Properties

Label 405.3.l.f
Level $405$
Weight $3$
Character orbit 405.l
Analytic conductor $11.035$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.0354507066\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{2} + ( 4 \zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} + ( -1 + 2 \zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( -3 - \zeta_{24} - \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{8} +O(q^{10})\) \( q + ( -\zeta_{24}^{2} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{2} + ( 4 \zeta_{24} + \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{4} + ( -1 + 2 \zeta_{24} - 3 \zeta_{24}^{2} - 2 \zeta_{24}^{3} + \zeta_{24}^{4} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{5} + ( -\zeta_{24}^{2} + 4 \zeta_{24}^{3} - \zeta_{24}^{4} + \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{7} + ( -3 - \zeta_{24} - \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{8} + ( 1 + 4 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} + 8 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{10} + ( 3 \zeta_{24} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{11} + ( 8 + 4 \zeta_{24} + 8 \zeta_{24}^{2} - 8 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{13} + ( -2 \zeta_{24} - 4 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{14} + ( 4 \zeta_{24} - 8 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 8 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{16} + ( 10 - 6 \zeta_{24}^{3} - 10 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{17} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} + 12 \zeta_{24}^{7} ) q^{19} + ( 6 \zeta_{24} + 17 \zeta_{24}^{2} - 14 \zeta_{24}^{3} - 9 \zeta_{24}^{4} - 12 \zeta_{24}^{5} - 17 \zeta_{24}^{6} + 7 \zeta_{24}^{7} ) q^{20} + ( -5 + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + 5 \zeta_{24}^{4} + 2 \zeta_{24}^{7} ) q^{22} + ( 14 + 4 \zeta_{24} + 14 \zeta_{24}^{2} - 14 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{23} + ( -14 \zeta_{24} - 3 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} + 28 \zeta_{24}^{5} + 3 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{25} + ( -22 - 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + 20 \zeta_{24}^{7} ) q^{26} + ( 11 - 2 \zeta_{24} - 2 \zeta_{24}^{5} + 11 \zeta_{24}^{6} ) q^{28} + ( -7 \zeta_{24} + 18 \zeta_{24}^{2} - 14 \zeta_{24}^{3} + 14 \zeta_{24}^{5} - 18 \zeta_{24}^{6} + 7 \zeta_{24}^{7} ) q^{29} + ( 4 + 12 \zeta_{24} - 6 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 6 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{31} + ( -19 + 19 \zeta_{24}^{2} + 7 \zeta_{24}^{3} + 19 \zeta_{24}^{4} + 7 \zeta_{24}^{7} ) q^{32} + ( -4 \zeta_{24} - 2 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{34} + ( 10 - 5 \zeta_{24} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 10 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{35} + ( 16 + 18 \zeta_{24}^{3} - 16 \zeta_{24}^{6} - 36 \zeta_{24}^{7} ) q^{37} + ( 18 \zeta_{24} + 24 \zeta_{24}^{2} - 24 \zeta_{24}^{4} - 36 \zeta_{24}^{5} - 24 \zeta_{24}^{6} ) q^{38} + ( -12 - 16 \zeta_{24} + 9 \zeta_{24}^{2} + 6 \zeta_{24}^{3} + 12 \zeta_{24}^{4} + 8 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{40} + ( -14 - 12 \zeta_{24} + 6 \zeta_{24}^{3} + 14 \zeta_{24}^{4} + 6 