Properties

Label 864.2.r.b
Level 864
Weight 2
Character orbit 864.r
Analytic conductor 6.899
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 864.r (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 72)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \beta_{4} - \beta_{5} - \beta_{11} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} ) q^{7} +O(q^{10})\) \( q + ( \beta_{4} - \beta_{5} - \beta_{11} ) q^{5} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} ) q^{7} + ( -\beta_{8} + \beta_{13} ) q^{11} + ( -\beta_{5} + \beta_{8} ) q^{13} + ( 2 - \beta_{7} ) q^{17} + ( \beta_{11} + 2 \beta_{12} + \beta_{13} - \beta_{14} ) q^{19} + ( -\beta_{1} - \beta_{7} - \beta_{10} - \beta_{15} ) q^{23} + ( \beta_{2} - \beta_{3} - \beta_{6} - \beta_{15} ) q^{25} + ( -\beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{12} + \beta_{13} ) q^{29} + ( \beta_{1} + 2 \beta_{6} - \beta_{7} - \beta_{10} - \beta_{15} ) q^{31} + ( \beta_{13} - \beta_{14} ) q^{35} + ( -\beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{37} + ( \beta_{1} - \beta_{6} + \beta_{7} + \beta_{10} - \beta_{15} ) q^{41} + \beta_{4} q^{43} + ( 1 - \beta_{1} - \beta_{3} - \beta_{6} + 2 \beta_{10} ) q^{47} + ( 2 \beta_{1} + 2 \beta_{6} - \beta_{7} - \beta_{10} ) q^{49} + ( -\beta_{11} + \beta_{13} + 2 \beta_{14} ) q^{53} + ( 1 + \beta_{2} + 2 \beta_{3} - \beta_{7} ) q^{55} + ( \beta_{4} - 2 \beta_{5} - 2 \beta_{9} - \beta_{11} + 2 \beta_{14} ) q^{59} + ( 2 \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} - \beta_{12} + \beta_{13} ) q^{61} + ( 1 - \beta_{1} - 2 \beta_{3} - 2 \beta_{6} - \beta_{10} ) q^{65} + ( 2 \beta_{4} - 4 \beta_{5} + \beta_{8} - 2 \beta_{11} ) q^{67} + ( 4 - \beta_{2} - \beta_{3} + \beta_{7} ) q^{71} + ( -2 + 2 \beta_{3} - \beta_{7} ) q^{73} + ( -3 \beta_{4} + \beta_{5} + 2 \beta_{9} + 3 \beta_{11} - 2 \beta_{14} ) q^{77} + ( 3 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{6} - \beta_{10} - \beta_{15} ) q^{79} + ( -2 \beta_{5} + \beta_{8} + \beta_{9} + 2 \beta_{12} - \beta_{13} ) q^{83} + ( \beta_{4} - 2 \beta_{5} - \beta_{8} + 2 \beta_{9} - \beta_{11} - 2 \beta_{14} ) q^{85} + ( -4 - \beta_{2} - \beta_{3} - \beta_{7} ) q^{89} + ( -3 \beta_{11} + \beta_{14} ) q^{91} + ( 6 \beta_{1} - 2 \beta_{6} + 2 \beta_{15} ) q^{95} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} - 3 \beta_{6} - \beta_{10} + \beta_{15} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{7} + O(q^{10}) \) \( 16q - 6q^{7} + 28q^{17} - 10q^{23} + 2q^{25} + 10q^{31} + 8q^{41} + 6q^{47} + 18q^{49} + 4q^{55} + 14q^{65} + 72q^{71} - 