Properties

Label 45.2.j.a
Level $45$
Weight $2$
Character orbit 45.j
Analytic conductor $0.359$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 45.j (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.359326809096\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( \zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{3} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{4} + ( \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{5} + ( -1 + \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{6} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{8} -3 \zeta_{24}^{2} q^{9} +O(q^{10})\) \( q + ( \zeta_{24} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24} + 2 \zeta_{24}^{5} ) q^{3} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{4} + ( \zeta_{24} + \zeta_{24}^{2} + \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{5} + ( -1 + \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{6} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( -\zeta_{24}^{5} - \zeta_{24}^{7} ) q^{8} -3 \zeta_{24}^{2} q^{9} + ( -1 - 2 \zeta_{24} + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} - \zeta_{24}^{7} ) q^{10} + ( -3 + \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{11} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{12} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{13} + ( -\zeta_{24}^{2} - \zeta_{24}^{6} ) q^{14} + ( 1 - \zeta_{24}^{2} - 3 \zeta_{24}^{3} + \zeta_{24}^{4} + 2 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{15} + ( 1 - 2 \zeta_{24}^{2} - \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{16} + ( -\zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{17} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{18} + ( -1 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{19} + ( 3 - 2 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{20} + ( 3 + 3 \zeta_{24}^{2} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{21} + ( \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{22} + ( -\zeta_{24}^{3} + \zeta_{24}^{5} ) q^{23} + ( 1 + 2 \zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{24} + ( -1 + 4 \zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{25} + ( 3 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{26} + ( 3 \zeta_{24}^{3} - 6 \zeta_{24}^{7} ) q^{27} + ( 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{28} + ( 3 - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{29} + ( -3 + \zeta_{24} + 3 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 3 \zeta_{24}^{7} ) q^{30} + ( -\zeta_{24}^{2} + \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{31} + ( -4 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{32} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{33} + ( -1 - \zeta_{24}^{2} + \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{34} + ( -3 - 2 \zeta_{24}^{2} - 3 \zeta_{24}^{5} + \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{35} + ( -3 + 6 \zeta_{24}^{4} ) q^{36} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{37} + ( -2 \zeta_{24} + \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{38} + ( -3 + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} ) q^{39} + ( 2 \zeta_{24} - \zeta_{24}^{2} + \zeta_{24}^{3} - \zeta_{24}^{4} - \zeta_{24}^{5} - \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{40} + ( -\zeta_{24}^{2} - 6 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{41} + ( 3 \zeta_{24}^{3} - 3 \zeta_{24}^{7} ) q^{42} + ( -4 \zeta_{24} + \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{43} + ( -3 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{44} + ( 3 - 3 \zeta_{24} - 3 \zeta_{24}^{3} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{5} ) q^{45} + ( -2 + 2 \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{46} + ( 3 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{47} + ( \zeta_{24} - 6 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{48} + ( 3 \zeta_{24}^{2} - \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{49} + ( -2 \zeta_{24}^{2} - \zeta_{24}^{3} + 6 \zeta_{24}^{4} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{50} + ( -4 - 5 \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{51} + ( -3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{52} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{53} + ( 3 + 3 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{54} + ( 3 - 