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{41} + ( 20 \zeta_{24} - 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 40 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{43} + ( 5 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 32 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{44} + ( -34 - 16 \zeta_{24} - 16 \zeta_{24}^{3} - 16 \zeta_{24}^{5} + 32 \zeta_{24}^{7} ) q^{46} + ( 32 \zeta_{24}^{2} - 20 \zeta_{24}^{3} + 32 \zeta_{24}^{4} - 32 \zeta_{24}^{6} + 10 \zeta_{24}^{7} ) q^{47} + ( -8 \zeta_{24} - 35 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{49} + ( 41 + 38 \zeta_{24} + 13 \zeta_{24}^{2} - 8 \zeta_{24}^{3} - 41 \zeta_{24}^{4} - 19 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{50} + ( 20 \zeta_{24}^{2} + 68 \zeta_{24}^{3} + 20 \zeta_{24}^{4} - 20 \zeta_{24}^{6} - 34 \zeta_{24}^{7} ) q^{52} + ( -14 - 12 \zeta_{24} - 12 \zeta_{24}^{5} - 14 \zeta_{24}^{6} ) q^{53} + ( -31 + 16 \zeta_{24} + 2 \zeta_{24}^{3} + 16 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{55} + ( 5 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{56} + ( 3 - 8 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{5} ) q^{58} + ( -62 \zeta_{24} - 36 \zeta_{24}^{2} - 31 \zeta_{24}^{3} + 31 \zeta_{24}^{5} - 31 \zeta_{24}^{7} ) q^{59} + ( 18 \zeta_{24} - 36 \zeta_{24}^{3} - 50 \zeta_{24}^{4} - 36 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{61} + ( -22 - 16 \zeta_{24}^{3} + 22 \zeta_{24}^{6} + 32 \zeta_{24}^{7} ) q^{62} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} - 79 \zeta_{24}^{6} - 20 \zeta_{24}^{7} ) q^{64} + ( 22 \zeta_{24} - 26 \zeta_{24}^{2} - 28 \zeta_{24}^{3} - 28 \zeta_{24}^{4} - 44 \zeta_{24}^{5} + 26 \zeta_{24}^{6} + 14 \zeta_{24}^{7} ) q^{65} + ( 50 - 50 \zeta_{24}^{2} - 4 \zeta_{24}^{3} - 50 \zeta_{24}^{4} - 4 \zeta_{24}^{7} ) q^{67} + ( -26 + 68 \zeta_{24} - 26 \zeta_{24}^{2} + 26 \zeta_{24}^{4} - 34 \zeta_{24}^{5} ) q^{68} + ( -10 \zeta_{24} - 5 \zeta_{24}^{2} + 15 \zeta_{24}^{4} + 20 \zeta_{24}^{5} + 5 \zeta_{24}^{6} ) q^{70} + 68 q^{71} + ( 19 - 48 \zeta_{24} - 48 \zeta_{24}^{5} + 19 \zeta_{24}^{6} ) q^{73} + ( -34 \zeta_{24} - 86 \zeta_{24}^{2} - 68 \zeta_{24}^{3} + 68 \zeta_{24}^{5} + 86 \zeta_{24}^{6} + 34 \zeta_{24}^{7} ) q^{74} + ( -78 - 36 \zeta_{24} + 18 \zeta_{24}^{3} + 78 \zeta_{24}^{4} + 18 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{76} + ( 22 - 22 \zeta_{24}^{2} + 14 \zeta_{24}^{3} - 22 \zeta_{24}^{4} + 14 \zeta_{24}^{7} ) q^{77} + ( -10 \zeta_{24} - 20 \zeta_{24}^{3} + 20 \zeta_{24}^{5} + 10 \zeta_{24}^{7} ) q^{79} + ( -41 + 21 \zeta_{24} + 2 \zeta_{24}^{3} + 21 \zeta_{24}^{5} - 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{80} + ( 32 + 26 \zeta_{24}^{3} - 32 \zeta_{24}^{6} - 52 \zeta_{24}^{7} ) q^{82} + ( 14 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 28 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{83} + ( -58 + 16 \zeta_{24} + 16 \zeta_{24}^{2} - 36 \zeta_{24}^{3} + 58 \zeta_{24}^{4} - 8 \zeta_{24}^{5} - 36 \zeta_{24}^{7} ) q^{85} + ( -56 - 36 \zeta_{24} + 18 \zeta_{24}^{3} + 56 \zeta_{24}^{4} + 18 \zeta_{24}^{5} + 18 \zeta_{24}^{7} ) q^{86} + ( -14 \zeta_{24} + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 28 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{88} + ( -36 \zeta_{24} + 36 \zeta_{24}^{3} - 36 \zeta_{24}^{5} - 