Properties

Label 729.2.e.r
Level $729$
Weight $2$
Character orbit 729.e
Analytic conductor $5.821$
Analytic rank $0$
Dimension $12$
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] = [729,2,Mod(82,729)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("729.82");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.e (of order \(9\), degree \(6\), not 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: \(5.82109430735\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{10} - \beta_{9}) q^{2} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{7} - \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} - \beta_{9}) q^{2} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{8} + \beta_{7} + \beta_{2}) q^{5} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{11} - 2 \beta_{9} + 2 \beta_{7} - \beta_{2}) q^{8} + (\beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_1) q^{10} + (4 \beta_{11} - 2 \beta_{10} - \beta_{9} - \beta_{8} + 2 \beta_{7}) q^{11} + ( - 3 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{13} + (\beta_{11} - \beta_{9} + \beta_{7} - 3 \beta_{2}) q^{14} + (2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 2) q^{16} + ( - 2 \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{2}) q^{17} + ( - \beta_{6} - \beta_{5} - 4 \beta_{4} - \beta_{3} + 2 \beta_1 + 4) q^{19} + (\beta_{11} + \beta_{9} - 3 \beta_{8} + 3 \beta_{2}) q^{20} + (2 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} - 1) q^{22} + (3 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 5 \beta_{7} - 5 \beta_{2}) q^{23} + (3 \beta_{5} + 2 \beta_{4} - 5 \beta_1 - 5) q^{25} + ( - 3 \beta_{11} + 4 \beta_{10} - 2 \beta_{9} + 2 \beta_{2}) q^{26} + (\beta_{6} + \beta_{5} - 2 \beta_{3} - 2 \beta_1 - 1) q^{28} + (\beta_{11} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{7} + \beta_{2}) q^{29} + (5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 5 \beta_1) q^{31} + (\beta_{11} - \beta_{10} + 2 \beta_{9} - 4 \beta_{8} + 3 \beta_{7} + \beta_{2}) q^{32} + (5 \beta_{5} - \beta_{4} - \beta_1 + 5) q^{34} + ( - 3 \beta_{11} + 5 \beta_{10} - 4 \beta_{9} + 4 \beta_{8} - \beta_{7} - 2 \beta_{2}) q^{35} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{37} + ( - 2 \beta_{11} + 7 \beta_{10} - 5 \beta_{9} + 6 \beta_{8} - \beta_{7} - 6 \beta_{2}) q^{38} + ( - \beta_{3} + \beta_1 - 1) q^{40} + (4 \beta_{11} - \beta_{10} - 4 \beta_{9} + \beta_{8} + 4 \beta_{7} - 3 \beta_{2}) q^{41} + (2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{43} + (\beta_{11} + \beta_{10} - \beta_{9} + 5 \beta_{7} - 2 \beta_{2}) q^{44} + ( - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 5 \beta_{3} - 3 \beta_1 - 1) q^{46} + (\beta_{11} + \beta_{8} + \beta_{7} - \beta_{2}) q^{47} + (3 \beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} + 4) q^{49} + (3 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} - 5 \beta_{8} + 3 \beta_{7} + 2 \beta_{2}) q^{50} + ( - \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_1 + 4) q^{52} + ( - 4 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{2}) q^{53} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_1 - 5) q^{55} + (2 \beta_{11} + 2 \beta_{10} - 4 \beta_{9} - 2 \beta_{8} + 3 \beta_{7} - 3 \beta_{2}) q^{56} + ( - \beta_{6} - 5 \beta_{5} - \beta_{4} + \beta_{3} + 5 \beta_1 + 1) q^{58} + ( - 5 \beta_{11} + 5 \beta_{10} - 5 \beta_{9} + 7 \beta_{8} - 2 \beta_{7} - 5 \beta_{2}) q^{59} + ( - 6 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 4 \beta_1 - 6) q^{61} + (5 \beta_{11} - 5 \beta_{10} - 3 \beta_{8} + 5 \beta_{7}) q^{62} + (4 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + 4 \beta_1) q^{64} + (8 \beta_{11} - 10 \beta_{10} + 5 \beta_{9} - 9 \beta_{8} + 4 \beta_{7} + 6 \beta_{2}) q^{65} + (5 \beta_{6} - 5 \beta_{4} - 8 \beta_{3} - \beta_1 - 3) q^{67} + ( - 3 \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 2 \beta_{8} - 3 \beta_{7} + 3 \beta_{2}) q^{68} + ( - 5 \beta_{6} - 7 \beta_{5} - 7 \beta_{4} + 3 \beta_1 + 4) q^{70} + (2 \beta_{11} - 2 \beta_{8} - 2 \beta_{7}) q^{71} + ( - 3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} + 6 \beta_1 - 2) q^{73} + ( - 2 \beta_{11} - \beta_{9} + \beta_{8} - \beta_{7} - \beta_{2}) q^{74} + ( - 3 \beta_{6} - 3 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} + 2) q^{76} + (2 \beta_{11} + 4 \beta_{10} - \beta_{9} + \beta_{8} - 3 \beta_{2}) q^{77} + ( - 8 \beta_{6} - 5 \beta_{5} + 3 \beta_{4} + 8 \beta_{3} + 2 \beta_1 + 2) q^{79} + (2 \beta_{11} - 6 \beta_{10} + 3 \beta_{9} - 3 \beta_{2}) q^{80} + (4 \beta_{6} - 3 \beta_{5} - \beta_{3} - \beta_1 - 2) q^{82} + ( - 4 \beta_{11} + \beta_{10} + 3 \beta_{9} + 4 \beta_{8} - 8 \beta_{7} + 8 \beta_{2}) q^{83} + (4 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} - 5 \beta_{3} - 3 \beta_1 - 4) q^{85} + (2 \beta_{9} + 2 \beta_{8} - 2 \beta_{7}) q^{86} + ( - 4 \beta_{4} + 7 \beta_{3} - 4 \beta_1) q^{88} + ( - 3 \beta_{11} - 2 \beta_{10} + 10 \beta_{9} + \beta_{8} - 8 \beta_{7} + 5 \beta_{2}) q^{89} + ( - 6 \beta_{6} - \beta_{5} - \beta_{3} - 6 \beta_1) q^{91} + ( - 4 \beta_{11} + 4 \beta_{10} + 3 \beta_{9} - \beta_{8} - 2 \beta_{7} - 2 \beta_{2}) q^{92} + ( - \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_1) q^{94} + (2 \beta_{11} + 5 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} + 2 \beta_{7} + 6 \beta_{2}) q^{95} + (2 \beta_{6} + 4 \beta_{5} + 4 \beta_{4} - 7 \beta_1 + 3) q^{97} + (2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} + 6 q^{7} + 12 q^{10} + 6 q^{13} - 18 q^{16} + 24 q^{19} + 6 q^{22} - 48 q^{25} - 12 q^{28} - 12 q^{31} + 54 q^{34} + 6 q^{37} - 12 q^{40} - 12 q^{43} - 6 q^{46} + 42 q^{49} + 54 q^{52} - 60 q^{55} + 6 q^{58} - 48 q^{61} - 6 q^{64} - 66 q^{67} + 6 q^{70} - 12 q^{73} - 6 q^{76} + 42 q^{79} - 24 q^{82} - 18 q^{85} - 24 q^{88} + 6 q^{94} + 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{36}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{36}^{3} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{36}^{4} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{36}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{36}^{8} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{36}^{10} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{36}^{11} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{36}^{5} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( \zeta_{36}^{9} - \zeta_{36}^{3} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{36}^{11} + \zeta_{36}^{5} + \zeta_{36} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{36}^{11} + \zeta_{36}^{7} \) Copy content Toggle raw display
