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Properties

Label 36.2.h.a
Level $36$
Weight $2$
Character orbit 36.h
Analytic conductor $0.287$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [36,2,Mod(11,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 36.h (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.170772624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_{5} + \beta_1 - 2) q^{5} + ( - 2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{8} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{4} - \beta_1 + 1) q^{2} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - 1) q^{4} + ( - \beta_{7} - \beta_{5} + \beta_1 - 2) q^{5} + ( - 2 \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{8} + (\beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1) q^{9} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1 - 1) q^{10} + ( - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_{2} - 1) q^{11} + ( - \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1) q^{12} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_1 + 2) q^{13} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{14} + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{15} + ( - \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - \beta_{3} - \beta_1 - 1) q^{16} + ( - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - 1) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{18} + (\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1 + 1) q^{19} + ( - 2 \beta_{7} + 2 \beta_{6} + 2) q^{20} + (2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_1) q^{21} + ( - 2 \beta_{6} - 2 \beta_{2} + \beta_1) q^{22} + (\beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - \beta_{2} + 1) q^{23} + (5 \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_1 + 5) q^{24} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - \beta_1 - 1) q^{25} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{26} + ( - \beta_{6} + 2 \beta_{5} + \beta_{3} - \beta_{2} + 3 \beta_1) q^{27} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{28} + (\beta_{6} + \beta_{4} - \beta_1 + 2) q^{29} + (2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{30} + ( - 3 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} - \beta_{2} + 2 \beta_1 - 3) q^{31} + ( - 4 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 7) q^{32} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_1) q^{33} + (\beta_{7} - \beta_{5} + \beta_{3} + \beta_1 + 1) q^{34} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 - 1) q^{35} + (\beta_{7} - 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{36} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_1 - 6) q^{37} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 - 5) q^{38} + ( - 3 \beta_{7} + \beta_{6} - 3 \beta_{4} + 2 \beta_{3} + 3 \beta_{2} - 3) q^{39} + (4 \beta_{4} - 2 \beta_1 + 4) q^{40} + (3 \beta_{7} + 2 \beta_{5} - 2 \beta_1 + 6) q^{41} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 3) q^{42} + (3 \beta_{7} + 2 \beta_{6} + \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 2 \beta_1 + 3) q^{43} + (3 \beta_{7} + 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{44} + ( - 5 \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_1) q^{45} + (\beta_{6} + 2 \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 + 3) q^{46} + (4 \beta_{7} - \beta_{6} - 3 \beta_{5} + 4 \beta_{4} - 3 \beta_{3} - 6 \beta_{2} - \beta_1 + 4) q^{47} + ( - 2 \beta_{7} - \beta_{6} + 3 \beta_{5} - \beta_{4} + \beta_{2} - 4 \beta_1 + 3) q^{48} + (2 \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{4} + \beta_1 - 2) q^{49} + ( - \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 1) q^{50} + (\beta_{7} - \beta_{6} + \beta_{4} - 2 \beta_{2} + \beta_1 + 1) q^{51} + (4 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1 + 4) q^{52} + (6 