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

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} - 4 T^{3} + 2 T^{4} - 8 T^{5} + 8 T^{6} + 16 T^{8} \)
$3$ 1
$5$ \( 1 + 16 T^{3} - 12 T^{4} - 48 T^{5} + 128 T^{6} + 32 T^{7} - 506 T^{8} + 160 T^{9} + 3200 T^{10} - 6000 T^{11} - 7500 T^{12} + 50000 T^{13} + 390625 T^{16} \)
$7$ \( 1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 119560 T^{10} + 701092 T^{12} - 2823576 T^{14} + 5764801 T^{16} \)
$11$ \( 1 - 8 T + 32 T^{2} - 88 T^{3} + 132 T^{4} - 344 T^{5} + 2400 T^{6} - 13000 T^{7} + 54374 T^{8} - 143000 T^{9} + 290400 T^{10} - 457864 T^{11} + 1932612 T^{12} - 14172488 T^{13} + 56689952 T^{14} - 155897368 T^{15} + 214358881 T^{16} \)
$13$ \( 1 - 64 T^{3} - 4 T^{4} + 704 T^{5} + 2048 T^{6} - 1408 T^{7} - 53466 T^{8} - 18304 T^{9} + 346112 T^{10} + 1546688 T^{11} - 114244 T^{12} - 23762752 T^{13} + 815730721 T^{16} \)
$17$ \( ( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 1088 T^{5} + 10404 T^{6} + 83521 T^{8} )^{2} \)
$19$ \( 1 + 8 T + 32 T^{2} + 120 T^{3} + 452 T^{4} + 2168 T^{5} + 10080 T^{6} + 37832 T^{7} + 138918 T^{8} + 718808 T^{9} + 3638880 T^{10} + 14870312 T^{11} + 58905092 T^{12} + 297131880 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 - 38 T^{2} + 529 T^{4} )^{4} \)
$29$ \( 1 - 16 T + 128 T^{2} - 928 T^{3} + 6580 T^{4} - 38208 T^{5} + 199680 T^{6} - 1073680 T^{7} + 5802054 T^{8} - 31136720 T^{9} + 167930880 T^{10} - 931854912 T^{11} + 4653908980 T^{12} - 19034346272 T^{13} + 76137385088 T^{14} - 275998020944 T^{15} + 500246412961 T^{16} \)
$31$ \( ( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 35340 T^{5} + 157604 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 5948 T^{4} + 15248 T^{5} - 9344 T^{6} - 717840 T^{7} - 7530650 T^{8} - 26560080 T^{9} - 12791936 T^{10} + 772356944 T^{11} + 11147509628 T^{12} + 69898708656 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 2092065016 T^{10} + 54966702972 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} \)
$43$ \( 1 + 8 T + 32 T^{2} + 56 T^{3} + 260 T^{4} + 504 T^{5} - 2720 T^{6} - 625528 T^{7} - 7635866 T^{8} - 26897704 T^{9} - 5029280 T^{10} + 40071528 T^{11} + 888888260 T^{12} + 8232472808 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \)
$47$ \( ( 1 + 86 T^{2} + 2209 T^{4} )^{4} \)
$53$ \( 1 + 16 T + 128 T^{2} + 928 T^{3} + 8564 T^{4} + 82496 T^{5} + 654336 T^{6} + 5021328 T^{7} + 38116486 T^{8} + 266130384 T^{9} + 1838029824 T^{10} + 12281756992 T^{11} + 67574079284 T^{12} + 388085417504 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 8 T + 32 T^{2} + 472 T^{3} + 3481 T^{4} )^{4} \)
$61$ \( 1 - 16 T + 128 T^{2} - 1392 T^{3} + 14204 T^{4} - 79760 T^{5} + 426880 T^{6} - 2945904 T^{7} + 19569574 T^{8} - 179700144 T^{9} + 1588420480 T^{10} - 18104004560 T^{11} + 196666325564 T^{12} - 1175678050992 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 16 T + 128 T^{2} + 304 T^{3} + 4388 T^{4} + 107696 T^{5} + 1207680 T^{6} + 4800272 T^{7} + 13154790 T^{8} + 321618224 T^{9} + 5421275520 T^{10} + 32390972048 T^{11} + 88423118948 T^{12} + 410438032528 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 58087201032 T^{10} + 2308498748764 T^{12} - 56364124925240 T^{14} + 645753531245761 T^{16} \)
$73$ \( 1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 19901939192 T^{10} + 1289393734364 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} \)
$79$ \( ( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 3476 T^{5} + 923668 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \)
$83$ \( 1 - 40 T + 800 T^{2} - 11000 T^{3} + 122436 T^{4} - 1297720 T^{5} + 14460000 T^{6} - 161033000 T^{7} + 1597489574 T^{8} - 13365739000 T^{9} + 99614940000 T^{10} - 742019425640 T^{11} + 5810606989956 T^{12} - 43329447073000 T^{13} + 261552298695200 T^{14} - 1085442039585080 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 36646538184 T^{10} + 2278547224156 T^{12} - 123251360158328 T^{14} + 3936588805702081 T^{16} \)
$97$ \( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 74496 T^{5} + 1543076 T^{6} + 88529281 T^{8} )^{2} \)
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