Properties

Label 50.2.d.b
Level $50$
Weight $2$
Character orbit 50.d
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.399252010106\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.58140625.2
Defining polynomial: \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{2} + \beta_{4} q^{3} -\beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + \beta_{5} q^{6} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{2} + \beta_{4} q^{3} -\beta_{2} q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} + \beta_{5} q^{6} + ( \beta_{1} - \beta_{4} - \beta_{5} ) q^{7} -\beta_{3} q^{8} + ( -\beta_{4} - \beta_{6} + \beta_{7} ) q^{9} + ( -\beta_{6} - \beta_{7} ) q^{10} + ( \beta_{1} - \beta_{4} ) q^{11} + ( \beta_{1} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{12} + ( -1 + \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{13} + ( -\beta_{1} + \beta_{4} ) q^{14} + ( -\beta_{1} - 4 \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{15} -\beta_{6} q^{16} + ( -\beta_{1} - \beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{17} + ( 1 - \beta_{4} - \beta_{5} ) q^{18} + ( \beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 4 \beta_{6} + \beta_{7} ) q^{19} + ( 1 + \beta_{4} ) q^{20} + ( -4 + 4 \beta_{2} + \beta_{4} + 4 \beta_{6} ) q^{21} + ( -\beta_{5} + \beta_{7} ) q^{22} + ( \beta_{1} - \beta_{4} + 2 \beta_{7} ) q^{23} -\beta_{1} q^{24} + ( 3 + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{25} + ( 4 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{26} + ( -\beta_{1} + 4 \beta_{3} + \beta_{4} + 4 \beta_{6} ) q^{27} + ( \beta_{5} - \beta_{7} ) q^{28} + ( -1 - 3 \beta_{2} + \beta_{4} + \beta_{6} ) q^{29} + ( \beta_{1} - 4 \beta_{3} ) q^{30} + ( -\beta_{1} - 4 \beta_{2} - \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{31} + q^{32} + ( 4 \beta_{2} - \beta_{5} + 4 \beta_{6} ) q^{33} + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - 4 \beta_{6} ) q^{34} + ( -4 + 4 \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{7} ) q^{35} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{36} + ( 1 - 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 2 \beta_{7} ) q^{37} + ( -4 + 4 \beta_{3} - \beta_{4} + 4 \beta_{6} + \beta_{7} ) q^{38} + ( 2 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} - \beta_{7} ) q^{39} + ( -1 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{40} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{41} + ( -4 \beta_{2} + \beta_{5} - 4 \beta_{6} ) q^{42} + ( 4 - \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + \beta_{4} + \beta_{5} ) q^{43} + ( -\beta_{1} + \beta_{5} - \beta_{7} ) q^{44} + ( 4 + 2 \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{45} + ( -2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{46} + ( \beta_{5} - \beta_{7} ) q^{47} -\beta_{7} q^{48} + ( 1 + \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{49} + ( -4 + \beta_{1} + 3 \beta_{2} + 4 \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{50} + ( 