Properties

Label 819.2.j.h
Level $819$
Weight $2$
Character orbit 819.j
Analytic conductor $6.540$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.53974792554\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - x^{9} + 8 x^{8} + 7 x^{7} + 41 x^{6} + 18 x^{5} + 58 x^{4} + 28 x^{3} + 64 x^{2} + 16 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -364 \nu^{9} + 176 \nu^{8} - 220 \nu^{7} - 5913 \nu^{6} + 880 \nu^{5} + 6908 \nu^{4} + 84549 \nu^{3} + 9416 \nu^{2} + 2376 \nu + 30518 \)\()/118350\)
\(\beta_{3}\)\(=\)\((\)\( -983 \nu^{9} - 7328 \nu^{8} + 9160 \nu^{7} - 87336 \nu^{6} - 36640 \nu^{5} - 287624 \nu^{4} - 39747 \nu^{3} - 392048 \nu^{2} - 98928 \nu - 22604 \)\()/118350\)
\(\beta_{4}\)\(=\)\((\)\( -1159 \nu^{9} - 9844 \nu^{8} + 12305 \nu^{7} - 109053 \nu^{6} - 49220 \nu^{5} - 386377 \nu^{4} + 25194 \nu^{3} - 526654 \nu^{2} - 132894 \nu - 348592 \)\()/118350\)
\(\beta_{5}\)\(=\)\((\)\( 916 \nu^{9} - 3044 \nu^{8} + 3805 \nu^{7} - 3978 \nu^{6} - 15220 \nu^{5} - 119477 \nu^{4} - 129531 \nu^{3} - 162854 \nu^{2} - 41094 \nu - 180842 \)\()/59175\)
\(\beta_{6}\)\(=\)\((\)\( 2249 \nu^{9} - 9541 \nu^{8} + 26720 \nu^{7} - 34617 \nu^{6} + 31195 \nu^{5} - 152578 \nu^{4} + 39066 \nu^{3} - 46906 \nu^{2} - 20316 \nu - 3988 \)\()/78900\)
\(\beta_{7}\)\(=\)\((\)\( -15259 \nu^{9} + 14531 \nu^{8} - 121720 \nu^{7} - 107253 \nu^{6} - 637445 \nu^{5} - 272902 \nu^{4} - 871206 \nu^{3} - 258154 \nu^{2} - 957744 \nu - 2692 \)\()/236700\)
\(\beta_{8}\)\(=\)\((\)\( 18139 \nu^{9} - 21776 \nu^{8} + 145570 \nu^{7} + 96813 \nu^{6} + 719570 \nu^{5} + 92092 \nu^{4} + 981276 \nu^{3} + 255184 \nu^{2} + 1362924 \nu + 532 \)\()/118350\)
\(\beta_{9}\)\(=\)\((\)\( 1058 \nu^{9} - 1552 \nu^{8} + 9041 \nu^{7} + 3726 \nu^{6} + 39580 \nu^{5} + 2993 \nu^{4} + 53832 \nu^{3} + 11648 \nu^{2} + 50058 \nu - 136 \)\()/4734\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + 3 \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 4\)
\(\nu^{4}\)\(=\)\(-8 \beta_{9} + 2 \beta_{8} - 19 \beta_{7} + 9 \beta_{6} - 13 \beta_{1}\)
\(\nu^{5}\)\(=\)\(-22 \beta_{9} + 9 \beta_{8} - 45 \beta_{7} + 23 \beta_{6} - 9 \beta_{5} + 22 \beta_{4} - 23 \beta_{3} - 47 \beta_{2} - 47 \beta_{1} + 45\)
\(\nu^{6}\)\(=\)\(-23 \beta_{5} + 70 \beta_{4} - 78 \beta_{3} - 128 \beta_{2} + 154\)
\(\nu^{7}\)\(=\)\(206 \beta_{9} - 78 \beta_{8} + 431 \beta_{7} - 221 \beta_{6} + 407 \beta_{1}\)
\(\nu^{8}\)\(=\)\(628 \beta_{9} - 221 \beta_{8} + 1349 \beta_{7} - 691 \beta_{6} + 221 \beta_{5} - 628 \beta_{4} + 691 \beta_{3} + 1187 \beta_{2} + 1187 \beta_{1} - 1349\)
\(\nu^{9}\)\(=\)\(691 \beta_{5} - 1878 \beta_{4} + 2036 \beta_{3} + 3634 \beta_{2} - 3968\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/819\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(379\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
