Properties

Label 637.2.h.h
Level $637$
Weight $2$
Character orbit 637.h
Analytic conductor $5.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.h (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - x^{7} + 7 x^{6} + 38 x^{4} - 16 x^{3} + 15 x^{2} + 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{1} - 2 \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{4} ) q^{6} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( -2 - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{3} q^{2} -\beta_{5} q^{3} + ( 1 - \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{1} - 2 \beta_{4} ) q^{5} + ( \beta_{1} - \beta_{4} ) q^{6} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( -2 - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + ( -\beta_{1} - 3 \beta_{4} - \beta_{7} ) q^{10} + \beta_{5} q^{11} + ( -2 \beta_{1} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{12} + ( \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{13} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{15} + ( 4 - \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{16} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{17} + ( 1 + 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{18} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{19} + ( -3 \beta_{1} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{20} + ( -\beta_{1} + \beta_{4} ) q^{22} + ( -\beta_{2} + \beta_{3} - \beta_{6} ) q^{23} + ( 4 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{24} + ( -2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - \beta_{7} ) q^{25} + ( -2 - 4 \beta_{1} - 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{26} + ( -7 + \beta_{3} + 3 \beta_{6} ) q^{27} + ( -\beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{29} + ( 1 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{30} + ( 1 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{31} + ( -7 + 2 \beta_{2} + 4 \beta_{3} - \beta_{6} ) q^{32} + ( 5 + \beta_{2} + 5 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{33} + ( -\beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{34} + ( 9 + 9 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{36} + ( -2 + \beta_{3} - \beta_{6} ) q^{37} + ( 5 - 4 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} + 5 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{38} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{39} + ( -4 \beta_{4} + \beta_{5} - \beta_{7} ) q^{40} + ( 6 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 6 \beta_{4} - \beta_{7} ) q^{41} + ( \beta_{1} + \beta_{4} ) q^{43} + ( 2 \beta_{1} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{44} + ( -3 - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{6} ) q^{45} + ( 6 - \beta_{2} + \beta_{3} + \beta_{6} ) q^{46} + ( 3 \beta_{1} + \beta_{4} - \beta_{5} ) q^{47} + ( -5 \beta_{1} + \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{48} + ( -11 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - 11 \beta_{4} + \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{50} + ( 2 \beta_{1} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{51} + ( -8 + 4 \beta_{1} + \beta_{2} - 7 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{52} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 4 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{53} + ( -\beta_{2} - 