Properties

Label 216.3.b.b
Level $216$
Weight $3$
Character orbit 216.b
Analytic conductor $5.886$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 2 x^{14} - 2 x^{12} - 56 x^{10} + 400 x^{8} - 896 x^{6} - 512 x^{4} - 8192 x^{2} + 65536\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{6} q^{5} -\beta_{14} q^{7} + \beta_{3} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} -\beta_{6} q^{5} -\beta_{14} q^{7} + \beta_{3} q^{8} + ( 1 + \beta_{8} ) q^{10} + ( \beta_{1} - \beta_{13} ) q^{11} + ( \beta_{8} - \beta_{12} ) q^{13} + ( -\beta_{10} - \beta_{13} ) q^{14} + ( 1 - \beta_{5} - \beta_{7} + \beta_{12} - \beta_{14} ) q^{16} + ( \beta_{3} - \beta_{9} - \beta_{11} - \beta_{13} ) q^{17} + ( -4 + \beta_{2} - \beta_{8} - \beta_{15} ) q^{19} + ( 2 \beta_{1} - \beta_{4} - 2 \beta_{6} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{20} + ( 5 + \beta_{2} - \beta_{7} + 2 \beta_{14} + \beta_{15} ) q^{22} + ( 3 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{9} - \beta_{11} ) q^{23} + ( -5 - 3 \beta_{2} + 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{15} ) q^{25} + ( \beta_{1} - 2 \beta_{6} - 2 \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{26} + ( -1 + \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{12} + \beta_{14} + \beta_{15} ) q^{28} + ( 2 \beta_{1} - 2 \beta_{10} + 2 \beta_{11} ) q^{29} + ( 2 - \beta_{2} + \beta_{7} - \beta_{8} - 2 \beta_{12} + \beta_{15} ) q^{31} + ( \beta_{4} + 2 \beta_{9} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{32} + ( 1 - 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + 2 \beta_{12} + 2 \beta_{14} ) q^{34} + ( \beta_{1} + 4 \beta_{3} + 2 \beta_{6} + 2 \beta_{10} + 2 \beta_{11} + \beta_{13} ) q^{35} + ( 4 - 5 \beta_{2} - 2 \beta_{5} + \beta_{7} - 2 \beta_{8} - \beta_{12} - 2 \beta_{14} + \beta_{15} ) q^{37} + ( -2 \beta_{1} + 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{10} + \beta_{11} + 2 \beta_{13} ) q^{38} + ( -6 + 2 \beta_{2} - 4 \beta_{7} + 2 \beta_{8} + 4 \beta_{14} ) q^{40} + ( 6 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{10} - 2 \beta_{11} ) q^{41} + ( -4 - 7 \beta_{2} + 4 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{15} ) q^{43} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 6 \beta_{6} + 3 \beta_{10} + \beta_{11} + \beta_{13} ) q^{44} + ( -15 + 3 \beta_{2} + 2 \beta_{5} - 5 \beta_{7} + 2 \beta_{12} - \beta_{15} ) q^{46} + ( -7 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{9} + 2 \beta_{10} - 3 \beta_{11} ) q^{47} + ( -8 + 2 \beta_{2} - 2 \beta_{7} - 2 \beta_{8} - 2 \beta_{15} ) q^{49} + ( -\beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{10} + 2 \beta_{13} ) q^{50} + ( 6 + \beta_{2} - 4 \beta_{5} + 4 \beta_{7} + 2 \beta_{8} + 4 \beta_{14} ) q^{52} + ( 2 \beta_{1} - 4 \beta_{3} - 4 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{53} + ( 2 - 7 \beta_{2} - 4 \beta_{5} - \beta_{7} + 5 \beta_{8} - 