Properties

Label 507.2.j.h
Level $507$
Weight $2$
Character orbit 507.j
Analytic conductor $4.048$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: 12.0.17213603549184.1
Defining polynomial: \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12} - 5 x^{10} + 19 x^{8} - 28 x^{6} + 31 x^{4} - 6 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -25 \nu^{11} + 95 \nu^{9} - 361 \nu^{7} + 155 \nu^{5} - 30 \nu^{3} - 1563 \nu \)\()/559\)
\(\beta_{3}\)\(=\)\((\)\( 25 \nu^{10} - 95 \nu^{8} + 361 \nu^{6} - 155 \nu^{4} + 30 \nu^{2} + 1004 \)\()/559\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{10} - 20 \nu^{8} + 76 \nu^{6} - 139 \nu^{4} + 124 \nu^{2} - 24 \)\()/43\)
\(\beta_{5}\)\(=\)\((\)\( 45 \nu^{10} - 171 \nu^{8} + 538 \nu^{6} - 279 \nu^{4} + 54 \nu^{2} + 242 \)\()/559\)
\(\beta_{6}\)\(=\)\((\)\( 70 \nu^{11} - 266 \nu^{9} + 899 \nu^{7} - 434 \nu^{5} + 84 \nu^{3} + 1246 \nu \)\()/559\)
\(\beta_{7}\)\(=\)\((\)\( 114 \nu^{10} - 545 \nu^{8} + 2071 \nu^{6} - 2831 \nu^{4} + 3379 \nu^{2} - 95 \)\()/559\)
\(\beta_{8}\)\(=\)\((\)\( 114 \nu^{11} - 545 \nu^{9} + 2071 \nu^{7} - 2831 \nu^{5} + 3379 \nu^{3} - 95 \nu \)\()/559\)
\(\beta_{9}\)\(=\)\((\)\( -128 \nu^{10} + 710 \nu^{8} - 2698 \nu^{6} + 4483 \nu^{4} - 4402 \nu^{2} + 852 \)\()/559\)
\(\beta_{10}\)\(=\)\((\)\( -242 \nu^{11} + 1255 \nu^{9} - 4769 \nu^{7} + 7314 \nu^{5} - 7781 \nu^{3} + 1506 \nu \)\()/559\)
\(\beta_{11}\)\(=\)\((\)\( -317 \nu^{11} + 1540 \nu^{9} - 5852 \nu^{7} + 8338 \nu^{5} - 9548 \nu^{3} + 1848 \nu \)\()/559\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{9} + \beta_{7} + \beta_{4} - \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{11} + 3 \beta_{8} + \beta_{2}\)
\(\nu^{4}\)\(=\)\(3 \beta_{9} + 2 \beta_{7} + 4 \beta_{4} - 2\)
\(\nu^{5}\)\(=\)\(4 \beta_{11} - \beta_{10} + 9 \beta_{8} - 9 \beta_{1}\)
\(\nu^{6}\)\(=\)\(-5 \beta_{5} + 9 \beta_{3} - 14\)
\(\nu^{7}\)\(=\)\(-5 \beta_{6} - 14 \beta_{2} - 28 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-28 \beta_{9} - 14 \beta_{7} - 19 \beta_{5} - 47 \beta_{4} + 28 \beta_{3} - 28\)
\(\nu^{9}\)\(=\)\(-47 \beta_{11} + 19 \beta_{10} - 89 \beta_{8} - 19 \beta_{6} - 47 \beta_{2}\)
\(\nu^{10}\)\(=\)\(-89 \beta_{9} - 42 \beta_{7} - 155 \beta_{4} + 42\)
\(\nu^{11}\)\(=\)\(-155 \beta_{11} + 66 \beta_{10} - 286 \beta_{8} + 286 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/507\mathbb{Z}\right)^\times\).