5 \beta_{3} ) q^{54} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{55} + ( -5 - 3 \beta_{2} - 4 \beta_{3} - \beta_{6} ) q^{57} + ( 3 + 2 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{58} + ( -1 + \beta_{2} + 2 \beta_{3} - 2 \beta_{6} ) q^{59} + ( 4 + 2 \beta_{1} + 2 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{60} + ( -2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{61} + ( 2 + 4 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{62} + ( 1 - 2 \beta_{2} - 10 \beta_{3} ) q^{64} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{65} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{66} + ( 6 \beta_{1} - \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{67} + ( -7 + 2 \beta_{2} + 4 \beta_{3} - \beta_{6} ) q^{68} + ( -4 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} ) q^{69} + ( -4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{72} + ( 2 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{73} + ( 4 - \beta_{2} - 4 \beta_{3} ) q^{74} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{6} ) q^{75} + ( -5 + 11 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} - 5 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{76} + ( -1 + 2 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{78} + ( -\beta_{1} - 6 \beta_{4} + 3 \beta_{5} - \beta_{7} ) q^{79} + ( -2 \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{80} + ( \beta_{1} + 8 \beta_{4} + 7 \beta_{5} ) q^{81} + ( 4 + \beta_{1} - 2 \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{82} + ( -1 - \beta_{2} - 4 \beta_{3} ) q^{83} + ( 2 \beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{85} + ( 3 \beta_{4} + \beta_{7} ) q^{86} + ( 3 + 2 \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{87} + ( -4 \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{88} + ( \beta_{2} - \beta_{3} - 2 \beta_{6} ) q^{89} + ( 7 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{6} ) q^{90} + ( 4 + \beta_{2} + 7 \beta_{3} + 3 \beta_{6} ) q^{92} + ( -2 + \beta_{2} + \beta_{3} + 2 \beta_{6} ) q^{93} + ( -\beta_{1} + 8 \beta_{4} + 3 \beta_{7} ) q^{94} + ( 5 - \beta_{2} + 2 \beta_{3} + 3 \beta_{6} ) q^{95} + ( 6 \beta_{1} - 13 \beta_{4} - \beta_{7} ) q^{96} + ( -2 \beta_{1} - \beta_{4} - 5 \beta_{5} - \beta_{7} ) q^{97} + ( 7 - \beta_{3} - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} - q^{3} + 10q^{4} + 7q^{5} + 5q^{6} - 12q^{8} - 7q^{9} + O(q^{10}) \) \( 8q - 2q^{2} - q^{3} + 10q^{4} + 7q^{5} + 5q^{6} - 12q^{8} - 7q^{9} + 11q^{10} + q^{11} - 12q^{12} + 4q^{13} - 3q^{15} + 38q^{16} - 8q^{17} + 3q^{18} - q^{19} + 2q^{20} - 5q^{22} - 4q^{23} + 3q^{24} - 5q^{25} - 15q^{26} - 52q^{27} - q^{29} + 4q^{30} + 4q^{31} - 66q^{32} + 19q^{33} + 6q^{34} + 34q^{36} - 20q^{37} + 23q^{38} - q^{39} + 17q^{40} + 22q^{41} - 3q^{43} + 12q^{44} - 22q^{45} + 48q^{46} - 2q^{47} - 11q^{48} - 43q^{50} - 7q^{51} - 31q^{52} - 2q^{53} + 10q^{54} + 3q^{55} - 34q^{57} + 11q^{58} - 16q^{59} + 11q^{60} - 8q^{61} + 5q^{62} + 28q^{64} - 11q^{65} - 6q^{66} + 6q^{67} - 66q^{68} + 18q^{69} + 14q^{71} - 5q^{72} + 8q^{73} + 40q^{74} - 14q^{75} - 32q^{76} - q^{78} + 26q^{79} - 7q^{80} - 24q^{81} + 14q^{82} - 5q^{85} - 12q^{86} + 26q^{87} - 3q^{88} - 2q^{89} + 52q^{90} + 24q^{92} - 14q^{93} - 33q^{94} + 42q^{95} + 58q^{96} - 3q^{97} + 46q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 7 x^{6} + 38 x^{4} - 16 x^{3} + 15 x^{2} + 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -12 \nu^{7} - 7 \nu^{6} - 76 \nu^{5} - 44 \nu^{4} - 602 \nu^{3} - 36 \nu^{2} - 8 \nu + 1249 \)\()/458\)