\(\zeta_{36}\)\(=\) \( ( \beta_{10} + \beta_{8} + \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{36}^{3}\)\(=\) \( ( -\beta_{10} - \beta_{8} - \beta_{7} + 3\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{4}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{36}^{5}\)\(=\) \( ( \beta_{10} - 2\beta_{8} + \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{6}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{36}^{7}\)\(=\) \( ( 3\beta_{11} - \beta_{10} - \beta_{8} + 2\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{8}\)\(=\) \( \beta_{5} \) Copy content Toggle raw display
\(\zeta_{36}^{9}\)\(=\) \( ( -2\beta_{10} + 3\beta_{9} - 2\beta_{8} - 2\beta_{7} + 3\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{36}^{10}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{36}^{11}\)\(=\) \( ( -\beta_{10} - \beta_{8} + 2\beta_{7} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/729\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{1} - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1
0.642788 + 0.766044i
−0.642788 0.766044i
−0.984808 0.173648i
0.984808 + 0.173648i
0.342020 0.939693i
−0.342020 + 0.939693i
0.342020 + 0.939693i
−0.342020 0.939693i
−0.984808 + 0.173648i
0.984808 0.173648i
0.642788 0.766044i
−0.642788 + 0.766044i
−0.223238 1.26604i 0 0.326352 0.118782i −0.342020 0.286989i 0 3.31908 + 1.20805i −1.50881 2.61334i 0 −0.286989 + 0.497079i
82.2 0.223238 + 1.26604i 0 0.326352 0.118782i 0.342020 + 0.286989i 0 3.31908 + 1.20805i 1.50881 + 2.61334i 0 −0.286989 + 0.497079i
163.1 −1.85083 + 0.673648i 0 1.43969 1.20805i −0.642788 3.64543i 0 −1.79813 1.50881i 0.118782 0.205737i 0 3.64543 + 6.31407i
163.2 1.85083 0.673648i 0 1.43969 1.20805i 0.642788 + 3.64543i 0 −1.79813 1.50881i −0.118782 + 0.205737i 0 3.64543 + 6.31407i
325.1 −0.524005 + 0.439693i 0 −0.266044 + 1.50881i 0.984808 0.358441i 0 −0.0209445 0.118782i −1.20805 2.09240i 0 −0.358441 + 0.620838i
325.2 0.524005 0.439693i 0 −0.266044 + 1.50881i −0.984808 + 0.358441i 0 −0.0209445 0.118782i 1.20805 + 2.09240i 0 −0.358441 + 0.620838i
406.1 −0.524005 0.439693i 0 −0.266044 1.50881i 0.984808 + 0.358441i 0 −0.0209445 + 0.118782i −1.20805 + 2.09240i 0 −0.358441 0.620838i
406.2 0.524005 + 0.439693i 0 −0.266044 1.50881i −0.984808 0.358441i 0 −0.0209445 + 0.118782i 1.20805 2.09240i 0 −0.358441 0.620838i
568.1 −1.85083 0.673648i 0 1.43969 + 1.20805i −0.642788 + 3.64543i 0 −1.79813 + 1.50881i 0.118782 + 0.205737i 0 3.64543 6.31407i
568.2 1.85083 + 0.673648i 0 1.43969 + 1.20805i 0.642788 3.64543i 0 −1.79813 + 1.50881i −0.118782 0.205737i 0 3.64543 6.31407i
649.1 −0.223238 + 1.26604i 0 0.326352 + 0.118782i −0.342020 + 0.286989i 0 3.31908 1.20805i −1.50881 + 2.61334i 0 −0.286989 0.497079i
649.2 0.223238 1.26604i 0 0.326352 + 0.118782i 0.342020 0.286989i 0 3.31908 1.20805i 1.50881 2.61334i 0 −0.286989 0.497079i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
27.e even 9 1 inner
27.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 729.2.e.r 12
3.b odd 2 1 inner 729.2.e.r 12
9.c even 3 1 729.2.e.m 12
9.c even 3 1 729.2.e.q 12
9.d odd 6 1 729.2.e.m 12
9.d odd 6 1 729.2.e.q 12
27.e even 9 1 729.2.a.c 6
27.e even 9 2 729.2.c.c 12
27.e even 9 1 729.2.e.m 12
