Properties

Label 144.2.k.b
Level $144$
Weight $2$
Character orbit 144.k
Analytic conductor $1.150$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.14984578911\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{6} q^{2} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{5} + ( \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{8} +O(q^{10})\) \( q -\beta_{6} q^{2} + ( -1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{4} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{5} + ( \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} ) q^{7} + ( 1 - \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{8} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{7} ) q^{10} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{11} + ( 1 + 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{13} + ( -1 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{14} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{16} + ( 2 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{17} + ( -2 - 2 \beta_{2} + 2 \beta_{6} ) q^{19} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{20} + ( -2 \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{22} -2 \beta_{7} q^{23} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{25} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{26} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{28} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 4 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{31} + ( 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + 2 \beta_{7} ) q^{32} + ( 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{34} + ( -4 - \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{35} + ( -3 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{7} ) q^{37} + ( 2 + 2 \beta_{1} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{7} ) q^{38} + ( 4 + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} + 2 \beta_{6} ) q^{40} + ( -\beta_{1} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{41} + ( -2 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + 2 \beta_{7} ) q^{43} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{7} ) q^{44} + ( 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{46} -2 \beta_{1} q^{47} + ( 1 - 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{49} + ( 5 + 4 \beta_{3} - \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{50} + ( -3 - 3 \beta_{2} + \beta_{3} - \beta_{4} + 5 \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{52} + ( -2 + 3 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{53} + ( -4 \beta_{5} - 4 \beta_{7} ) q^{55} + ( 2 + 4 \beta_{2} + 6 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{56} + ( -3 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} + 5 \beta_{5} - 3 \beta_{6} ) q^{58} + ( -4 - 4 \beta_{5} ) q^{59} + ( 1 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 1 + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{62} + ( -2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{64} + ( -5 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + \beta_{7} ) q^{65} + ( -4 - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{7} ) q^{67} + ( -6 - 2 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} ) q^{68} + ( 6 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + 4 \beta_{6} - 2 \beta_{7} ) q^{70} + ( 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{71} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - 4 \beta_{7} ) q^{73} + ( -5 - 2 \beta_{1} + 4 \beta_{2} + \beta_{4} + 3 \beta_{6} - 2 \beta_{7} ) q^{74} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{76} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{77} + ( -4 \beta_{1} + 3 \beta_{2} + 5 \beta_{3} - 3 \beta_{4} + 5 \beta_{6} - 4 \beta_{7} ) q^{79} + ( 2 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{80} + ( 4 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{5} - 2 \beta_{7} ) q^{82} + ( 4 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{83} + ( -4 