44q^{73} + 30q^{79} - 64q^{89} + 44q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - x^{15} + x^{14} + 2 x^{12} - 4 x^{11} - 8 x^{9} + 4 x^{8} - 16 x^{7} - 32 x^{5} + 32 x^{4} + 64 x^{2} - 128 x + 256\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{15} - \nu^{14} - 3 \nu^{13} - 4 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} - 8 \nu^{9} - 4 \nu^{8} + 12 \nu^{7} + 24 \nu^{6} + 32 \nu^{5} + 72 \nu^{4} + 96 \nu^{3} + 128 \nu^{2} + 96 \nu + 256 \)\()/192\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{14} - 3 \nu^{13} + 3 \nu^{12} - 8 \nu^{11} - 6 \nu^{10} + 8 \nu^{8} + 60 \nu^{6} + 32 \nu^{5} + 96 \nu^{4} + 240 \nu^{3} + 128 \nu^{2} \)\()/192\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{14} + \nu^{11} + 8 \nu^{8} + 12 \nu^{7} - 12 \nu^{6} - 4 \nu^{5} + 24 \nu^{3} - 16 \nu^{2} - 96 \nu - 96 \)\()/96\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{14} + \nu^{13} + 2 \nu^{12} - \nu^{11} + 5 \nu^{10} + 4 \nu^{9} + 10 \nu^{8} - 8 \nu^{7} - 4 \nu^{6} - 32 \nu^{5} - 20 \nu^{4} - 64 \nu^{3} - 80 \nu^{2} - 80 \nu + 32 \)\()/96\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{15} + 3 \nu^{14} + 5 \nu^{13} + 12 \nu^{12} + 18 \nu^{11} + 28 \nu^{10} + 24 \nu^{9} + 24 \nu^{8} + 20 \nu^{7} - 64 \nu^{6} - 96 \nu^{5} - 160 \nu^{4} - 256 \nu^{3} - 384 \nu^{2} - 64 \nu - 512 \)\()/384\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{15} - 3 \nu^{14} - 5 \nu^{13} - 12 \nu^{12} - 18 \nu^{11} - 28 \nu^{10} - 24 \nu^{9} - 24 \nu^{8} - 20 \nu^{7} + 64 \nu^{6} + 96 \nu^{5} + 160 \nu^{4} + 256 \nu^{3} + 384 \nu^{2} + 832 \nu + 512 \)\()/384\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{15} + 3 \nu^{14} + \nu^{13} + \nu^{12} + 2 \nu^{10} - 28 \nu^{9} - 24 \nu^{8} - 32 \nu^{7} - 28 \nu^{6} - 32 \nu^{4} - 64 \nu^{3} + 256 \nu + 192 \)\()/192\)
\(\beta_{8}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{14} + 6 \nu^{13} + 7 \nu^{12} + 4 \nu^{11} + 8 \nu^{9} + 20 \nu^{8} + 12 \nu^{7} - 28 \nu^{6} - 64 \nu^{5} + 24 \nu^{4} - 112 \nu^{3} - 160 \nu^{2} - 288 \nu \)\()/192\)
\(\beta_{9}\)\(=\)\((\)\( 7 \nu^{15} + \nu^{14} + 7 \nu^{13} + 16 \nu^{12} + 38 \nu^{11} + 20 \nu^{10} + 32 \nu^{9} - 8 \nu^{8} + 28 \nu^{7} - 48 \nu^{6} - 224 \nu^{5} - 416 \nu^{4} - 288 \nu^{3} - 704 \nu^{2} - 320 \nu - 1408 \)\()/384\)
\(\beta_{10}\)\(=\)\((\)\( 7 \nu^{15} + \nu^{14} + 7 \nu^{13} + 16 \nu^{12} + 38 \nu^{11} + 20 \nu^{10} + 32 \nu^{9} - 8 \nu^{8} + 28 \nu^{7} - 48 \nu^{6} - 224 \nu^{5} - 416 \nu^{4} - 288 \nu^{3} + 64 \nu^{2} - 320 \nu - 1408 \)\()/384\)
