Properties

Label 363.3.b.j
Level $363$
Weight $3$
Character orbit 363.b
Analytic conductor $9.891$
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] = [363,3,Mod(122,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.122");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 363.b (of order \(2\), degree \(1\), 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: \(9.89103359628\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 21x^{4} + 111x^{2} + 47 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{4} + \beta_{2} - 2) q^{6} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + 1) q^{3} + (\beta_{5} - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{4} + \beta_{2} - 2) q^{6} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 2 \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} - 2 \beta_1 - 3) q^{9} + ( - 2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{10} + ( - 2 \beta_{5} + \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - \beta_1 + 2) q^{12} + (\beta_{4} + \beta_{3} - 7) q^{13} + (\beta_{4} - 5 \beta_{3} - 2 \beta_{2} - 4 \beta_1 + 2) q^{14} + ( - 4 \beta_{5} + 3 \beta_{3} - 3 \beta_{2} + 7 \beta_1 + 5) q^{15} + (2 \beta_{4} + 2 \beta_{3} + 5) q^{16} + (3 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{17} + ( - 3 \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 15) q^{18} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 + 14) q^{19} + (2 \beta_{4} - 10 \beta_{3} - 4 \beta_{2} + 7 \beta_1 + 4) q^{20} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 4) q^{21} + (3 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1 - 2) q^{23} + ( - 4 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 7 \beta_1 + 9) q^{24} + (\beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 - 18) q^{25} + (\beta_{5} + \beta_{4} - 6 \beta_{3} - 3 \beta_{2} - 8 \beta_1 + 2) q^{26} + ( - 3 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} - 6) q^{27} + (3 \beta_{5} - 3 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 22) q^{28} + ( - 2 \beta_{4} + 10 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 4) q^{29} + (6 \beta_{5} - 5 \beta_{3} + 12 \beta_1 - 25) q^{30} + ( - 4 \beta_{5} + 3 \beta_{4} + 7 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{31} + ( - 2 \beta_{5} + 2 \beta_{4} - 8 \beta_{3} - 2 \beta_{2} - 5 \beta_1 + 4) q^{32} + (\beta_{5} + \beta_{4} + \beta_{2} - \beta_1 + 11) q^{34} + ( - 6 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 8 \beta_1 + 6) q^{35} + ( - 6 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 5 \beta_{2} + 21 \beta_1 + 9) q^{36} + (2 \beta_{5} + 3 \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 7) q^{37} + (2 \beta_{5} + \beta_{4} - 7 \beta_{3} - 4 \beta_{2} + 16 \beta_1 + 2) q^{38} + (\beta_{5} + \beta_{4} + 7 \beta_{3} + 2 \beta_{2} + 5 \beta_1 - 16) q^{39} + (5 \beta_{5} - 6 \beta_{4} - 11 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 49) q^{40} + (2 \beta_{5} - \beta_{4} + 3 \beta_{3} - 19 \beta_1 - 2) q^{41} + (\beta_{5} + 5 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 13 \beta_1 - 31) q^{42} + ( - 5 \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 26) q^{43} + (4 \beta_{5} - 