Properties

Label 91.2.f.c
Level $91$
Weight $2$
Character orbit 91.f
Analytic conductor $0.727$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.726638658394\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -\beta_{5} q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{6} + ( 1 + \beta_{4} ) q^{7} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{6} ) q^{8} + ( -2 - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -\beta_{5} q^{3} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} ) q^{4} + ( -2 - \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{6} + ( 1 + \beta_{4} ) q^{7} + ( -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} + ( 3 - \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{12} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{13} -\beta_{3} q^{14} + ( -\beta_{1} + \beta_{4} + 2 \beta_{5} ) q^{15} + ( -4 \beta_{1} + 4 \beta_{4} + \beta_{5} + \beta_{7} ) q^{16} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( -1 - 2 \beta_{3} - \beta_{6} ) q^{18} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{19} + ( 1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{20} -\beta_{6} q^{21} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{22} + ( \beta_{1} + \beta_{5} + \beta_{7} ) q^{23} + ( 4 \beta_{1} - \beta_{5} - 2 \beta_{7} ) q^{24} + ( 2 - \beta_{2} + 3 \beta_{3} ) q^{25} + ( -3 \beta_{1} - \beta_{2} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{6} ) q^{26} + ( -7 + \beta_{3} + 3 \beta_{6} ) q^{27} + ( \beta_{1} - \beta_{4} - \beta_{7} ) q^{28} + ( -\beta_{5} - \beta_{7} ) q^{29} + ( 1 - \beta_{2} + \beta_{4} + \beta_{7} ) q^{30} + ( -1 + \beta_{2} ) q^{31} + ( 7 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} ) 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} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{35} + ( -9 \beta_{4} - 2 \beta_{5} ) q^{36} + ( \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{37} + ( -5 + 3 \beta_{2} + 4 \beta_{3} + \beta_{6} ) q^{38} + ( 2 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{39} + ( -4 + \beta_{2} - \beta_{6} ) q^{40} + ( -2 \beta_{1} - 6 \beta_{4} + \beta_{7} ) q^{41} + ( -\beta_{1} + \beta_{4} ) q^{42} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{43} + ( -3 + \beta_{2} + 2 \beta_{3} + 2 \beta_{6} ) q^{44} + ( 3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{45} + ( -6 - \beta_{1} + \beta_{2} - \beta_{3} - 6 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{46} + ( 1 + 3 \beta_{3} + \beta_{6} ) q^{47} + ( -1 + 5 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{48} + \beta_{4} q^{49} + ( 2 \beta_{1} + 11 \beta_{4} - \beta_{5} + 3 \beta_{7} ) q^{50} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + \beta_{6} ) q^{51} + ( 8 + 4 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{52} + ( 1 - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{6} ) q^{53} + ( -5 \beta_{1} + \beta_{7} ) q^{54} + ( \beta_{1} - \beta_{4} - 2 \beta_{5} ) q^{55} + ( -1 + 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{56} + ( -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 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{59} + ( -4 - 2 \beta_{3} + 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} + ( -4 \beta_{1} - 2 \beta_{4} + \beta_{5} ) q^{62} + ( -2 \beta_{4} - \beta_{5} + \beta_{7} ) q^{63} + ( 1 - 2 \beta_{2} - 10 \beta_{3} ) q^{64} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{65} + ( 1 - \beta_{3} + \beta_{6} ) q^{66} + ( 6 \beta_{1} - \beta_{4} - 4 \beta_{5} - \beta_{7} ) q^{67} + ( 4 \beta_{1} - 7 \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{68} + ( 4 + 2 \beta_{2} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 3 - \beta_{2} + \beta_{3} ) q^{70} + ( 4 - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{72} + ( -2 - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{6} ) q^{73} + ( -4 + 4 \beta_{1} + \beta_{2} + 4 \beta_{3} - 4 \beta_{4} - \beta_{7} ) q^{74} + ( 2 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{75} + ( -11 \beta_{1} + 5 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{76} + \beta_{6} q^{77} + ( 2 - 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{78} + ( -6 + \beta_{2} - \beta_{3} - 3 \beta_{6} ) 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} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{84} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{85} + ( 3 - \beta_{2} ) q^{86} + ( -3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{87} + ( -4 \beta_{1} + \beta_{5} + 2 \beta_{7} ) q^{88} + ( -\beta_{1} + 2 \beta_{5} - \beta_{7} ) q^{89} + ( 7 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{6} ) q^{90} + ( -\beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} ) q^{91} + ( 4 + \beta_{2} + 7 \beta_{3} + 3 \beta_{6} ) q^{92} + ( \beta_{1} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} ) q^{93} + ( -\beta_{1} + 8 \beta_{4} + 3 \beta_{7} ) q^{94} + ( -5 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{95} + ( -13 + \beta_{2} + 6 \beta_{3} ) q^{96} + ( 1 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 5 \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{97} + ( -\beta_{1} - \beta_{3} ) q^{98} + ( 7 - \beta_{3} - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + q^{2} - q^{3} - 5q^{4} - 14q^{5} + 5q^{6} + 4q^{7} - 12q^{8} - 7q^{9} + O(q^{10}) \) \( 8q + q^{2} - q^{3} - 5q^{4} - 14q^{5} + 5q^{6} + 4q^{7} - 12q^{8} - 7q^{9} + 11q^{10} + q^{11} + 24q^{12} + 4q^{13} + 2q^{14} - 3q^{15} - 19q^{16} + 4q^{17} - 6q^{18} - q^{19} + 2q^{20} - 2q^{21} - 5q^{22} + 2q^{23} + 3q^{24} + 10q^{25} + 12q^{26} - 52q^{27} + 5q^{28} - q^{29} + 4q^{30} - 8q^{31} + 33q^{32} + 19q^{33} + 6q^{34} - 7q^{35} + 34q^{36} + 10q^{37} - 46q^{38} + 20q^{39} - 34q^{40} + 22q^{41} - 5q^{42} - 3q^{43} - 24q^{44} + 11q^{45} - 