Properties

Label 1386.2.g.a
Level $1386$
Weight $2$
Character orbit 1386.g
Analytic conductor $11.067$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 4 x^{14} + 8 x^{12} + 80 x^{10} + 1189 x^{8} - 2028 x^{6} + 1800 x^{4} + 1080 x^{2} + 324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} - q^{4} -\beta_{4} q^{5} + \beta_{7} q^{7} -\beta_{1} q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} - q^{4} -\beta_{4} q^{5} + \beta_{7} q^{7} -\beta_{1} q^{8} + \beta_{3} q^{10} -\beta_{1} q^{11} + \beta_{11} q^{14} + q^{16} + ( -\beta_{1} - \beta_{4} + \beta_{10} - \beta_{11} ) q^{17} + \beta_{14} q^{19} + \beta_{4} q^{20} + q^{22} + ( \beta_{12} - \beta_{13} ) q^{23} + ( 1 - \beta_{2} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{25} -\beta_{7} q^{28} + ( \beta_{1} + \beta_{6} + \beta_{10} + \beta_{11} ) q^{29} + ( \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - \beta_{14} ) q^{31} + \beta_{1} q^{32} + ( 1 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{34} + ( 2 \beta_{4} + \beta_{12} ) q^{35} + ( -1 + \beta_{8} - \beta_{9} ) q^{37} + \beta_{15} q^{38} -\beta_{3} q^{40} + ( -\beta_{1} + \beta_{10} - \beta_{11} + \beta_{15} ) q^{41} + ( 3 + \beta_{5} - \beta_{7} ) q^{43} + \beta_{1} q^{44} + ( -1 - \beta_{8} + \beta_{9} ) q^{46} + ( -\beta_{1} + 2 \beta_{4} + \beta_{10} - \beta_{11} + \beta_{15} ) q^{47} + ( 1 + \beta_{5} - \beta_{7} - \beta_{9} + \beta_{14} ) q^{49} + ( \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{50} + ( -5 \beta_{1} + \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{53} -\beta_{3} q^{55} -\beta_{11} q^{56} + ( -1 + \beta_{2} + \beta_{5} - \beta_{7} ) q^{58} -2 \beta_{4} q^{59} + ( 2 - 4 \beta_{3} + 2 \beta_{5} + 2 \beta_{7} ) q^{61} + ( \beta_{1} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{15} ) q^{62} - q^{64} + ( 1 - \beta_{2} + \beta_{5} - \beta_{7} ) q^{67} + ( \beta_{1} + \beta_{4} - \beta_{10} + \beta_{11} ) q^{68} + ( -2 \beta_{3} - \beta_{8} ) q^{70} + ( 2 \beta_{6} + \beta_{12} - \beta_{13} ) q^{71} + ( -1 + 2 \beta_{3} - \beta_{5} - \beta_{7} + \beta_{14} ) q^{73} + ( -2 \beta_{1} + \beta_{12} - \beta_{13} ) q^{74} -\beta_{14} q^{76} -\beta_{11} q^{77} + ( -1 + \beta_{2} + 2 \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{79} -\beta_{4} q^{80} + ( 1 + \beta_{5} + \beta_{7} - \beta_{14} ) q^{82} + ( \beta_{1} - \beta_{4} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{83} + ( 7 - \beta_{2} + \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{85} + ( 3 \beta_{1} - \beta_{10} - \beta_{11} ) q^{86} - q^{88} + ( -\beta_{1} + \beta_{10} - \beta_{11} + 2 \beta_{15} ) q^{89} + ( -\beta_{12} + \beta_{13} ) q^{92} + ( 1 - 2 \beta_{3} + \beta_{5} + \beta_{7} - \beta_{14} ) q^{94} + ( -\beta_{1} - \beta_{6} - \beta_{10} - \beta_{11} ) q^{95} + ( 2 \beta_{3} + 2 \beta_{14} ) q^{97} + ( -\beta_{10} - \beta_{11} - \beta_{13} + \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{4} - 