\(n\) \(170\) \(340\)
\(\chi(n)\) \(1\) \(\beta_{7}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
−1.56052 0.900969i
−1.07992 0.623490i
−0.385418 0.222521i
0.385418 + 0.222521i
1.07992 + 0.623490i
1.56052 + 0.900969i
−1.56052 + 0.900969i
−1.07992 + 0.623490i
−0.385418 + 0.222521i
0.385418 0.222521i
1.07992 0.623490i
1.56052 0.900969i
−1.56052 0.900969i −0.500000 + 0.866025i 0.623490 + 1.07992i 1.44504i 1.56052 0.900969i −2.98349 + 1.72252i 1.35690i −0.500000 0.866025i 1.30194 2.25502i
316.2 −1.07992 0.623490i −0.500000 + 0.866025i −0.222521 0.385418i 2.80194i 1.07992 0.623490i 4.15860 2.40097i 3.04892i −0.500000 0.866025i −1.74698 + 3.02586i
316.3 −0.385418 0.222521i −0.500000 + 0.866025i −0.900969 1.56052i 0.246980i 0.385418 0.222521i −1.51816 + 0.876510i 1.69202i −0.500000 0.866025i −0.0549581 + 0.0951903i
316.4 0.385418 + 0.222521i −0.500000 + 0.866025i −0.900969 1.56052i 0.246980i −0.385418 + 0.222521i 1.51816 0.876510i 1.69202i −0.500000 0.866025i −0.0549581 + 0.0951903i
316.5 1.07992 + 0.623490i −0.500000 + 0.866025i −0.222521 0.385418i 2.80194i −1.07992 + 0.623490i −4.15860 + 2.40097i 3.04892i −0.500000 0.866025i −1.74698 + 3.02586i
316.6 1.56052 + 0.900969i −0.500000 + 0.866025i 0.623490 + 1.07992i 1.44504i −1.56052 + 0.900969i 2.98349 1.72252i 1.35690i −0.500000 0.866025i 1.30194 2.25502i
361.1 −1.56052 + 0.900969i −0.500000 0.866025i 0.623490 1.07992i 1.44504i 1.56052 + 0.900969i −2.98349 1.72252i 1.35690i −0.500000 + 0.866025i 1.30194 + 2.25502i
361.2 −1.07992 + 0.623490i −0.500000 0.866025i −0.222521 + 0.385418i 2.80194i 1.07992 + 0.623490i 4.15860 + 2.40097i 3.04892i −0.500000 + 0.866025i −1.74698 3.02586i
361.3 −0.385418 + 0.222521i −0.500000 0.866025i −0.900969 + 1.56052i 0.246980i 0.385418 + 0.222521i −1.51816 0.876510i 1.69202i −0.500000 + 0.866025i −0.0549581 0.0951903i
361.4 0.385418 0.222521i −0.500000 0.866025i −0.900969 + 1.56052i 0.246980i −0.385418 0.222521i 1.51816 + 0.876510i 1.69202i −0.500000 + 0.866025i −0.0549581 0.0951903i
361.5 1.07992 0.623490i −0.500000 0.866025i −0.222521 + 0.385418i 2.80194i −1.07992 0.623490i −4.15860 2.40097i 3.04892i −0.500000 + 0.866025i −1.74698 3.02586i
361.6 1.56052 0.900969i −0.500000 0.866025i 0.623490 1.07992i 1.44504i −1.56052 0.900969i 2.98349 + 1.72252i 1.35690i −0.500000 + 0.866025i 1.30194 + 2.25502i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner
13.c even 3 1 inner
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 507.2.j.h 12
13.b even 2 1 inner 507.2.j.h 12
13.c even 3 1 507.2.b.g 6
13.c even 3 1 inner 507.2.j.h 12
13.d odd 4 1 507.2.e.j 6
13.d odd 4 1 507.2.e.k 6
13.e even 6 1 507.2.b.g 6
13.e even 6 1 inner 507.2.j.h 12
13.f odd 12 1 507.2.a.j 3
13.f odd 12 1 507.2.a.k yes 3
13.f odd 12 1 507.2.e.j 6
13.f odd 12 1 507.2.e.k 6
39.h odd 6 1 1521.2.b.m 6
39.i odd 6 1 1521.2.b.m 6