4 - 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{51} + ( -4 + \beta_{1} + 4 \beta_{2} + 3 \beta_{3} - \beta_{4} + 3 \beta_{6} + \beta_{7} ) q^{52} + ( 1 - 4 \beta_{2} - 3 \beta_{4} - \beta_{6} ) q^{53} + ( -4 + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{54} + ( -4 + \beta_{1} + 4 \beta_{2} - \beta_{5} + \beta_{7} ) q^{55} + ( \beta_{1} - \beta_{5} + \beta_{7} ) q^{56} + ( -\beta_{1} + 4 \beta_{2} + 4 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{57} + ( -\beta_{2} - 4 \beta_{3} + \beta_{5} - \beta_{6} ) q^{58} + ( 4 - 2 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{59} + ( -4 \beta_{6} + \beta_{7} ) q^{60} + ( -2 \beta_{1} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{61} + ( 4 - \beta_{1} - 4 \beta_{3} + 2 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{62} + ( -\beta_{1} - \beta_{4} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{63} + ( -1 + \beta_{2} + \beta_{3} + \beta_{6} ) q^{64} + ( -3 + 3 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{65} + ( -4 - \beta_{1} + 4 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{66} + ( 3 \beta_{1} - 4 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{67} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{68} + ( -2 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 4 - \beta_{1} - 4 \beta_{2} + \beta_{5} - \beta_{7} ) q^{70} + ( -4 + 8 \beta_{2} + \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{71} + ( -\beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{72} + ( -4 - 2 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 3 \beta_{6} - 3 \beta_{7} ) q^{73} + ( 3 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{74} + ( 4 + 4 \beta_{4} + 4 \beta_{6} - \beta_{7} ) q^{75} + ( -4 \beta_{2} - 4 \beta_{3} - \beta_{4} - \beta_{5} ) q^{76} + ( 8 + \beta_{1} - 8 \beta_{2} - 4 \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{77} + ( 4 + \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{78} + ( -4 + 4 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} ) q^{79} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{80} + ( \beta_{1} - 4 \beta_{2} + 3 \beta_{3} + \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{81} + ( 4 - 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{82} + ( -\beta_{1} + 4 \beta_{2} + 8 \beta_{3} + 3 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{83} + ( 4 + \beta_{1} - 4 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{84} + ( -3 + 2 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} - \beta_{4} + 2 \beta_{5} + 8 \beta_{6} + \beta_{7} ) q^{85} + ( -4 + \beta_{1} + 4 \beta_{2} - \beta_{4} ) q^{86} + ( 3 \beta_{1} - 5 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} + 5 \beta_{7} ) q^{87} + ( \beta_{1} - \beta_{5} ) q^{88} + ( 1 - 3 \beta_{1} - \beta_{2} - 4 \beta_{3} + 3 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{89} + ( -\beta_{1} + 4 \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{90} + ( -5 \beta_{1} + 