Properties

Label 882.2.f.n
Level $882$
Weight $2$
Character orbit 882.f
Analytic conductor $7.043$
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] = [882,2,Mod(295,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.295");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.f (of order \(3\), 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: \(7.04280545828\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} + 1) q^{2} + (\beta_{5} - 1) q^{3} + \beta_{4} q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} - q^{8} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 1) q^{2} + (\beta_{5} - 1) q^{3} + \beta_{4} q^{4} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} - q^{8} + ( - \beta_{4} - 2 \beta_{3} - \beta_{2} - 2) q^{9} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{10} + ( - \beta_{5} + \beta_{2}) q^{11} + ( - \beta_{5} + \beta_{2} + \beta_1) q^{12} + (\beta_{5} - 4 \beta_{4} + \beta_{3} - 3 \beta_1 + 1) q^{13} + (2 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{15} + ( - \beta_{4} - 1) q^{16} + (2 \beta_{3} + 2 \beta_{2} + 2) q^{17} + (\beta_{5} - 2 \beta_{4} - \beta_{3} - \beta_{2} - 2 \beta_1) q^{18} + (3 \beta_{3} + 3 \beta_{2} + 2) q^{19} + (\beta_{5} - \beta_{2}) q^{20} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_1 + 1) q^{22} + (\beta_{5} + 2 \beta_{4} - \beta_1) q^{23} + ( - \beta_{5} + 1) q^{24} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{25} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} - \beta_1 + 4) q^{26} + ( - 2 \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_1 + 2) q^{27} + ( - 5 \beta_{5} + 3 \beta_{3} + 5 \beta_{2} + 3 \beta_1) q^{29} + (4 \beta_{4} - \beta_{3} + 2 \beta_1) q^{30} + ( - 2 \beta_{5} - 7 \beta_{4} + \beta_{3} + 1) q^{31} - \beta_{4} q^{32} + ( - 2 \beta_{4} + 2 \beta_{3} - \beta_1 + 3) q^{33} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{34} + (\beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_1 + 2) q^{36} - q^{37} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 2) q^{38} + (3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 5 \beta_{2} - 1) q^{39} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{40} + (3 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_1 - 2) q^{41} + ( - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 2) q^{43} + ( - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{44} + (2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 4) q^{45} + (\beta_{3} + \beta_{2} - 2) q^{46} + ( - 3 \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 3) q^{47} + ( - \beta_{2} - \beta_1 + 1) q^{48} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 2) q^{50} + (4 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_{2} - 2) q^{51} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 3) q^{52} + ( - \beta_{4} + 2 \beta_{3} - \beta_1 - 4) q^{53} + ( - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 1) q^{54} + (2 \beta_{4} - 3 \beta_{3} + \beta_{2} + 2 \beta_1 - 6) q^{55} + (5 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} - 3 \beta_{2} - 2) q^{57} + ( - 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_1 + 2) q^{58} + ( - \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 3 \beta_1 + 2) q^{59} + (2 \beta_{4} - 2 \beta_{3} + \beta_1 - 3) q^{60} + ( - \beta_{5} + 3 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{61} + ( - \beta_{4} - 2 \beta_{2} - \beta_1 + 7) q^{62} + q^{64} + ( - 6 \beta_{5} - 2 \beta_{4} + \beta_{3} + 6 \beta_{2} + \beta_1 - 2) q^{65} + (2 \beta_{4} + \beta_{3} + \beta_1 + 3) q^{66} + (5 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 3 \beta_1 - 4) q^{67} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{68} + ( - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} + \beta_{2} + 3 \beta_1 - 2) q^{69} + ( - 2 \beta_{4} + 9 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 6) q^{71} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 2) q^{72} + ( - 