24q^{46} + 4q^{47} - 11q^{48} - 4q^{49} - 43q^{50} + 14q^{51} + 65q^{52} + 4q^{53} - 5q^{54} + 3q^{55} - 6q^{56} - 34q^{57} + 11q^{58} + 8q^{59} - 22q^{60} - 8q^{61} + 5q^{62} + 7q^{63} + 28q^{64} + 7q^{65} + 12q^{66} + 6q^{67} + 33q^{68} + 18q^{69} + 22q^{70} + 14q^{71} - 5q^{72} - 16q^{73} - 20q^{74} + 7q^{75} - 32q^{76} + 2q^{77} - q^{78} - 52q^{79} - 7q^{80} - 24q^{81} + 14q^{82} + 12q^{84} - 5q^{85} + 24q^{86} - 13q^{87} - 3q^{88} + q^{89} + 52q^{90} - 4q^{91} + 24q^{92} + 7q^{93} - 33q^{94} - 21q^{95} - 116q^{96} - 3q^{97} + q^{98} + 46q^{99} + O(q^{100}) \)

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 \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−1.11000 + 1.92258i
−0.115680 + 0.200364i
0.355143 0.615126i
1.37054 2.37385i
−1.11000 1.92258i
−0.115680 0.200364i
0.355143 + 0.615126i
1.37054 + 2.37385i
−1.11000 + 1.92258i −0.274776 + 0.475925i −1.46422 2.53610i −4.22001 −0.610004 1.05656i 0.500000 + 0.866025i 2.06113 1.34900 + 2.33653i 4.68423 8.11332i
22.2 −0.115680 + 0.200364i 1.66113 2.87716i 0.973236 + 1.68569i −2.23136 0.384320 + 0.665661i 0.500000 + 0.866025i −0.913059 −4.01868 6.96056i 0.258125 0.447085i
22.3 0.355143 0.615126i −1.20394 + 2.08529i 0.747746 + 1.29513i −1.28971 0.855143 + 1.48115i 0.500000 + 0.866025i 2.48280 −1.39895 2.42305i −0.458033 + 0.793337i
22.4 1.37054 2.37385i −0.682410 + 1.18197i −2.75677 4.77486i 0.741082 1.87054 + 3.23987i 0.500000 + 0.866025i −9.63087 0.568634 + 0.984903i 1.01568 1.75921i
29.1 −1.11000 1.92258i −0.274776 0.475925i −1.46422 + 2.53610i −4.22001 −0.610004 + 1.05656i 0.500000 0.866025i 2.06113 1.34900 2.33653i 4.68423 + 8.11332i
29.2 −0.115680 0.200364i 1.66113 + 2.87716i 0.973236 1.68569i −2.23136 0.384320 0.665661i 0.500000 0.866025i −0.913059 −4.01868 + 6.96056i 0.258125 + 0.447085i
29.3 0.355143 + 0.615126i −1.20394 2.08529i 0.747746 1.29513i −1.28971 0.855143 1.48115i 0.500000 0.866025i 2.48280 −1.39895 + 2.42305i −0.458033 0.793337i
29.4 1.37054 + 2.37385i −0.682410 1.18197i −2.75677 + 4.77486i 0.741082 1.87054 3.23987i 0.500000 0.866025i −9.63087 0.568634 0.984903i 1.01568 + 1.75921i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.2.f.c 8
3.b odd 2 1 819.2.o.h 8
4.b odd 2 1 1456.2.s.q 8
7.b odd 2 1 637.2.f.i 8
7.c even 3 1 637.2.g.k 8
7.c even 3 1 637.2.h.h 8
7.d odd 6 1 637.2.g.j 8
7.d odd 6 1 637.2.h.i 8
13.c even 3 1 inner 91.2.f.c 8
13.c even 3 1 1183.2.a.k 4
13.e even 6 1 1183.2.a.l 4
13.f odd 12 2 1183.2.c.g 8
39.i odd 6 1 819.2.o.h 8
52.j odd 6 1 1456.2.s.q 8
91.g even 3 1 637.2.h.h 8
91.h even 3 1 637.2.g.k 8
91.m odd 6 1 637.2.h.i 8
91.n odd 6 1 637.2.f.i 8
91.n odd 6 1 8281.2.a.bp 4
91.t odd 6 1 8281.2.a.bt 4
91.v odd 6 1 637.2.g.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.c 8 1.a even 1 1 trivial
91.2.f.c 8 13.c even 3 1 inner
637.2.f.i 8 7.b odd 2 1
637.2.f.i 8 91.n odd 6 1
637.2.g.j 8 7.d odd 6 1
637.2.g.j 8 91.v odd 6 1
637.2.g.k 8 7.c even 3 1