235.1
1.50426 + 2.60546i
0.597828 + 1.03547i
−0.132804 0.230024i
−0.606661 1.05077i
−0.862625 1.49411i
1.50426 2.60546i
0.597828 1.03547i
−0.132804 + 0.230024i
−0.606661 + 1.05077i
−0.862625 + 1.49411i
−1.00426 1.73943i 0 −1.01709 + 1.76164i 0.452861 + 0.784378i 0 0.237709 2.63505i 0.0686323 0 0.909582 1.57544i
235.2 −0.0978281 0.169443i 0 0.980859 1.69890i 1.96625 + 3.40565i 0 1.12324 + 2.39548i −0.775135 0 0.384710 0.666337i
235.3 0.632804 + 1.09605i 0 0.199118 0.344882i −1.45130 2.51373i 0 −1.29536 + 2.30696i 3.03523 0 1.83678 3.18139i
235.4 1.10666 + 1.91679i 0 −1.44940 + 2.51043i −1.06140 1.83839i 0 2.63169 + 0.272389i −1.98932 0 2.34921 4.06896i
235.5 1.36263 + 2.36014i 0 −2.71349 + 4.69991i 1.09358 + 1.89414i 0 −2.19729 1.47375i −9.33940 0 −2.98028 + 5.16200i
352.1 −1.00426 + 1.73943i 0 −1.01709 1.76164i 0.452861 0.784378i 0 0.237709 + 2.63505i 0.0686323 0 0.909582 + 1.57544i
352.2 −0.0978281 + 0.169443i 0 0.980859 + 1.69890i 1.96625 3.40565i 0 1.12324 2.39548i −0.775135 0 0.384710 + 0.666337i
352.3 0.632804 1.09605i 0 0.199118 + 0.344882i −1.45130 + 2.51373i 0 −1.29536 2.30696i 3.03523 0 1.83678 + 3.18139i
352.4 1.10666 1.91679i 0 −1.44940 2.51043i −1.06140 + 1.83839i 0 2.63169 0.272389i −1.98932 0 2.34921 + 4.06896i
352.5 1.36263 2.36014i 0 −2.71349 4.69991i 1.09358 1.89414i 0 −2.19729 + 1.47375i −9.33940 0 −2.98028 5.16200i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 352.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 819.2.j.h 10
3.b odd 2 1 91.2.e.c 10
7.c even 3 1 inner 819.2.j.h 10
7.c even 3 1 5733.2.a.bl 5
7.d odd 6 1 5733.2.a.bm 5
12.b even 2 1 1456.2.r.p 10
21.c even 2 1 637.2.e.m 10
21.g even 6 1 637.2.a.k 5
21.g even 6 1 637.2.e.m 10
21.h odd 6 1 91.2.e.c 10
21.h odd 6 1 637.2.a.l 5
39.d odd 2 1 1183.2.e.f 10
84.n even 6 1 1456.2.r.p 10
273.w odd 6 1 1183.2.e.f 10
273.w odd 6 1 8281.2.a.bw 5
273.ba even 6 1 8281.2.a.bx 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.e.c 10 3.b odd 2 1
91.2.e.c 10 21.h odd 6 1
637.2.a.k 5 21.g even 6 1
637.2.a.l 5 21.h odd 6 1
637.2.e.m 10 21.c even 2 1
637.2.e.m 10 21.g even 6 1
819.2.j.h 10 1.a even 1 1 trivial
819.2.j.h 10 7.c even 3 1 inner
1183.2.e.f 10 39.d odd 2 1
1183.2.e.f 10 273.w odd 6 1
1456.2.r.p 10 12.b even 2 1
1456.2.r.p 10 84.n even 6 1
5733.2.a.bl 5 7.c even 3 1
5733.2.a.bm 5 7.d odd 6 1
8281.2.a.bw 5 273.w odd 6 1
8281.2.a.bx 5 273.ba even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{10} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(819, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 9 + 36 T + 195 T^{2} - 210 T^{3} + 265 T^{4} - 116 T^{5} + 81 T^{6} - 30 T^{7} + 17 T^{8} - 4 T^{9} + T^{10} \)
$3$ \( T^{10} \)