7 \beta_{4} - 7 \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 21) q^{45} + (4 \beta_{5} + \beta_{4} - 3 \beta_{3} + 4 \beta_{2} - 4 \beta_1 - 10) q^{46} + ( - \beta_{4} + 5 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{47} + (2 \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 4 \beta_{2} + 10 \beta_1 - 13) q^{48} + (5 \beta_{5} - 6 \beta_{4} - 11 \beta_{3} + 5 \beta_{2} - 5 \beta_1 + 23) q^{49} + ( - 3 \beta_{5} - 2 \beta_{4} + 13 \beta_{3} + 7 \beta_{2} - 19 \beta_1 - 4) q^{50} + (11 \beta_{5} + \beta_{4} - 3 \beta_{3} + 5 \beta_{2} - 8 \beta_1 + 28) q^{51} + ( - 3 \beta_{5} - 5 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 19) q^{52} + ( - 7 \beta_{5} + 7 \beta_{3} + 7 \beta_{2} + 22 \beta_1) q^{53} + (4 \beta_{5} - \beta_{4} - 16 \beta_{3} - 11 \beta_{2} + 5 \beta_1 + 27) q^{54} + ( - 6 \beta_{5} + \beta_{4} + \beta_{3} + 4 \beta_{2} + 2) q^{56} + (3 \beta_{5} - 15 \beta_{3} + 6 \beta_1) q^{57} + ( - 3 \beta_{5} + 14 \beta_{4} + 17 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 9) q^{58} + ( - 7 \beta_{5} + 7 \beta_{3} + 7 \beta_{2} - 7 \beta_1) q^{59} + (2 \beta_{5} - 5 \beta_{4} - 12 \beta_{3} + 11 \beta_{2} - 26 \beta_1 - 92) q^{60} + ( - 4 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 44) q^{61} + (7 \beta_{5} + 3 \beta_{4} - 22 \beta_{3} - 13 \beta_{2} + 9 \beta_1 + 6) q^{62} + ( - 13 \beta_{5} - 2 \beta_{4} + 13 \beta_{3} + 5 \beta_{2} + 22 \beta_1 - 21) q^{63} + ( - 3 \beta_{5} - 2 \beta_{4} + \beta_{3} - 3 \beta_{2} + 3 \beta_1 + 49) q^{64} + (4 \beta_{5} + \beta_{4} - 9 \beta_{3} - 6 \beta_{2} - 15 \beta_1 + 2) q^{65} + (8 \beta_{5} - 9 \beta_{4} - 17 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 4) q^{67} + (12 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 6 \beta_{2} - \beta_1 - 6) q^{68} + (11 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 8 \beta_{2} - 8 \beta_1 + 22) q^{69} + ( - 11 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} - 11 \beta_{2} + 11 \beta_1 + 56) q^{70} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 11 \beta_1) q^{71} + (10 \beta_{5} - 5 \beta_{4} - 15 \beta_{3} - \beta_{2} + 5 \beta_1 - 51) q^{72} + ( - 17 \beta_{5} + 2 \beta_{4} + 19 \beta_{3} - 17 \beta_{2} + 17 \beta_1 + 12) q^{73} + (\beta_{5} + 3 \beta_{4} - 16 \beta_{3} - 7 \beta_{2} - 16 \beta_1 + 6) q^{74} + ( - 4 \beta_{5} - \beta_{4} + 19 \beta_{3} - 2 \beta_{2} - 11 \beta_1 + 5) q^{75} + (19 \beta_{5} - 7 \beta_{4} - 26 \beta_{3} + 19 \beta_{2} - 19 \beta_1 - 68) q^{76} + (5 \beta_{5} + 9 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} - 20 \beta_1 - 28) q^{78} + ( - 16 \beta_{5} - 3 \beta_{4} + 13 \beta_{3} - 16 \beta_{2} + 16 \beta_1) q^{79} + ( - 11 \beta_{5} + 2 \beta_{4} + \beta_{3} + 7 \beta_{2} - 30 \beta_1 + 4) q^{80} + (13 \beta_{5} + 2 \beta_{4} - \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 45) q^{81} + ( - 18 \beta_{5} + 3 \beta_{4} + 21 \beta_{3} - 18 \beta_{2} + 18 \beta_1 + 133) q^{82} + ( - 9 \beta_{5} + 3 \beta_{4} - 6 \beta_{3} + 3 \beta_{2} + 27 \beta_1 + 6) q^{83} + ( - 9 \beta_{5} - 19 \beta_{3} - 18 \beta_1 + 64) q^{84} + (20 \beta_{5} + 3 \beta_{4} - 17 \beta_{3} + 20 \beta_{2} - 20 \beta_1 + 