Properties

Label 546.2.j.c
Level $546$
Weight $2$
Character orbit 546.j
Analytic conductor $4.360$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 546.j (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.35983195036\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.447703281.1
Defining polynomial: \(x^{8} - x^{7} - 2 x^{6} + 2 x^{5} + 3 x^{4} + 4 x^{3} - 8 x^{2} - 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} - 2 x^{6} + 2 x^{5} + 3 x^{4} + 4 x^{3} - 8 x^{2} - 8 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{7} - \nu^{6} + 4 \nu^{5} + 2 \nu^{4} + \nu^{3} - 10 \nu^{2} + 16 \)\()/8\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + \nu^{5} - 2 \nu^{3} - \nu^{2} + 6 \nu + 8 \)\()/4\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} - \nu^{6} + 2 \nu^{5} + 2 \nu^{4} - \nu^{3} - 16 \nu^{2} - 8 \nu + 8 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{7} + \nu^{6} - 6 \nu^{5} - 6 \nu^{4} + \nu^{3} + 16 \nu^{2} + 4 \nu - 24 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - \nu^{6} - 4 \nu^{5} + 3 \nu^{3} + 8 \nu^{2} - 2 \nu - 20 \)\()/4\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{7} + \nu^{6} - 6 \nu^{5} - 6 \nu^{4} + \nu^{3} + 24 \nu^{2} + 12 \nu - 32 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} + \nu^{6} - 9 \nu^{5} - 6 \nu^{4} + 6 \nu^{3} + 27 \nu^{2} + 4 \nu - 40 \)\()/4\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)\()/3\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{6} - \beta_{5} - 3 \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 3\)\()/3\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{7} - \beta_{6} - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 4 \beta_{1}\)\()/3\)
\(\nu^{4}\)\(=\)\(-\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} - 1\)
\(\nu^{5}\)\(=\)\((\)\(2 \beta_{6} - 4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + \beta_{1} - 9\)\()/3\)
\(\nu^{6}\)\(=\)\((\)\(6 \beta_{7} - 8 \beta_{6} - 5 \beta_{5} - 6 \beta_{4} + \beta_{3} + 5 \beta_{2} - 7 \beta_{1} - 12\)\()/3\)
\(\nu^{7}\)\(=\)\((\)\(-3 \beta_{7} - 11 \beta_{6} + 4 \beta_{5} + 15 \beta_{4} - 11 \beta_{3} + 2 \beta_{2} + 5 \beta_{1} - 12\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(365\) \(379\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(-1 + \beta_{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
1.19003 0.764088i
−0.571299 + 1.29368i
1.26359 + 0.635098i
−1.38232 0.298668i
1.19003 + 0.764088i
−0.571299 1.29368i
1.26359 0.635098i
−1.38232 + 0.298668i
1.00000 −0.500000 + 0.866025i 1.00000 −0.924396 + 1.60110i −0.500000 + 0.866025i 2.61442 + 0.405935i 1.00000 −0.500000 0.866025i −0.924396 + 1.60110i
289.2 1.00000 −0.500000 + 0.866025i 1.00000 −0.441221 + 0.764218i −0.500000 + 0.866025i 0.369922 + 2.61976i 1.00000 −0.500000 0.866025i −0.441221 + 0.764218i
289.3 1.00000 −0.500000 + 0.866025i 1.00000 0.611519 1.05918i −0.500000 + 0.866025i 1.15207 2.38175i 1.00000 −0.500000 0.866025i 0.611519 1.05918i
289.4 1.00000 −0.500000 + 0.866025i 1.00000 1.75410 3.03819i −0.500000 + 0.866025i −2.63641 + 0.222079i 1.00000 −0.500000 0.866025i 1.75410 3.03819i
529.1 1.00000 −0.500000 0.866025i 1.00000 −0.924396 1.60110i −0.500000 0.866025i 2.61442 0.405935i 1.00000 −0.500000 + 0.866025i −0.924396 1.60110i
529.2 1.00000 −0.500000 0.866025i 1.00000 −0.441221 0.764218i −0.500000 0.866025i 0.369922 2.61976i 1.00000 −0.500000 + 0.866025i −0.441221 0.764218i