6 \zeta_{24}^{6} - 72 \zeta_{24}^{7} ) q^{89} + ( -4 + 14 \zeta_{24} + 14 \zeta_{24}^{3} + 14 \zeta_{24}^{5} - 28 \zeta_{24}^{7} ) q^{91} + ( 26 \zeta_{24}^{2} + 116 \zeta_{24}^{3} + 26 \zeta_{24}^{4} - 26 \zeta_{24}^{6} - 58 \zeta_{24}^{7} ) q^{92} + ( -44 \zeta_{24} - 34 \zeta_{24}^{2} - 22 \zeta_{24}^{3} + 22 \zeta_{24}^{5} - 22 \zeta_{24}^{7} ) q^{94} + ( 36 + 48 \zeta_{24} + 48 \zeta_{24}^{2} - 18 \zeta_{24}^{3} - 36 \zeta_{24}^{4} - 24 \zeta_{24}^{5} - 18 \zeta_{24}^{7} ) q^{95} + ( 5 \zeta_{24}^{2} - 32 \zeta_{24}^{3} + 5 \zeta_{24}^{4} - 5 \zeta_{24}^{6} + 16 \zeta_{24}^{7} ) q^{97} + ( 47 + 43 \zeta_{24} + 43 \zeta_{24}^{5} + 47 \zeta_{24}^{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} - 4q^{5} - 4q^{7} - 24q^{8} + O(q^{10}) \) \( 8q - 4q^{2} - 4q^{5} - 4q^{7} - 24q^{8} + 8q^{10} + 16q^{11} + 32q^{13} + 20q^{16} + 80q^{17} - 36q^{20} - 20q^{22} + 56q^{23} - 16q^{25} - 176q^{26} + 88q^{28} + 16q^{31} - 76q^{32} + 80q^{35} + 128q^{37} - 96q^{38} - 48q^{40} - 56q^{41} + 8q^{43} - 272q^{46} + 128q^{47} + 164q^{50} + 80q^{52} - 112q^{53} - 248q^{55} + 12q^{58} - 200q^{61} - 176q^{62} - 112q^{65} + 200q^{67} - 104q^{68} + 60q^{70} + 544q^{71} + 152q^{73} - 312q^{76} + 88q^{77} - 328q^{80} + 256q^{82} - 16q^{83} - 232q^{85} - 224q^{86} - 12q^{88} - 32q^{91} + 104q^{92} + 144q^{95} + 20q^{97} + 376q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-\zeta_{24}^{4}\) \(-1 + \zeta_{24}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.965926 + 0.258819i
−0.965926 0.258819i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.258819 0.965926i
−3.03906 0.814313i 0 5.10867 + 2.94949i −2.32162 4.42833i 0 1.98004 + 0.530550i −4.22474 4.22474i 0 3.44949 + 15.3485i
28.2 0.307007 + 0.0822623i 0 −3.37662 1.94949i −3.87453 + 3.16038i 0 −4.71209 1.26260i −1.77526 1.77526i 0 −1.44949 + 0.651531i
217.1 −3.03906 + 0.814313i 0 5.10867 2.94949i −2.32162 + 4.42833i 0 1.98004 0.530550i −4.22474 + 4.22474i 0 3.44949 15.3485i
217.2 0.307007 0.0822623i 0 −3.37662 + 1.94949i −3.87453 3.16038i 0 −4.71209 + 1.26260i −1.77526 + 1.77526i 0 −1.44949 0.651531i
298.1 −0.0822623 0.307007i 0 3.37662 1.94949i −0.799701 4.93563i 0 1.26260 + 4.71209i −1.77526 1.77526i 0 −1.44949 + 0.651531i
298.2 0.814313 + 3.03906i 0 −5.10867 + 2.94949i 4.99585 + 0.203583i 0 −0.530550 1.98004i −4.22474 4.22474i 0 3.44949 + 15.3485i
352.1 −0.0822623 + 0.307007i 0 3.37662 + 1.94949i −0.799701 + 4.93563i 0 1.26260 4.71209i −1.77526 + 1.77526i 0 −1.44949 0.651531i
352.2 0.814313 3.03906i 0 −5.10867 2.94949i 4.99585 0.203583i 0 −0.530550 + 1.98004i −4.22474 + 4.22474i 0 3.44949 15.3485i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 352.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.c even 3 1 inner
45.k odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.l.f 8
3.b odd 2 1 405.3.l.h 8
5.c odd 4 1 inner 405.3.l.f 8
9.c even 3 1 45.3.g.b 4
9.c even 3 1 inner 405.3.l.f 8
9.d odd 6 1 15.3.f.a 4
9.d odd 6 1 405.3.l.h 8
15.e even 4 1 405.3.l.h 8
36.f odd 6 1 720.3.bh.k 4
36.h even 6 1 240.3.bg.a 4
45.h odd 6 1 75.3.f.c 4
45.j even 6 1 225.3.g.a 4
45.k odd 12 1 45.3.g.b 4
45.k odd 12 1 225.3.g.a 4
45.k odd 12 1 inner 405.3.l.f 8
45.l even 12 1 15.3.f.a 4
45.l even 12 1 75.3.f.c 4
45.l even 12 1 405.3.l.h 8
72.j odd 6 1 960.3.bg.i 4
72.l even 6 1 960.3.bg.h 4
