Properties

Label 605.2.m.b
Level $605$
Weight $2$
Character orbit 605.m
Analytic conductor $4.831$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $16$

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Newspace parameters

Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.m (of order \(20\), degree \(8\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\Q(\zeta_{40})\)
Defining polynomial: \(x^{16} - x^{12} + x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{4} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{40}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{40}^{3} + \zeta_{40}^{7} - \zeta_{40}^{11} - \zeta_{40}^{15} ) q^{2} + ( \zeta_{40}^{2} + \zeta_{40}^{12} ) q^{3} + ( 3 \zeta_{40}^{2} - 3 \zeta_{40}^{6} + 3 \zeta_{40}^{10} - 3 \zeta_{40}^{14} ) q^{4} + ( 2 - 2 \zeta_{40}^{4} - \zeta_{40}^{6} + 2 \zeta_{40}^{8} - 2 \zeta_{40}^{12} ) q^{5} + ( \zeta_{40} + 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} - \zeta_{40}^{11} - 2 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{6} + ( \zeta_{40} + \zeta_{40}^{5} - \zeta_{40}^{9} - \zeta_{40}^{13} ) q^{8} -\zeta_{40}^{14} q^{9} +O(q^{10})\) \( q + ( \zeta_{40}^{3} + \zeta_{40}^{7} - \zeta_{40}^{11} - \zeta_{40}^{15} ) q^{2} + ( \zeta_{40}^{2} + \zeta_{40}^{12} ) q^{3} + ( 3 \zeta_{40}^{2} - 3 \zeta_{40}^{6} + 3 \zeta_{40}^{10} - 3 \zeta_{40}^{14} ) q^{4} + ( 2 - 2 \zeta_{40}^{4} - \zeta_{40}^{6} + 2 \zeta_{40}^{8} - 2 \zeta_{40}^{12} ) q^{5} + ( \zeta_{40} + 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} - \zeta_{40}^{11} - 2 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{6} + ( \zeta_{40} + \zeta_{40}^{5} - \zeta_{40}^{9} - \zeta_{40}^{13} ) q^{8} -\zeta_{40}^{14} q^{9} + ( -2 \zeta_{40} + 4 \zeta_{40}^{3} + \zeta_{40}^{5} - 4 \zeta_{40}^{7} - 2 \zeta_{40}^{9} - 2 \zeta_{40}^{15} ) q^{10} + ( 3 + 3 \zeta_{40}^{10} ) q^{12} + ( -4 \zeta_{40}^{5} + 2 \zeta_{40}^{9} - 4 \zeta_{40}^{13} ) q^{13} + ( 3 \zeta_{40}^{2} - 3 \zeta_{40}^{6} + \zeta_{40}^{8} + 3 \zeta_{40}^{10} - 3 \zeta_{40}^{14} ) q^{15} + ( -1 + \zeta_{40}^{4} - \zeta_{40}^{8} + \zeta_{40}^{12} ) q^{16} + ( 4 \zeta_{40}^{7} - 2 \zeta_{40}^{11} + 4 \zeta_{40}^{15} ) q^{17} + ( 2 \zeta_{40} - 2 \zeta_{40}^{5} - \zeta_{40}^{13} ) q^{18} + ( -2 \zeta_{40} + 4 \zeta_{40}^{3} - 2 \zeta_{40}^{5} - 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} + 4 \zeta_{40}^{11} + 2 \zeta_{40}^{13} ) q^{19} + ( -3 \zeta_{40}^{4} - 6 \zeta_{40}^{14} ) q^{20} + ( -1 + \zeta_{40}^{10} ) q^{23} + ( \zeta_{40}^{3} + 2 \zeta_{40}^{5} + \zeta_{40}^{7} - \zeta_{40}^{9} - \zeta_{40}^{11} + 2 \zeta_{40}^{13} - \zeta_{40}^{15} ) q^{24} + ( -4 \zeta_{40}^{2} - 3 \zeta_{40}^{12} ) q^{25} -10 \zeta_{40}^{8} q^{26} + ( 4 - 4 \zeta_{40}^{4} + 4 \zeta_{40}^{6} + 4 \zeta_{40}^{8} - 4 \zeta_{40}^{12} ) q^{27} + ( -4 \zeta_{40} + 2 \zeta_{40}^{3} + 4 \zeta_{40}^{5} + 4 \zeta_{40}^{11} + 2 \zeta_{40}^{13} - 4 \zeta_{40}^{15} ) q^{29} + ( 3 \zeta_{40} + 2 \zeta_{40}^{3} + 3 \zeta_{40}^{5} - \zeta_{40}^{7} - 3 \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 3 \zeta_{40}^{13} ) q^{30} -2 \zeta_{40}^{4} q^{31} + ( -6 \zeta_{40}^{3} + 6 \zeta_{40}^{7} + 3 \zeta_{40}^{15} ) q^{32} + 10 \zeta_{40}^{10} q^{34} -3 \zeta_{40}^{12} q^{36} + ( -3 \zeta_{40}^{2} + 3 \zeta_{40}^{6} + 3 \zeta_{40}^{8} - 3 \zeta_{40}^{10} + 3 \zeta_{40}^{14} ) q^{37} + ( -10 + 10 \zeta_{40}^{4} + 10 \zeta_{40}^{6} - 10 \zeta_{40}^{8} + 10 \zeta_{40}^{12} ) q^{38} + ( 2 \zeta_{40} - 4 \zeta_{40}^{7} + 4 \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 4 \zeta_{40}^{13} - 4 \zeta_{40}^{15} ) q^{39} + ( 4 \zeta_{40} - \zeta_{40}^{3} - 4 \zeta_{40}^{5} - 2 \zeta_{40}^{11} - 2 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{40} + ( -2 \zeta_{40} - 4 \zeta_{40}^{3} - 2 \zeta_{40}^{5} + 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} - 4 \zeta_{40}^{11} + 2 \zeta_{40}^{13} ) q^{41} + ( -1 - 2 \zeta_{40}^{10} ) q^{45} + ( -\zeta_{40}^{3} + 2 \zeta_{40}^{5} - \zeta_{40}^{7} - \zeta_{40}^{9} + \zeta_{40}^{11} + 2 \zeta_{40}^{13} + \zeta_{40}^{15} ) q^{46} + ( 3 \zeta_{40}^{2} - 3 \zeta_{40}^{12} ) q^{47} + ( -\zeta_{40}^{2} + \zeta_{40}^{6} - \zeta_{40}^{8} - \zeta_{40}^{10} + \zeta_{40}^{14} ) q^{48} -7 \zeta_{40}^{6} q^{49} + ( -4 \zeta_{40} - 6 \zeta_{40}^{7} - 8 \zeta_{40}^{9} + 3 \zeta_{40}^{11} + 8 \zeta_{40}^{13} - 6 \zeta_{40}^{15} ) q^{50} + ( -4 \zeta_{40} - 2 \zeta_{40}^{3} + 4 \zeta_{40}^{5} - 4 \zeta_{40}^{11} + 2 \zeta_{40}^{13} + 4 \zeta_{40}^{15} ) q^{51} + ( -12 \zeta_{40}^{3} + 6 \zeta_{40}^{7} - 12 \zeta_{40}^{11} ) q^{52} + ( \zeta_{40}^{4} - \zeta_{40}^{14} ) q^{53} + ( 8 \zeta_{40} + 8 \zeta_{40}^{3} - 4 \zeta_{40}^{5} - 8 \zeta_{40}^{7} + 8 \zeta_{40}^{9} - 4 \zeta_{40}^{15} ) q^{54} + ( -4 \zeta_{40}^{3} - 4 \zeta_{40}^{7} + 4 \zeta_{40}^{11} + 4 \zeta_{40}^{15} ) q^{57} + ( 10 \zeta_{40}^{2} + 