4 \zeta_{24} - 6 \zeta_{24}^{2} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{55} + ( -3 - 2 \zeta_{24}^{2} + 3 \zeta_{24}^{4} + 4 \zeta_{24}^{6} ) q^{56} + ( \zeta_{24} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{57} + ( -2 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{58} + ( 2 \zeta_{24}^{2} + 6 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{59} + ( 3 + 3 \zeta_{24} + 3 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{60} + ( -2 + 5 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 10 \zeta_{24}^{6} ) q^{61} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{62} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{63} + ( 4 - 2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{64} + ( 2 \zeta_{24}^{2} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 3 \zeta_{24}^{7} ) q^{65} + ( 9 - 9 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{66} + ( -5 \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{67} + ( 5 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{68} + ( 2 - 2 \zeta_{24}^{2} - \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{69} + ( 3 - 2 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{4} + \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{70} + ( 9 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{6} ) q^{71} + ( -3 \zeta_{24} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{72} + ( 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} ) q^{73} + ( -3 + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{74} + ( -6 - \zeta_{24} - \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{75} + ( \zeta_{24}^{2} - 3 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{76} + ( 3 \zeta_{24} - 3 \zeta_{24}^{7} ) q^{77} + ( -3 \zeta_{24} + 6 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{78} + ( 4 + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{6} ) q^{79} + ( -6 + 3 \zeta_{24} + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{80} + 9 \zeta_{24}^{4} q^{81} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{82} + ( \zeta_{24} - 6 \zeta_{24}^{3} - 7 \zeta_{24}^{5} + 7 \zeta_{24}^{7} ) q^{83} + ( -3 - 6 \zeta_{24}^{2} - 3 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{84} + ( -5 \zeta_{24} + 2 \zeta_{24}^{2} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} + 2 \zeta_{24}^{6} + 5 \zeta_{24}^{7} ) q^{85} + ( 5 \zeta_{24}^{2} - 9 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{86} + ( 3 \zeta_{24} - 6 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{87} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{88} + ( -3 - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{89} + ( -3 - 3 \zeta_{24} + 3 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 3 \zeta_{24}^{4} - 3 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{90} + ( -3 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{91} + ( -\zeta_{24} + \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{92} + ( -2 \zeta_{24} + 3 \zeta_{24}^{3} + \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{93} + ( -2 \zeta_{24}^{2} + 5 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{94} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{3} + 3 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{95} + ( -6 + 6 \zeta_{24}^{2} - 3 \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{96} + ( 2 \zeta_{24} + 10 \zeta_{24}^{3} + 8 \zeta_{24}^{5} - 8 \zeta_{24}^{7} ) q^{97} + ( -7 \zeta_{24} + 7 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{98} + ( -6 + 9 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 9 \zeta_{24}^{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 12 q^{6} + O(q^{10}) \) \( 8 q - 12 q^{6} - 8 q^{10} - 12 q^{11} + 12 q^{15} + 4 q^{16} - 8 q^{19} + 12 q^{20} + 12 q^{21} + 12 q^{24} - 4 q^{25} + 24 q^{26} + 12 q^{29} - 12 q^{30} + 4 q^{31} - 4 q^{34} - 24 q^{35} - 12 q^{39} - 4 q^{40} - 24 q^{41} - 24 q^{44} - 16 q^{46} - 4 q^{49} + 24 q^{50} - 36 q^{51} + 24 q^{55} - 12 q^{56} + 24 q^{59} + 24 q^{60} - 8 q^{61} + 32 q^{64} + 60 q^{66} + 12 q^{69} + 12 q^{70} + 72 q^{71} - 12 q^{74} - 48 q^{75} - 12 q^{76} + 16 q^{79} - 48 q^{80} + 36 q^{81} - 36 q^{84} + 16 q^{85} - 36 q^{86} - 24 q^{89} - 36 q^{90} - 24 q^{91} + 20 q^{94} + 12 q^{95} - 60 q^{96} - 36 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/45\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-\zeta_{24}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−1.67303 + 0.965926i 1.67303 0.448288i 0.866025 1.50000i 0.358719 + 2.20711i −2.36603 + 2.36603i −0.776457 + 0.448288i 0.517638i 2.59808 1.50000i −2.73205 3.34607i