2 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 2 \beta_{7} ) q^{91} + ( -\beta_{1} - \beta_{5} - \beta_{7} ) q^{92} + ( -4 + 5 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{93} + ( \beta_{1} - \beta_{5} + \beta_{7} ) q^{94} + ( -4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} + \beta_{4} + \beta_{5} - 8 \beta_{6} - 4 \beta_{7} ) q^{95} + \beta_{4} q^{96} + ( -5 - 3 \beta_{2} + 5 \beta_{6} ) q^{97} + ( -1 - \beta_{1} + \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} ) q^{98} + ( -4 - \beta_{1} + 2 \beta_{4} + 2 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 2q^{2} - 3q^{3} - 2q^{4} - 3q^{6} + 4q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 8q - 2q^{2} - 3q^{3} - 2q^{4} - 3q^{6} + 4q^{7} - 2q^{8} - q^{9} + q^{11} + 2q^{12} - 13q^{13} - q^{14} - 10q^{15} - 2q^{16} - 11q^{17} + 14q^{18} + 20q^{19} + 5q^{20} - 19q^{21} + q^{22} - 3q^{23} + 2q^{24} + 30q^{25} + 22q^{26} + 15q^{27} - q^{28} - 15q^{29} - 10q^{30} - 9q^{31} + 8q^{32} + 19q^{33} - q^{34} - 15q^{35} - q^{36} - 6q^{37} - 15q^{38} - 12q^{39} - 5q^{40} - 9q^{41} - 19q^{42} + 12q^{43} + q^{44} + 15q^{45} + 7q^{46} - q^{47} + 2q^{48} - 4q^{49} - 5q^{50} + 26q^{51} - 13q^{52} + 7q^{53} - 25q^{54} - 25q^{55} - q^{56} - 15q^{58} + 10q^{59} - 10q^{60} + 6q^{61} + 21q^{62} - 8q^{63} - 2q^{64} - 10q^{65} - 26q^{66} - 11q^{67} + 24q^{68} + 43q^{69} + 25q^{70} - 9q^{71} - 6q^{72} - 8q^{73} + 24q^{74} + 30q^{75} - 10q^{76} + 33q^{77} + 23q^{78} - 10q^{79} - 17q^{81} + 26q^{82} + 27q^{83} + 26q^{84} + 5q^{85} - 23q^{86} + q^{88} - 15q^{89} + 20q^{90} + q^{91} + 7q^{92} - 46q^{93} - q^{94} - 30q^{95} - 3q^{96} - 36q^{97} - 19q^{98} - 42q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} + 4 x^{6} - 7 x^{5} + 11 x^{4} + 5 x^{3} - 10 x^{2} - 25 x + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -64 \nu^{7} + 206 \nu^{6} - 318 \nu^{5} + 399 \nu^{4} - 732 \nu^{3} - 126 \nu^{2} + 2175 \nu + 235 \)\()/1355\)
\(\beta_{2}\)\(=\)\((\)\( 406 \nu^{7} - 714 \nu^{6} + 747 \nu^{5} - 1896 \nu^{4} + 2103 \nu^{3} + 4949 \nu^{2} + 1065 \nu - 7800 \)\()/1355\)
\(\beta_{3}\)\(=\)\((\)\( -420 \nu^{7} + 776 \nu^{6} - 698 \nu^{5} + 1924 \nu^{4} - 2297 \nu^{3} - 5129 \nu^{2} - 1055 \nu + 10265 \)\()/1355\)
\(\beta_{4}\)\(=\)\((\)\( 504 \nu^{7} - 877 \nu^{6} + 946 \nu^{5} - 2363 \nu^{4} + 2919 \nu^{3} + 5125 \nu^{2} + 3705 \nu - 11505 \)\()/1355\)
\(\beta_{5}\)\(=\)\((\)\( -613 \nu^{7} + 1321 \nu^{6} - 1513 \nu^{5} + 3394 \nu^{4} - 4352 \nu^{3} - 6178 \nu^{2} - 530 \nu + 12985 \)\()/1355\)
\(\beta_{6}\)\(=\)\((\)\( -714 \nu^{7} + 1265 \nu^{6} - 1295 \nu^{5} + 3325 \nu^{4} - 3390 \nu^{3} - 8367 \nu^{2} - 2200 \nu + 14605 \)\()/1355\)
\(\beta_{7}\)\(=\)\((\)\( -1020 \nu^{7} + 1962 \nu^{6} - 2121 \nu^{5} + 5563 \nu^{4} - 6314 \nu^{3} - 10443 \nu^{2} - 3530 \nu + 21445 \)\()/1355\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_{1} + 3\)\()/5\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 11 \beta_{2} + 6 \beta_{1} - 4\)\()/5\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{7} + 16 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 17 \beta_{2} - 3 \beta_{1} + 2\)\()/5\)