8q^{7} + O(q^{10}) \) \( 16q - 16q^{4} - 8q^{7} + 16q^{16} + 16q^{22} + 16q^{25} + 8q^{28} - 16q^{37} + 48q^{43} - 16q^{46} + 8q^{49} - 16q^{58} - 16q^{64} + 16q^{67} - 8q^{70} - 16q^{79} + 112q^{85} - 16q^{88} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 4 x^{14} + 8 x^{12} + 80 x^{10} + 1189 x^{8} - 2028 x^{6} + 1800 x^{4} + 1080 x^{2} + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -96575 \nu^{14} + 438755 \nu^{12} - 939814 \nu^{10} - 7435564 \nu^{8} - 110537567 \nu^{6} + 264348225 \nu^{4} - 229027878 \nu^{2} - 51358644 \)\()/86730966\)
\(\beta_{2}\)\(=\)\((\)\( -3833 \nu^{14} + 11990 \nu^{12} - 16306 \nu^{10} - 332152 \nu^{8} - 4851221 \nu^{6} + 3917118 \nu^{4} + 2396970 \nu^{2} - 7232976 \)\()/1867860\)
\(\beta_{3}\)\(=\)\((\)\( -3079634 \nu^{15} + 11597979 \nu^{13} - 22413962 \nu^{11} - 248772494 \nu^{9} - 3735647454 \nu^{7} + 5354283043 \nu^{5} - 4822618578 \nu^{3} - 5487054570 \nu \)\()/ 2601928980 \)
\(\beta_{4}\)\(=\)\((\)\( 12591283 \nu^{15} - 57579817 \nu^{13} + 129110888 \nu^{11} + 952830506 \nu^{9} + 14397572539 \nu^{7} - 34397644005 \nu^{5} + 38419343964 \nu^{3} + 3202119162 \nu \)\()/ 7805786940 \)
\(\beta_{5}\)\(=\)\((\)\(14545077 \nu^{15} - 3765719 \nu^{14} - 55196585 \nu^{13} + 11685650 \nu^{12} + 105758504 \nu^{11} - 14001178 \nu^{10} + 1189121858 \nu^{9} - 333984496 \nu^{8} + 17514628229 \nu^{7} - 4702927103 \nu^{6} - 25836351497 \nu^{5} + 3800759034 \nu^{4} + 22083943080 \nu^{3} + 2325320910 \nu^{2} + 25498762434 \nu - 9136826568\)\()/ 5203857960 \)
\(\beta_{6}\)\(=\)\((\)\( -11395 \nu^{14} + 45868 \nu^{12} - 96938 \nu^{10} - 893276 \nu^{8} - 13578607 \nu^{6} + 23130228 \nu^{4} - 27991926 \nu^{2} - 6275124 \)\()/2256660\)
\(\beta_{7}\)\(=\)\((\)\(14545077 \nu^{15} + 3765719 \nu^{14} - 55196585 \nu^{13} - 11685650 \nu^{12} + 105758504 \nu^{11} + 14001178 \nu^{10} + 1189121858 \nu^{9} + 333984496 \nu^{8} + 17514628229 \nu^{7} + 4702927103 \nu^{6} - 25836351497 \nu^{5} - 3800759034 \nu^{4} + 22083943080 \nu^{3} - 2325320910 \nu^{2} + 25498762434 \nu + 3932968608\)\()/ 5203857960 \)
\(\beta_{8}\)\(=\)\((\)\(18506049 \nu^{15} - 1441114 \nu^{14} - 72148880 \nu^{13} + 4446220 \nu^{12} + 136369838 \nu^{11} - 4803548 \nu^{10} + 1512241796 \nu^{9} - 147925256 \nu^{8} + 22114467473 \nu^{7} - 1782429418 \nu^{6} - 35425496144 \nu^{5} + 1441445244 \nu^{4} + 25614327990 \nu^{3} + 881758260 \nu^{2} + 31120049628 \nu - 13615971048\)\()/ 5203857960 \)
\(\beta_{9}\)\(=\)\((\)\(18506049 \nu^{15} + 1441114 \nu^{14} - 72148880 \nu^{13} - 4446220 \nu^{12} + 136369838 \nu^{11} + 4803548 \nu^{10} + 1512241796 \nu^{9} + 147925256 \nu^{8} + 22114467473 \nu^{7} + 1782429418 \nu^{6} - 35425496144 \nu^{5} - 1441445244 \nu^{4} + 25614327990 \nu^{3} - 881758260 \nu^{2} + 31120049628 \nu + 18819829008\)\()/ 5203857960 \)