637.2.g.k 8 91.h even 3 1
637.2.h.h 8 7.c even 3 1
637.2.h.h 8 91.g even 3 1
637.2.h.i 8 7.d odd 6 1
637.2.h.i 8 91.m odd 6 1
819.2.o.h 8 3.b odd 2 1
819.2.o.h 8 39.i odd 6 1
1183.2.a.k 4 13.c even 3 1
1183.2.a.l 4 13.e even 6 1
1183.2.c.g 8 13.f odd 12 2
1456.2.s.q 8 4.b odd 2 1
1456.2.s.q 8 52.j odd 6 1
8281.2.a.bp 4 91.n odd 6 1
8281.2.a.bt 4 91.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - T_{2}^{7} + 7 T_{2}^{6} + 38 T_{2}^{4} - 16 T_{2}^{3} + 15 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(91, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 15 T^{2} - 16 T^{3} + 38 T^{4} + 7 T^{6} - T^{7} + T^{8} \)
$3$ \( 36 + 96 T + 202 T^{2} + 156 T^{3} + 103 T^{4} + 23 T^{5} + 10 T^{6} + T^{7} + T^{8} \)
$5$ \( ( -9 - T + 12 T^{2} + 7 T^{3} + T^{4} )^{2} \)
$7$ \( ( 1 - T + T^{2} )^{4} \)
$11$ \( 36 - 96 T + 202 T^{2} - 156 T^{3} + 103 T^{4} - 23 T^{5} + 10 T^{6} - T^{7} + T^{8} \)
$13$ \( 28561 - 8788 T + 2704 T^{2} - 416 T^{3} + 17 T^{4} - 32 T^{5} + 16 T^{6} - 4 T^{7} + T^{8} \)
$17$ \( 2809 - 3180 T + 2964 T^{2} - 1144 T^{3} + 437 T^{4} - 72 T^{5} + 28 T^{6} - 4 T^{7} + T^{8} \)
$19$ \( 250000 + 27500 T^{2} - 1000 T^{3} + 2525 T^{4} - 55 T^{5} + 56 T^{6} + T^{7} + T^{8} \)
$23$ \( 5184 + 4032 T + 5872 T^{2} - 1840 T^{3} + 1484 T^{4} - 36 T^{5} + 42 T^{6} - 2 T^{7} + T^{8} \)
$29$ \( 25 - 105 T + 331 T^{2} - 452 T^{3} + 468 T^{4} - 64 T^{5} + 23 T^{6} + T^{7} + T^{8} \)
$31$ \( ( 54 - 26 T - 13 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$37$ \( 256 - 704 T + 2208 T^{2} + 428 T^{3} + 745 T^{4} - 258 T^{5} + 83 T^{6} - 10 T^{7} + T^{8} \)
$41$ \( 318096 + 127464 T + 135112 T^{2} - 58490 T^{3} + 17793 T^{4} - 2826 T^{5} + 335 T^{6} - 22 T^{7} + T^{8} \)
$43$ \( 4 + 16 T + 58 T^{2} + 36 T^{3} + 35 T^{4} + 7 T^{5} + 12 T^{6} + 3 T^{7} + T^{8} \)
$47$ \( ( 100 - 12 T - 51 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$53$ \( ( -1389 + 890 T - 140 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$59$ \( 498436 - 388300 T + 248138 T^{2} - 53646 T^{3} + 11035 T^{4} - 484 T^{5} + 141 T^{6} - 8 T^{7} + T^{8} \)
$61$ \( 10000 + 17600 T + 29476 T^{2} + 4240 T^{3} + 1733 T^{4} + 232 T^{5} + 79 T^{6} + 8 T^{7} + T^{8} \)
$67$ \( 121220100 + 6980340 T + 2725066 T^{2} - 1654 T^{3} + 37315 T^{4} - 2 T^{5} + 247 T^{6} - 6 T^{7} + T^{8} \)
$71$ \( 4129024 - 1446784 T + 474432 T^{2} - 68288 T^{3} + 12256 T^{4} - 1200 T^{5} + 212 T^{6} - 14 T^{7} + T^{8} \)
$73$ \( ( 1772 - 88 T - 119 T^{2} + 8 T^{3} + T^{4} )^{2} \)
$79$ \( ( -7680 - 1024 T + 134 T^{2} + 26 T^{3} + T^{4} )^{2} \)
$83$ \( ( -426 - 442 T - 97 T^{2} + T^{4} )^{2} \)
$89$ \( 11664 - 19872 T + 26188 T^{2} - 13280 T^{3} + 5333 T^{4} - 297 T^{5} + 72 T^{6} - T^{7} + T^{8} \)
$97$ \( 246238864 - 10482256 T + 5232284 T^{2} + 109588 T^{3} + 79337 T^{4} + 421 T^{5} + 314 T^{6} + 3 T^{7} + T^{8} \)
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