4.2 −0.448288 + 0.258819i 0.448288 + 1.67303i −0.866025 + 1.50000i −0.358719 2.20711i −0.633975 0.633975i 2.89778 1.67303i 1.93185i −2.59808 + 1.50000i 0.732051 + 0.896575i
4.3 0.448288 0.258819i −0.448288 1.67303i −0.866025 + 1.50000i 2.09077 0.792893i −0.633975 0.633975i −2.89778 + 1.67303i 1.93185i −2.59808 + 1.50000i 0.732051 0.896575i
4.4 1.67303 0.965926i −1.67303 + 0.448288i 0.866025 1.50000i −2.09077 + 0.792893i −2.36603 + 2.36603i 0.776457 0.448288i 0.517638i 2.59808 1.50000i −2.73205 + 3.34607i
34.1 −1.67303 0.965926i 1.67303 + 0.448288i 0.866025 + 1.50000i 0.358719 2.20711i −2.36603 2.36603i −0.776457 0.448288i 0.517638i 2.59808 + 1.50000i −2.73205 + 3.34607i
34.2 −0.448288 0.258819i 0.448288 1.67303i −0.866025 1.50000i −0.358719 + 2.20711i −0.633975 + 0.633975i 2.89778 + 1.67303i 1.93185i −2.59808 1.50000i 0.732051 0.896575i
34.3 0.448288 + 0.258819i −0.448288 + 1.67303i −0.866025 1.50000i 2.09077 + 0.792893i −0.633975 + 0.633975i −2.89778 1.67303i 1.93185i −2.59808 1.50000i 0.732051 + 0.896575i
34.4 1.67303 + 0.965926i −1.67303 0.448288i 0.866025 + 1.50000i −2.09077 0.792893i −2.36603 2.36603i 0.776457 + 0.448288i 0.517638i 2.59808 + 1.50000i −2.73205 3.34607i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.2.j.a 8
3.b odd 2 1 135.2.j.a 8
4.b odd 2 1 720.2.by.d 8
5.b even 2 1 inner 45.2.j.a 8
5.c odd 4 2 225.2.e.d 8
9.c even 3 1 inner 45.2.j.a 8
9.c even 3 1 405.2.b.d 4
9.d odd 6 1 135.2.j.a 8
9.d odd 6 1 405.2.b.c 4
12.b even 2 1 2160.2.by.c 8
15.d odd 2 1 135.2.j.a 8
15.e even 4 2 675.2.e.d 8
20.d odd 2 1 720.2.by.d 8
36.f odd 6 1 720.2.by.d 8
36.h even 6 1 2160.2.by.c 8
45.h odd 6 1 135.2.j.a 8
45.h odd 6 1 405.2.b.c 4
45.j even 6 1 inner 45.2.j.a 8
45.j even 6 1 405.2.b.d 4
45.k odd 12 2 225.2.e.d 8
45.k odd 12 2 2025.2.a.t 4
45.l even 12 2 675.2.e.d 8
45.l even 12 2 2025.2.a.r 4
60.h even 2 1 2160.2.by.c 8
180.n even 6 1 2160.2.by.c 8
180.p odd 6 1 720.2.by.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a 8 1.a even 1 1 trivial
45.2.j.a 8 5.b even 2 1 inner
45.2.j.a 8 9.c even 3 1 inner
45.2.j.a 8 45.j even 6 1 inner
135.2.j.a 8 3.b odd 2 1
135.2.j.a 8 9.d odd 6 1
135.2.j.a 8 15.d odd 2 1
135.2.j.a 8 45.h odd 6 1
225.2.e.d 8 5.c odd 4 2
225.2.e.d 8 45.k odd 12 2
405.2.b.c 4 9.d odd 6 1
405.2.b.c 4 45.h odd 6 1
405.2.b.d 4 9.c even 3 1
405.2.b.d 4 45.j even 6 1
675.2.e.d 8 15.e even 4 2
675.2.e.d 8 45.l even 12 2
720.2.by.d 8 4.b odd 2 1
720.2.by.d 8 20.d odd 2 1
720.2.by.d 8 36.f odd 6 1
720.2.by.d 8 180.p odd 6 1
2025.2.a.r 4 45.l even 12 2
2025.2.a.t 4 45.k odd 12 2
2160.2.by.c 8 12.b even 2 1
2160.2.by.c 8 36.h even 6 1
2160.2.by.c 8 60.h even 2 1
2160.2.by.c 8 180.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(45, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} \)
$3$ \( 81 - 9 T^{4} + T^{8} \)
$5$ \( 625 + 50 T^{2} - 21 T^{4} + 2 T^{6} + T^{8} \)
$7$ \( 81 - 108 T^{2} + 135 T^{4} - 12 T^{6} + T^{8} \)
$11$ \( ( 36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$13$ \( ( 36 - 6 T^{2} + T^{4} )^{2} \)
$17$ \( ( 4 + 28 T^{2} + T^{4} )^{2} \)
$19$ \( ( -2 + 2 T + T^{2} )^{4} \)
$23$ \( 1 - 4 T^{2} + 15 T^{4} - 4 T^{6} + T^{8} \)
$29$ \( ( 9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$31$ \( ( 4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4} )^{2} \)
$37$ \( ( 18 + T^{2} )^{4} \)
$41$ \( ( 1089 + 396 T + 111 T^{2} + 12 T^{3} + T^{4} )^{2} \)
$43$ \( 1296 - 3024 T^{2} + 7020 T^{4} - 84 T^{6} + T^{8} \)
$47$ \( 28561 - 4732 T^{2} + 615 T^{4} - 28 T^{6} + T^{8} \)
$53$ \( ( 16 + 16 T^{2} + T^{4} )^{2} \)
$59$ \( ( 576 - 288 T + 120 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$61$ \( ( 5041 - 284 T + 87 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$67$ \( 2313441 - 127764 T^{2} + 5535 T^{4} - 84 T^{6} + T^{8} \)
$71$ \( ( 54 - 18 T + T^{2} )^{4} \)
$73$ \( ( 72 + T^{2} )^{4} \)
$79$ \( ( 16 - 32 T + 60 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$83$ \( 47458321 - 1184908 T^{2} + 22695 T^{4} - 172 T^{6} + T^{8} \)
$89$ \( ( -99 + 6 T + T^{2} )^{4} \)
$97$ \( 592240896 - 8176896 T^{2} + 88560 T^{4} - 336 T^{6} + T^{8} \)
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