\(\nu^{4}\)\(=\)\((\)\(13 \beta_{7} - 3 \beta_{6} - 6 \beta_{5} + 13 \beta_{4} - 7 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} + 14\)\()/5\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{7} - 6 \beta_{6} - 22 \beta_{5} + 6 \beta_{4} + 51 \beta_{3} + 43 \beta_{2} + 8 \beta_{1} - 67\)\()/5\)
\(\nu^{6}\)\(=\)\((\)\(-23 \beta_{7} + 63 \beta_{6} + 46 \beta_{5} + 12 \beta_{4} + 52 \beta_{3} + 156 \beta_{2} - 34 \beta_{1} - 144\)\()/5\)
\(\nu^{7}\)\(=\)\((\)\(-31 \beta_{7} - 9 \beta_{6} + 137 \beta_{5} + 84 \beta_{4} - 31 \beta_{3} - 33 \beta_{2} - 168 \beta_{1} + 62\)\()/5\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-\beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
1.66637 + 0.917186i
−0.357358 1.86824i
−0.983224 0.644389i
1.17421 + 0.0566033i
−0.983224 + 0.644389i
1.17421 0.0566033i
1.66637 0.917186i
−0.357358 + 1.86824i
−0.809017 + 0.587785i −0.720859 + 2.21858i 0.309017 0.951057i −2.02373 + 0.951057i −0.720859 2.21858i 3.77447 0.309017 + 0.951057i −1.97539 1.43521i 1.07822 1.95894i
11.2 −0.809017 + 0.587785i 0.529876 1.63079i 0.309017 0.951057i 2.02373 + 0.951057i 0.529876 + 1.63079i −2.77447 0.309017 + 0.951057i 0.0483405 + 0.0351215i −2.19625 + 0.420099i
21.1 0.309017 0.951057i −2.39991 1.74363i −0.809017 0.587785i 2.15743 + 0.587785i −2.39991 + 1.74363i 1.83337 −0.809017 + 0.587785i 1.79224 + 5.51595i 1.22570 1.87020i
21.2 0.309017 0.951057i 1.09089 + 0.792578i −0.809017 0.587785i −2.15743 + 0.587785i 1.09089 0.792578i −0.833366 −0.809017 + 0.587785i −0.365190 1.12394i −0.107666 + 2.23347i
31.1 0.309017 + 0.951057i −2.39991 + 1.74363i −0.809017 + 0.587785i 2.15743 0.587785i −2.39991 1.74363i 1.83337 −0.809017 0.587785i 1.79224 5.51595i 1.22570 + 1.87020i
31.2 0.309017 + 0.951057i 1.09089 0.792578i −0.809017 + 0.587785i −2.15743 0.587785i 1.09089 + 0.792578i −0.833366 −0.809017 0.587785i −0.365190 + 1.12394i −0.107666 2.23347i
41.1 −0.809017 0.587785i −0.720859 2.21858i 0.309017 + 0.951057i −2.02373 0.951057i −0.720859 + 2.21858i 3.77447 0.309017 0.951057i −1.97539 + 1.43521i 1.07822 + 1.95894i
41.2 −0.809017 0.587785i 0.529876 + 1.63079i 0.309017 + 0.951057i 2.02373 0.951057i 0.529876 1.63079i −2.77447 0.309017 0.951057i 0.0483405 0.0351215i −2.19625 0.420099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.2.d.b 8
3.b odd 2 1 450.2.h.e 8
4.b odd 2 1 400.2.u.d 8
5.b even 2 1 250.2.d.d 8
5.c odd 4 2 250.2.e.c 16
25.d even 5 1 inner 50.2.d.b 8
25.d even 5 1 1250.2.a.l 4
25.e even 10 1 250.2.d.d 8
25.e even 10 1 1250.2.a.f 4
25.f odd 20 2 250.2.e.c 16
25.f odd 20 2 1250.2.b.e 8
75.j odd 10 1 450.2.h.e 8
100.h odd 10 1 10000.2.a.x 4
100.j odd 10 1 400.2.u.d 8
100.j odd 10 1 10000.2.a.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b 8 1.a even 1 1 trivial
50.2.d.b 8 25.d even 5 1 inner
250.2.d.d 8 5.b even 2 1