39.k even 12 1 1521.2.a.p 3
39.k even 12 1 1521.2.a.q 3
52.l even 12 1 8112.2.a.by 3
52.l even 12 1 8112.2.a.cf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
507.2.a.j 3 13.f odd 12 1
507.2.a.k yes 3 13.f odd 12 1
507.2.b.g 6 13.c even 3 1
507.2.b.g 6 13.e even 6 1
507.2.e.j 6 13.d odd 4 1
507.2.e.j 6 13.f odd 12 1
507.2.e.k 6 13.d odd 4 1
507.2.e.k 6 13.f odd 12 1
507.2.j.h 12 1.a even 1 1 trivial
507.2.j.h 12 13.b even 2 1 inner
507.2.j.h 12 13.c even 3 1 inner
507.2.j.h 12 13.e even 6 1 inner
1521.2.a.p 3 39.k even 12 1
1521.2.a.q 3 39.k even 12 1
1521.2.b.m 6 39.h odd 6 1
1521.2.b.m 6 39.i odd 6 1
8112.2.a.by 3 52.l even 12 1
8112.2.a.cf 3 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(507, [\chi])\):

\( T_{2}^{12} - 5 T_{2}^{10} + 19 T_{2}^{8} - 28 T_{2}^{6} + 31 T_{2}^{4} - 6 T_{2}^{2} + 1 \)
\( T_{5}^{6} + 10 T_{5}^{4} + 17 T_{5}^{2} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$3$ \( ( 1 + T + T^{2} )^{6} \)
$5$ \( ( 1 + 17 T^{2} + 10 T^{4} + T^{6} )^{2} \)
$7$ \( 707281 - 320421 T^{2} + 113203 T^{4} - 12796 T^{6} + 1063 T^{8} - 38 T^{10} + T^{12} \)
$11$ \( 3418801 - 1823114 T^{2} + 859407 T^{4} - 56448 T^{6} + 2735 T^{8} - 61 T^{10} + T^{12} \)
$13$ \( T^{12} \)
$17$ \( ( 49 + 98 T + 147 T^{2} + 84 T^{3} + 35 T^{4} + 7 T^{5} + T^{6} )^{2} \)
$19$ \( 163047361 - 33020634 T^{2} + 5397727 T^{4} - 235648 T^{6} + 7615 T^{8} - 101 T^{10} + T^{12} \)
$23$ \( ( 6889 + 3569 T + 1683 T^{2} + 252 T^{3} + 47 T^{4} - 2 T^{5} + T^{6} )^{2} \)
$29$ \( ( 1849 + 215 T + 369 T^{2} - 126 T^{3} + 59 T^{4} - 8 T^{5} + T^{6} )^{2} \)
$31$ \( ( 38809 + 3681 T^{2} + 110 T^{4} + T^{6} )^{2} \)
$37$ \( 68574961 - 11767301 T^{2} + 1439571 T^{4} - 82908 T^{6} + 3479 T^{8} - 70 T^{10} + T^{12} \)
$41$ \( 1 - 6 T^{2} + 31 T^{4} - 28 T^{6} + 19 T^{8} - 5 T^{10} + T^{12} \)
$43$ \( ( 841 - 725 T + 538 T^{2} - 133 T^{3} + 34 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$47$ \( ( 829921 + 30798 T^{2} + 321 T^{4} + T^{6} )^{2} \)
$53$ \( ( 29 + 40 T + 13 T^{2} + T^{3} )^{4} \)
$59$ \( 9834496 - 4917248 T^{2} + 1843968 T^{4} - 301056 T^{6} + 36848 T^{8} - 196 T^{10} + T^{12} \)
$61$ \( ( 49729 + 2676 T + 3043 T^{2} - 602 T^{3} + 157 T^{4} - 13 T^{5} + T^{6} )^{2} \)
$67$ \( 88529281 - 13680686 T^{2} + 1464895 T^{4} - 81508 T^{6} + 3307 T^{8} - 69 T^{10} + T^{12} \)
$71$ \( 45165175441 - 2502009733 T^{2} + 97374455 T^{4} - 1858920 T^{6} + 25863 T^{8} - 194 T^{10} + T^{12} \)
$73$ \( ( 27889 + 4189 T^{2} + 122 T^{4} + T^{6} )^{2} \)
$79$ \( ( -169 - 22 T + 9 T^{2} + T^{3} )^{4} \)
$83$ \( ( 1849 + 4401 T^{2} + 146 T^{4} + T^{6} )^{2} \)
$89$ \( 1 - 54 T^{2} + 2875 T^{4} - 2212 T^{6} + 1627 T^{8} - 41 T^{10} + T^{12} \)
$97$ \( 1998607065841 - 57186428171 T^{2} + 1123102678 T^{4} - 11856271 T^{6} + 91318 T^{8} - 363 T^{10} + T^{12} \)
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