\(\beta_{11}\)\(=\)\((\)\( 4 \nu^{15} + \nu^{14} + 3 \nu^{13} + \nu^{12} + 8 \nu^{11} + 6 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} - 12 \nu^{6} - 32 \nu^{5} - 96 \nu^{4} - 144 \nu^{3} - 128 \nu^{2} - 448 \)\()/192\)
\(\beta_{12}\)\(=\)\((\)\( 2 \nu^{15} + \nu^{14} + 4 \nu^{13} + 6 \nu^{12} + 11 \nu^{11} + 8 \nu^{10} + 12 \nu^{9} - 8 \nu^{8} + 4 \nu^{7} - 44 \nu^{6} - 44 \nu^{5} - 128 \nu^{4} - 104 \nu^{3} - 176 \nu^{2} - 32 \nu - 352 \)\()/96\)
\(\beta_{13}\)\(=\)\((\)\( 2 \nu^{15} - 3 \nu^{14} + 5 \nu^{13} + 13 \nu^{12} + 18 \nu^{11} + 10 \nu^{10} + 20 \nu^{9} + 24 \nu^{8} + 56 \nu^{7} - 4 \nu^{6} - 72 \nu^{5} - 160 \nu^{4} - 208 \nu^{3} - 288 \nu^{2} - 448 \nu - 960 \)\()/192\)
\(\beta_{14}\)\(=\)\((\)\( 6 \nu^{15} + \nu^{14} + 11 \nu^{13} + 11 \nu^{12} + 32 \nu^{11} + 22 \nu^{10} + 28 \nu^{9} - 8 \nu^{8} + 32 \nu^{7} - 52 \nu^{6} - 128 \nu^{5} - 352 \nu^{4} - 256 \nu^{3} - 512 \nu^{2} - 256 \nu - 1216 \)\()/192\)
\(\beta_{15}\)\(=\)\((\)\( -12 \nu^{15} - 5 \nu^{14} - 15 \nu^{13} - 21 \nu^{12} - 64 \nu^{11} - 30 \nu^{10} - 48 \nu^{9} + 16 \nu^{8} - 48 \nu^{7} + 156 \nu^{6} + 160 \nu^{5} + 768 \nu^{4} + 624 \nu^{3} + 736 \nu^{2} + 384 \nu + 2304 \)\()/192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} + \beta_{5}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{10} - \beta_{9}\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(\beta_{14} - \beta_{13} - \beta_{11} + \beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(\beta_{15} - \beta_{14} + 3 \beta_{11} + \beta_{9} + \beta_{8} - \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-\beta_{15} + \beta_{13} - 2 \beta_{10} - 3 \beta_{9} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} + \beta_{2} - 2 \beta_{1} + 2\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(3 \beta_{14} + \beta_{13} - 4 \beta_{12} - \beta_{11} - 5 \beta_{3} + \beta_{2} + 2\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-\beta_{15} + \beta_{14} + \beta_{11} - \beta_{9} + 3 \beta_{8} - \beta_{6} + 4 \beta_{5} - \beta_{4} + 18 \beta_{1}\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(-3 \beta_{15} - \beta_{13} - 8 \beta_{12} - 3 \beta_{9} + \beta_{8} + 9 \beta_{6} + 8 \beta_{5} + 5 \beta_{4} + 9 \beta_{3} + 3 \beta_{2} - 10 \beta_{1} + 10\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-5 \beta_{14} - 3 \beta_{13} + 8 \beta_{12} + 7 \beta_{11} - 12 \beta_{7} - 5 \beta_{3} + \beta_{2} + 6\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(7 \beta_{15} + 13 \beta_{14} + \beta_{11} + 4 \beta_{10} - 13 \beta_{9} - 13 \beta_{8} + 4 \beta_{7} - 13 \beta_{6} + 16 \beta_{5} - \beta_{4} + 2 \beta_{1}\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(\beta_{15} - 5 \beta_{13} - 24 \beta_{12} + 4 \beta_{10} + 13 \beta_{9} + 5 \beta_{8} - 3 \beta_{6} + 24 \beta_{5} - 39 \beta_{4} - 3 \beta_{3} - \beta_{2} - 50 \beta_{1} + 50\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(-21 \beta_{14} + 21 \beta_{13} + 24 \beta_{12} - 9 \beta_{11} + 12 \beta_{7} - 13 \beta_{3} + 17 \beta_{2} + 14\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(7 \beta_{15} + 61 \beta_{14} - 15 \beta_{11} + 4 \beta_{10} - 61 \beta_{9} + 19 \beta_{8} + 4 \beta_{7} + 27 \beta_{6} - 24 \beta_{5} + 15 \beta_{4} - 14 \beta_{1}\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(-31 \beta_{15} - 53 \beta_{13} - 40 \beta_{12} - 4 \beta_{10} + 5 \beta_{9} + 53 \beta_{8} - 35 \beta_{6} + 40 \beta_{5} - 7 \beta_{4} - 35 \beta_{3} + 31 \beta_{2} + 94 \beta_{1} - 94\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(-29 \beta_{14} - 3 \beta_{13} + 8 \beta_{12} + 127 \beta_{11} + 12 \beta_{7} + 59 \beta_{3} + 41 \beta_{2} + 174\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.41411 + 0.0174668i
0.587625 1.28635i
−0.179748 + 1.40274i
−1.34532 0.436011i
1.05026 + 0.947078i
−1.12494 + 0.857038i
0.820200 1.15207i
−0.722180 1.21592i
1.41411 0.0174668i
0.587625 + 1.28635i
−0.179748 1.40274i
−1.34532 + 0.436011i
1.05026 0.947078i
−1.12494 0.857038i
0.820200 + 1.15207i
−0.722180 + 1.21592i
0 0 0 −3.17262 1.83171i 0 0.191926 + 0.332426i 0 0 0
145.2 0 0 0 −1.97542 1.14051i 0 0.907824 + 1.57240i 0 0 0
145.3 0 0 0 −1.19115 0.687709i 0 −1.80469 3.12581i 0 0 0
145.4 0 0 0 −0.602794 0.348023i 0 −0.795065 1.37709i 0 0 0
145.5 0 0 0 0.602794 + 0.348023i 0 −0.795065 1.37709i 0 0 0
145.6 0 0 0 1.19115 + 0.687709i 0 −1.80469 3.12581i 0 0 0
145.7 0 0 0 1.97542 + 1.14051i 0 0.907824 + 1.57240i 0 0 0
145.8 0 0 0 3.17262 + 1.83171i 0 0.191926 + 0.332426i 0 0 0
721.1 0 0 0 −3.17262 + 1.83171i 0 0.191926 0.332426i 0 0 0
721.2 0 0 0 −1.97542 + 1.14051i 0 0.907824 1.57240i 0 0 0
721.3 0 0 0 −1.19115 + 0.687709i 0 −1.80469 + 3.12581i 0 0 0
721.4 0 0 0 −0.602794 + 0.348023i 0 −0.795065 + 1.37709i 0 0 0
721.5 0 0 0 0.602794 0.348023i 0 −0.795065 + 1.37709i 0 0 0
721.6 0 0 0 1.19115 0.687709i 0 −1.80469 + 3.12581i 0 0 0
721.7 0 0 0 1.97542 1.14051i 0 0.907824 1.57240i 0 0 0
721.8 0 0 0 3.17262 1.83171i 0 0.191926 0.332426i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.c even 3 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.r.b 16
3.b odd 2 1 288.2.r.b 16
4.b odd 2 1 216.2.n.b 16
8.b even 2 1 inner 864.2.r.b 16
8.d odd 2 1 216.2.n.b 16