27.e even 9 1 729.2.e.q 12
27.e even 9 1 inner 729.2.e.r 12
27.f odd 18 1 729.2.a.c 6
27.f odd 18 2 729.2.c.c 12
27.f odd 18 1 729.2.e.m 12
27.f odd 18 1 729.2.e.q 12
27.f odd 18 1 inner 729.2.e.r 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
729.2.a.c 6 27.e even 9 1
729.2.a.c 6 27.f odd 18 1
729.2.c.c 12 27.e even 9 2
729.2.c.c 12 27.f odd 18 2
729.2.e.m 12 9.c even 3 1
729.2.e.m 12 9.d odd 6 1
729.2.e.m 12 27.e even 9 1
729.2.e.m 12 27.f odd 18 1
729.2.e.q 12 9.c even 3 1
729.2.e.q 12 9.d odd 6 1
729.2.e.q 12 27.e even 9 1
729.2.e.q 12 27.f odd 18 1
729.2.e.r 12 1.a even 1 1 trivial
729.2.e.r 12 3.b odd 2 1 inner
729.2.e.r 12 27.e even 9 1 inner
729.2.e.r 12 27.f odd 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(729, [\chi])\):

\( T_{2}^{12} - 3T_{2}^{10} + 30T_{2}^{6} + 36T_{2}^{4} + 9 \) Copy content Toggle raw display
\( T_{5}^{12} + 24T_{5}^{10} + 144T_{5}^{8} - 294T_{5}^{6} + 252T_{5}^{4} - 27T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} - 6T_{7}^{4} + 8T_{7}^{3} + 69T_{7}^{2} + 3T_{7} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 3 T^{10} + 30 T^{6} + 36 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 24 T^{10} + 144 T^{8} - 294 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T^{6} - 3 T^{5} - 6 T^{4} + 8 T^{3} + 69 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 39 T^{10} + 648 T^{8} + \cdots + 1172889 \) Copy content Toggle raw display
$13$ \( (T^{6} - 3 T^{5} - 6 T^{4} - 28 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 81 T^{10} + 4806 T^{8} + \cdots + 95004009 \) Copy content Toggle raw display
$19$ \( (T^{6} - 12 T^{5} + 105 T^{4} - 430 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 33 T^{10} - 2772 T^{8} + \cdots + 751689 \) Copy content Toggle raw display
$29$ \( T^{12} - 3 T^{10} + 468 T^{8} + \cdots + 16867449 \) Copy content Toggle raw display
$31$ \( (T^{6} + 6 T^{5} + 84 T^{4} + 179 T^{3} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 3 T^{5} + 15 T^{4} - 16 T^{3} + \cdots + 289)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 15 T^{10} + 2745 T^{8} + \cdots + 71014329 \) Copy content Toggle raw display
$43$ \( (T^{6} + 6 T^{5} + 12 T^{4} - 64 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} - 3 T^{10} - 36 T^{8} + 435 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( (T^{6} - 108 T^{4} + 2592 T^{2} + \cdots - 15552)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 330 T^{10} + 30465 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( (T^{6} + 24 T^{5} + 336 T^{4} + \cdots + 87616)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 33 T^{5} + 471 T^{4} + \cdots + 63001)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 72 T^{10} + 3888 T^{8} + \cdots + 2985984 \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + 105 T^{4} - 592 T^{3} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 21 T^{5} + 228 T^{4} + \cdots + 11449)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 285 T^{10} + \cdots + 2405238079689 \) Copy content Toggle raw display
$89$ \( T^{12} + 387 T^{10} + \cdots + 60886809 \) Copy content Toggle raw display
$97$ \( (T^{6} - 30 T^{5} + 417 T^{4} + \cdots + 418609)^{2} \) Copy content Toggle raw display
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