Properties

Label 441.2.g.d
Level $441$
Weight $2$
Character orbit 441.g
Analytic conductor $3.521$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,2,Mod(67,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.g (of order \(3\), degree \(2\), 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: \(3.52140272914\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - 2 \beta_1 + 1) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{5} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{6} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 - 1) q^{8} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} - \beta_{3} - \beta_{2} - \beta_1) q^{2} + (\beta_{2} + \beta_1 - 1) q^{3} + (\beta_{3} - 2 \beta_1 + 1) q^{4} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{5} + ( - 2 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{6} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 - 1) q^{8} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{9} + (3 \beta_{5} - \beta_{4} - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 - 1) q^{10} + (\beta_{4} - 3 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{11} + (3 \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{12} + (\beta_{4} + 1) q^{13} + ( - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{15} + ( - 3 \beta_{5} + 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1) q^{16} + ( - 4 \beta_{4} - \beta_{3} - \beta_1 - 4) q^{17} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{18} + (\beta_{3} - 2 \beta_1 + 1) q^{19} + (\beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_1 + 1) q^{20} + (5 \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 5) q^{22} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{23} + (\beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{24} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 1) q^{25} + (\beta_{5} - \beta_1) q^{26} + (2 \beta_{5} + \beta_{4} - 4 \beta_{3} + 2 \beta_1 - 2) q^{27} + (\beta_{5} - \beta_1) q^{29} + (\beta_{5} - 6 \beta_{4} - 3 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 6) q^{30} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{31} + (\beta_{5} - 3 \beta_{4} - \beta_1) q^{32} + ( - 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} - \beta_{2}) q^{33} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{34} + ( - 4 \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 2) q^{36} + ( - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 3 \beta_1 + 3) q^{37} + ( - \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{38} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{39} + ( - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_1 - 5) q^{40} + ( - \beta_{5} - 7 \beta_{4} + \beta_{2} - 7) q^{41} + ( - 3 \beta_{5} + \beta_{4} - \beta_{3} + 5 \beta_1 - 1) q^{43} + (5 \beta_{5} + 6 \beta_{4} - 5 \beta_1) q^{44} + (4 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{45} + (3 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 - 5) q^{46} + ( - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 3) q^{47} + (6 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 7) q^{48} + (5 \beta_{5} - 5 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 5) q^{50} + (4 \beta_{5} - 5 \beta_{2} - 2 \beta_1 + 1) q^{51} + ( - \beta_{4} + 2 \beta_{3} - \beta_1) q^{52} + (3 \beta_{5} + 5 \beta_{4} - \beta_{3} - 3 \beta_{2} - \beta_1 + 5) q^{53} + (2 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 5 \beta_{2} - 4 \beta_1) q^{54} + ( - \beta_{4} - 4 \beta_{3} - 6 \beta_{2} - \beta_1 + 1) q^{55} + (3 \beta_{4} - 3 \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{57} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 - 2) q^{58} + ( - 3 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - \beta_1 + 2) q^{59} + ( - 3 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 5) q^{60} + ( - 3 \beta_{5} - \beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{61} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 5) q^{62} + (\beta_{4} + \beta_{3} + 3 \beta_{2} + \beta_1 - 