\(\beta_{3}\)\(=\)\((\)\( 57 \nu^{7} - 24 \nu^{6} + 361 \nu^{5} + 209 \nu^{4} + 2287 \nu^{3} + 171 \nu^{2} + 38 \nu + 193 \)\()/916\)
\(\beta_{4}\)\(=\)\((\)\( 193 \nu^{7} - 250 \nu^{6} + 1375 \nu^{5} - 361 \nu^{4} + 7125 \nu^{3} - 5375 \nu^{2} + 2724 \nu - 375 \)\()/916\)
\(\beta_{5}\)\(=\)\((\)\( 174 \nu^{7} - 242 \nu^{6} + 1331 \nu^{5} - 507 \nu^{4} + 6897 \nu^{3} - 5203 \nu^{2} + 5383 \nu - 363 \)\()/458\)
\(\beta_{6}\)\(=\)\((\)\( -375 \nu^{7} + 182 \nu^{6} - 2375 \nu^{5} - 1375 \nu^{4} - 13889 \nu^{3} - 1125 \nu^{2} - 250 \nu - 2933 \)\()/916\)
\(\beta_{7}\)\(=\)\((\)\( -273 \nu^{7} + 356 \nu^{6} - 1958 \nu^{5} + 602 \nu^{4} - 10146 \nu^{3} + 7654 \nu^{2} - 3617 \nu + 534 \)\()/458\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{7} - 3 \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + 7 \beta_{3} + \beta_{2} - 1\)
\(\nu^{4}\)\(=\)\(7 \beta_{7} + \beta_{5} + 18 \beta_{4} - 10 \beta_{1}\)
\(\nu^{5}\)\(=\)\(10 \beta_{7} - 7 \beta_{6} + 7 \beta_{5} + 15 \beta_{4} - 48 \beta_{3} - 10 \beta_{2} - 48 \beta_{1} + 15\)
\(\nu^{6}\)\(=\)\(-10 \beta_{6} - 86 \beta_{3} - 48 \beta_{2} + 117\)
\(\nu^{7}\)\(=\)\(-86 \beta_{7} - 48 \beta_{5} - 152 \beta_{4} + 337 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/637\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(248\)
\(\chi(n)\) \(\beta_{4}\) \(\beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
1.37054 + 2.37385i
0.355143 + 0.615126i
−0.115680 0.200364i
−1.11000 1.92258i
1.37054 2.37385i
0.355143 0.615126i
−0.115680 + 0.200364i
−1.11000 + 1.92258i
−2.74108 −0.682410 1.18197i 5.51353 −0.370541 0.641796i 1.87054 + 3.23987i 0 −9.63087 0.568634 0.984903i 1.01568 + 1.75921i
165.2 −0.710287 −1.20394 2.08529i −1.49549 0.644857 + 1.11692i 0.855143 + 1.48115i 0 2.48280 −1.39895 + 2.42305i −0.458033 0.793337i
165.3 0.231361 1.66113 + 2.87716i −1.94647 1.11568 + 1.93242i 0.384320 + 0.665661i 0 −0.913059 −4.01868 + 6.96056i 0.258125 + 0.447085i
165.4 2.22001 −0.274776 0.475925i 2.92843 2.11000 + 3.65463i −0.610004 1.05656i 0 2.06113 1.34900 2.33653i 4.68423 + 8.11332i
471.1 −2.74108 −0.682410 + 1.18197i 5.51353 −0.370541 + 0.641796i 1.87054 3.23987i 0 −9.63087 0.568634 + 0.984903i 1.01568 1.75921i
471.2 −0.710287 −1.20394 + 2.08529i −1.49549 0.644857 1.11692i 0.855143 1.48115i 0 2.48280 −1.39895 2.42305i −0.458033 + 0.793337i
471.3 0.231361 1.66113 2.87716i −1.94647 1.11568 1.93242i 0.384320 0.665661i 0 −0.913059 −4.01868 6.96056i 0.258125 0.447085i
471.4 2.22001 −0.274776 + 0.475925i 2.92843 2.11000 3.65463i −0.610004 + 1.05656i 0 2.06113 1.34900 + 2.33653i 4.68423 8.11332i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 471.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.h.h 8
7.b odd 2 1 637.2.h.i 8
7.c even 3 1 91.2.f.c 8
7.c even 3 1 637.2.g.k 8
7.d odd 6 1 637.2.f.i 8
7.d odd 6 1 637.2.g.j 8
13.c even 3 1 637.2.g.k 8
21.h odd 6 1 819.2.o.h 8
28.g odd 6 1 1456.2.s.q 8
91.g even 3 1 91.2.f.c 8
91.h even 3 1 inner 637.2.h.h 8
91.h even 3 1 1183.2.a.k 4
91.k even 6 1 1183.2.a.l 4
91.l odd 6 1 8281.2.a.bt 4
91.m odd 6 1 637.2.f.i 8