\(\beta_{10}\)\(=\)\((\)\(60462680 \nu^{15} - 34694355 \nu^{14} - 274567721 \nu^{13} + 143608152 \nu^{12} + 617221546 \nu^{11} - 309169302 \nu^{10} + 4576799302 \nu^{9} - 2703731964 \nu^{8} + 69289811864 \nu^{7} - 40803930363 \nu^{6} - 161200544481 \nu^{5} + 75040099632 \nu^{4} + 178659278442 \nu^{3} - 84253473054 \nu^{2} + 14747147118 \nu - 18889526916\)\()/ 15611573880 \)
\(\beta_{11}\)\(=\)\((\)\(-60462680 \nu^{15} - 17310855 \nu^{14} + 274567721 \nu^{13} + 64632252 \nu^{12} - 617221546 \nu^{11} - 140002782 \nu^{10} - 4576799302 \nu^{9} - 1365330444 \nu^{8} - 69289811864 \nu^{7} - 20907168303 \nu^{6} + 161200544481 \nu^{5} + 27457419132 \nu^{4} - 178659278442 \nu^{3} - 43028455014 \nu^{2} - 14747147118 \nu - 9644970996\)\()/ 15611573880 \)
\(\beta_{12}\)\(=\)\((\)\(-81296765 \nu^{15} + 47163030 \nu^{14} + 357230942 \nu^{13} - 210264972 \nu^{12} - 785602642 \nu^{11} + 450833772 \nu^{10} - 6207754144 \nu^{9} + 3640498104 \nu^{8} - 94072523573 \nu^{7} + 54294184518 \nu^{6} + 202259564982 \nu^{5} - 121172690412 \nu^{4} - 218620670994 \nu^{3} + 112411745244 \nu^{2} - 17473193856 \nu + 25206770376\)\()/ 15611573880 \)
\(\beta_{13}\)\(=\)\((\)\(-81296765 \nu^{15} - 47163030 \nu^{14} + 357230942 \nu^{13} + 210264972 \nu^{12} - 785602642 \nu^{11} - 450833772 \nu^{10} - 6207754144 \nu^{9} - 3640498104 \nu^{8} - 94072523573 \nu^{7} - 54294184518 \nu^{6} + 202259564982 \nu^{5} + 121172690412 \nu^{4} - 218620670994 \nu^{3} - 112411745244 \nu^{2} - 17473193856 \nu - 25206770376\)\()/ 15611573880 \)
\(\beta_{14}\)\(=\)\((\)\(23298266 \nu^{15} - 93544963 \nu^{13} + 176425470 \nu^{11} + 1897812810 \nu^{9} + 27613656994 \nu^{7} - 48610532639 \nu^{5} + 28420002066 \nu^{3} + 37162486386 \nu\)\()/ 2601928980 \)
\(\beta_{15}\)\(=\)\((\)\(-15573353 \nu^{15} + 65944679 \nu^{13} - 141046900 \nu^{11} - 1206672370 \nu^{9} - 18241907237 \nu^{7} + 35778301767 \nu^{5} - 37432034028 \nu^{3} - 2860280478 \nu\)\()/ 1115112420 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{15} - \beta_{13} - \beta_{12} - 3 \beta_{3}\)\()/6\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{11} + \beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{14} + \beta_{11} - \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} - \beta_{5} + 5 \beta_{4} + \beta_{1} - 3\)\()/2\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{13} + 7 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 32 \beta_{1}\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(-5 \beta_{14} + \beta_{13} + \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} - 9 \beta_{7} - 9 \beta_{5} - 27 \beta_{4} - 2 \beta_{3} - 7 \beta_{1} - 22\)\()/2\)
\(\nu^{6}\)\(=\)\((\)\(-11 \beta_{13} + 11 \beta_{12} + 45 \beta_{11} + 45 \beta_{10} + 11 \beta_{9} - 11 \beta_{8} - 45 \beta_{7} - 29 \beta_{6} + 45 \beta_{5} - 29 \beta_{2} + 57 \beta_{1} - 68\)\()/2\)
\(\nu^{7}\)\(=\)\((\)\(-31 \beta_{15} - 2 \beta_{14} + 87 \beta_{13} + 87 \beta_{12} - 67 \beta_{11} + 67 \beta_{10} + 13 \beta_{9} + 13 \beta_{8} - 45 \beta_{7} - 45 \beta_{5} - 28 \beta_{4} - 149 \beta_{3} - 67 \beta_{1} - 58\)\()/2\)