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 4 \beta_{7} ) q^{85} + ( -2 \beta_{1} - 4 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} + 6 \beta_{7} ) q^{86} + ( 2 - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 6 \beta_{6} ) q^{88} + ( -2 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} + 2 \beta_{5} - 6 \beta_{7} ) q^{89} + ( -2 - 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{7} ) q^{91} + ( 2 + 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{92} + ( -2 \beta_{2} - 2 \beta_{3} - 2 \beta_{5} ) q^{94} + ( 8 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{95} + ( 6 \beta_{1} - 4 \beta_{3} - 4 \beta_{6} + 2 \beta_{7} ) q^{97} + ( 2 - 4 \beta_{2} - 4 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{4} + 12q^{8} + O(q^{10}) \) \( 8q - 4q^{4} + 12q^{8} - 8q^{10} + 8q^{11} - 12q^{14} - 8q^{19} - 16q^{20} - 20q^{26} + 8q^{28} + 16q^{29} + 24q^{31} - 24q^{35} - 16q^{37} + 8q^{38} + 16q^{40} - 8q^{43} + 40q^{44} - 8q^{46} - 8q^{49} + 36q^{50} - 16q^{52} - 16q^{53} - 16q^{58} - 32q^{59} + 16q^{61} + 12q^{62} + 8q^{64} + 16q^{65} - 16q^{67} - 32q^{68} + 32q^{70} - 52q^{74} + 8q^{76} - 16q^{77} - 24q^{79} - 8q^{80} + 40q^{82} + 40q^{83} - 16q^{85} + 16q^{86} + 32q^{88} - 8q^{91} + 16q^{92} + 8q^{94} + 48q^{95} + 40q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} + 14 x^{6} - 28 x^{5} + 43 x^{4} - 44 x^{3} + 30 x^{2} - 12 x + 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{6} - 3 \nu^{5} + 10 \nu^{4} - 15 \nu^{3} + 19 \nu^{2} - 12 \nu + 4 \)
\(\beta_{2}\)\(=\)\( -2 \nu^{7} + 7 \nu^{6} - 24 \nu^{5} + 42 \nu^{4} - 59 \nu^{3} + 48 \nu^{2} - 24 \nu + 4 \)
\(\beta_{3}\)\(=\)\( -3 \nu^{7} + 10 \nu^{6} - 35 \nu^{5} + 60 \nu^{4} - 87 \nu^{3} + 73 \nu^{2} - 42 \nu + 11 \)
\(\beta_{4}\)\(=\)\( -7 \nu^{7} + 25 \nu^{6} - 87 \nu^{5} + 158 \nu^{4} - 231 \nu^{3} + 206 \nu^{2} - 118 \nu + 31 \)
\(\beta_{5}\)\(=\)\( -8 \nu^{7} + 28 \nu^{6} - 98 \nu^{5} + 175 \nu^{4} - 256 \nu^{3} + 223 \nu^{2} - 126 \nu + 31 \)
\(\beta_{6}\)\(=\)\( 8 \nu^{7} - 28 \nu^{6} + 98 \nu^{5} - 175 \nu^{4} + 257 \nu^{3} - 224 \nu^{2} + 130 \nu - 32 \)
\(\beta_{7}\)\(=\)\( 10 \nu^{7} - 35 \nu^{6} + 123 \nu^{5} - 220 \nu^{4} + 325 \nu^{3} - 285 \nu^{2} + 166 \nu - 42 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-2 \beta_{7} + 3 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{7} + \beta_{6} + 5 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + 4 \beta_{1} - 5\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(12 \beta_{7} - 11 \beta_{6} + 11 \beta_{5} - 5 \beta_{4} - \beta_{3} - 9 \beta_{2} + 2 \beta_{1} + 5\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(6 \beta_{7} - 17 \beta_{6} - 13 \beta_{5} + 13 \beta_{4} - 23 \beta_{3} + 3 \beta_{2} - 18 \beta_{1} + 21\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-46 \beta_{7} + 29 \beta_{6} - 67 \beta_{5} + 37 \beta_{4} - 15 \beta_{3} + 47 \beta_{2} - 24 \beta_{1} - 1\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-76 \beta_{7} + 105 \beta_{6} - 7 \beta_{5} - 31 \beta_{4} + 91 \beta_{3} + 39 \beta_{2} + 68 \beta_{1} - 83\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 0.0297061i
0.500000 + 0.691860i
0.500000 + 1.44392i
0.500000 2.10607i
0.500000 + 0.0297061i
0.500000 0.691860i
0.500000 1.44392i
0.500000 + 2.10607i
−0.874559 + 1.11137i 0 −0.470294 1.94392i 0.334904 + 0.334904i 0 4.55765i 2.57172 + 1.17740i 0 −0.665096 + 0.0793096i
37.2 −0.635665 1.26330i 0 −1.19186 + 1.60607i 2.68554 + 2.68554i 0 2.15894i 2.78658 + 0.484753i 0 1.68554 5.09976i
37.3 0.167452 1.40426i 0 −1.94392 0.470294i −1.74912 1.74912i 0 2.55765i −0.985930 + 2.65103i 0 −2.74912 + 2.16333i
37.4 1.34277 0.443806i 0 1.60607 1.19186i −1.27133 1.27133i 0 0.158942i 1.62764 2.31318i 0 −2.27133 1.14288i
109.1 −0.874559 1.11137i 0 −0.470294 + 1.94392i 0.334904 0.334904i 0 4.55765i 2.57172 1.17740i 0 −0.665096 0.0793096i
109.2 −0.635665 + 1.26330i 0 −1.19186 1.60607i 2.68554 2.68554i 0 2.15894i 2.78658 0.484753i 0 1.68554 + 5.09976i