180.n even 6 1 1200.3.bg.k 4
180.v odd 12 1 240.3.bg.a 4
180.v odd 12 1 1200.3.bg.k 4
180.x even 12 1 720.3.bh.k 4
360.br even 12 1 960.3.bg.i 4
360.bt odd 12 1 960.3.bg.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 9.d odd 6 1
15.3.f.a 4 45.l even 12 1
45.3.g.b 4 9.c even 3 1
45.3.g.b 4 45.k odd 12 1
75.3.f.c 4 45.h odd 6 1
75.3.f.c 4 45.l even 12 1
225.3.g.a 4 45.j even 6 1
225.3.g.a 4 45.k odd 12 1
240.3.bg.a 4 36.h even 6 1
240.3.bg.a 4 180.v odd 12 1
405.3.l.f 8 1.a even 1 1 trivial
405.3.l.f 8 5.c odd 4 1 inner
405.3.l.f 8 9.c even 3 1 inner
405.3.l.f 8 45.k odd 12 1 inner
405.3.l.h 8 3.b odd 2 1
405.3.l.h 8 9.d odd 6 1
405.3.l.h 8 15.e even 4 1
405.3.l.h 8 45.l even 12 1
720.3.bh.k 4 36.f odd 6 1
720.3.bh.k 4 180.x even 12 1
960.3.bg.h 4 72.l even 6 1
960.3.bg.h 4 360.bt odd 12 1
960.3.bg.i 4 72.j odd 6 1
960.3.bg.i 4 360.br even 12 1
1200.3.bg.k 4 180.n even 6 1
1200.3.bg.k 4 180.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} + \cdots\) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T + 8 T^{2} - 40 T^{3} + 79 T^{4} + 40 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$3$ \( T^{8} \)
$5$ \( 390625 + 62500 T + 10000 T^{2} - 5000 T^{3} - 1025 T^{4} - 200 T^{5} + 16 T^{6} + 4 T^{7} + T^{8} \)
$7$ \( 10000 - 4000 T + 800 T^{2} - 1120 T^{3} + 124 T^{4} + 112 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} \)
$11$ \( ( 1444 + 304 T + 102 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$13$ \( 181063936 - 49948672 T + 6889472 T^{2} - 1039360 T^{3} + 129904 T^{4} - 8960 T^{5} + 512 T^{6} - 32 T^{7} + T^{8} \)
$17$ \( ( 8464 - 3680 T + 800 T^{2} - 40 T^{3} + T^{4} )^{2} \)
$19$ \( ( 32400 + 504 T^{2} + T^{4} )^{2} \)
$23$ \( 20851360000 - 3072832000 T + 226419200 T^{2} - 17194240 T^{3} + 1122544 T^{4} - 45248 T^{5} + 1568 T^{6} - 56 T^{7} + T^{8} \)
$29$ \( 810000 - 1112400 T^{2} + 1526796 T^{4} - 1236 T^{6} + T^{8} \)
$31$ \( ( 40000 + 1600 T + 264 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$37$ \( ( 211600 + 29440 T + 2048 T^{2} - 64 T^{3} + T^{4} )^{2} \)
$41$ \( ( 400 - 560 T + 804 T^{2} + 28 T^{3} + T^{4} )^{2} \)
$43$ \( 2018854506496 + 13549359104 T + 45467648 T^{2} + 23038976 T^{3} - 1343552 T^{4} - 19328 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} \)
$47$ \( 9336104694016 - 683650686976 T + 25030688768 T^{2} - 1050701824 T^{3} + 35414128 T^{4} - 601088 T^{5} + 8192 T^{6} - 128 T^{7} + T^{8} \)
$53$ \( ( 1600 - 2240 T + 1568 T^{2} + 56 T^{3} + T^{4} )^{2} \)
$59$ \( 399236364810000 - 282210231600 T^{2} + 179506476 T^{4} - 14124 T^{6} + T^{8} \)
$61$ \( ( 309136 + 55600 T + 9444 T^{2} + 100 T^{3} + T^{4} )^{2} \)
$67$ \( 601343393468416 - 24286889881600 T + 490446080000 T^{2} - 9999078400 T^{3} + 177397696 T^{4} - 2019200 T^{5} + 20000 T^{6} - 200 T^{7} + T^{8} \)
$71$ \( ( -68 + T )^{8} \)
$73$ \( ( 38316100 + 470440 T + 2888 T^{2} - 76 T^{3} + T^{4} )^{2} \)
$79$ \( ( 360000 - 600 T^{2} + T^{4} )^{2} \)
$83$ \( 95565066496 - 2750073856 T + 39569408 T^{2} - 11031040 T^{3} - 150416 T^{4} + 19840 T^{5} + 128 T^{6} + 16 T^{7} + T^{8} \)
$89$ \( ( 59907600 + 15624 T^{2} + T^{4} )^{2} \)
$97$ \( 265764994576 + 7402924640 T + 103104800 T^{2} + 23492960 T^{3} - 188324 T^{4} - 32720 T^{5} + 200 T^{6} - 20 T^{7} + T^{8} \)
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