Properties

Label 273.2.cd.c
Level $273$
Weight $2$
Character orbit 273.cd
Analytic conductor $2.180$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.cd (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/273\mathbb{Z}\right)^\times\).

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(-1\) \(-\zeta_{24}^{6}\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 + 0.965926i
−0.965926 + 0.258819i 1.62484 0.599900i −0.866025 + 0.500000i 0.565826 0.151613i −1.41421 + 1.00000i −2.38014 1.15539i 2.12132 2.12132i 2.28024 1.94949i −0.507306 + 0.292893i
44.2 0.965926 0.258819i 1.10721 + 1.33195i −0.866025 + 0.500000i −3.29788 + 0.883663i 1.41421 + 1.00000i 2.38014 + 1.15539i −2.12132 + 2.12132i −0.548188 + 2.94949i −2.95680 + 1.70711i
86.1 −0.258819 0.965926i 0.599900 + 1.62484i 0.866025 0.500000i 0.883663 + 3.29788i 1.41421 1.00000i 1.15539 2.38014i −2.12132 2.12132i −2.28024 + 1.94949i 2.95680 1.70711i
86.2 0.258819 + 0.965926i −1.33195 + 1.10721i 0.866025 0.500000i −0.151613 0.565826i −1.41421 1.00000i −1.15539 + 2.38014i 2.12132 + 2.12132i 0.548188 2.94949i 0.507306 0.292893i
200.1 −0.258819 + 0.965926i 0.599900 1.62484i 0.866025 + 0.500000i 0.883663 3.29788i 1.41421 + 1.00000i 1.15539 + 2.38014i −2.12132 + 2.12132i −2.28024 1.94949i 2.95680 + 1.70711i
200.2 0.258819 0.965926i −1.33195 1.10721i 0.866025 + 0.500000i −0.151613 + 0.565826i −1.41421 + 1.00000i −1.15539 2.38014i 2.12132 2.12132i 0.548188 + 2.94949i 0.507306 + 0.292893i
242.1 −0.965926 0.258819i 1.62484 + 0.599900i −0.866025 0.500000i 0.565826 + 0.151613i −1.41421 1.00000i −2.38014 + 1.15539i 2.12132 + 2.12132i 2.28024 + 1.94949i −0.507306 0.292893i
242.2 0.965926 + 0.258819i 1.10721 1.33195i −0.866025 0.500000i −3.29788 0.883663i 1.41421 1.00000i 2.38014 1.15539i −2.12132 2.12132i −0.548188 2.94949i −2.95680 1.70711i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 242.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
39.f even 4 1 inner
273.cd even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.cd.c 8
3.b odd 2 1 273.2.cd.d yes 8
7.c even 3 1 inner 273.2.cd.c 8
13.d odd 4 1 273.2.cd.d yes 8
21.h odd 6 1 273.2.cd.d yes 8
39.f even 4 1 inner 273.2.cd.c 8
91.z odd 12 1 273.2.cd.d yes 8
273.cd even 12 1 inner 273.2.cd.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cd.c 8 1.a even 1 1 trivial
273.2.cd.c 8 7.c even 3 1 inner
273.2.cd.c 8 39.f even 4 1 inner
273.2.cd.c 8 273.cd even 12 1 inner
273.2.cd.d yes 8 3.b odd 2 1
273.2.cd.d yes 8 13.d odd 4 1
273.2.cd.d yes 8 21.h odd 6 1
273.2.cd.d yes 8 91.z odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\):

\( T_{2}^{8} - T_{2}^{4} + 1 \)
\(T_{5}^{8} + \cdots\)
\(T_{19}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( 81 - 108 T + 72 T^{2} - 24 T^{3} + 7 T^{4} - 8 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} \)
$5$ \( 16 - 32 T + 32 T^{2} - 96 T^{3} + 92 T^{4} + 48 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( 2401 + 23 T^{4} + T^{8} \)
$11$ \( 1 + 8 T + 32 T^{2} + 272 T^{3} + 1087 T^{4} - 272 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( ( 13 - 4 T + T^{2} )^{4} \)
$17$ \( ( 49 + 14 T + 11 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$19$ \( 923521 + 476656 T + 123008 T^{2} + 32736 T^{3} + 7487 T^{4} + 1056 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$23$ \( ( 3844 - 992 T + 194 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$29$ \( ( 1 + T^{2} )^{4} \)
$31$ \( 256 + 512 T + 512 T^{2} + 768 T^{3} + 752 T^{4} + 192 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$37$ \( 65536 - 256 T^{4} + T^{8} \)
$41$ \( ( 32 - 8 T + T^{2} )^{4} \)
$43$ \( ( 36 + T^{2} )^{4} \)
$47$ \( 2401 + 5488 T + 6272 T^{2} + 12768 T^{3} + 14543 T^{4} + 1824 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$53$ \( 279841 - 43378 T^{2} + 6195 T^{4} - 82 T^{6} + T^{8} \)
$59$ \( 25411681 + 8589864 T + 1451808 T^{2} + 248784 T^{3} + 37007 T^{4} + 3504 T^{5} + 288 T^{6} + 24 T^{7} + T^{8} \)
$61$ \( ( 1 - T + T^{2} )^{4} \)
$67$ \( 4879681 + 2491752 T + 636192 T^{2} + 218832 T^{3} + 53663 T^{4} + 4656 T^{5} + 288 T^{6} + 24 T^{7} + T^{8} \)
$71$ \( ( 49 - 112 T + 128 T^{2} - 16 T^{3} + T^{4} )^{2} \)
$73$ \( 71639296 + 6229504 T + 270848 T^{2} + 158976 T^{3} - 1552 T^{4} - 1728 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$79$ \( ( 4 + 2 T^{2} + T^{4} )^{2} \)
$83$ \( ( 4624 - 1088 T + 128 T^{2} + 16 T^{3} + T^{4} )^{2} \)
$89$ \( 9834496 + 4214784 T + 903168 T^{2} + 236544 T^{3} + 47552 T^{4} + 4224 T^{5} + 288 T^{6} + 24 T^{7} + T^{8} \)
$97$ \( ( 98 + 14 T + T^{2} )^{4} \)
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