$5$ \( 2304 - 2304 T + 3264 T^{2} - 480 T^{3} + 1024 T^{4} - 156 T^{5} + 217 T^{6} - 10 T^{7} + 19 T^{8} - 2 T^{9} + T^{10} \)
$7$ \( 16807 - 2401 T + 2058 T^{2} - 833 T^{3} + 119 T^{4} - 204 T^{5} + 17 T^{6} - 17 T^{7} + 6 T^{8} - T^{9} + T^{10} \)
$11$ \( 1089 - 1485 T + 2751 T^{2} - 1386 T^{3} + 2467 T^{4} - 1749 T^{5} + 1099 T^{6} - 352 T^{7} + 85 T^{8} - 11 T^{9} + T^{10} \)
$13$ \( ( -1 + T )^{10} \)
$17$ \( 184041 + 39897 T + 54123 T^{2} + 9018 T^{3} + 11137 T^{4} + 1831 T^{5} + 921 T^{6} + 102 T^{7} + 47 T^{8} + 5 T^{9} + T^{10} \)
$19$ \( 49729 + 38579 T + 69177 T^{2} - 24204 T^{3} + 31391 T^{4} - 427 T^{5} + 1607 T^{6} + 226 T^{7} + 95 T^{8} + 9 T^{9} + T^{10} \)
$23$ \( 144 - 144 T + 456 T^{2} - 432 T^{3} + 1168 T^{4} - 1034 T^{5} + 713 T^{6} - 258 T^{7} + 69 T^{8} - 10 T^{9} + T^{10} \)
$29$ \( ( 108 + 144 T + 19 T^{2} - 25 T^{3} - 3 T^{4} + T^{5} )^{2} \)
$31$ \( 126736 + 180848 T + 221752 T^{2} + 95248 T^{3} + 43528 T^{4} + 230 T^{5} + 3825 T^{6} + 162 T^{7} + 97 T^{8} - 6 T^{9} + T^{10} \)
$37$ \( 49505296 + 4643760 T + 5206008 T^{2} + 1114512 T^{3} + 504800 T^{4} + 77014 T^{5} + 14373 T^{6} + 912 T^{7} + 127 T^{8} + 4 T^{9} + T^{10} \)
$41$ \( ( 1584 - 2544 T - 940 T^{2} - 28 T^{3} + 14 T^{4} + T^{5} )^{2} \)
$43$ \( ( 64 - 288 T + 308 T^{2} - 72 T^{3} - 2 T^{4} + T^{5} )^{2} \)
$47$ \( 26718561 - 14530059 T + 8036115 T^{2} - 1208826 T^{3} + 344071 T^{4} - 2771 T^{5} + 12591 T^{6} + 72 T^{7} + 125 T^{8} - T^{9} + T^{10} \)
$53$ \( 398361681 - 254656881 T + 114371547 T^{2} - 27999402 T^{3} + 5280613 T^{4} - 593371 T^{5} + 59477 T^{6} - 3594 T^{7} + 363 T^{8} - 17 T^{9} + T^{10} \)
$59$ \( 1089 - 1485 T + 2751 T^{2} - 1386 T^{3} + 2467 T^{4} - 1749 T^{5} + 1099 T^{6} - 352 T^{7} + 85 T^{8} - 11 T^{9} + T^{10} \)
$61$ \( 71588521 + 49759141 T + 28105035 T^{2} + 6569330 T^{3} + 1397309 T^{4} + 44391 T^{5} + 17429 T^{6} - 190 T^{7} + 243 T^{8} - 11 T^{9} + T^{10} \)
$67$ \( 515244601 - 13415109 T + 49379121 T^{2} + 8631036 T^{3} + 4274771 T^{4} + 387985 T^{5} + 54915 T^{6} + 2214 T^{7} + 331 T^{8} + 13 T^{9} + T^{10} \)
$71$ \( ( 6336 - 456 T - 853 T^{2} - 25 T^{3} + 15 T^{4} + T^{5} )^{2} \)
$73$ \( 506944 + 498400 T + 519904 T^{2} + 77400 T^{3} + 54264 T^{4} + 3862 T^{5} + 4925 T^{6} + 84 T^{7} + 75 T^{8} + T^{10} \)
$79$ \( 1000000 + 1500000 T + 2060000 T^{2} + 559000 T^{3} + 239600 T^{4} - 31030 T^{5} + 16889 T^{6} - 654 T^{7} + 141 T^{8} + 2 T^{9} + T^{10} \)
$83$ \( ( 7488 + 2688 T - 308 T^{2} - 124 T^{3} + 6 T^{4} + T^{5} )^{2} \)
$89$ \( 59166864 + 16522416 T + 9952152 T^{2} + 893808 T^{3} + 783808 T^{4} + 98078 T^{5} + 24653 T^{6} + 768 T^{7} + 171 T^{8} + 4 T^{9} + T^{10} \)
$97$ \( ( -2384 - 2240 T - 612 T^{2} - 16 T^{3} + 12 T^{4} + T^{5} )^{2} \)
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