10 \zeta_{40}^{12} ) q^{58} + ( 6 \zeta_{40}^{2} - 6 \zeta_{40}^{6} + 6 \zeta_{40}^{10} - 6 \zeta_{40}^{14} ) q^{59} + ( 9 - 9 \zeta_{40}^{4} + 3 \zeta_{40}^{6} + 9 \zeta_{40}^{8} - 9 \zeta_{40}^{12} ) q^{60} + ( -2 \zeta_{40} - 4 \zeta_{40}^{7} - 4 \zeta_{40}^{9} + 2 \zeta_{40}^{11} + 4 \zeta_{40}^{13} - 4 \zeta_{40}^{15} ) q^{61} + ( -2 \zeta_{40}^{3} - 4 \zeta_{40}^{11} + 4 \zeta_{40}^{15} ) q^{62} + 13 \zeta_{40}^{14} q^{64} + ( -8 \zeta_{40} - 4 \zeta_{40}^{3} + 4 \zeta_{40}^{5} + 4 \zeta_{40}^{7} - 8 \zeta_{40}^{9} + 2 \zeta_{40}^{15} ) q^{65} + ( 3 + 3 \zeta_{40}^{10} ) q^{67} + ( 12 \zeta_{40}^{5} - 6 \zeta_{40}^{9} + 12 \zeta_{40}^{13} ) q^{68} -2 \zeta_{40}^{2} q^{69} + ( 8 - 8 \zeta_{40}^{4} + 8 \zeta_{40}^{8} - 8 \zeta_{40}^{12} ) q^{71} + ( -2 \zeta_{40}^{7} + \zeta_{40}^{11} - 2 \zeta_{40}^{15} ) q^{72} + ( 4 \zeta_{40} - 4 \zeta_{40}^{5} - 2 \zeta_{40}^{13} ) q^{73} + ( -3 \zeta_{40} + 6 \zeta_{40}^{3} - 3 \zeta_{40}^{5} - 3 \zeta_{40}^{7} + 3 \zeta_{40}^{9} + 6 \zeta_{40}^{11} + 3 \zeta_{40}^{13} ) q^{74} + ( -\zeta_{40}^{4} - 7 \zeta_{40}^{14} ) q^{75} + ( 12 \zeta_{40} - 12 \zeta_{40}^{3} - 6 \zeta_{40}^{5} + 12 \zeta_{40}^{7} + 12 \zeta_{40}^{9} + 6 \zeta_{40}^{15} ) q^{76} + ( 10 - 10 \zeta_{40}^{10} ) q^{78} + ( -2 \zeta_{40}^{3} - 4 \zeta_{40}^{5} - 2 \zeta_{40}^{7} + 2 \zeta_{40}^{9} + 2 \zeta_{40}^{11} - 4 \zeta_{40}^{13} + 2 \zeta_{40}^{15} ) q^{79} + ( \zeta_{40}^{2} + 2 \zeta_{40}^{12} ) q^{80} + 5 \zeta_{40}^{8} q^{81} + ( -10 + 10 \zeta_{40}^{4} - 10 \zeta_{40}^{6} - 10 \zeta_{40}^{8} + 10 \zeta_{40}^{12} ) q^{82} + ( -4 \zeta_{40} - 8 \zeta_{40}^{9} + 8 \zeta_{40}^{13} ) q^{83} + ( 2 \zeta_{40} + 8 \zeta_{40}^{3} + 2 \zeta_{40}^{5} - 4 \zeta_{40}^{7} - 2 \zeta_{40}^{9} + 8 \zeta_{40}^{11} - 2 \zeta_{40}^{13} ) q^{85} + ( -8 \zeta_{40}^{3} + 8 \zeta_{40}^{7} + 4 \zeta_{40}^{15} ) q^{87} -6 \zeta_{40}^{10} q^{89} + ( -\zeta_{40}^{3} - 4 \zeta_{40}^{5} - \zeta_{40}^{7} + 2 \zeta_{40}^{9} + \zeta_{40}^{11} - 4 \zeta_{40}^{13} + \zeta_{40}^{15} ) q^{90} + ( -3 \zeta_{40}^{2} + 3 \zeta_{40}^{6} + 3 \zeta_{40}^{8} - 3 \zeta_{40}^{10} + 3 \zeta_{40}^{14} ) q^{92} + ( 2 - 2 \zeta_{40}^{4} - 2 \zeta_{40}^{6} + 2 \zeta_{40}^{8} - 2 \zeta_{40}^{12} ) q^{93} + ( 3 \zeta_{40} - 6 \zeta_{40}^{7} + 6 \zeta_{40}^{9} + 3 \zeta_{40}^{11} - 6 \zeta_{40}^{13} - 6 \zeta_{40}^{15} ) q^{94} + ( -4 \zeta_{40} + 6 \zeta_{40}^{3} + 4 \zeta_{40}^{5} + 12 \zeta_{40}^{11} + 2 \zeta_{40}^{13} - 12 \zeta_{40}^{15} ) q^{95} + ( -3 \zeta_{40} - 6 \zeta_{40}^{3} - 3 \zeta_{40}^{5} + 3 \zeta_{40}^{7} + 3 \zeta_{40}^{9} - 6 \zeta_{40}^{11} + 3 \zeta_{40}^{13} ) q^{96} + ( 7 \zeta_{40}^{4} + 7 \zeta_{40}^{14} ) q^{97} + ( -14 \zeta_{40} + 7 \zeta_{40}^{5} - 14 \zeta_{40}^{9} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 4q^{3} + 8q^{5} + O(q^{10}) \) \( 16q + 4q^{3} + 8q^{5} + 48q^{12} - 4q^{15} - 4q^{16} - 12q^{20} - 16q^{23} - 12q^{25} + 40q^{26} + 16q^{27} - 8q^{31} - 12q^{36} - 12q^{37} - 40q^{38} - 16q^{45} - 12q^{47} + 4q^{48} + 4q^{53} + 40q^{58} + 36q^{60} + 48q^{67} + 32q^{71} - 4q^{75} + 160q^{78} + 8q^{80} - 20q^{81} - 40q^{82} - 12q^{92} + 8q^{93} + 28q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(\zeta_{40}^{5}\) \(-\zeta_{40}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
112.1
0.453990 0.891007i
−0.453990 + 0.891007i
−0.987688 + 0.156434i
0.987688 0.156434i
0.891007 + 0.453990i
−0.891007 0.453990i
−0.987688 0.156434i
0.987688 + 0.156434i
−0.156434 + 0.987688i
0.156434 0.987688i
0.891007 0.453990i
−0.891007 + 0.453990i
0.453990 + 0.891007i
−0.453990 0.891007i
−0.156434 0.987688i
0.156434 + 0.987688i
−1.01515 1.99235i 0.221232 1.39680i −1.76336 + 2.42705i −1.56909 1.59310i −3.00750 + 0.977198i 0 2.20854 + 0.349798i 0.951057 + 0.309017i −1.58114 + 4.74342i
112.2 1.01515 + 1.99235i 0.221232 1.39680i −1.76336 + 2.42705i −1.56909 1.59310i 3.00750 0.977198i 0 −2.20854 0.349798i 0.951057 + 0.309017i 1.58114 4.74342i
118.1 −2.20854 0.349798i 0.642040 1.26007i 2.85317 + 0.927051i 1.03025 + 1.98459i −1.85874 + 2.55834i 0 −1.99235 1.01515i 0.587785 + 0.809017i −1.58114 4.74342i
118.2 2.20854 + 0.349798i 0.642040 1.26007i 2.85317 + 0.927051i 1.03025 + 1.98459i 1.85874 2.55834i 0 1.99235 + 1.01515i 0.587785 + 0.809017i 1.58114 + 4.74342i
233.1 −1.99235 + 1.01515i 1.39680 + 0.221232i 1.76336 2.42705i 0.333023 2.21113i −3.00750 + 0.977198i 0 −0.349798 + 2.20854i −0.951057 0.309017i 1.58114 + 4.74342i
233.2 1.99235 1.01515i 1.39680 + 0.221232i 1.76336 2.42705i 0.333023 2.21113i 3.00750 0.977198i 0 0.349798 2.20854i −0.951057 0.309017i −1.58114 4.74342i
282.1 −2.20854 + 0.349798i 0.642040 + 1.26007i 2.85317 0.927051i 1.03025 1.98459i −1.85874 2.55834i 0 −1.99235 + 1.01515i 0.587785 0.809017i −1.58114 + 4.74342i
282.2 2.20854 0.349798i 0.642040 + 1.26007i 2.85317 0.927051i 1.03025 1.98459i 1.85874 + 2.55834i 0 1.99235 1.01515i 0.587785 0.809017i 1.58114 4.74342i