4) q^{64} + ( - \beta_{5} + 2 \beta_{4} + \beta_{2} + 2) q^{65} + ( - 5 \beta_{5} + 3 \beta_{2} + 9 \beta_1 + 2) q^{66} + ( - 6 \beta_{5} + 3 \beta_{4} - \beta_{3} + 8 \beta_1 - 1) q^{67} + (4 \beta_{4} - 5 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 5) q^{68} + (\beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} - 2 \beta_1 + 7) q^{69} + ( - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 4) q^{71} + ( - 4 \beta_{5} - 3 \beta_{4} - \beta_{2} + 5 \beta_1 - 1) q^{72} + (6 \beta_{5} - \beta_{4} - \beta_{3} - 6 \beta_{2} - \beta_1 - 1) q^{73} + (3 \beta_{3} + 3 \beta_{2} + 3) q^{74} + (\beta_{5} - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 5) q^{75} + ( - 3 \beta_{5} - 6 \beta_{4} + 4 \beta_{3} + 3 \beta_{2} + 4 \beta_1 - 6) q^{76} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{78} + ( - 6 \beta_{5} - 3 \beta_{4} + 5 \beta_{3} + 6 \beta_{2} + 5 \beta_1 - 3) q^{79} + ( - 7 \beta_{5} + 6 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + 7 \beta_1 + 6) q^{80} + (\beta_{5} - 10 \beta_{4} + 4 \beta_{3} - 2 \beta_1 - 1) q^{81} + ( - 6 \beta_{5} - \beta_{4} + 6 \beta_1) q^{82} + (5 \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{83} + (6 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1 - 5) q^{85} + (4 \beta_{4} - 9 \beta_{3} - \beta_{2} + 4 \beta_1 + 7) q^{86} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2}) q^{87} + ( - \beta_{4} + 8 \beta_{3} + 6 \beta_{2} - \beta_1 - 4) q^{88} + (5 \beta_{5} - 2 \beta_{4} + \beta_{3} - 7 \beta_1 + 1) q^{89} + (3 \beta_{5} - 7 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 6 \beta_1 - 11) q^{90} + (7 \beta_{4} - 2 \beta_{3} + 4 \beta_1 - 2) q^{92} + ( - 4 \beta_{5} + 6 \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{93} + ( - 6 \beta_{5} - 3 \beta_{4} - 3 \beta_{3} + 12 \beta_1 - 3) q^{94} + (\beta_{5} + 4 \beta_{4} + \beta_{3} - 3 \beta_1 + 1) q^{95} + (4 \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - 3) q^{96} + (3 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} - 7 \beta_1 + 2) q^{97} + (\beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - 2 q^{3} - 3 q^{4} + 10 q^{5} - q^{6} - 12 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - 2 q^{3} - 3 q^{4} + 10 q^{5} - q^{6} - 12 q^{8} + 8 q^{9} - 4 q^{11} + 11 q^{12} + 3 q^{13} + 11 q^{15} - 3 q^{16} - 12 q^{17} - 23 q^{18} - 3 q^{19} - 16 q^{20} + 15 q^{22} + 12 q^{25} - q^{26} + 7 q^{27} - q^{29} - 5 q^{30} - 3 q^{31} + 8 q^{32} - 5 q^{33} - 3 q^{34} - 11 q^{36} + 3 q^{37} - 16 q^{38} + 2 q^{39} - 42 q^{40} - 22 q^{41} + 3 q^{43} - 23 q^{44} + 4 q^{45} - 12 q^{46} - 9 q^{47} + 14 q^{48} - 10 q^{50} + 3 q^{51} - 6 q^{52} + 18 q^{53} - 4 q^{54} + 12 q^{55} + 11 q^{57} - 18 q^{58} - 9 q^{59} + 37 q^{60} - 6 q^{61} + 36 q^{62} - 24 q^{64} + 5 q^{65} + 32 q^{66} - 12 q^{68} + 39 q^{69} + 18 q^{71} + 9 q^{72} + 3 q^{73} + 12 q^{74} + 35 q^{75} - 21 q^{76} + 10 q^{78} - 15 q^{79} + 11 q^{80} + 8 q^{81} + 9 q^{82} - 12 q^{83} - 9 q^{85} + 68 q^{86} + 10 q^{87} - 42 q^{88} - 2 q^{89} - 73 q^{90} - 15 q^{92} - 15 q^{93} + 24 q^{94} - 16 q^{95} - 2 q^{96} + 3 q^{97} - 46 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(\beta_{4}\) \(-1 - \beta_{4}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
0.500000 + 0.224437i
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
−0.849814 + 1.47192i 1.64400 + 0.545231i −0.444368 0.769668i 3.58836 −2.19963 + 1.95649i 0 −1.88874 2.40545 + 1.79272i −3.04944 + 5.28179i
67.2 0.119562 0.207087i −1.71053 0.272169i 0.971410 + 1.68253i −1.18194 −0.260877 + 0.321688i 0 0.942820 2.85185 + 0.931107i −0.141315 + 0.244765i
67.3 1.23025 2.13086i −0.933463 + 1.45899i −2.02704 3.51094i 2.59358 1.96050 + 3.78400i 0 −5.05408 −1.25729 2.72382i 3.19076 5.52655i
79.1 −0.849814 1.47192i 1.64400 0.545231i −0.444368 + 0.769668i 3.58836 −2.19963 1.95649i 0 −1.88874 2.40545 1.79272i −3.04944 5.28179i
79.2 0.119562 + 0.207087i −1.71053 + 0.272169i 0.971410 1.68253i −1.18194 −0.260877 0.321688i 0 0.942820 2.85185 0.931107i −0.141315 0.244765i