9.c even 3 1 inner 864.2.r.b 16
9.c even 3 1 2592.2.d.k 8
9.d odd 6 1 288.2.r.b 16
9.d odd 6 1 2592.2.d.j 8
12.b even 2 1 72.2.n.b 16
24.f even 2 1 72.2.n.b 16
24.h odd 2 1 288.2.r.b 16
36.f odd 6 1 216.2.n.b 16
36.f odd 6 1 648.2.d.k 8
36.h even 6 1 72.2.n.b 16
36.h even 6 1 648.2.d.j 8
72.j odd 6 1 288.2.r.b 16
72.j odd 6 1 2592.2.d.j 8
72.l even 6 1 72.2.n.b 16
72.l even 6 1 648.2.d.j 8
72.n even 6 1 inner 864.2.r.b 16
72.n even 6 1 2592.2.d.k 8
72.p odd 6 1 216.2.n.b 16
72.p odd 6 1 648.2.d.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.n.b 16 12.b even 2 1
72.2.n.b 16 24.f even 2 1
72.2.n.b 16 36.h even 6 1
72.2.n.b 16 72.l even 6 1
216.2.n.b 16 4.b odd 2 1
216.2.n.b 16 8.d odd 2 1
216.2.n.b 16 36.f odd 6 1
216.2.n.b 16 72.p odd 6 1
288.2.r.b 16 3.b odd 2 1
288.2.r.b 16 9.d odd 6 1
288.2.r.b 16 24.h odd 2 1
288.2.r.b 16 72.j odd 6 1
648.2.d.j 8 36.h even 6 1
648.2.d.j 8 72.l even 6 1
648.2.d.k 8 36.f odd 6 1
648.2.d.k 8 72.p odd 6 1
864.2.r.b 16 1.a even 1 1 trivial
864.2.r.b 16 8.b even 2 1 inner
864.2.r.b 16 9.c even 3 1 inner
864.2.r.b 16 72.n even 6 1 inner
2592.2.d.j 8 9.d odd 6 1
2592.2.d.j 8 72.j odd 6 1
2592.2.d.k 8 9.c even 3 1
2592.2.d.k 8 72.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 + 19 T^{2} + 176 T^{4} + 1031 T^{6} + 3893 T^{8} + 4928 T^{10} - 61926 T^{12} - 631122 T^{14} - 3717344 T^{16} - 15778050 T^{18} - 38703750 T^{20} + 77000000 T^{22} + 1520703125 T^{24} + 10068359375 T^{26} + 42968750000 T^{28} + 115966796875 T^{30} + 152587890625 T^{32} \)
$7$ \( ( 1 + 3 T - 14 T^{2} - 39 T^{3} + 139 T^{4} + 252 T^{5} - 1208 T^{6} - 666 T^{7} + 9424 T^{8} - 4662 T^{9} - 59192 T^{10} + 86436 T^{11} + 333739 T^{12} - 655473 T^{13} - 1647086 T^{14} + 2470629 T^{15} + 5764801 T^{16} )^{2} \)
$11$ \( 1 + 48 T^{2} + 1090 T^{4} + 17592 T^{6} + 242041 T^{8} + 2632140 T^{10} + 21031138 T^{12} + 158095260 T^{14} + 1558598596 T^{16} + 19129526460 T^{18} + 307916891458 T^{20} + 4662996570540 T^{22} + 51883637916121 T^{24} + 456291173580792 T^{26} + 3420886930625890 T^{28} + 18227992011995568 T^{30} + 45949729863572161 T^{32} \)
$13$ \( 1 + 51 T^{2} + 1288 T^{4} + 16911 T^{6} + 73645 T^{8} - 1059264 T^{10} - 5456606 T^{12} + 411982806 T^{14} + 9084740848 T^{16} + 69625094214 T^{18} - 155846123966 T^{20} - 5112865008576 T^{22} + 60074488948045 T^{24} + 2331324955658439 T^{26} + 30007933637755528 T^{28} + 200806195670663739 T^{30} + 665416609183179841 T^{32} \)
$17$ \( ( 1 - 7 T + 66 T^{2} - 309 T^{3} + 1702 T^{4} - 5253 T^{5} + 19074 T^{6} - 34391 T^{7} + 83521 T^{8} )^{4} \)