403.1 −0.349798 2.20854i −1.26007 + 0.642040i −2.85317 + 0.927051i 2.20582 0.366554i 1.85874 + 2.55834i 0 1.01515 + 1.99235i −0.587785 + 0.809017i −1.58114 4.74342i
403.2 0.349798 + 2.20854i −1.26007 + 0.642040i −2.85317 + 0.927051i 2.20582 0.366554i −1.85874 2.55834i 0 −1.01515 1.99235i −0.587785 + 0.809017i 1.58114 + 4.74342i
457.1 −1.99235 1.01515i 1.39680 0.221232i 1.76336 + 2.42705i 0.333023 + 2.21113i −3.00750 0.977198i 0 −0.349798 2.20854i −0.951057 + 0.309017i 1.58114 4.74342i
457.2 1.99235 + 1.01515i 1.39680 0.221232i 1.76336 + 2.42705i 0.333023 + 2.21113i 3.00750 + 0.977198i 0 0.349798 + 2.20854i −0.951057 + 0.309017i −1.58114 + 4.74342i
578.1 −1.01515 + 1.99235i 0.221232 + 1.39680i −1.76336 2.42705i −1.56909 + 1.59310i −3.00750 0.977198i 0 2.20854 0.349798i 0.951057 0.309017i −1.58114 4.74342i
578.2 1.01515 1.99235i 0.221232 + 1.39680i −1.76336 2.42705i −1.56909 + 1.59310i 3.00750 + 0.977198i 0 −2.20854 + 0.349798i 0.951057 0.309017i 1.58114 + 4.74342i
602.1 −0.349798 + 2.20854i −1.26007 0.642040i −2.85317 0.927051i 2.20582 + 0.366554i 1.85874 2.55834i 0 1.01515 1.99235i −0.587785 0.809017i −1.58114 + 4.74342i
602.2 0.349798 2.20854i −1.26007 0.642040i −2.85317 0.927051i 2.20582 + 0.366554i −1.85874 + 2.55834i 0 −1.01515 + 1.99235i −0.587785 0.809017i 1.58114 4.74342i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 602.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.e even 4 1 inner
55.k odd 20 3 inner
55.l even 20 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.m.b 16
5.c odd 4 1 inner 605.2.m.b 16
11.b odd 2 1 inner 605.2.m.b 16
11.c even 5 1 55.2.e.a 4
11.c even 5 3 inner 605.2.m.b 16
11.d odd 10 1 55.2.e.a 4
11.d odd 10 3 inner 605.2.m.b 16
33.f even 10 1 495.2.k.b 4
33.h odd 10 1 495.2.k.b 4
44.g even 10 1 880.2.bd.e 4
44.h odd 10 1 880.2.bd.e 4
55.e even 4 1 inner 605.2.m.b 16
55.h odd 10 1 275.2.e.b 4
55.j even 10 1 275.2.e.b 4
55.k odd 20 1 55.2.e.a 4
55.k odd 20 1 275.2.e.b 4
55.k odd 20 3 inner 605.2.m.b 16
55.l even 20 1 55.2.e.a 4
55.l even 20 1 275.2.e.b 4
55.l even 20 3 inner 605.2.m.b 16
165.u odd 20 1 495.2.k.b 4
165.v even 20 1 495.2.k.b 4
220.v even 20 1 880.2.bd.e 4
220.w odd 20 1 880.2.bd.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.e.a 4 11.c even 5 1
55.2.e.a 4 11.d odd 10 1
55.2.e.a 4 55.k odd 20 1
55.2.e.a 4 55.l even 20 1
275.2.e.b 4 55.h odd 10 1