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{55} + ( -4 \beta_{1} + \beta_{3} - 3 \beta_{4} + 8 \beta_{6} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{56} + ( -6 + 2 \beta_{2} + 6 \beta_{7} - 4 \beta_{14} + 2 \beta_{15} ) q^{58} + ( -13 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{9} + 4 \beta_{11} + \beta_{13} ) q^{59} + ( -4 + 8 \beta_{2} + 4 \beta_{5} + \beta_{8} - \beta_{12} - 4 \beta_{14} ) q^{61} + ( -2 \beta_{1} + 2 \beta_{4} + 8 \beta_{6} - 4 \beta_{9} ) q^{62} + ( 22 + 4 \beta_{5} - 6 \beta_{7} + 2 \beta_{15} ) q^{64} + ( -14 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - \beta_{9} - 2 \beta_{10} - 3 \beta_{11} + 3 \beta_{13} ) q^{65} + ( 8 + \beta_{7} ) q^{67} + ( -2 \beta_{1} + 3 \beta_{4} + 2 \beta_{6} + 4 \beta_{9} + \beta_{10} - \beta_{11} + 3 \beta_{13} ) q^{68} + ( 15 + \beta_{2} - 8 \beta_{5} - 9 \beta_{7} + 2 \beta_{8} - 6 \beta_{14} + \beta_{15} ) q^{70} + ( -4 \beta_{1} + 8 \beta_{6} + 4 \beta_{10} - 4 \beta_{11} ) q^{71} + ( 5 + 8 \beta_{2} - 4 \beta_{5} + 4 \beta_{7} ) q^{73} + ( -3 \beta_{1} - 4 \beta_{3} + 6 \beta_{4} + 10 \beta_{6} - 2 \beta_{9} - 3 \beta_{10} + \beta_{11} - \beta_{13} ) q^{74} + ( 17 - 2 \beta_{2} + \beta_{5} + 8 \beta_{7} + 3 \beta_{8} + \beta_{12} - 7 \beta_{14} - \beta_{15} ) q^{76} + ( -26 \beta_{1} + 8 \beta_{4} + \beta_{6} - 6 \beta_{10} + 6 \beta_{11} ) q^{77} + ( -6 + 15 \beta_{2} + 8 \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{12} + 3 \beta_{14} + \beta_{15} ) q^{79} + ( -4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{6} + 6 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} ) q^{80} + ( 12 + 6 \beta_{2} + 8 \beta_{5} + 10 \beta_{7} - 2 \beta_{8} + 4 \beta_{14} - 2 \beta_{15} ) q^{82} + ( 18 \beta_{1} + 4 \beta_{4} + 2 \beta_{6} + 4 \beta_{9} + 2 \beta_{10} + 6 \beta_{11} + 4 \beta_{13} ) q^{83} + ( -3 \beta_{2} - 2 \beta_{5} - \beta_{7} + \beta_{8} + 2 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{85} + ( 2 \beta_{1} - 8 \beta_{3} - 6 \beta_{4} - 4 \beta_{6} - 2 \beta_{10} + 2 \beta_{13} ) q^{86} + ( -4 + 2 \beta_{2} - 6 \beta_{5} - 10 \beta_{7} - 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{14} ) q^{88} + ( -8 \beta_{1} + 3 \beta_{3} - 3 \beta_{9} - 3 \beta_{11} + 5 \beta_{13} ) q^{89} + ( 12 + 15 \beta_{2} - 8 \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{15} ) q^{91} + ( -10 \beta_{1} + 2 \beta_{3} - 5 \beta_{4} - 6 \beta_{6} + 4 \beta_{9} - \beta_{10} + 5 \beta_{11} + \beta_{13} ) q^{92} + ( 21 - 7 \beta_{2} + 2 \beta_{5} - 11 \beta_{7} - 2 \beta_{8} + 2 \beta_{12} + 4 \beta_{14} - 3 \beta_{15} ) q^{94} + ( 25 \beta_{1} + 3 \beta_{3} - 7 \beta_{4} + 9 \beta_{6} + 3 \beta_{9} - 3 \beta_{11} ) q^{95} + ( 5 - 15 \beta_{2} + 6 \beta_{5} - 7 \beta_{7} + 3 \beta_{8} + 3 \beta_{15} ) q^{97} + ( -4 \beta_{1} + 4 \beta_{4} - 8 \beta_{6} - 4 \beta_{10} + 4 \beta_{11} + 4 \beta_{13} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{4} + O(q^{10}) \) \( 16q + 