Properties

Label 273.2.k.a
Level $273$
Weight $2$
Character orbit 273.k
Analytic conductor $2.180$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(92\) \(106\) \(157\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{18}^{3}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
22.1
−0.173648 + 0.984808i
0.939693 0.342020i
−0.766044 0.642788i
−0.173648 0.984808i
0.939693 + 0.342020i
−0.766044 + 0.642788i
−0.766044 + 1.32683i −0.500000 + 0.866025i −0.173648 0.300767i 0.120615 −0.766044 1.32683i 0.500000 + 0.866025i −2.53209 −0.500000 0.866025i −0.0923963 + 0.160035i
22.2 −0.173648 + 0.300767i −0.500000 + 0.866025i 0.939693 + 1.62760i 3.53209 −0.173648 0.300767i 0.500000 + 0.866025i −1.34730 −0.500000 0.866025i −0.613341 + 1.06234i
22.3 0.939693 1.62760i −0.500000 + 0.866025i −0.766044 1.32683i 2.34730 0.939693 + 1.62760i 0.500000 + 0.866025i 0.879385 −0.500000 0.866025i 2.20574 3.82045i
211.1 −0.766044 1.32683i −0.500000 0.866025i −0.173648 + 0.300767i 0.120615 −0.766044 + 1.32683i 0.500000 0.866025i −2.53209 −0.500000 + 0.866025i −0.0923963 0.160035i
211.2 −0.173648 0.300767i −0.500000 0.866025i 0.939693 1.62760i 3.53209 −0.173648 + 0.300767i 0.500000 0.866025i −1.34730 −0.500000 + 0.866025i −0.613341 1.06234i
211.3 0.939693 + 1.62760i −0.500000 0.866025i −0.766044 + 1.32683i 2.34730 0.939693 1.62760i 0.500000 0.866025i 0.879385 −0.500000 + 0.866025i 2.20574 + 3.82045i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.k.a 6
3.b odd 2 1 819.2.o.g 6
13.c even 3 1 inner 273.2.k.a 6
13.c even 3 1 3549.2.a.o 3
13.e even 6 1 3549.2.a.n 3
39.i odd 6 1 819.2.o.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.k.a 6 1.a even 1 1 trivial
273.2.k.a 6 13.c even 3 1 inner
819.2.o.g 6 3.b odd 2 1
819.2.o.g 6 39.i odd 6 1
3549.2.a.n 3 13.e even 6 1
3549.2.a.o 3 13.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 3 T_{2}^{4} + 2 T_{2}^{3} + 9 T_{2}^{2} + 3 T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{6} \)
$3$ \( ( 1 + T + T^{2} )^{3} \)
$5$ \( ( -1 + 9 T - 6 T^{2} + T^{3} )^{2} \)
$7$ \( ( 1 - T + T^{2} )^{3} \)
$11$ \( 5041 + 639 T + 507 T^{2} + 88 T^{3} + 45 T^{4} + 6 T^{5} + T^{6} \)
$13$ \( 2197 - 19 T^{3} + T^{6} \)
$17$ \( 25281 + 4293 T + 1683 T^{2} + 156 T^{3} + 63 T^{4} + 6 T^{5} + T^{6} \)
$19$ \( 289 - 255 T + 378 T^{2} + 169 T^{3} + 66 T^{4} + 9 T^{5} + T^{6} \)
$23$ \( 3249 + 2052 T + 1467 T^{2} + 6 T^{3} + 45 T^{4} + 3 T^{5} + T^{6} \)
$29$ \( 104329 + 15504 T + 5211 T^{2} + 214 T^{3} + 129 T^{4} + 9 T^{5} + T^{6} \)
$31$ \( ( 27 - 9 T^{2} + T^{3} )^{2} \)
$37$ \( 106929 + 2943 T + 4005 T^{2} + 546 T^{3} + 153 T^{4} + 12 T^{5} + T^{6} \)
$41$ \( 732736 + 112992 T + 22560 T^{2} + 920 T^{3} + 168 T^{4} + 6 T^{5} + T^{6} \)
$43$ \( 5329 + 3504 T + 2961 T^{2} - 286 T^{3} + 129 T^{4} + 9 T^{5} + T^{6} \)
$47$ \( ( 3 - 18 T - 3 T^{2} + T^{3} )^{2} \)
$53$ \( ( 867 - 153 T - 3 T^{2} + T^{3} )^{2} \)
$59$ \( 1671849 + 93096 T + 24579 T^{2} + 1506 T^{3} + 297 T^{4} + 15 T^{5} + T^{6} \)
$61$ \( 16129 + 4953 T + 3426 T^{2} - 839 T^{3} + 186 T^{4} - 15 T^{5} + T^{6} \)
$67$ \( 2505889 + 275442 T + 44523 T^{2} + 1600 T^{3} + 255 T^{4} + 9 T^{5} + T^{6} \)
$71$ \( 12321 + 999 T + 1413 T^{2} - 330 T^{3} + 135 T^{4} - 12 T^{5} + T^{6} \)
$73$ \( ( -807 + 279 T - 30 T^{2} + T^{3} )^{2} \)
$79$ \( ( 271 + 138 T + 21 T^{2} + T^{3} )^{2} \)
$83$ \( ( -111 + 153 T - 24 T^{2} + T^{3} )^{2} \)
$89$ \( 1 + 15 T + 234 T^{2} - 137 T^{3} + 66 T^{4} - 9 T^{5} + T^{6} \)
$97$ \( 3249 + 7182 T + 16047 T^{2} - 264 T^{3} + 135 T^{4} + 3 T^{5} + T^{6} \)
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