T^{13} + 78125 T^{14} - 312500 T^{15} + 390625 T^{16} \)
$7$ \( ( 1 - T + T^{2} )^{4} \)
$11$ \( 1 - 6 T + T^{2} + 24 T^{3} + 49 T^{4} + 306 T^{5} - 2153 T^{6} + 1236 T^{7} + 7063 T^{8} + 13596 T^{9} - 260513 T^{10} + 407286 T^{11} + 717409 T^{12} + 3865224 T^{13} + 1771561 T^{14} - 116923026 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 3 T - 16 T^{2} + 39 T^{3} + 337 T^{4} - 720 T^{5} + 2222 T^{6} + 11778 T^{7} - 44096 T^{8} + 153114 T^{9} + 375518 T^{10} - 1581840 T^{11} + 9625057 T^{12} + 14480427 T^{13} - 77228944 T^{14} + 188245551 T^{15} + 815730721 T^{16} \)
$17$ \( ( 1 + 8 T + 35 T^{2} + 167 T^{3} + 886 T^{4} + 2839 T^{5} + 10115 T^{6} + 39304 T^{7} + 83521 T^{8} )^{2} \)
$19$ \( ( 1 - 2 T + 55 T^{2} - 41 T^{3} + 1309 T^{4} - 779 T^{5} + 19855 T^{6} - 13718 T^{7} + 130321 T^{8} )^{2} \)
$23$ \( 1 - 5 T - 28 T^{2} + 177 T^{3} - 25 T^{4} + 494 T^{5} - 12137 T^{6} - 44331 T^{7} + 708646 T^{8} - 1019613 T^{9} - 6420473 T^{10} + 6010498 T^{11} - 6996025 T^{12} + 1139232711 T^{13} - 4145004892 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - T - 49 T^{2} + 294 T^{3} + 1136 T^{4} - 11204 T^{5} + 49096 T^{6} + 227337 T^{7} - 2198075 T^{8} + 6592773 T^{9} + 41289736 T^{10} - 273254356 T^{11} + 803471216 T^{12} + 6030277806 T^{13} - 29146342729 T^{14} - 17249876309 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 + 11 T - 39 T^{2} - 356 T^{3} + 5954 T^{4} + 25902 T^{5} - 215174 T^{6} - 3835 T^{7} + 11112081 T^{8} - 118885 T^{9} - 206782214 T^{10} + 771646482 T^{11} + 5498644034 T^{12} - 10191977756 T^{13} - 34612643559 T^{14} + 302638755221 T^{15} + 852891037441 T^{16} \)
$37$ \( ( 1 - 27 T + 412 T^{2} - 4104 T^{3} + 29436 T^{4} - 151848 T^{5} + 564028 T^{6} - 1367631 T^{7} + 1874161 T^{8} )^{2} \)
$41$ \( 1 - 2 T - 37 T^{2} + 432 T^{3} - 841 T^{4} - 13042 T^{5} + 100489 T^{6} + 72294 T^{7} - 3776099 T^{8} + 2964054 T^{9} + 168922009 T^{10} - 898867682 T^{11} - 2376465001 T^{12} + 50049878832 T^{13} - 175753856917 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 11 T - 27 T^{2} + 596 T^{3} + 140 T^{4} - 2688 T^{5} - 152864 T^{6} - 330797 T^{7} + 13009029 T^{8} - 14224271 T^{9} - 282645536 T^{10} - 213714816 T^{11} + 478632140 T^{12} + 87617032028 T^{13} - 170676802323 T^{14} - 2990004722177 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 7 T - 61 T^{2} + 174 T^{3} + 5108 T^{4} - 21898 T^{5} - 62918 T^{6} + 808359 T^{7} - 1538879 T^{8} + 37992873 T^{9} - 138985862 T^{10} - 2273516054 T^{11} + 24925410548 T^{12} + 39906031218 T^{13} - 657532135069 T^{14} + 3546361843241 T^{15} + 23811286661761 T^{16} \)
$53$ \( ( 1 + 4 T + 83 T^{2} - 197 T^{3} + 2722 T^{4} - 10441 T^{5} + 233147 T^{6} + 595508 T^{7} + 7890481 T^{8} )^{2} \)
$59$ \( 1 + 9 T - 119 T^{2} - 1116 T^{3} + 10336 T^{4} + 79866 T^{5} - 564434 T^{6} - 2176353 T^{7} + 30131725 T^{8} - 128404827 T^{9} - 1964794754 T^{10} + 16402799214 T^{11} + 125245043296 T^{12} - 797855517684 T^{13} - 5019483503279 T^{14} + 22397863363371 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 7 T - 144 T^{2} - 1315 T^{3} + 11783 T^{4} + 112692 T^{5} - 470051 T^{6} - 3262925 T^{7} + 21742764 T^{8} - 199038425 T^{9} - 1749059771 T^{10} + 25578942852 T^{11} + 163145544503 T^{12} - 1110644135815 T^{13} - 7418933907984 T^{14} + 21999199852147 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 - 12 T - 139 T^{2} + 1338 T^{3} + 22729 T^{4} - 133506 T^{5} - 1954153 T^{6} + 2368722 T^{7} + 170018875 T^{8} + 158704374 T^{9} - 8772192817 T^{10} - 40153665078 T^{11} + 458014829209 T^{12} + 1806467393166 T^{13} - 12573715121491 T^{14} - 72728539263876 T^{15} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 12 T + 167 T^{2} - 591 T^{3} + 7587 T^{4} - 41961 T^{5} + 841847 T^{6} - 4294932 T^{7} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 - 13 T + 232 T^{2} - 1504 T^{3} + 18820 T^{4} - 109792 T^{5} + 1236328 T^{6} - 5057221 T^{7} + 28398241 T^{8} )^{2} \)
$79$ \( 1 - 22 T + 237 T^{2} - 1616 T^{3} + 2882 T^{4} + 87342 T^{5} - 574370 T^{6} - 1050079 T^{7} + 29800944 T^{8} - 82956241 T^{9} - 3584643170 T^{10} + 43063012338 T^{11} + 112254133442 T^{12} - 4972523140784 T^{13} + 57611726958477 T^{14} - 422485997695498 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 6 T - 113 T^{2} + 1704 T^{3} + 3151 T^{4} - 166296 T^{5} + 1416277 T^{6} + 7918434 T^{7} - 172589093 T^{8} + 657230022 T^{9} + 9756732253 T^{10} - 95085890952 T^{11} + 149541169471 T^{12} + 6712125255672 T^{13} - 36944262190697 T^{14} - 162816305937762 T^{15} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 14 T + 323 T^{2} + 2963 T^{3} + 41152 T^{4} + 263707 T^{5} + 2558483 T^{6} + 9869566 T^{7} + 62742241 T^{8} )^{2} \)
$97$ \( 1 + T - 123 T^{2} - 856 T^{3} - 4138 T^{4} + 70242 T^{5} + 167026 T^{6} - 386291 T^{7} + 107246157 T^{8} - 37470227 T^{9} + 1571547634 T^{10} + 64107976866 T^{11} - 366334164778 T^{12} - 7350763259992 T^{13} - 102455556606267 T^{14} + 80798284478113 T^{15} + 7837433594376961 T^{16} \)
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