$19$ \( ( 1 - 69 T^{2} + 2306 T^{4} - 53763 T^{6} + 1069146 T^{8} - 19408443 T^{10} + 300520226 T^{12} - 3246165789 T^{14} + 16983563041 T^{16} )^{2} \)
$23$ \( ( 1 + 5 T - 32 T^{2} + 45 T^{3} + 1385 T^{4} - 3040 T^{5} - 11142 T^{6} + 48640 T^{7} - 123716 T^{8} + 1118720 T^{9} - 5894118 T^{10} - 36987680 T^{11} + 387579785 T^{12} + 289635435 T^{13} - 4737148448 T^{14} + 17024127235 T^{15} + 78310985281 T^{16} )^{2} \)
$29$ \( 1 + 123 T^{2} + 7912 T^{4} + 340647 T^{6} + 10067965 T^{8} + 144056448 T^{10} - 3705324014 T^{12} - 328820871018 T^{14} - 12137186779472 T^{16} - 276538352526138 T^{18} - 2620705273945934 T^{20} + 85688134810823808 T^{22} + 5036463377066894365 T^{24} + \)\(14\!\cdots\!47\)\( T^{26} + \)\(27\!\cdots\!92\)\( T^{28} + \)\(36\!\cdots\!63\)\( T^{30} + \)\(25\!\cdots\!21\)\( T^{32} \)
$31$ \( ( 1 - 5 T - 48 T^{2} + 155 T^{3} + 1121 T^{4} + 2040 T^{5} - 44678 T^{6} - 79040 T^{7} + 1805724 T^{8} - 2450240 T^{9} - 42935558 T^{10} + 60773640 T^{11} + 1035267041 T^{12} + 4437518405 T^{13} - 42600176688 T^{14} - 137563070555 T^{15} + 852891037441 T^{16} )^{2} \)
$37$ \( ( 1 - 144 T^{2} + 12668 T^{4} - 733824 T^{6} + 31784838 T^{8} - 1004605056 T^{10} + 23741871548 T^{12} - 369464602896 T^{14} + 3512479453921 T^{16} )^{2} \)
$41$ \( ( 1 - 4 T - 74 T^{2} - 288 T^{3} + 4517 T^{4} + 22796 T^{5} - 11538 T^{6} - 738968 T^{7} - 2462804 T^{8} - 30297688 T^{9} - 19395378 T^{10} + 1571123116 T^{11} + 12763962437 T^{12} - 33366585888 T^{13} - 351507713834 T^{14} - 779017095524 T^{15} + 7984925229121 T^{16} )^{2} \)
$43$ \( 1 + 324 T^{2} + 58258 T^{4} + 7305024 T^{6} + 705172105 T^{8} + 54954910764 T^{10} + 3558033934834 T^{12} + 194436327001344 T^{14} + 9048929543300068 T^{16} + 359512768625485056 T^{18} + 12164209974444414034 T^{20} + \)\(34\!\cdots\!36\)\( T^{22} + \)\(82\!\cdots\!05\)\( T^{24} + \)\(15\!\cdots\!76\)\( T^{26} + \)\(23\!\cdots\!58\)\( T^{28} + \)\(23\!\cdots\!76\)\( T^{30} + \)\(13\!\cdots\!01\)\( T^{32} \)
$47$ \( ( 1 - 3 T - 98 T^{2} + 219 T^{3} + 3523 T^{4} + 1116 T^{5} - 253856 T^{6} - 226218 T^{7} + 18909448 T^{8} - 10632246 T^{9} - 560767904 T^{10} + 115866468 T^{11} + 17191116163 T^{12} + 50226556533 T^{13} - 1056363102242 T^{14} - 1519869361389 T^{15} + 23811286661761 T^{16} )^{2} \)
$53$ \( ( 1 - 264 T^{2} + 35516 T^{4} - 3123720 T^{6} + 194863110 T^{8} - 8774529480 T^{10} + 280238323196 T^{12} - 5851391338056 T^{14} + 62259690411361 T^{16} )^{2} \)