4q^{4} + 24q^{10} + 16q^{16} - 64q^{19} + 80q^{22} - 80q^{25} - 12q^{28} + 8q^{34} - 72q^{40} - 64q^{43} - 192q^{46} - 128q^{49} + 84q^{52} - 96q^{58} + 376q^{64} + 128q^{67} + 192q^{70} + 80q^{73} + 308q^{76} + 272q^{82} - 136q^{88} + 192q^{91} + 336q^{94} + 80q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{14} - 2 x^{12} - 56 x^{10} + 400 x^{8} - 896 x^{6} - 512 x^{4} - 8192 x^{2} + 65536\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)
\(\beta_{3}\)\(=\)\( \nu^{3} \)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{15} + 2 \nu^{13} + 2 \nu^{11} + 56 \nu^{9} - 400 \nu^{7} + 896 \nu^{5} + 512 \nu^{3} + 8192 \nu \)\()/4096\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{14} - 2 \nu^{12} - 2 \nu^{10} - 56 \nu^{8} + 400 \nu^{6} - 896 \nu^{4} - 512 \nu^{2} - 6144 \)\()/2048\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{15} - 10 \nu^{13} + 78 \nu^{11} - 168 \nu^{9} + 720 \nu^{7} - 3584 \nu^{5} + 15872 \nu^{3} - 36864 \nu \)\()/8192\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{14} - 22 \nu^{12} + 90 \nu^{10} - 264 \nu^{8} + 944 \nu^{6} - 6528 \nu^{4} + 19968 \nu^{2} - 28672 \)\()/4096\)
\(\beta_{8}\)\(=\)\((\)\( \nu^{14} - 10 \nu^{12} + 14 \nu^{10} - 40 \nu^{8} + 336 \nu^{6} - 2048 \nu^{4} + 3584 \nu^{2} + 7168 \)\()/1024\)
\(\beta_{9}\)\(=\)\((\)\( -3 \nu^{15} - 10 \nu^{13} + 166 \nu^{11} - 568 \nu^{9} + 464 \nu^{7} - 1664 \nu^{5} + 46592 \nu^{3} - 139264 \nu \)\()/8192\)
\(\beta_{10}\)\(=\)\((\)\( -3 \nu^{15} - 6 \nu^{13} + 62 \nu^{11} - 128 \nu^{9} - 80 \nu^{7} - 1344 \nu^{5} + 18944 \nu^{3} - 38912 \nu \)\()/4096\)
\(\beta_{11}\)\(=\)\((\)\( -3 \nu^{15} + 22 \nu^{13} - 90 \nu^{11} + 264 \nu^{9} - 944 \nu^{7} + 6528 \nu^{5} - 19968 \nu^{3} + 28672 \nu \)\()/4096\)
\(\beta_{12}\)\(=\)\((\)\( -\nu^{14} - 6 \nu^{12} + 82 \nu^{10} - 312 \nu^{8} + 432 \nu^{6} - 1280 \nu^{4} + 23040 \nu^{2} - 73728 \)\()/2048\)
\(\beta_{13}\)\(=\)\((\)\( -\nu^{15} + 5 \nu^{13} + 4 \nu^{11} - 30 \nu^{9} - 200 \nu^{7} + 752 \nu^{5} + 2048 \nu^{3} - 15872 \nu \)\()/1024\)
\(\beta_{14}\)\(=\)\((\)\( -7 \nu^{14} + 14 \nu^{12} + 78 \nu^{10} - 248 \nu^{8} - 880 \nu^{6} + 1664 \nu^{4} + 27136 \nu^{2} - 102400 \)\()/4096\)
\(\beta_{15}\)\(=\)\((\)\( 5 \nu^{14} - 58 \nu^{12} + 278 \nu^{10} - 568 \nu^{8} + 3280 \nu^{6} - 16000 \nu^{4} + 61952 \nu^{2} - 106496 \)\()/4096\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\)
\(\nu^{3}\)\(=\)\(\beta_{3}\)
\(\nu^{4}\)\(=\)\(-\beta_{14} + \beta_{12} - \beta_{7} - \beta_{5} + 1\)
\(\nu^{5}\)\(=\)\(-\beta_{13} + \beta_{11} - \beta_{10} + 2 \beta_{9} + \beta_{4}\)
\(\nu^{6}\)\(=\)\(2 \beta_{15} - 6 \beta_{7} + 4 \beta_{5} + 22\)
\(\nu^{7}\)\(=\)\(-2 \beta_{13} + 6 \beta_{11} + 2 \beta_{10} + 12 \beta_{6} - 10 \beta_{4} + 2 \beta_{3} + 20 \beta_{1}\)
\(\nu^{8}\)\(=\)\(8 \beta_{15} - 2 \beta_{14} - 6 \beta_{12} - 4 \beta_{8} - 10 \beta_{7} - 10 \beta_{5} + 20 \beta_{2} - 130\)