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 4) q^{53} + (3 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + \cdots + 9) q^{54}+ \cdots + (3 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 3 \beta_{4} - 5 \beta_{3} - 4 \beta_{2} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3 q^{2} - q^{4} - 6 q^{5} - 3 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 3 q^{2} - q^{4} - 6 q^{5} - 3 q^{6} - 6 q^{9} - 8 q^{10} + 6 q^{12} - 2 q^{13} + 12 q^{14} - q^{16} + 18 q^{18} + 18 q^{20} - 6 q^{21} + 3 q^{22} + 3 q^{24} - 6 q^{25} - 12 q^{28} + 6 q^{29} - 18 q^{30} - 33 q^{32} + 24 q^{33} + 7 q^{34} - 33 q^{36} - 8 q^{37} - 27 q^{38} + 10 q^{40} + 24 q^{41} - 18 q^{42} + 6 q^{45} + 12 q^{46} + 21 q^{48} - 10 q^{49} + 21 q^{50} + 16 q^{52} + 39 q^{54} + 18 q^{56} + 6 q^{57} + 4 q^{58} + 6 q^{60} - 2 q^{61} + 26 q^{64} - 30 q^{65} - 24 q^{66} - 15 q^{68} - 30 q^{69} - 6 q^{70} - 21 q^{72} + 4 q^{73} - 30 q^{74} - 3 q^{76} - 30 q^{77} - 12 q^{78} - 30 q^{81} + 10 q^{82} + 30 q^{84} + 8 q^{85} + 21 q^{86} - 21 q^{88} + 6 q^{90} + 24 q^{92} + 30 q^{93} - 18 q^{94} + 12 q^{96} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} - \nu^{5} - 2\nu^{3} - 4\nu^{2} + 4\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + \nu^{5} - \nu^{4} + 2\nu^{3} + 4\nu - 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 5\nu^{5} - 2\nu^{4} + 2\nu^{3} - 8\nu^{2} + 20\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 3\nu^{6} - 5\nu^{5} + 4\nu^{4} - 6\nu^{3} + 16\nu^{2} - 12\nu + 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{6} + 5\nu^{5} - 6\nu^{4} + 6\nu^{3} - 12\nu^{2} + 20\nu - 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 5\nu^{6} + 7\nu^{5} - 6\nu^{4} + 10\nu^{3} - 24\nu^{2} + 28\nu - 32 ) / 8 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{5} + \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 3\beta_{6} + \beta_{5} + \beta_{4} - \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} - \beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} - \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 3\beta_{3} - 3\beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{7} - \beta_{6} - 7\beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 6\beta _1 - 7 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.774115 − 1.18353i
1.41203 − 0.0786378i
−1.02187 − 0.977642i
0.335728 + 1.37379i
0.774115 + 1.18353i
1.41203 + 0.0786378i
−1.02187 + 0.977642i
0.335728 − 1.37379i
−1.41203 − 0.0786378i 0.637910 + 1.61030i 1.98763 + 0.222077i 0.686141 − 0.396143i −0.774115 − 2.32395i −2.35143 − 1.35760i −2.78912 − 0.469882i −2.18614 + 2.05446i −1.00000 + 0.505408i
11.2 −0.774115 − 1.18353i −0.637910 − 1.61030i −0.801492 + 1.83238i 0.686141 − 0.396143i −1.41203 + 2.00155i 2.35143 + 1.35760i 2.78912 − 0.469882i −2.18614 + 2.05446i −1.00000 − 0.505408i
11.3 −0.335728 + 1.37379i 1.35760 − 1.07561i −1.77457 − 0.922437i −2.18614 + 1.26217i 1.02187 + 2.22616i −1.10489 − 0.637910i 1.86301 − 2.12819i 0.686141 − 2.92048i −1.00000 − 3.42703i
11.4 1.02187 − 0.977642i −1.35760 + 1.07561i 0.0884324 − 1.99804i −2.18614 + 1.26217i −0.335728 + 2.42637i 1.10489 + 0.637910i −1.86301 − 2.12819i 0.686141 − 2.92048i −1.00000 + 3.42703i
23.1 −1.41203 + 0.0786378i 0.637910 − 1.61030i 1.98763 − 0.222077i 0.686141 + 0.396143i −0.774115 + 2.32395i −2.35143 + 1.35760i −2.78912 + 0.469882i −2.18614 − 2.05446i −1.00000 − 0.505408i
23.2 −0.774115 + 1.18353i −0.637910 + 1.61030i −0.801492 − 1.83238i 0.686141 + 0.396143i −1.41203 − 2.00155i 2.35143 − 1.35760i 2.78912 + 0.469882i −2.18614 − 2.05446i −1.00000 + 0.505408i
23.3 −0.335728 − 1.37379i 1.35760 + 1.07561i −1.77457 + 0.922437i −2.18614 − 1.26217i 1.02187 − 2.22616i −1.10489 + 0.637910i 1.86301 + 2.12819i 0.686141 + 2.92048i −1.00000 + 3.42703i