\(\nu^{8}\)\(=\)\((\)\(289 \beta_{9} - 289 \beta_{8} - 190 \beta_{7} + 190 \beta_{5} - 56 \beta_{2} - 1541\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-205 \beta_{15} + 28 \beta_{14} + 589 \beta_{13} + 589 \beta_{12} - 479 \beta_{11} + 479 \beta_{10} - 123 \beta_{9} - 123 \beta_{8} + 289 \beta_{7} + 289 \beta_{5} - 274 \beta_{4} + 835 \beta_{3} - 479 \beta_{1} + 412\)\()/2\)
\(\nu^{10}\)\(=\)\((\)\(753 \beta_{13} - 753 \beta_{12} - 1877 \beta_{11} - 1877 \beta_{10} + 753 \beta_{9} - 753 \beta_{8} - 1877 \beta_{7} + 1109 \beta_{6} + 1877 \beta_{5} - 1109 \beta_{2} - 3521 \beta_{1} - 4274\)\()/2\)
\(\nu^{11}\)\(=\)\((\)\(-274 \beta_{15} + 1383 \beta_{14} + 1027 \beta_{13} + 1027 \beta_{12} - 1877 \beta_{11} + 1877 \beta_{10} - 4013 \beta_{9} - 4013 \beta_{8} + 3383 \beta_{7} + 3383 \beta_{5} - 4753 \beta_{4} + 2328 \beta_{3} - 1877 \beta_{1} + 7396\)\()/2\)
\(\nu^{12}\)\(=\)\((\)\(12341 \beta_{13} - 12341 \beta_{12} - 11422 \beta_{11} - 11422 \beta_{10} + 4656 \beta_{6} - 53908 \beta_{1}\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(2328 \beta_{15} + 9409 \beta_{14} - 8039 \beta_{13} - 8039 \beta_{12} + 12341 \beta_{11} - 12341 \beta_{10} - 27461 \beta_{9} - 27461 \beta_{8} + 23763 \beta_{7} + 23763 \beta_{5} + 27519 \beta_{4} + 18406 \beta_{3} + 12341 \beta_{1} + 51224\)\()/2\)
\(\nu^{14}\)\(=\)\((\)\(42169 \beta_{13} - 42169 \beta_{12} - 82029 \beta_{11} - 82029 \beta_{10} - 42169 \beta_{9} + 42169 \beta_{8} + 82029 \beta_{7} + 45925 \beta_{6} - 82029 \beta_{5} + 45925 \beta_{2} - 192465 \beta_{1} + 234634\)\()/2\)
\(\nu^{15}\)\(=\)\((\)\(64331 \beta_{15} + 18406 \beta_{14} - 188529 \beta_{13} - 188529 \beta_{12} + 166367 \beta_{11} - 166367 \beta_{10} - 60575 \beta_{9} - 60575 \beta_{8} + 82029 \beta_{7} + 82029 \beta_{5} + 139556 \beta_{4} + 162277 \beta_{3} + 166367 \beta_{1} + 142604\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times\).

\(n\) \(155\) \(199\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
−0.876932 2.11710i
2.43348 1.00798i
−0.221383 0.534465i
1.12256 0.464978i
−1.12256 + 0.464978i
0.221383 + 0.534465i
−2.43348 + 1.00798i
0.876932 + 2.11710i
−0.876932 + 2.11710i
2.43348 + 1.00798i
−0.221383 + 0.534465i
1.12256 + 0.464978i
−1.12256 0.464978i
0.221383 0.534465i
−2.43348 1.00798i
0.876932 2.11710i
1.00000i 0 −1.00000 −4.23420 0 −2.59178 + 0.531652i 1.00000i 0 4.23420i
881.2 1.00000i 0 −1.00000 −2.01596 0 1.78450 + 1.95335i 1.00000i 0 2.01596i
881.3 1.00000i 0 −1.00000 −1.06893 0 0.884677 2.49346i 1.00000i 0 1.06893i
881.4 1.00000i 0 −1.00000 −0.929956 0 −2.07739 1.63843i 1.00000i 0 0.929956i
881.5 1.00000i 0 −1.00000 0.929956 0 −2.07739 + 1.63843i 1.00000i 0 0.929956i
881.6 1.00000i 0 −1.00000 1.06893 0 0.884677 + 2.49346i 1.00000i 0 1.06893i
881.7 1.00000i 0 −1.00000 2.01596 0 1.78450 1.95335i 1.00000i 0 2.01596i
