Properties

Label 1323.2.g.f
Level $1323$
Weight $2$
Character orbit 1323.g
Analytic conductor $10.564$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.5642081874\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.991381711347.1
Defining polynomial: \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 + \beta_{2} - \beta_{9} ) q^{5} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( -\beta_{3} + \beta_{6} + \beta_{7} ) q^{4} + ( -1 + \beta_{2} - \beta_{9} ) q^{5} + ( 1 - \beta_{4} - \beta_{8} ) q^{8} + ( 2 - \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{10} + ( 1 - \beta_{3} - \beta_{4} - \beta_{8} ) q^{11} + ( 1 + \beta_{1} + \beta_{2} + \beta_{6} ) q^{13} + ( \beta_{2} + \beta_{4} + \beta_{7} ) q^{16} + ( 3 - \beta_{1} + 3 \beta_{6} + \beta_{7} ) q^{17} + ( -\beta_{3} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{19} + ( \beta_{3} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} ) q^{20} + ( -1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} ) q^{22} + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{8} - \beta_{9} ) q^{23} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{9} ) q^{25} + ( \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{26} + ( 2 \beta_{3} + \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} ) q^{29} + ( -\beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{31} + ( -\beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{32} + ( -\beta_{3} - 4 \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} ) q^{34} + ( -2 \beta_{5} - 2 \beta_{8} ) q^{37} + ( -5 + 2 \beta_{1} - 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{8} ) q^{38} + ( -1 - \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} ) q^{40} + ( \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{7} ) q^{41} + ( \beta_{3} - 2 \beta_{5} + 3 \beta_{6} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{43} + ( -\beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{44} + ( 4 - 5 \beta_{1} - 3 \beta_{2} - \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{46} + ( 4 + \beta_{1} + \beta_{2} + \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -6 + 3 \beta_{1} + 2 \beta_{2} + \beta_{4} - 6 \beta_{6} - \beta_{7} ) q^{50} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{8} ) q^{52} + ( 5 - \beta_{1} - 2 \beta_{2} + 5 \beta_{6} ) q^{53} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{55} + ( 3 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} + \beta_{9} ) q^{58} + ( \beta_{3} - \beta_{5} - 6 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{61} + ( -3 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{8} - 2 \beta_{9} ) q^{62} + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{9} ) q^{64} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{65} + ( -2 \beta_{3} + 4 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{67} + ( -7 + 2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{5} - \beta_{9} ) q^{68} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 3 \beta_{8} - 2 \beta_{9} ) q^{71} + ( -4 + 3 \beta_{4} - 4 \beta_{6} + \beta_{7} ) q^{73} + ( -10 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} - 2 \beta_{9} ) q^{74} + ( -3 + 5 \beta_{1} - \beta_{2} - 3 \beta_{6} - \beta_{7} ) q^{76} + ( -3 + 4 \beta_{1} + \beta_{2} - 3 \beta_{6} - 2 \beta_{7} ) q^{79} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{6} ) q^{80} + ( -4 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - 3 \beta_{9} ) q^{82} + ( 2 \beta_{3} - \beta_{6} - 2 \beta_{7} + 4 \beta_{8} + 2 \beta_{9} ) q^{83} + ( -2 + \beta_{2} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{85} + ( 1 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - 3 \beta_{8} + 4 \beta_{9} ) q^{86} + ( 4 + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - \beta_{9} ) q^{88} + ( 2 \beta_{5} - 7 \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{89} + ( -2 \beta_{5} + 5 \beta_{6} - 2 \beta_{8} + 2 \beta_{9} ) q^{92} + ( -4 \beta_{5} + 3 \beta_{6} + 4 \beta_{8} - 2 \beta_{9} ) q^{94} + ( -2 \beta_{3} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{95} + ( 4 \beta_{3} - \beta_{5} - 2 \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + O(q^{10}) \) \( 10q - 2q^{2} - 4q^{4} - 8q^{5} + 6q^{8} + 7q^{10} + 8q^{11} + 8q^{13} + 2q^{16} + 12q^{17} - q^{19} + 5q^{20} - q^{22} + 6q^{23} + 2q^{25} + 11q^{26} - 7q^{29} + 3q^{31} + 2q^{32} - 3q^{34} - 40q^{38} - 6q^{40} + 5q^{41} - 7q^{43} + 10q^{44} + 3q^{46} + 27q^{47} - 19q^{50} - 20q^{52} + 21q^{53} - 4q^{55} + 20q^{58} + 30q^{59} + 14q^{61} - 12q^{62} - 50q^{64} + 11q^{65} - 2q^{67} - 54q^{68} + 6q^{71} - 15q^{73} - 72q^{74} - 5q^{76} - 4q^{79} + 20q^{80} + 5q^{82} + 9q^{83} - 6q^{85} - 16q^{86} + 36q^{88} + 28q^{89} - 27q^{92} + 3q^{94} + 14q^{95} + 12q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 2 x^{9} + 9 x^{8} - 8 x^{7} + 40 x^{6} - 36 x^{5} + 90 x^{4} - 3 x^{3} + 36 x^{2} - 9 x + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{9} - 9 \nu^{8} - 3 \nu^{7} - 61 \nu^{6} - 72 \nu^{5} - 282 \nu^{4} - 204 \nu^{3} - 387 \nu^{2} - 873 \nu - 117 \)\()/189\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{9} - 12 \nu^{8} + 48 \nu^{7} - 23 \nu^{6} + 204 \nu^{5} - 240 \nu^{4} + 303 \nu^{3} - 108 \nu^{2} + 36 \nu - 1557 \)\()/567\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{9} - \nu^{8} + 12 \nu^{7} + 8 \nu^{6} + 68 \nu^{5} + 30 \nu^{4} + 123 \nu^{3} + 204 \nu^{2} + 270 \nu + 63 \)\()/63\)