4 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} - 4 \beta_1 - 4) q^{73} + ( - \beta_{4} - 1) q^{74} + ( - \beta_{5} - 5 \beta_{4} + 5 \beta_{3} + \beta_{2} + 3) q^{75} + ( - 3 \beta_{5} + 2 \beta_{4} + 3 \beta_1) q^{76} + (5 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{78} + (5 \beta_{5} - 4 \beta_{3} - 5 \beta_{2} - 4 \beta_1) q^{79} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{80} + ( - \beta_{5} + 8 \beta_{4} - 2 \beta_{3} + \beta_{2} + 5 \beta_1 + 6) q^{81} + (2 \beta_{4} - \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{82} + ( - \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{83} + ( - 4 \beta_{5} + 2 \beta_{3} + 2) q^{85} + (\beta_{4} - 3 \beta_{3} + 6 \beta_1 - 3) q^{86} + (3 \beta_{5} - \beta_{4} + \beta_{3} - 5 \beta_1 + 12) q^{87} + (\beta_{5} - \beta_{2}) q^{88} + (\beta_{4} - 5 \beta_{3} - 3 \beta_{2} + \beta_1 + 1) q^{89} + ( - 2 \beta_{5} + 4 \beta_{2} + \beta_1 - 2) q^{90} + ( - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 2) q^{92} + (6 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} - 5 \beta_{2} - 6 \beta_1 + 5) q^{93} + ( - 3 \beta_{3} + 6 \beta_1 - 3) q^{94} + ( - 5 \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 2) q^{95} + (\beta_{5} - \beta_{2} - \beta_1) q^{96} + (4 \beta_{5} + 8 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 8) q^{97} + (4 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{2} - 4 q^{3} - 3 q^{4} + q^{5} - 2 q^{6} - 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{2} - 4 q^{3} - 3 q^{4} + q^{5} - 2 q^{6} - 6 q^{8} - 4 q^{9} + 2 q^{10} - q^{11} + 2 q^{12} + 8 q^{13} + 12 q^{15} - 3 q^{16} + 8 q^{17} + 4 q^{18} + 6 q^{19} + q^{20} + q^{22} - 7 q^{23} + 4 q^{24} + 2 q^{25} + 16 q^{26} - 7 q^{27} - 5 q^{29} - 3 q^{30} + 20 q^{31} + 3 q^{32} + 15 q^{33} + 4 q^{34} + 8 q^{36} - 6 q^{37} + 3 q^{38} + 4 q^{39} - q^{40} - 6 q^{43} + 2 q^{44} - 12 q^{45} - 14 q^{46} - 9 q^{47} + 2 q^{48} - 2 q^{50} + 12 q^{51} + 8 q^{52} - 30 q^{53} - 8 q^{54} - 26 q^{55} + 22 q^{57} + 5 q^{58} - 14 q^{59} - 15 q^{60} + 8 q^{61} + 40 q^{62} + 6 q^{64} - 12 q^{65} + 12 q^{66} + q^{67} - 4 q^{68} + 3 q^{69} + 14 q^{71} + 4 q^{72} - 38 q^{73} - 3 q^{74} + 17 q^{75} - 3 q^{76} + 5 q^{78} + 5 q^{79} - 2 q^{80} + 32 q^{81} + 2 q^{83} - 2 q^{85} + 6 q^{86} + 63 q^{87} + q^{88} + 18 q^{89} - 9 q^{90} - 7 q^{92} + q^{93} + 9 q^{94} - 4 q^{95} - 2 q^{96} + 28 q^{97} - 3 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
295.1
0.500000 + 2.05195i
0.500000 1.41036i
0.500000 + 0.224437i
0.500000 2.05195i
0.500000 + 1.41036i
0.500000 0.224437i
0.500000 0.866025i −1.73025 0.0789082i −0.500000 0.866025i −0.296790 0.514055i −0.933463 + 1.45899i 0 −1.00000 2.98755 + 0.273062i −0.593579
295.2 0.500000 0.866025i −0.619562 1.61745i −0.500000 0.866025i 1.59097 + 2.75564i −1.71053 0.272169i 0 −1.00000 −2.23229 + 2.00422i 3.18194
295.3 0.500000 0.866025i 0.349814 + 1.69636i −0.500000 0.866025i −0.794182 1.37556i 1.64400 + 0.545231i 0 −1.00000 −2.75526 + 1.18682i −1.58836
589.1 0.500000 + 0.866025i −1.73025 + 0.0789082i −0.500000 + 0.866025i −0.296790 + 0.514055i −0.933463 1.45899i 0 −1.00000 2.98755 0.273062i −0.593579
589.2 0.500000 + 0.866025i −0.619562 + 1.61745i −0.500000 + 0.866025i 1.59097 2.75564i −1.71053 + 0.272169i 0 −1.00000 −2.23229 2.00422i 3.18194
589.3 0.500000 + 0.866025i 0.349814 1.69636i −0.500000 + 0.866025i −0.794182 + 1.37556i 1.64400 0.545231i 0 −1.00000 −2.75526 1.18682i −1.58836
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 295.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.f.n 6
3.b odd 2 1 2646.2.f.l 6
7.b odd 2 1 882.2.f.o 6
7.c even 3 1 126.2.e.c 6
7.c even 3 1 126.2.h.d yes 6
7.d odd 6 1 882.2.e.o 6
7.d odd 6 1 882.2.h.p 6
9.c even 3 1 inner 882.2.f.n 6
9.c even 3 1 7938.2.a.bv 3
9.d odd 6 1 2646.2.f.l 6
9.d odd 6 1 7938.2.a.ca 3
21.c even 2 1 2646.2.f.m 6
21.g even 6 1 2646.2.e.p 6
21.g even 6 1 2646.2.h.o 6
21.h odd 6 1 378.2.e.d 6
21.h odd 6 1 378.2.h.c 6