23.4 1.02187 + 0.977642i −1.35760 − 1.07561i 0.0884324 + 1.99804i −2.18614 − 1.26217i −0.335728 − 2.42637i 1.10489 − 0.637910i −1.86301 + 2.12819i 0.686141 + 2.92048i −1.00000 − 3.42703i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.d odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.2.h.a ✓ 8
3.b odd 2 1 108.2.h.a 8
4.b odd 2 1 inner 36.2.h.a ✓ 8
5.b even 2 1 900.2.r.c 8
5.c odd 4 2 900.2.o.a 16
8.b even 2 1 576.2.s.f 8
8.d odd 2 1 576.2.s.f 8
9.c even 3 1 108.2.h.a 8
9.c even 3 1 324.2.b.b 8
9.d odd 6 1 inner 36.2.h.a ✓ 8
9.d odd 6 1 324.2.b.b 8
12.b even 2 1 108.2.h.a 8
20.d odd 2 1 900.2.r.c 8
20.e even 4 2 900.2.o.a 16
24.f even 2 1 1728.2.s.f 8
24.h odd 2 1 1728.2.s.f 8
36.f odd 6 1 108.2.h.a 8
36.f odd 6 1 324.2.b.b 8
36.h even 6 1 inner 36.2.h.a ✓ 8
36.h even 6 1 324.2.b.b 8
45.h odd 6 1 900.2.r.c 8
45.l even 12 2 900.2.o.a 16
72.j odd 6 1 576.2.s.f 8
72.j odd 6 1 5184.2.c.j 8
72.l even 6 1 576.2.s.f 8
72.l even 6 1 5184.2.c.j 8
72.n even 6 1 1728.2.s.f 8
72.n even 6 1 5184.2.c.j 8
72.p odd 6 1 1728.2.s.f 8
72.p odd 6 1 5184.2.c.j 8
180.n even 6 1 900.2.r.c 8
180.v odd 12 2 900.2.o.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.h.a ✓ 8 1.a even 1 1 trivial
36.2.h.a ✓ 8 4.b odd 2 1 inner
36.2.h.a ✓ 8 9.d odd 6 1 inner
36.2.h.a ✓ 8 36.h even 6 1 inner
108.2.h.a 8 3.b odd 2 1
108.2.h.a 8 9.c even 3 1
108.2.h.a 8 12.b even 2 1
108.2.h.a 8 36.f odd 6 1
324.2.b.b 8 9.c even 3 1
324.2.b.b 8 9.d odd 6 1
324.2.b.b 8 36.f odd 6 1
324.2.b.b 8 36.h even 6 1
576.2.s.f 8 8.b even 2 1
576.2.s.f 8 8.d odd 2 1
576.2.s.f 8 72.j odd 6 1
576.2.s.f 8 72.l even 6 1
900.2.o.a 16 5.c odd 4 2
900.2.o.a 16 20.e even 4 2
900.2.o.a 16 45.l even 12 2
900.2.o.a 16 180.v odd 12 2
900.2.r.c 8 5.b even 2 1
900.2.r.c 8 20.d odd 2 1
900.2.r.c 8 45.h odd 6 1
900.2.r.c 8 180.n even 6 1
1728.2.s.f 8 24.f even 2 1
1728.2.s.f 8 24.h odd 2 1
1728.2.s.f 8 72.n even 6 1
1728.2.s.f 8 72.p odd 6 1
5184.2.c.j 8 72.j odd 6 1
5184.2.c.j 8 72.l even 6 1
5184.2.c.j 8 72.n even 6 1
5184.2.c.j 8 72.p odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(36, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{7} + 5 T^{6} + 6 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{6} + 12 T^{4} + 27 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( (T^{4} + 3 T^{3} + T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 9 T^{6} + 69 T^{4} - 108 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{6} + 141 T^{4} + 36 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( (T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 7 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 27 T^{2} + 108)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 15 T^{6} + 177 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} + T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 69 T^{6} + 4569 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 32)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} + 49 T^{2} - 12 T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689 \) Copy content Toggle raw display
$47$ \( T^{8} + 135 T^{6} + 18033 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$53$ \( (T^{4} + 76 T^{2} + 256)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 180 T^{6} + \cdots + 16867449 \) Copy content Toggle raw display
$61$ \( (T^{4} + T^{3} + 9 T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 108 T^{6} + 8781 T^{4} + \cdots + 8311689 \) Copy content Toggle raw display
$71$ \( (T^{4} - 144 T^{2} + 432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - T - 8)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} - 201 T^{6} + \cdots + 101848464 \) Copy content Toggle raw display
$83$ \( T^{8} + 111 T^{6} + 9249 T^{4} + \cdots + 9437184 \) Copy content Toggle raw display
$89$ \( (T^{4} + 172 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 135 T^{2} + 262 T + 17161)^{2} \) Copy content Toggle raw display
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