\(\beta_{5}\)\(=\)\((\)\( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} - 351 \nu^{2} + 684 \nu - 180 \)\()/567\)
\(\beta_{6}\)\(=\)\((\)\( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + 1620 \nu^{2} + 369 \nu - 63 \)\()/567\)
\(\beta_{7}\)\(=\)\((\)\( -53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} - 4401 \nu^{2} - 1071 \nu - 1368 \)\()/567\)
\(\beta_{8}\)\(=\)\((\)\( -82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} - 432 \nu^{2} - 2898 \nu + 720 \)\()/567\)
\(\beta_{9}\)\(=\)\((\)\( -91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + 648 \nu^{2} - 3222 \nu + 801 \)\()/567\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + 3 \beta_{6} - \beta_{3}\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 4 \beta_{5} + \beta_{4} - 4 \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(-5 \beta_{7} - 14 \beta_{6} + \beta_{4} + \beta_{2} - 14\)
\(\nu^{5}\)\(=\)\(2 \beta_{9} - 7 \beta_{8} - \beta_{7} - 9 \beta_{6} - 17 \beta_{5} + \beta_{3}\)
\(\nu^{6}\)\(=\)\(9 \beta_{9} - 10 \beta_{8} - \beta_{5} - 10 \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + \beta_{1} + 70\)
\(\nu^{7}\)\(=\)\(11 \beta_{7} + 65 \beta_{6} - 43 \beta_{4} - 19 \beta_{2} + 75 \beta_{1} + 65\)
\(\nu^{8}\)\(=\)\(-62 \beta_{9} + 73 \beta_{8} + 118 \beta_{7} + 360 \beta_{6} + 14 \beta_{5} - 118 \beta_{3}\)
\(\nu^{9}\)\(=\)\(-135 \beta_{9} + 253 \beta_{8} + 343 \beta_{5} + 253 \beta_{4} - 87 \beta_{3} + 135 \beta_{2} - 343 \beta_{1} - 430\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1 - \beta_{6}\) \(\beta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
1.19343 2.06709i
0.920620 1.59456i
0.247934 0.429435i
−0.335166 + 0.580525i
−1.02682 + 1.77851i
1.19343 + 2.06709i
0.920620 + 1.59456i
0.247934 + 0.429435i
−0.335166 0.580525i
−1.02682 1.77851i
−1.19343 + 2.06709i 0 −1.84857 3.20182i −2.92087 0 0 4.05086 0 3.48586 6.03769i
361.2 −0.920620 + 1.59456i 0 −0.695084 1.20392i 1.33475 0 0 −1.12285 0 −1.22880 + 2.12835i
361.3 −0.247934 + 0.429435i 0 0.877057 + 1.51911i −3.69258 0 0 −1.86155 0 0.915516 1.58572i
361.4 0.335166 0.580525i 0 0.775327 + 1.34291i 1.42494 0 0 2.38012 0 0.477591 0.827212i
361.5 1.02682 1.77851i 0 −1.10873 1.92038i −0.146246 0 0 −0.446582 0 −0.150168 + 0.260099i
667.1 −1.19343 2.06709i 0 −1.84857 + 3.20182i −2.92087 0 0 4.05086 0 3.48586 + 6.03769i
667.2 −0.920620 1.59456i 0 −0.695084 + 1.20392i 1.33475 0 0 −1.12285 0 −1.22880 2.12835i
667.3 −0.247934 0.429435i 0 0.877057 1.51911i −3.69258 0 0 −1.86155 0 0.915516 + 1.58572i
667.4 0.335166 + 0.580525i 0 0.775327 1.34291i 1.42494 0 0 2.38012 0 0.477591 + 0.827212i
667.5 1.02682 + 1.77851i 0 −1.10873 + 1.92038i −0.146246 0 0 −0.446582 0 −0.150168 0.260099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 667.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.g.f 10