250.2.d.d 8 25.e even 10 1
250.2.e.c 16 5.c odd 4 2
250.2.e.c 16 25.f odd 20 2
400.2.u.d 8 4.b odd 2 1
400.2.u.d 8 100.j odd 10 1
450.2.h.e 8 3.b odd 2 1
450.2.h.e 8 75.j odd 10 1
1250.2.a.f 4 25.e even 10 1
1250.2.a.l 4 25.d even 5 1
1250.2.b.e 8 25.f odd 20 2
10000.2.a.t 4 100.j odd 10 1
10000.2.a.x 4 100.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{3}^{8} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(50, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \)
$3$ \( 256 - 192 T + 128 T^{2} - 24 T^{3} + 25 T^{4} + 6 T^{5} + 8 T^{6} + 3 T^{7} + T^{8} \)
$5$ \( 625 - 375 T^{2} + 105 T^{4} - 15 T^{6} + T^{8} \)
$7$ \( ( 16 + 12 T - 11 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$11$ \( 256 - 64 T + 352 T^{2} + 128 T^{3} + 105 T^{4} + 7 T^{5} - 3 T^{6} - T^{7} + T^{8} \)
$13$ \( 39601 + 36218 T + 21988 T^{2} + 8426 T^{3} + 2325 T^{4} + 476 T^{5} + 93 T^{6} + 13 T^{7} + T^{8} \)
$17$ \( 11881 + 14279 T + 13302 T^{2} + 6657 T^{3} + 2025 T^{4} + 383 T^{5} + 72 T^{6} + 11 T^{7} + T^{8} \)
$19$ \( 6400 - 11200 T + 12000 T^{2} - 8000 T^{3} + 3585 T^{4} - 965 T^{5} + 190 T^{6} - 20 T^{7} + T^{8} \)
$23$ \( 256 - 192 T - 192 T^{2} + 296 T^{3} + 705 T^{4} - 49 T^{5} + 63 T^{6} + 3 T^{7} + T^{8} \)
$29$ \( 25 - 300 T + 10150 T^{2} + 5700 T^{3} + 1735 T^{4} + 390 T^{5} + 105 T^{6} + 15 T^{7} + T^{8} \)
$31$ \( 1597696 + 257856 T + 56432 T^{2} + 228 T^{3} + 805 T^{4} + 237 T^{5} + 87 T^{6} + 9 T^{7} + T^{8} \)
$37$ \( 5041 - 4686 T + 3162 T^{2} - 1878 T^{3} + 2380 T^{4} + 348 T^{5} + 47 T^{6} + 6 T^{7} + T^{8} \)
$41$ \( 7921 - 11214 T + 9162 T^{2} - 4232 T^{3} + 1425 T^{4} - 128 T^{5} + 27 T^{6} + 9 T^{7} + T^{8} \)
$43$ \( ( 176 + 64 T - 39 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$47$ \( 256 - 256 T + 32 T^{2} + 132 T^{3} + 125 T^{4} + 33 T^{5} + 17 T^{6} + T^{7} + T^{8} \)
$53$ \( 65536 + 73728 T + 232448 T^{2} - 12304 T^{3} - 1695 T^{4} + 206 T^{5} + 108 T^{6} - 7 T^{7} + T^{8} \)
$59$ \( 102400 - 281600 T + 320000 T^{2} - 105600 T^{3} + 17360 T^{4} - 1320 T^{5} + 260 T^{6} - 10 T^{7} + T^{8} \)
$61$ \( 2920681 - 984384 T + 570837 T^{2} - 56572 T^{3} + 9180 T^{4} + 332 T^{5} - 18 T^{6} - 6 T^{7} + T^{8} \)
$67$ \( 891136 + 1204544 T + 730272 T^{2} + 193072 T^{3} + 40505 T^{4} + 3883 T^{5} + 237 T^{6} + 11 T^{7} + T^{8} \)
$71$ \( 4096 + 19456 T + 32512 T^{2} - 21432 T^{3} + 5225 T^{4} + 297 T^{5} + 217 T^{6} + 9 T^{7} + T^{8} \)
$73$ \( 1175056 + 988608 T + 471808 T^{2} + 141726 T^{3} + 27875 T^{4} + 3231 T^{5} + 208 T^{6} + 8 T^{7} + T^{8} \)
$79$ \( 102400 + 25600 T + 25600 T^{2} - 3200 T^{3} - 240 T^{4} + 320 T^{5} + 120 T^{6} + 10 T^{7} + T^{8} \)
$83$ \( 13424896 - 6111552 T + 1867728 T^{2} - 352404 T^{3} + 46405 T^{4} - 4359 T^{5} + 403 T^{6} - 27 T^{7} + T^{8} \)
$89$ \( 9610000 + 2015000 T + 1025500 T^{2} + 30750 T^{3} + 8525 T^{4} - 750 T^{5} + 70 T^{6} + 15 T^{7} + T^{8} \)
$97$ \( ( 1 + 7 T + 124 T^{2} + 18 T^{3} + T^{4} )^{2} \)
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