55) q^{85} + (3 \beta_{5} - 2 \beta_{4} + 7 \beta_{3} + \beta_{2} - 9 \beta_1 - 4) q^{86} + ( - 2 \beta_{5} - 5 \beta_{4} + 12 \beta_{3} - \beta_{2} + 26 \beta_1 + 72) q^{87} + ( - 3 \beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - 12 \beta_1 + 2) q^{89} + (7 \beta_{5} - 12 \beta_{4} + 19 \beta_{3} + 30 \beta_{2} - 22 \beta_1 - 51) q^{90} + ( - 8 \beta_{5} - 13 \beta_{4} - 5 \beta_{3} - 8 \beta_{2} + 8 \beta_1 + 50) q^{91} + (9 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} - 19 \beta_1 - 6) q^{92} + (11 \beta_{5} - \beta_{4} - 4 \beta_{3} - 2 \beta_{2} + 19 \beta_1 - 47) q^{93} + ( - 5 \beta_{5} + 7 \beta_{4} + 12 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 20) q^{94} + ( - 14 \beta_{5} - \beta_{4} + 19 \beta_{3} + 16 \beta_{2} - 22 \beta_1 - 2) q^{95} + ( - 6 \beta_{5} + 7 \beta_{4} - 6 \beta_{3} - 7 \beta_{2} - 12 \beta_1 - 58) q^{96} + (25 \beta_{5} + 2 \beta_{4} - 23 \beta_{3} + 25 \beta_{2} - 25 \beta_1 + 21) q^{97} + ( - 11 \beta_{5} - 6 \beta_{4} + 41 \beta_{3} + 23 \beta_{2} + 14 \beta_1 - 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 18 q^{4} - 10 q^{6} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 18 q^{4} - 10 q^{6} - 22 q^{9} + 18 q^{10} + 14 q^{12} - 42 q^{13} + 28 q^{15} + 30 q^{16} + 94 q^{18} + 84 q^{19} - 28 q^{21} + 48 q^{24} - 108 q^{25} - 38 q^{27} + 132 q^{28} - 148 q^{30} + 66 q^{34} + 46 q^{36} - 42 q^{37} - 82 q^{39} - 294 q^{40} - 206 q^{42} - 156 q^{43} - 118 q^{45} - 60 q^{46} - 88 q^{48} + 138 q^{49} + 182 q^{51} + 114 q^{52} + 140 q^{54} - 24 q^{57} - 54 q^{58} - 562 q^{60} + 264 q^{61} - 122 q^{63} + 294 q^{64} + 24 q^{67} + 152 q^{69} + 336 q^{70} - 306 q^{72} + 72 q^{73} + 62 q^{75} - 408 q^{76} - 194 q^{78} - 250 q^{81} + 798 q^{82} + 328 q^{84} + 330 q^{85} + 462 q^{87} - 230 q^{90} + 300 q^{91} - 266 q^{93} + 120 q^{94} - 386 q^{96} + 126 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 21x^{4} + 111x^{2} + 47 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} + 11\nu + 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 16\nu^{3} + 10\nu^{2} - 55\nu + 1 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 5\nu^{4} + 16\nu^{3} + 62\nu^{2} + 55\nu + 65 ) / 12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 22\nu^{3} + 16\nu^{2} - 109\nu + 43 ) / 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} + \beta_{2} - \beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{3} + \beta_{2} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{5} + 2\beta_{4} + 14\beta_{3} - 12\beta_{2} + 12\beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{5} + 2\beta_{4} - 24\beta_{3} - 18\beta_{2} + 107\beta _1 + 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
122.1
3.50091i
2.87759i
0.680517i
0.680517i
2.87759i
3.50091i
3.50091i −1.38788 2.65966i −8.25635 7.89925i −9.31122 + 4.85883i 1.81458 14.9011i −5.14758 + 7.38257i 27.6545
122.2 2.87759i 2.10316 + 2.13932i −4.28055 4.94788i 6.15610 6.05206i −11.1801 0.807314i −0.153394 + 8.99869i −14.2380
122.3 0.680517i 1.28471 2.71100i 3.53690 6.49002i −1.84488 0.874269i 9.36551 5.12898i −5.69902 6.96571i −4.41657
122.4 0.680517i 1.28471 + 2.71100i 3.53690 6.49002i −1.84488 + 0.874269i 9.36551 5.12898i −5.69902 + 6.96571i −4.41657