\(\nu^{9}\)\(=\)\(-6 \beta_{13} + 14 \beta_{11} + 2 \beta_{10} - 12 \beta_{9} + 56 \beta_{6} + 22 \beta_{4} + 28 \beta_{3} - 168 \beta_{1}\)
\(\nu^{10}\)\(=\)\(20 \beta_{15} - 28 \beta_{14} + 12 \beta_{12} - 40 \beta_{8} + 8 \beta_{7} - 68 \beta_{5} - 168 \beta_{2} + 384\)
\(\nu^{11}\)\(=\)\(-8 \beta_{13} + 32 \beta_{11} - 48 \beta_{10} + 24 \beta_{9} + 200 \beta_{6} + 144 \beta_{4} - 148 \beta_{3} + 216 \beta_{1}\)
\(\nu^{12}\)\(=\)\(40 \beta_{15} + 84 \beta_{14} - 132 \beta_{12} - 216 \beta_{8} + 188 \beta_{7} + 212 \beta_{5} + 216 \beta_{2} + 1852\)
\(\nu^{13}\)\(=\)\(260 \beta_{13} + 28 \beta_{11} - 92 \beta_{10} - 264 \beta_{9} + 672 \beta_{6} - 116 \beta_{4} + 256 \beta_{3} + 1728 \beta_{1}\)
\(\nu^{14}\)\(=\)\(-232 \beta_{15} - 896 \beta_{14} + 320 \beta_{12} - 736 \beta_{8} + 1336 \beta_{7} - 720 \beta_{5} + 1728 \beta_{2} - 4568\)
\(\nu^{15}\)\(=\)\(72 \beta_{13} - 600 \beta_{11} - 1864 \beta_{10} + 640 \beta_{9} + 80 \beta_{6} + 2088 \beta_{4} + 1496 \beta_{3} - 5328 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
163.1
−1.98451 0.248465i
−1.98451 + 0.248465i
−1.77866 0.914534i
−1.77866 + 0.914534i
−0.862987 1.80423i
−0.862987 + 1.80423i
−0.808307 1.82938i
−0.808307 + 1.82938i
0.808307 1.82938i
0.808307 + 1.82938i
0.862987 1.80423i
0.862987 + 1.80423i
1.77866 0.914534i
1.77866 + 0.914534i
1.98451 0.248465i
1.98451 + 0.248465i
−1.98451 0.248465i 0 3.87653 + 0.986159i 8.47512i 0 7.86423i −7.44797 2.92022i 0 2.10577 16.8189i
163.2 −1.98451 + 0.248465i 0 3.87653 0.986159i 8.47512i 0 7.86423i −7.44797 + 2.92022i 0 2.10577 + 16.8189i
163.3 −1.77866 0.914534i 0 2.32725 + 3.25329i 3.96500i 0 3.99887i −1.16415 7.91484i 0 −3.62613 + 7.05238i
163.4 −1.77866 + 0.914534i 0 2.32725 3.25329i 3.96500i 0 3.99887i −1.16415 + 7.91484i 0 −3.62613 7.05238i
163.5 −0.862987 1.80423i 0 −2.51051 + 3.11406i 1.42436i 0 4.94379i 7.78502 + 1.84214i 0 −2.56987 + 1.22920i
163.6 −0.862987 + 1.80423i 0 −2.51051 3.11406i 1.42436i 0 4.94379i 7.78502 1.84214i 0 −2.56987 1.22920i
163.7 −0.808307 1.82938i 0 −2.69328 + 2.95740i 5.51565i 0 11.2126i 7.58722 + 2.53655i 0 10.0902 4.45834i
163.8 −0.808307 + 1.82938i 0 −2.69328 2.95740i 5.51565i 0 11.2126i 7.58722 2.53655i 0 10.0902 + 4.45834i
163.9 0.808307 1.82938i 0 −2.69328 2.95740i 5.51565i 0 11.2126i −7.58722 + 2.53655i 0 10.0902 + 4.45834i
163.10 0.808307 + 1.82938i 0 −2.69328 + 2.95740i 5.51565i 0 11.2126i −7.58722 2.53655i 0 10.0902 4.45834i
163.11 0.862987 1.80423i 0 −2.51051 3.11406i 1.42436i 0 4.94379i −7.78502 + 1.84214i 0 −2.56987 1.22920i
163.12 0.862987 + 1.80423i 0 −2.51051 + 3.11406i 1.42436i 0 4.94379i −7.78502 1.84214i 0 −2.56987 + 1.22920i
163.13 1.77866 0.914534i 0 2.32725 3.25329i 3.96500i 0 3.99887i 1.16415 7.91484i 0 −3.62613 7.05238i
163.14 1.77866 + 0.914534i 0 2.32725 + 3.25329i 3.96500i 0 3.99887i 1.16415 + 7.91484i 0 −3.62613 + 7.05238i
163.15 1.98451 0.248465i 0 3.87653 0.986159i 8.47512i 0 7.86423i 7.44797 2.92022i 0 2.10577 + 16.8189i
