Properties

Label 546.2.k.c
Level $546$
Weight $2$
Character orbit 546.k
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.k (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.

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

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 \(\left(\mathbb{Z}/546\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
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
−0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.924396 + 1.60110i −0.500000 0.866025i −1.65876 + 2.06119i 1.00000 1.00000 1.84879
373.2 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i −0.441221 + 0.764218i −0.500000 0.866025i −2.45374 0.989520i 1.00000 1.00000 0.882443
373.3 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.611519 1.05918i −0.500000 0.866025i 1.48662 + 2.18860i 1.00000 1.00000 −1.22304
373.4 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 1.75410 3.03819i −0.500000 0.866025i 1.12588 2.39424i 1.00000 1.00000 −3.50820
445.1 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.924396 1.60110i −0.500000 + 0.866025i −1.65876 2.06119i 1.00000 1.00000 1.84879
445.2 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i −0.441221 0.764218i −0.500000 + 0.866025i −2.45374 + 0.989520i 1.00000 1.00000 0.882443
445.3 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0.611519 + 1.05918i −0.500000 + 0.866025i 1.48662 2.18860i 1.00000 1.00000 −1.22304
445.4 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 1.75410 + 3.03819i −0.500000 + 0.866025i 1.12588 + 2.39424i 1.00000 1.00000 −3.50820
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 445.4
Significant digits:
Format:

Inner twists

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

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{4} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( 49 + 28 T + 65 T^{2} + 50 T^{4} + 6 T^{5} + 11 T^{6} - 2 T^{7} + T^{8} \)
$7$ \( 2401 + 1029 T + 392 T^{2} + 231 T^{3} + 123 T^{4} + 33 T^{5} + 8 T^{6} + 3 T^{7} + T^{8} \)
$11$ \( ( -1 + 9 T - 14 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$13$ \( 28561 - 6591 T + 3380 T^{2} - 195 T^{3} + 231 T^{4} - 15 T^{5} + 20 T^{6} - 3 T^{7} + T^{8} \)
$17$ \( 729 + 4131 T + 22221 T^{2} + 6840 T^{3} + 2269 T^{4} + 218 T^{5} + 48 T^{6} + 2 T^{7} + T^{8} \)
$19$ \( ( -153 + 210 T - 41 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$23$ \( 9 + 243 T + 6663 T^{2} - 2730 T^{3} + 1477 T^{4} - 26 T^{5} + 50 T^{6} - 4 T^{7} + T^{8} \)
$29$ \( 1121481 + 244629 T + 150789 T^{2} - 17016 T^{3} + 7867 T^{4} - 278 T^{5} + 96 T^{6} - 2 T^{7} + T^{8} \)
$31$ \( 219961 - 97083 T + 55981 T^{2} - 7336 T^{3} + 4151 T^{4} - 806 T^{5} + 168 T^{6} - 14 T^{7} + T^{8} \)
$37$ \( 1 + T + 9 T^{2} + 4 T^{3} + 71 T^{4} + 50 T^{5} + 28 T^{6} + 6 T^{7} + T^{8} \)
$41$ \( 881721 - 397197 T + 165783 T^{2} - 28458 T^{3} + 6211 T^{4} - 678 T^{5} + 158 T^{6} - 12 T^{7} + T^{8} \)
$43$ \( 169 + 949 T + 4887 T^{2} + 2482 T^{3} + 1169 T^{4} + 146 T^{5} + 34 T^{6} + T^{8} \)
$47$ \( 3969 - 7182 T + 10980 T^{2} - 4530 T^{3} + 1885 T^{4} - 4 T^{5} + 81 T^{6} - 7 T^{7} + T^{8} \)
$53$ \( 2283121 + 625554 T + 392002 T^{2} - 63466 T^{3} + 19391 T^{4} - 974 T^{5} + 147 T^{6} + T^{7} + T^{8} \)
$59$ \( 1164241 - 513604 T + 244919 T^{2} - 26436 T^{3} + 8984 T^{4} - 1224 T^{5} + 239 T^{6} - 16 T^{7} + T^{8} \)
$61$ \( ( -432 + 1224 T - 176 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$67$ \( ( -3701 - 1013 T + 17 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$71$ \( 79762761 - 20041164 T + 4722951 T^{2} - 435780 T^{3} + 55036 T^{4} - 3788 T^{5} + 435 T^{6} - 20 T^{7} + T^{8} \)
$73$ \( 388129 + 290941 T + 176348 T^{2} + 40011 T^{3} + 8381 T^{4} + 465 T^{5} + 116 T^{6} + 7 T^{7} + T^{8} \)
$79$ \( 4313929 - 2297162 T + 1370703 T^{2} - 21170 T^{3} + 33662 T^{4} - 3916 T^{5} + 505 T^{6} - 24 T^{7} + T^{8} \)
$83$ \( ( 2429 + 1683 T + 364 T^{2} + 32 T^{3} + T^{4} )^{2} \)
$89$ \( 8696601 - 1486296 T + 560712 T^{2} - 12462 T^{3} + 13411 T^{4} - 136 T^{5} + 225 T^{6} + 11 T^{7} + T^{8} \)
$97$ \( 4239481 - 2112534 T + 887956 T^{2} - 127378 T^{3} + 19745 T^{4} - 1172 T^{5} + 201 T^{6} - 11 T^{7} + T^{8} \)
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