3.b odd 2 1 441.2.g.f 10
7.b odd 2 1 189.2.g.b 10
7.c even 3 1 1323.2.f.f 10
7.c even 3 1 1323.2.h.f 10
7.d odd 6 1 189.2.h.b 10
7.d odd 6 1 1323.2.f.e 10
9.c even 3 1 1323.2.h.f 10
9.d odd 6 1 441.2.h.f 10
21.c even 2 1 63.2.g.b 10
21.g even 6 1 63.2.h.b yes 10
21.g even 6 1 441.2.f.e 10
21.h odd 6 1 441.2.f.f 10
21.h odd 6 1 441.2.h.f 10
28.d even 2 1 3024.2.t.i 10
28.f even 6 1 3024.2.q.i 10
63.g even 3 1 inner 1323.2.g.f 10
63.g even 3 1 3969.2.a.bb 5
63.h even 3 1 1323.2.f.f 10
63.i even 6 1 441.2.f.e 10
63.i even 6 1 567.2.e.f 10
63.j odd 6 1 441.2.f.f 10
63.k odd 6 1 189.2.g.b 10
63.k odd 6 1 3969.2.a.bc 5
63.l odd 6 1 189.2.h.b 10
63.l odd 6 1 567.2.e.e 10
63.n odd 6 1 441.2.g.f 10
63.n odd 6 1 3969.2.a.ba 5
63.o even 6 1 63.2.h.b yes 10
63.o even 6 1 567.2.e.f 10
63.s even 6 1 63.2.g.b 10
63.s even 6 1 3969.2.a.z 5
63.t odd 6 1 567.2.e.e 10
63.t odd 6 1 1323.2.f.e 10
84.h odd 2 1 1008.2.t.i 10
84.j odd 6 1 1008.2.q.i 10
252.n even 6 1 3024.2.t.i 10
252.s odd 6 1 1008.2.q.i 10
252.bi even 6 1 3024.2.q.i 10
252.bn odd 6 1 1008.2.t.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 21.c even 2 1
63.2.g.b 10 63.s even 6 1
63.2.h.b yes 10 21.g even 6 1
63.2.h.b yes 10 63.o even 6 1
189.2.g.b 10 7.b odd 2 1
189.2.g.b 10 63.k odd 6 1
189.2.h.b 10 7.d odd 6 1
189.2.h.b 10 63.l odd 6 1
441.2.f.e 10 21.g even 6 1
441.2.f.e 10 63.i even 6 1
441.2.f.f 10 21.h odd 6 1
441.2.f.f 10 63.j odd 6 1
441.2.g.f 10 3.b odd 2 1
441.2.g.f 10 63.n odd 6 1
441.2.h.f 10 9.d odd 6 1
441.2.h.f 10 21.h odd 6 1
567.2.e.e 10 63.l odd 6 1
567.2.e.e 10 63.t odd 6 1
567.2.e.f 10 63.i even 6 1
567.2.e.f 10 63.o even 6 1
1008.2.q.i 10 84.j odd 6 1
1008.2.q.i 10 252.s odd 6 1
1008.2.t.i 10 84.h odd 2 1
1008.2.t.i 10 252.bn odd 6 1
1323.2.f.e 10 7.d odd 6 1
1323.2.f.e 10 63.t odd 6 1
1323.2.f.f 10 7.c even 3 1
1323.2.f.f 10 63.h even 3 1
1323.2.g.f 10 1.a even 1 1 trivial
1323.2.g.f 10 63.g even 3 1 inner
1323.2.h.f 10 7.c even 3 1
1323.2.h.f 10 9.c even 3 1
3024.2.q.i 10 28.f even 6 1
3024.2.q.i 10 252.bi even 6 1
3024.2.t.i 10 28.d even 2 1
3024.2.t.i 10 252.n even 6 1
3969.2.a.z 5 63.s even 6 1
3969.2.a.ba 5 63.n odd 6 1
3969.2.a.bb 5 63.g even 3 1
3969.2.a.bc 5 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1323, [\chi])\):

\(T_{2}^{10} + \cdots\)
\( T_{5}^{5} + 4 T_{5}^{4} - 5 T_{5}^{3} - 18 T_{5}^{2} + 18 T_{5} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 9 T + 36 T^{2} + 3 T^{3} + 90 T^{4} + 36 T^{5} + 40 T^{6} + 8 T^{7} + 9 T^{8} + 2 T^{9} + T^{10} \)
$3$ \( T^{10} \)
$5$ \( ( 3 + 18 T - 18 T^{2} - 5 T^{3} + 4 T^{4} + T^{5} )^{2} \)
$7$ \( T^{10} \)
$11$ \( ( -15 + 12 T + 15 T^{2} - 8 T^{3} - 4 T^{4} + T^{5} )^{2} \)