109.3 0.167452 + 1.40426i 0 −1.94392 + 0.470294i −1.74912 + 1.74912i 0 2.55765i −0.985930 2.65103i 0 −2.74912 2.16333i
109.4 1.34277 + 0.443806i 0 1.60607 + 1.19186i −1.27133 + 1.27133i 0 0.158942i 1.62764 + 2.31318i 0 −2.27133 + 1.14288i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.2.k.b 8
3.b odd 2 1 48.2.j.a 8
4.b odd 2 1 576.2.k.b 8
8.b even 2 1 1152.2.k.c 8
8.d odd 2 1 1152.2.k.f 8
12.b even 2 1 192.2.j.a 8
16.e even 4 1 inner 144.2.k.b 8
16.e even 4 1 1152.2.k.c 8
16.f odd 4 1 576.2.k.b 8
16.f odd 4 1 1152.2.k.f 8
24.f even 2 1 384.2.j.a 8
24.h odd 2 1 384.2.j.b 8
32.g even 8 1 9216.2.a.y 4
32.g even 8 1 9216.2.a.bo 4
32.h odd 8 1 9216.2.a.x 4
32.h odd 8 1 9216.2.a.bn 4
48.i odd 4 1 48.2.j.a 8
48.i odd 4 1 384.2.j.b 8
48.k even 4 1 192.2.j.a 8
48.k even 4 1 384.2.j.a 8
96.o even 8 1 3072.2.a.n 4
96.o even 8 1 3072.2.a.o 4
96.o even 8 2 3072.2.d.i 8
96.p odd 8 1 3072.2.a.i 4
96.p odd 8 1 3072.2.a.t 4
96.p odd 8 2 3072.2.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.j.a 8 3.b odd 2 1
48.2.j.a 8 48.i odd 4 1
144.2.k.b 8 1.a even 1 1 trivial
144.2.k.b 8 16.e even 4 1 inner
192.2.j.a 8 12.b even 2 1
192.2.j.a 8 48.k even 4 1
384.2.j.a 8 24.f even 2 1
384.2.j.a 8 48.k even 4 1
384.2.j.b 8 24.h odd 2 1
384.2.j.b 8 48.i odd 4 1
576.2.k.b 8 4.b odd 2 1
576.2.k.b 8 16.f odd 4 1
1152.2.k.c 8 8.b even 2 1
1152.2.k.c 8 16.e even 4 1
1152.2.k.f 8 8.d odd 2 1
1152.2.k.f 8 16.f odd 4 1
3072.2.a.i 4 96.p odd 8 1
3072.2.a.n 4 96.o even 8 1
3072.2.a.o 4 96.o even 8 1
3072.2.a.t 4 96.p odd 8 1
3072.2.d.f 8 96.p odd 8 2
3072.2.d.i 8 96.o even 8 2
9216.2.a.x 4 32.h odd 8 1
9216.2.a.y 4 32.g even 8 1
9216.2.a.bn 4 32.h odd 8 1
9216.2.a.bo 4 32.g even 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 16 T_{5}^{5} + 128 T_{5}^{4} + 192 T_{5}^{3} + 128 T_{5}^{2} - 128 T_{5} + 64 \) acting on \(S_{2}^{\mathrm{new}}(144, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + 8 T^{2} - 8 T^{3} + 2 T^{4} - 4 T^{5} + 2 T^{6} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 64 - 128 T + 128 T^{2} + 192 T^{3} + 128 T^{4} + 16 T^{5} + T^{8} \)
$7$ \( 16 + 640 T^{2} + 264 T^{4} + 32 T^{6} + T^{8} \)
$11$ \( 1024 + 2048 T + 2048 T^{2} - 256 T^{3} + 32 T^{6} - 8 T^{7} + T^{8} \)
$13$ \( 16 + 256 T + 2048 T^{2} - 1792 T^{3} + 776 T^{4} - 64 T^{5} + T^{8} \)
$17$ \( ( 16 - 64 T - 32 T^{2} + T^{4} )^{2} \)
$19$ \( 256 - 1536 T + 4608 T^{2} + 1408 T^{3} + 224 T^{4} - 32 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$23$ \( ( 8 + T^{2} )^{4} \)
$29$ \( 61504 - 35712 T + 10368 T^{2} - 1088 T^{3} + 896 T^{4} - 464 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} \)
$31$ \( ( -28 - 24 T + 40 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$37$ \( 1106704 + 639616 T + 184832 T^{2} + 24128 T^{3} + 2248 T^{4} + 416 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$41$ \( 12544 + 22528 T^{2} + 3872 T^{4} + 128 T^{6} + T^{8} \)
$43$ \( 12544 + 39424 T + 61952 T^{2} + 29056 T^{3} + 6624 T^{4} - 288 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} \)
$47$ \( ( -8 + T^{2} )^{4} \)
$53$ \( 18496 - 36992 T + 36992 T^{2} + 1088 T^{3} - 128 T^{4} + 80 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$59$ \( ( 32 + 8 T + T^{2} )^{4} \)
$61$ \( 1106704 - 639616 T + 184832 T^{2} - 24128 T^{3} + 2248 T^{4} - 416 T^{5} + 128 T^{6} - 16 T^{7} + T^{8} \)
$67$ \( 65536 - 65536 T + 32768 T^{2} + 12288 T^{3} + 3584 T^{4} - 768 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$71$ \( 4096 + 40960 T^{2} + 4224 T^{4} + 128 T^{6} + T^{8} \)
$73$ \( 4096 + 16384 T^{2} + 8320 T^{4} + 256 T^{6} + T^{8} \)
$79$ \( ( -10108 - 2888 T - 168 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$83$ \( 1024 + 10240 T + 51200 T^{2} + 60160 T^{3} + 33792 T^{4} - 7680 T^{5} + 800 T^{6} - 40 T^{7} + T^{8} \)
$89$ \( 3625216 + 1901824 T^{2} + 62304 T^{4} + 464 T^{6} + T^{8} \)
$97$ \( ( 512 + 768 T - 224 T^{2} + T^{4} )^{2} \)
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