275.2.e.b 4 55.j even 10 1
275.2.e.b 4 55.k odd 20 1
275.2.e.b 4 55.l even 20 1
495.2.k.b 4 33.f even 10 1
495.2.k.b 4 33.h odd 10 1
495.2.k.b 4 165.u odd 20 1
495.2.k.b 4 165.v even 20 1
605.2.m.b 16 1.a even 1 1 trivial
605.2.m.b 16 5.c odd 4 1 inner
605.2.m.b 16 11.b odd 2 1 inner
605.2.m.b 16 11.c even 5 3 inner
605.2.m.b 16 11.d odd 10 3 inner
605.2.m.b 16 55.e even 4 1 inner
605.2.m.b 16 55.k odd 20 3 inner
605.2.m.b 16 55.l even 20 3 inner
880.2.bd.e 4 44.g even 10 1
880.2.bd.e 4 44.h odd 10 1
880.2.bd.e 4 220.v even 20 1
880.2.bd.e 4 220.w odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 25 T_{2}^{12} + 625 T_{2}^{8} - 15625 T_{2}^{4} + 390625 \) acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 390625 - 15625 T^{4} + 625 T^{8} - 25 T^{12} + T^{16} \)
$3$ \( ( 16 - 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$5$ \( ( 625 - 500 T + 275 T^{2} - 120 T^{3} + 41 T^{4} - 24 T^{5} + 11 T^{6} - 4 T^{7} + T^{8} )^{2} \)
$7$ \( T^{16} \)
$11$ \( T^{16} \)
$13$ \( 25600000000 - 64000000 T^{4} + 160000 T^{8} - 400 T^{12} + T^{16} \)
$17$ \( 25600000000 - 64000000 T^{4} + 160000 T^{8} - 400 T^{12} + T^{16} \)
$19$ \( ( 2560000 + 64000 T^{2} + 1600 T^{4} + 40 T^{6} + T^{8} )^{2} \)
$23$ \( ( 2 + 2 T + T^{2} )^{8} \)
$29$ \( ( 2560000 + 64000 T^{2} + 1600 T^{4} + 40 T^{6} + T^{8} )^{2} \)
$31$ \( ( 16 + 8 T + 4 T^{2} + 2 T^{3} + T^{4} )^{4} \)
$37$ \( ( 104976 + 34992 T + 5832 T^{2} - 324 T^{4} + 18 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$41$ \( ( 2560000 - 64000 T^{2} + 1600 T^{4} - 40 T^{6} + T^{8} )^{2} \)
$43$ \( T^{16} \)
$47$ \( ( 104976 + 34992 T + 5832 T^{2} - 324 T^{4} + 18 T^{6} + 6 T^{7} + T^{8} )^{2} \)
$53$ \( ( 16 - 16 T + 8 T^{2} - 4 T^{4} + 2 T^{6} - 2 T^{7} + T^{8} )^{2} \)
$59$ \( ( 1679616 - 46656 T^{2} + 1296 T^{4} - 36 T^{6} + T^{8} )^{2} \)
$61$ \( ( 2560000 - 64000 T^{2} + 1600 T^{4} - 40 T^{6} + T^{8} )^{2} \)
$67$ \( ( 18 - 6 T + T^{2} )^{8} \)
$71$ \( ( 4096 - 512 T + 64 T^{2} - 8 T^{3} + T^{4} )^{4} \)
$73$ \( 25600000000 - 64000000 T^{4} + 160000 T^{8} - 400 T^{12} + T^{16} \)
$79$ \( ( 2560000 + 64000 T^{2} + 1600 T^{4} + 40 T^{6} + T^{8} )^{2} \)
$83$ \( 1677721600000000 - 262144000000 T^{4} + 40960000 T^{8} - 6400 T^{12} + T^{16} \)
$89$ \( ( 36 + T^{2} )^{8} \)
$97$ \( ( 92236816 - 13176688 T + 941192 T^{2} - 9604 T^{4} + 98 T^{6} - 14 T^{7} + T^{8} )^{2} \)
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