529.3 1.00000 −0.500000 0.866025i 1.00000 0.611519 + 1.05918i −0.500000 0.866025i 1.15207 + 2.38175i 1.00000 −0.500000 + 0.866025i 0.611519 + 1.05918i
529.4 1.00000 −0.500000 0.866025i 1.00000 1.75410 + 3.03819i −0.500000 0.866025i −2.63641 0.222079i 1.00000 −0.500000 + 0.866025i 1.75410 + 3.03819i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 529.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.j.c 8
3.b odd 2 1 1638.2.m.h 8
7.c even 3 1 546.2.k.c yes 8
13.c even 3 1 546.2.k.c yes 8
21.h odd 6 1 1638.2.p.h 8
39.i odd 6 1 1638.2.p.h 8
91.h even 3 1 inner 546.2.j.c 8
273.s odd 6 1 1638.2.m.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.j.c 8 1.a even 1 1 trivial
546.2.j.c 8 91.h even 3 1 inner
546.2.k.c yes 8 7.c even 3 1
546.2.k.c yes 8 13.c even 3 1
1638.2.m.h 8 3.b odd 2 1
1638.2.m.h 8 273.s odd 6 1
1638.2.p.h 8 21.h odd 6 1
1638.2.p.h 8 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2 T_{5}^{7} + 11 T_{5}^{6} + 6 T_{5}^{5} + 50 T_{5}^{4} + 65 T_{5}^{2} + 28 T_{5} + 49 \) acting on \(S_{2}^{\mathrm{new}}(546, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( 49 + 28 T + 65 T^{2} + 50 T^{4} + 6 T^{5} + 11 T^{6} - 2 T^{7} + T^{8} \)
$7$ \( 2401 - 1029 T + 98 T^{2} + 147 T^{3} - 117 T^{4} + 21 T^{5} + 2 T^{6} - 3 T^{7} + T^{8} \)
$11$ \( 1 + 9 T + 67 T^{2} + 118 T^{3} + 161 T^{4} + 74 T^{5} + 30 T^{6} - 4 T^{7} + T^{8} \)
$13$ \( 28561 - 6591 T + 3380 T^{2} - 195 T^{3} + 231 T^{4} - 15 T^{5} + 20 T^{6} - 3 T^{7} + T^{8} \)
$17$ \( ( -27 + 153 T - 44 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$19$ \( 23409 + 32130 T + 37827 T^{2} + 9834 T^{3} + 2674 T^{4} + 256 T^{5} + 57 T^{6} + 4 T^{7} + T^{8} \)
$23$ \( ( 3 - 81 T - 34 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$29$ \( 1121481 + 244629 T + 150789 T^{2} - 17016 T^{3} + 7867 T^{4} - 278 T^{5} + 96 T^{6} - 2 T^{7} + T^{8} \)
$31$ \( 219961 - 97083 T + 55981 T^{2} - 7336 T^{3} + 4151 T^{4} - 806 T^{5} + 168 T^{6} - 14 T^{7} + T^{8} \)
$37$ \( ( -1 + T + 8 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$41$ \( 881721 - 397197 T + 165783 T^{2} - 28458 T^{3} + 6211 T^{4} - 678 T^{5} + 158 T^{6} - 12 T^{7} + T^{8} \)
$43$ \( 169 + 949 T + 4887 T^{2} + 2482 T^{3} + 1169 T^{4} + 146 T^{5} + 34 T^{6} + T^{8} \)
$47$ \( 3969 - 7182 T + 10980 T^{2} - 4530 T^{3} + 1885 T^{4} - 4 T^{5} + 81 T^{6} - 7 T^{7} + T^{8} \)
$53$ \( 2283121 + 625554 T + 392002 T^{2} - 63466 T^{3} + 19391 T^{4} - 974 T^{5} + 147 T^{6} + T^{7} + T^{8} \)
$59$ \( ( -1079 - 476 T + 17 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$61$ \( 186624 + 528768 T + 1422144 T^{2} + 218880 T^{3} + 36304 T^{4} + 1744 T^{5} + 192 T^{6} + 4 T^{7} + T^{8} \)
$67$ \( 13697401 - 3749113 T + 1089086 T^{2} - 123417 T^{3} + 23237 T^{4} - 2349 T^{5} + 344 T^{6} - 19 T^{7} + T^{8} \)
$71$ \( 79762761 - 20041164 T + 4722951 T^{2} - 435780 T^{3} + 55036 T^{4} - 3788 T^{5} + 435 T^{6} - 20 T^{7} + T^{8} \)
$73$ \( 388129 + 290941 T + 176348 T^{2} + 40011 T^{3} + 8381 T^{4} + 465 T^{5} + 116 T^{6} + 7 T^{7} + T^{8} \)
$79$ \( 4313929 - 2297162 T + 1370703 T^{2} - 21170 T^{3} + 33662 T^{4} - 3916 T^{5} + 505 T^{6} - 24 T^{7} + T^{8} \)
$83$ \( ( 2429 + 1683 T + 364 T^{2} + 32 T^{3} + T^{4} )^{2} \)
$89$ \( ( 2949 + 504 T - 104 T^{2} - 11 T^{3} + T^{4} )^{2} \)
$97$ \( 4239481 - 2112534 T + 887956 T^{2} - 127378 T^{3} + 19745 T^{4} - 1172 T^{5} + 201 T^{6} - 11 T^{7} + T^{8} \)
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