91.n odd 6 1 637.2.g.j 8
91.v odd 6 1 637.2.h.i 8
91.v odd 6 1 8281.2.a.bp 4
91.x odd 12 2 1183.2.c.g 8
273.bm odd 6 1 819.2.o.h 8
364.q odd 6 1 1456.2.s.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 7.c even 3 1
91.2.f.c 8 91.g even 3 1
637.2.f.i 8 7.d odd 6 1
637.2.f.i 8 91.m odd 6 1
637.2.g.j 8 7.d odd 6 1
637.2.g.j 8 91.n odd 6 1
637.2.g.k 8 7.c even 3 1
637.2.g.k 8 13.c even 3 1
637.2.h.h 8 1.a even 1 1 trivial
637.2.h.h 8 91.h even 3 1 inner
637.2.h.i 8 7.b odd 2 1
637.2.h.i 8 91.v odd 6 1
819.2.o.h 8 21.h odd 6 1
819.2.o.h 8 273.bm odd 6 1
1183.2.a.k 4 91.h even 3 1
1183.2.a.l 4 91.k even 6 1
1183.2.c.g 8 91.x odd 12 2
1456.2.s.q 8 28.g odd 6 1
1456.2.s.q 8 364.q odd 6 1
8281.2.a.bp 4 91.v odd 6 1
8281.2.a.bt 4 91.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{4} + T_{2}^{3} - 6 T_{2}^{2} - 3 T_{2} + 1 \)
\(T_{3}^{8} + \cdots\)
\(T_{5}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 3 T - 6 T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( 36 + 96 T + 202 T^{2} + 156 T^{3} + 103 T^{4} + 23 T^{5} + 10 T^{6} + T^{7} + T^{8} \)
$5$ \( 81 - 9 T + 109 T^{2} - 114 T^{3} + 160 T^{4} - 86 T^{5} + 37 T^{6} - 7 T^{7} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 36 - 96 T + 202 T^{2} - 156 T^{3} + 103 T^{4} - 23 T^{5} + 10 T^{6} - T^{7} + T^{8} \)
$13$ \( 28561 - 8788 T + 2704 T^{2} - 416 T^{3} + 17 T^{4} - 32 T^{5} + 16 T^{6} - 4 T^{7} + T^{8} \)
$17$ \( ( -53 - 60 T - 12 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$19$ \( 250000 + 27500 T^{2} - 1000 T^{3} + 2525 T^{4} - 55 T^{5} + 56 T^{6} + T^{7} + T^{8} \)
$23$ \( ( 72 - 56 T - 38 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$29$ \( 25 - 105 T + 331 T^{2} - 452 T^{3} + 468 T^{4} - 64 T^{5} + 23 T^{6} + T^{7} + T^{8} \)
$31$ \( 2916 + 1404 T + 1378 T^{2} + 94 T^{3} + 219 T^{4} + 29 T^{6} - 4 T^{7} + T^{8} \)
$37$ \( ( -16 - 44 T + 17 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$41$ \( 318096 + 127464 T + 135112 T^{2} - 58490 T^{3} + 17793 T^{4} - 2826 T^{5} + 335 T^{6} - 22 T^{7} + T^{8} \)
$43$ \( 4 + 16 T + 58 T^{2} + 36 T^{3} + 35 T^{4} + 7 T^{5} + 12 T^{6} + 3 T^{7} + T^{8} \)
$47$ \( 10000 + 1200 T + 5244 T^{2} - 1012 T^{3} + 2477 T^{4} - 126 T^{5} + 55 T^{6} + 2 T^{7} + T^{8} \)
$53$ \( 1929321 + 1236210 T + 597640 T^{2} + 130156 T^{3} + 22769 T^{4} + 1500 T^{5} + 144 T^{6} + 2 T^{7} + T^{8} \)
$59$ \( ( -706 - 550 T - 77 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$61$ \( 10000 + 17600 T + 29476 T^{2} + 4240 T^{3} + 1733 T^{4} + 232 T^{5} + 79 T^{6} + 8 T^{7} + T^{8} \)
$67$ \( 121220100 + 6980340 T + 2725066 T^{2} - 1654 T^{3} + 37315 T^{4} - 2 T^{5} + 247 T^{6} - 6 T^{7} + T^{8} \)
$71$ \( 4129024 - 1446784 T + 474432 T^{2} - 68288 T^{3} + 12256 T^{4} - 1200 T^{5} + 212 T^{6} - 14 T^{7} + T^{8} \)
$73$ \( 3139984 + 155936 T + 218612 T^{2} + 17880 T^{3} + 13093 T^{4} + 776 T^{5} + 183 T^{6} - 8 T^{7} + T^{8} \)
$79$ \( 58982400 - 7864320 T + 2077696 T^{2} - 262144 T^{3} + 52260 T^{4} - 5532 T^{5} + 542 T^{6} - 26 T^{7} + T^{8} \)
$83$ \( ( -426 - 442 T - 97 T^{2} + T^{4} )^{2} \)
$89$ \( ( -108 - 184 T - 71 T^{2} + T^{3} + T^{4} )^{2} \)
$97$ \( 246238864 - 10482256 T + 5232284 T^{2} + 109588 T^{3} + 79337 T^{4} + 421 T^{5} + 314 T^{6} + 3 T^{7} + T^{8} \)
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