79.3 1.23025 + 2.13086i −0.933463 1.45899i −2.02704 + 3.51094i 2.59358 1.96050 3.78400i 0 −5.05408 −1.25729 + 2.72382i 3.19076 + 5.52655i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.g.d 6
3.b odd 2 1 1323.2.g.b 6
7.b odd 2 1 441.2.g.e 6
7.c even 3 1 441.2.f.d 6
7.c even 3 1 441.2.h.b 6
7.d odd 6 1 63.2.f.b 6
7.d odd 6 1 441.2.h.c 6
9.c even 3 1 441.2.h.b 6
9.d odd 6 1 1323.2.h.e 6
21.c even 2 1 1323.2.g.c 6
21.g even 6 1 189.2.f.a 6
21.g even 6 1 1323.2.h.d 6
21.h odd 6 1 1323.2.f.c 6
21.h odd 6 1 1323.2.h.e 6
28.f even 6 1 1008.2.r.k 6
63.g even 3 1 inner 441.2.g.d 6
63.g even 3 1 3969.2.a.m 3
63.h even 3 1 441.2.f.d 6
63.i even 6 1 189.2.f.a 6
63.j odd 6 1 1323.2.f.c 6
63.k odd 6 1 441.2.g.e 6
63.k odd 6 1 567.2.a.d 3
63.l odd 6 1 441.2.h.c 6
63.n odd 6 1 1323.2.g.b 6
63.n odd 6 1 3969.2.a.p 3
63.o even 6 1 1323.2.h.d 6
63.s even 6 1 567.2.a.g 3
63.s even 6 1 1323.2.g.c 6
63.t odd 6 1 63.2.f.b 6
84.j odd 6 1 3024.2.r.g 6
252.n even 6 1 9072.2.a.bq 3
252.r odd 6 1 3024.2.r.g 6
252.bj even 6 1 1008.2.r.k 6
252.bn odd 6 1 9072.2.a.cd 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 7.d odd 6 1
63.2.f.b 6 63.t odd 6 1
189.2.f.a 6 21.g even 6 1
189.2.f.a 6 63.i even 6 1
441.2.f.d 6 7.c even 3 1
441.2.f.d 6 63.h even 3 1
441.2.g.d 6 1.a even 1 1 trivial
441.2.g.d 6 63.g even 3 1 inner
441.2.g.e 6 7.b odd 2 1
441.2.g.e 6 63.k odd 6 1
441.2.h.b 6 7.c even 3 1
441.2.h.b 6 9.c even 3 1
441.2.h.c 6 7.d odd 6 1
441.2.h.c 6 63.l odd 6 1
567.2.a.d 3 63.k odd 6 1
567.2.a.g 3 63.s even 6 1
1008.2.r.k 6 28.f even 6 1
1008.2.r.k 6 252.bj even 6 1
1323.2.f.c 6 21.h odd 6 1
1323.2.f.c 6 63.j odd 6 1
1323.2.g.b 6 3.b odd 2 1
1323.2.g.b 6 63.n odd 6 1
1323.2.g.c 6 21.c even 2 1
1323.2.g.c 6 63.s even 6 1
1323.2.h.d 6 21.g even 6 1
1323.2.h.d 6 63.o even 6 1
1323.2.h.e 6 9.d odd 6 1
1323.2.h.e 6 21.h odd 6 1
3024.2.r.g 6 84.j odd 6 1
3024.2.r.g 6 252.r odd 6 1
3969.2.a.m 3 63.g even 3 1
3969.2.a.p 3 63.n odd 6 1
9072.2.a.bq 3 252.n even 6 1
9072.2.a.cd 3 252.bn odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(441, [\chi])\):

\( T_{2}^{6} - T_{2}^{5} + 5T_{2}^{4} + 2T_{2}^{3} + 17T_{2}^{2} - 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 5T_{5}^{2} + 2T_{5} + 11 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{5} + 5 T^{4} + 2 T^{3} + 17 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} - 2 T^{4} - 9 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{3} - 5 T^{2} + 2 T + 11)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} + 2 T^{2} - 19 T - 47)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$19$ \( T^{6} + 3 T^{5} + 15 T^{4} - 4 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( (T^{3} - 33 T - 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + T^{5} + 5 T^{4} - 2 T^{3} + 17 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$31$ \( T^{6} + 3 T^{5} + 33 T^{4} - 126 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + 63 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$41$ \( T^{6} + 22 T^{5} + 329 T^{4} + \cdots + 124609 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + 75 T^{4} + \cdots + 14641 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( T^{6} - 18 T^{5} + 249 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( T^{6} + 9 T^{5} + 87 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( T^{6} + 6 T^{5} + 57 T^{4} + \cdots + 4489 \) Copy content Toggle raw display
$67$ \( T^{6} + 207 T^{4} + 1366 T^{3} + \cdots + 466489 \) Copy content Toggle raw display
$71$ \( (T^{3} - 9 T^{2} - 6 T + 81)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 3 T^{5} + 177 T^{4} + \cdots + 59049 \) Copy content Toggle raw display
$79$ \( T^{6} + 15 T^{5} + 273 T^{4} + \cdots + 591361 \) Copy content Toggle raw display
$83$ \( T^{6} + 12 T^{5} + 105 T^{4} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( T^{6} + 2 T^{5} + 155 T^{4} + \cdots + 143641 \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + 123 T^{4} + \cdots + 363609 \) Copy content Toggle raw display
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