$59$ \( 1 + 364 T^{2} + 70130 T^{4} + 9549968 T^{6} + 1028964713 T^{8} + 92991430988 T^{10} + 7291973853618 T^{12} + 506327334867240 T^{14} + 31497451193778532 T^{16} + 1762525452672862440 T^{18} + 88359479586850462098 T^{20} + \)\(39\!\cdots\!08\)\( T^{22} + \)\(15\!\cdots\!73\)\( T^{24} + \)\(48\!\cdots\!68\)\( T^{26} + \)\(12\!\cdots\!30\)\( T^{28} + \)\(22\!\cdots\!04\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( 1 + 323 T^{2} + 52152 T^{4} + 6035887 T^{6} + 584536925 T^{8} + 49803739968 T^{10} + 3801157073266 T^{12} + 264102902456582 T^{14} + 16823476283802768 T^{16} + 982726900040941622 T^{18} + 52630216452466386706 T^{20} + \)\(25\!\cdots\!48\)\( T^{22} + \)\(11\!\cdots\!25\)\( T^{24} + \)\(43\!\cdots\!87\)\( T^{26} + \)\(13\!\cdots\!92\)\( T^{28} + \)\(31\!\cdots\!43\)\( T^{30} + \)\(36\!\cdots\!61\)\( T^{32} \)
$67$ \( 1 + 296 T^{2} + 39450 T^{4} + 3796936 T^{6} + 362561921 T^{8} + 33090030780 T^{10} + 2631719645962 T^{12} + 193036342550180 T^{14} + 13423375489686516 T^{16} + 866540141707758020 T^{18} + 53032101023857423402 T^{20} + \)\(29\!\cdots\!20\)\( T^{22} + \)\(14\!\cdots\!61\)\( T^{24} + \)\(69\!\cdots\!64\)\( T^{26} + \)\(32\!\cdots\!50\)\( T^{28} + \)\(10\!\cdots\!84\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 - 18 T + 332 T^{2} - 3582 T^{3} + 36198 T^{4} - 254322 T^{5} + 1673612 T^{6} - 6442398 T^{7} + 25411681 T^{8} )^{4} \)
$73$ \( ( 1 + 11 T + 278 T^{2} + 2325 T^{3} + 29966 T^{4} + 169725 T^{5} + 1481462 T^{6} + 4279187 T^{7} + 28398241 T^{8} )^{4} \)
$79$ \( ( 1 - 15 T - 80 T^{2} + 1305 T^{3} + 13273 T^{4} - 59400 T^{5} - 1907150 T^{6} + 913560 T^{7} + 190569148 T^{8} + 72171240 T^{9} - 11902523150 T^{10} - 29286516600 T^{11} + 516984425113 T^{12} + 4015558600695 T^{13} - 19446996441680 T^{14} - 288058634792385 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( 1 + 559 T^{2} + 168260 T^{4} + 35967119 T^{6} + 6055752221 T^{8} + 843016361960 T^{10} + 99713806040442 T^{12} + 10182388433060610 T^{14} + 904920089638581976 T^{16} + 70146473915354542290 T^{18} + \)\(47\!\cdots\!82\)\( T^{20} + \)\(27\!\cdots\!40\)\( T^{22} + \)\(13\!\cdots\!61\)\( T^{24} + \)\(55\!\cdots\!31\)\( T^{26} + \)\(17\!\cdots\!60\)\( T^{28} + \)\(41\!\cdots\!11\)\( T^{30} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( ( 1 + 16 T + 408 T^{2} + 4116 T^{3} + 56278 T^{4} + 366324 T^{5} + 3231768 T^{6} + 11279504 T^{7} + 62742241 T^{8} )^{4} \)
$97$ \( ( 1 - 254 T^{2} + 1560 T^{3} + 35209 T^{4} - 273780 T^{5} - 2055806 T^{6} + 15575820 T^{7} + 104325124 T^{8} + 1510854540 T^{9} - 19343078654 T^{10} - 249871613940 T^{11} + 3117027454729 T^{12} + 13396250800920 T^{13} - 211574889251966 T^{14} + 7837433594376961 T^{16} )^{2} \)
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