163.16 1.98451 + 0.248465i 0 3.87653 + 0.986159i 8.47512i 0 7.86423i 7.44797 + 2.92022i 0 2.10577 16.8189i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.3.b.b 16
3.b odd 2 1 inner 216.3.b.b 16
4.b odd 2 1 864.3.b.b 16
8.b even 2 1 864.3.b.b 16
8.d odd 2 1 inner 216.3.b.b 16
12.b even 2 1 864.3.b.b 16
24.f even 2 1 inner 216.3.b.b 16
24.h odd 2 1 864.3.b.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.3.b.b 16 1.a even 1 1 trivial
216.3.b.b 16 3.b odd 2 1 inner
216.3.b.b 16 8.d odd 2 1 inner
216.3.b.b 16 24.f even 2 1 inner
864.3.b.b 16 4.b odd 2 1
864.3.b.b 16 8.b even 2 1
864.3.b.b 16 12.b even 2 1
864.3.b.b 16 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 120 T_{5}^{6} + 4032 T_{5}^{4} + 42048 T_{5}^{2} + 69696 \) acting on \(S_{3}^{\mathrm{new}}(216, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 65536 - 8192 T^{2} - 512 T^{4} - 896 T^{6} + 400 T^{8} - 56 T^{10} - 2 T^{12} - 2 T^{14} + T^{16} \)
$3$ \( T^{16} \)
$5$ \( ( 69696 + 42048 T^{2} + 4032 T^{4} + 120 T^{6} + T^{8} )^{2} \)
$7$ \( ( 3038913 + 387684 T^{2} + 15750 T^{4} + 228 T^{6} + T^{8} )^{2} \)
$11$ \( ( 31294528 - 4924480 T^{2} + 94272 T^{4} - 568 T^{6} + T^{8} )^{2} \)
$13$ \( ( 300808737 + 14194332 T^{2} + 172134 T^{4} + 732 T^{6} + T^{8} )^{2} \)
$17$ \( ( 33082432 - 5713600 T^{2} + 141504 T^{4} - 1000 T^{6} + T^{8} )^{2} \)
$19$ \( ( 59233 - 4208 T - 750 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$23$ \( ( 655769664 + 55841472 T^{2} + 643392 T^{4} + 1704 T^{6} + T^{8} )^{2} \)
$29$ \( ( 89942409216 + 1532030976 T^{2} + 3806208 T^{4} + 3360 T^{6} + T^{8} )^{2} \)
$31$ \( ( 64927957248 + 1718615808 T^{2} + 4694112 T^{4} + 4080 T^{6} + T^{8} )^{2} \)
$37$ \( ( 225812702097 + 12209829516 T^{2} + 17860662 T^{4} + 7596 T^{6} + T^{8} )^{2} \)
$41$ \( ( 11517064118272 - 33478574080 T^{2} + 27561984 T^{4} - 8896 T^{6} + T^{8} )^{2} \)
$43$ \( ( 417856 + 40576 T - 3936 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$47$ \( ( 54006041664 + 3802859712 T^{2} + 10146240 T^{4} + 7464 T^{6} + T^{8} )^{2} \)
$53$ \( ( 13649064247296 + 117537288192 T^{2} + 85349376 T^{4} + 17568 T^{6} + T^{8} )^{2} \)
$59$ \( ( 12544953873472 - 131868157504 T^{2} + 117047616 T^{4} - 21496 T^{6} + T^{8} )^{2} \)
$61$ \( ( 199870871553 + 20005796988 T^{2} + 32736006 T^{4} + 11388 T^{6} + T^{8} )^{2} \)
$67$ \( ( 37 - 16 T + T^{2} )^{8} \)
$71$ \( ( 516742447104 + 163902652416 T^{2} + 110149632 T^{4} + 19584 T^{6} + T^{8} )^{2} \)
$73$ \( ( 3184753 + 73228 T - 4458 T^{2} - 20 T^{3} + T^{4} )^{4} \)
$79$ \( ( 593133122547153 + 814704489108 T^{2} + 326673846 T^{4} + 33780 T^{6} + T^{8} )^{2} \)
$83$ \( ( 4297326592 - 984087887872 T^{2} + 423075840 T^{4} - 38464 T^{6} + T^{8} )^{2} \)
$89$ \( ( 4479298104603712 - 2398120782016 T^{2} + 448601280 T^{4} - 35368 T^{6} + T^{8} )^{2} \)
$97$ \( ( 34042441 + 981436 T - 23394 T^{2} - 20 T^{3} + T^{4} )^{4} \)
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