$13$ \( 25 + 115 T + 594 T^{2} - 169 T^{3} + 428 T^{4} - 204 T^{5} + 296 T^{6} - 130 T^{7} + 51 T^{8} - 8 T^{9} + T^{10} \)
$17$ \( 81 + 162 T + 864 T^{2} - 1890 T^{3} + 2898 T^{4} - 2259 T^{5} + 1287 T^{6} - 420 T^{7} + 99 T^{8} - 12 T^{9} + T^{10} \)
$19$ \( 185761 - 111629 T + 86907 T^{2} - 23428 T^{3} + 13166 T^{4} - 2835 T^{5} + 1376 T^{6} - 133 T^{7} + 42 T^{8} + T^{9} + T^{10} \)
$23$ \( ( 1611 + 1080 T + 51 T^{2} - 63 T^{3} - 3 T^{4} + T^{5} )^{2} \)
$29$ \( 81 - 1215 T + 19710 T^{2} + 22635 T^{3} + 24462 T^{4} + 5199 T^{5} + 1690 T^{6} + 190 T^{7} + 69 T^{8} + 7 T^{9} + T^{10} \)
$31$ \( 81225 - 26505 T + 26889 T^{2} - 6018 T^{3} + 5194 T^{4} - 1071 T^{5} + 540 T^{6} - 65 T^{7} + 30 T^{8} - 3 T^{9} + T^{10} \)
$37$ \( 82944 + 110592 T + 228096 T^{2} - 52224 T^{3} + 115264 T^{4} + 27168 T^{5} + 8832 T^{6} + 560 T^{7} + 96 T^{8} + T^{10} \)
$41$ \( 2025 - 7695 T + 31536 T^{2} + 4761 T^{3} + 9900 T^{4} - 579 T^{5} + 2020 T^{6} + 118 T^{7} + 69 T^{8} - 5 T^{9} + T^{10} \)
$43$ \( 687241 + 1214485 T + 1541055 T^{2} + 921888 T^{3} + 408318 T^{4} + 84651 T^{5} + 14496 T^{6} + 837 T^{7} + 138 T^{8} + 7 T^{9} + T^{10} \)
$47$ \( 43758225 - 26671680 T + 16872219 T^{2} - 2443014 T^{3} + 1046070 T^{4} - 230922 T^{5} + 46890 T^{6} - 5565 T^{7} + 516 T^{8} - 27 T^{9} + T^{10} \)
$53$ \( 178929 - 178929 T + 267759 T^{2} - 25380 T^{3} + 110088 T^{4} - 45693 T^{5} + 14238 T^{6} - 2415 T^{7} + 306 T^{8} - 21 T^{9} + T^{10} \)
$59$ \( 31640625 - 607500 T + 6272289 T^{2} - 3322296 T^{3} + 1440567 T^{4} - 341433 T^{5} + 60354 T^{6} - 6954 T^{7} + 594 T^{8} - 30 T^{9} + T^{10} \)
$61$ \( 1 - 14 T + 189 T^{2} - 166 T^{3} + 539 T^{4} - 153 T^{5} + 1268 T^{6} - 490 T^{7} + 162 T^{8} - 14 T^{9} + T^{10} \)
$67$ \( 50708641 + 48308864 T + 48443796 T^{2} + 584566 T^{3} + 1478510 T^{4} + 49005 T^{5} + 35105 T^{6} + 274 T^{7} + 207 T^{8} + 2 T^{9} + T^{10} \)
$71$ \( ( 81 + 1053 T + 567 T^{2} - 168 T^{3} - 3 T^{4} + T^{5} )^{2} \)
$73$ \( 772641 - 1052163 T + 1048686 T^{2} - 533637 T^{3} + 211336 T^{4} - 34167 T^{5} + 5394 T^{6} + 784 T^{7} + 231 T^{8} + 15 T^{9} + T^{10} \)
$79$ \( 37249 + 4246 T + 43716 T^{2} - 41598 T^{3} + 48858 T^{4} - 21297 T^{5} + 8151 T^{6} - 828 T^{7} + 111 T^{8} + 4 T^{9} + T^{10} \)
$83$ \( 218123361 + 40275063 T + 38540043 T^{2} - 11237130 T^{3} + 3795093 T^{4} - 455571 T^{5} + 56277 T^{6} - 2538 T^{7} + 267 T^{8} - 9 T^{9} + T^{10} \)
$89$ \( 7080921 - 4502412 T + 3980484 T^{2} - 540030 T^{3} + 648528 T^{4} - 190791 T^{5} + 45157 T^{6} - 5740 T^{7} + 549 T^{8} - 28 T^{9} + T^{10} \)
$97$ \( 2307745521 - 121346514 T + 94916553 T^{2} - 9179814 T^{3} + 3183925 T^{4} - 252807 T^{5} + 40326 T^{6} - 1958 T^{7} + 288 T^{8} - 12 T^{9} + T^{10} \)
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