881.8 1.00000i 0 −1.00000 4.23420 0 −2.59178 0.531652i 1.00000i 0 4.23420i
881.9 1.00000i 0 −1.00000 −4.23420 0 −2.59178 0.531652i 1.00000i 0 4.23420i
881.10 1.00000i 0 −1.00000 −2.01596 0 1.78450 1.95335i 1.00000i 0 2.01596i
881.11 1.00000i 0 −1.00000 −1.06893 0 0.884677 + 2.49346i 1.00000i 0 1.06893i
881.12 1.00000i 0 −1.00000 −0.929956 0 −2.07739 + 1.63843i 1.00000i 0 0.929956i
881.13 1.00000i 0 −1.00000 0.929956 0 −2.07739 1.63843i 1.00000i 0 0.929956i
881.14 1.00000i 0 −1.00000 1.06893 0 0.884677 2.49346i 1.00000i 0 1.06893i
881.15 1.00000i 0 −1.00000 2.01596 0 1.78450 + 1.95335i 1.00000i 0 2.01596i
881.16 1.00000i 0 −1.00000 4.23420 0 −2.59178 + 0.531652i 1.00000i 0 4.23420i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1386.2.g.a 16
3.b odd 2 1 inner 1386.2.g.a 16
7.b odd 2 1 inner 1386.2.g.a 16
21.c even 2 1 inner 1386.2.g.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1386.2.g.a 16 1.a even 1 1 trivial
1386.2.g.a 16 3.b odd 2 1 inner
1386.2.g.a 16 7.b odd 2 1 inner
1386.2.g.a 16 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 24 T_{5}^{6} + 118 T_{5}^{4} - 168 T_{5}^{2} + 72 \) acting on \(S_{2}^{\mathrm{new}}(1386, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{8} \)
$3$ \( T^{16} \)
$5$ \( ( 72 - 168 T^{2} + 118 T^{4} - 24 T^{6} + T^{8} )^{2} \)
$7$ \( ( 2401 + 1372 T + 294 T^{2} + 196 T^{3} + 122 T^{4} + 28 T^{5} + 6 T^{6} + 4 T^{7} + T^{8} )^{2} \)
$11$ \( ( 1 + T^{2} )^{8} \)
$13$ \( T^{16} \)
$17$ \( ( 83232 - 25872 T^{2} + 2386 T^{4} - 84 T^{6} + T^{8} )^{2} \)
$19$ \( ( 72 + 1320 T^{2} + 2710 T^{4} + 104 T^{6} + T^{8} )^{2} \)
$23$ \( ( 331776 + 69120 T^{2} + 4624 T^{4} + 120 T^{6} + T^{8} )^{2} \)
$29$ \( ( 5184 + 7776 T^{2} + 2932 T^{4} + 108 T^{6} + T^{8} )^{2} \)
$31$ \( ( 6139008 + 727008 T^{2} + 23458 T^{4} + 268 T^{6} + T^{8} )^{2} \)
$37$ \( ( 544 - 128 T - 52 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$41$ \( ( 23328 - 18576 T^{2} + 3154 T^{4} - 132 T^{6} + T^{8} )^{2} \)
$43$ \( ( -16 + 64 T + 26 T^{2} - 12 T^{3} + T^{4} )^{4} \)
$47$ \( ( 288 - 3120 T^{2} + 7474 T^{4} - 196 T^{6} + T^{8} )^{2} \)
$53$ \( ( 8714304 + 776736 T^{2} + 22516 T^{4} + 260 T^{6} + T^{8} )^{2} \)
$59$ \( ( 18432 - 10752 T^{2} + 1888 T^{4} - 96 T^{6} + T^{8} )^{2} \)
$61$ \( ( 294912 + 4706304 T^{2} + 89152 T^{4} + 528 T^{6} + T^{8} )^{2} \)
$67$ \( ( 16 + 176 T - 70 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$71$ \( ( 82944 + 232704 T^{2} + 25552 T^{4} + 312 T^{6} + T^{8} )^{2} \)
$73$ \( ( 288 + 3120 T^{2} + 7474 T^{4} + 196 T^{6} + T^{8} )^{2} \)
$79$ \( ( -292 - 840 T - 176 T^{2} + 4 T^{3} + T^{4} )^{4} \)
$83$ \( ( 1492992 - 549504 T^{2} + 33282 T^{4} - 348 T^{6} + T^{8} )^{2} \)
$89$ \( ( 1936512 - 889152 T^{2} + 40708 T^{4} - 420 T^{6} + T^{8} )^{2} \)
$97$ \( ( 18874368 + 3047424 T^{2} + 69760 T^{4} + 480 T^{6} + T^{8} )^{2} \)
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