28.g odd 6 1 1008.2.q.g 6
28.g odd 6 1 1008.2.t.h 6
63.g even 3 1 126.2.e.c 6
63.g even 3 1 1134.2.g.m 6
63.h even 3 1 126.2.h.d yes 6
63.h even 3 1 1134.2.g.m 6
63.i even 6 1 2646.2.h.o 6
63.j odd 6 1 378.2.h.c 6
63.j odd 6 1 1134.2.g.l 6
63.k odd 6 1 882.2.e.o 6
63.l odd 6 1 882.2.f.o 6
63.l odd 6 1 7938.2.a.bw 3
63.n odd 6 1 378.2.e.d 6
63.n odd 6 1 1134.2.g.l 6
63.o even 6 1 2646.2.f.m 6
63.o even 6 1 7938.2.a.bz 3
63.s even 6 1 2646.2.e.p 6
63.t odd 6 1 882.2.h.p 6
84.n even 6 1 3024.2.q.g 6
84.n even 6 1 3024.2.t.h 6
252.o even 6 1 3024.2.q.g 6
252.u odd 6 1 1008.2.t.h 6
252.bb even 6 1 3024.2.t.h 6
252.bl odd 6 1 1008.2.q.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.c 6 7.c even 3 1
126.2.e.c 6 63.g even 3 1
126.2.h.d yes 6 7.c even 3 1
126.2.h.d yes 6 63.h even 3 1
378.2.e.d 6 21.h odd 6 1
378.2.e.d 6 63.n odd 6 1
378.2.h.c 6 21.h odd 6 1
378.2.h.c 6 63.j odd 6 1
882.2.e.o 6 7.d odd 6 1
882.2.e.o 6 63.k odd 6 1
882.2.f.n 6 1.a even 1 1 trivial
882.2.f.n 6 9.c even 3 1 inner
882.2.f.o 6 7.b odd 2 1
882.2.f.o 6 63.l odd 6 1
882.2.h.p 6 7.d odd 6 1
882.2.h.p 6 63.t odd 6 1
1008.2.q.g 6 28.g odd 6 1
1008.2.q.g 6 252.bl odd 6 1
1008.2.t.h 6 28.g odd 6 1
1008.2.t.h 6 252.u odd 6 1
1134.2.g.l 6 63.j odd 6 1
1134.2.g.l 6 63.n odd 6 1
1134.2.g.m 6 63.g even 3 1
1134.2.g.m 6 63.h even 3 1
2646.2.e.p 6 21.g even 6 1
2646.2.e.p 6 63.s even 6 1
2646.2.f.l 6 3.b odd 2 1
2646.2.f.l 6 9.d odd 6 1
2646.2.f.m 6 21.c even 2 1
2646.2.f.m 6 63.o even 6 1
2646.2.h.o 6 21.g even 6 1
2646.2.h.o 6 63.i even 6 1
3024.2.q.g 6 84.n even 6 1
3024.2.q.g 6 252.o even 6 1
3024.2.t.h 6 84.n even 6 1
3024.2.t.h 6 252.bb even 6 1
7938.2.a.bv 3 9.c even 3 1
7938.2.a.bw 3 63.l odd 6 1
7938.2.a.bz 3 63.o even 6 1
7938.2.a.ca 3 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{6} - T_{5}^{5} + 7T_{5}^{4} + 12T_{5}^{3} + 33T_{5}^{2} + 18T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{6} + T_{11}^{5} + 7T_{11}^{4} - 12T_{11}^{3} + 33T_{11}^{2} - 18T_{11} + 9 \) Copy content Toggle raw display
\( T_{13}^{6} - 8T_{13}^{5} + 63T_{13}^{4} - 146T_{13}^{3} + 553T_{13}^{2} + 69T_{13} + 4761 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{5} + 10 T^{4} + 21 T^{3} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} + 7 T^{4} + 12 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + 7 T^{4} - 12 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + 63 T^{4} + \cdots + 4761 \) Copy content Toggle raw display
$17$ \( (T^{3} - 4 T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} - 3 T^{2} - 36 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 7 T^{5} + 37 T^{4} + 78 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 5 T^{5} + 91 T^{4} + \cdots + 131769 \) Copy content Toggle raw display
$31$ \( T^{6} - 20 T^{5} + 279 T^{4} + \cdots + 40401 \) Copy content Toggle raw display
$37$ \( (T + 1)^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 33 T^{4} + 18 T^{3} + 1089 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
$47$ \( T^{6} + 9 T^{5} + 135 T^{4} + \cdots + 35721 \) Copy content Toggle raw display
$53$ \( (T^{3} + 15 T^{2} + 66 T + 81)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 14 T^{5} + 157 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$61$ \( T^{6} - 8 T^{5} + 69 T^{4} + \cdots + 8649 \) Copy content Toggle raw display
$67$ \( T^{6} - T^{5} + 113 T^{4} + \cdots + 44521 \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} - 198 T + 1593)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 19 T^{2} + 8 T - 631)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 5 T^{5} + 99 T^{4} + \cdots + 103041 \) Copy content Toggle raw display
$83$ \( T^{6} - 2 T^{5} + 67 T^{4} + \cdots + 21609 \) Copy content Toggle raw display
$89$ \( (T^{3} - 9 T^{2} - 42 T - 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + 572 T^{4} + \cdots + 61504 \) Copy content Toggle raw display
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