122.5 2.87759i 2.10316 2.13932i −4.28055 4.94788i 6.15610 + 6.05206i −11.1801 0.807314i −0.153394 8.99869i −14.2380
122.6 3.50091i −1.38788 + 2.65966i −8.25635 7.89925i −9.31122 4.85883i 1.81458 14.9011i −5.14758 7.38257i 27.6545
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 122.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.3.b.j 6
3.b odd 2 1 inner 363.3.b.j 6
11.b odd 2 1 363.3.b.k yes 6
11.c even 5 4 363.3.h.q 24
11.d odd 10 4 363.3.h.p 24
33.d even 2 1 363.3.b.k yes 6
33.f even 10 4 363.3.h.p 24
33.h odd 10 4 363.3.h.q 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.3.b.j 6 1.a even 1 1 trivial
363.3.b.j 6 3.b odd 2 1 inner
363.3.b.k yes 6 11.b odd 2 1
363.3.b.k yes 6 33.d even 2 1
363.3.h.p 24 11.d odd 10 4
363.3.h.p 24 33.f even 10 4
363.3.h.q 24 11.c even 5 4
363.3.h.q 24 33.h odd 10 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):

\( T_{2}^{6} + 21T_{2}^{4} + 111T_{2}^{2} + 47 \) Copy content Toggle raw display
\( T_{5}^{6} + 129T_{5}^{4} + 5187T_{5}^{2} + 64343 \) Copy content Toggle raw display
\( T_{7}^{3} - 108T_{7} + 190 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 21 T^{4} + 111 T^{2} + \cdots + 47 \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + 19 T^{4} - 42 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( T^{6} + 129 T^{4} + 5187 T^{2} + \cdots + 64343 \) Copy content Toggle raw display
$7$ \( (T^{3} - 108 T + 190)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( (T^{3} + 21 T^{2} + 69 T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 909 T^{4} + 12003 T^{2} + \cdots + 47 \) Copy content Toggle raw display
$19$ \( (T^{3} - 42 T^{2} + 468 T - 1458)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 900 T^{4} + 98736 T^{2} + \cdots + 2752508 \) Copy content Toggle raw display
$29$ \( T^{6} + 3669 T^{4} + \cdots + 593987175 \) Copy content Toggle raw display
$31$ \( (T^{3} - 1350 T - 18334)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 21 T^{2} - 663 T + 2115)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 8205 T^{4} + \cdots + 11934438575 \) Copy content Toggle raw display
$43$ \( (T^{3} + 78 T^{2} + 876 T - 180)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 1080 T^{4} + \cdots + 18183548 \) Copy content Toggle raw display
$53$ \( T^{6} + 13713 T^{4} + \cdots + 16518150423 \) Copy content Toggle raw display
$59$ \( T^{6} + 8232 T^{4} + \cdots + 2211801200 \) Copy content Toggle raw display
$61$ \( (T^{3} - 132 T^{2} + 4872 T - 54576)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 12 T^{2} - 9006 T + 259810)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 2868 T^{4} + \cdots + 1522800 \) Copy content Toggle raw display
$73$ \( (T^{3} - 36 T^{2} - 10488 T + 150820)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 9630 T + 356894)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 21600 T^{4} + \cdots + 280649740572 \) Copy content Toggle raw display
$89$ \( T^{6} + 4101 T^{4} + \cdots + 1393457175 \) Copy content Toggle raw display
$97$ \( (T^{3} - 63 T^{2} - 21189 T - 563825)^{2} \) Copy content Toggle raw display
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