LMFDB - Newform orbit 720.2.q.j

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,2,Mod(241,720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("720.241"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	720.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,1,0,-3,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$5.74922894553$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
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_{2}) q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - 2) q^{7} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 1) q^{9} + (\beta_{2} - \beta_1) q^{11}+ \cdots + (5 \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 4) q^{99}+O(q^{100}) $ q + (-b5 + b2) * q^3 + b3 * q^5 + (-b5 - b4 - 2b3 - 2) q^7 + (-2b4 - b3 + b2 - b1 + 1) q^9 + (b2 - b1) * q^11 + (b5 + b4 + b2 + 2b1) q^13 - b2 * q^15 + (b5 - 3b4 + 2b2 - b1 + 2) * q^17 + (b5 - b4 - b1 + 2) * q^19 + (b5 - b4 + 4b3 + b2 - 2b1 + 5) * q^21 + (-4b5 + b4 + 4b3 + b2 - 3b1) q^23 + (-b3 - 1) * q^25 + (-b4 + 4b3 + 2b2 + b1 + 2) * q^27 + (-2b5 - 2b4 + 3b3 + 2b2 - 2b1 + 3) q^29 + (-b5 + b4 + 6b3 + b2) q^31 + (b5 - b4 - 2b3 + b1 - 4) q^33 + (b5 - b2 - b1 + 2) * q^35 + (-b5 + b4 + b1) * q^37 + (2b5 - 6b3 - b2 + 3b1) q^39 + (b5 - b4 - b3 - b2) * q^41 + (-2b5 - 2b4 + 2b3 + 3b2 - 3b1 + 2) q^43 + (-b5 + b4 + 2b3 - b1 + 1) q^45 + (b5 + b4 - 2b3 + 3b2 - 3b1 - 2) q^47 + (-3b5 + 5b4 + 6b3 + 5b2 + 2b1) q^49 + (b5 - b4 + 4b3 + b2 + 4b1 + 2) * q^51 + (2b4 - 2b2 - 4) * q^53 + (-b5 + b4 + b1) * q^55 + (-b5 + b4 + 2b3 + b2 + 2b1 - 2) * q^57 + (4b5 - 2b4 - 2b2 + 2b1) * q^59 + (4b5 + 4b4 + 3b3 - 3b2 + 3b1 + 3) q^61 + (-3b5 + 3b1 - 6) * q^63 + (-2b5 - 2b4 + b2 - b1) * q^65 + (5b5 - 2b3 + 5b1) q^67 + (-3b5 - 5b4 - b3 - 7b1 - 5) q^69 + (3b5 - b4 - 2b2 - 3b1 + 6) q^71 + (4b4 - 4b2 + 4) * q^73 + b5 * q^75 + (3b5 - 3b4 - 6b3 - 3b2) * q^77 + (4b5 + 4b4 - 2b3 - 2b2 + 2b1 - 2) q^79 + (-2b4 - b3 - 2b2 + 2b1 + 4) q^81 + (-b5 - b4 - 6b3 + 2b2 - 2b1 - 6) q^83 + (-3b5 + b4 + 2b3 + b2 - 2b1) q^85 + (-3b5 - 4b4 + 4b3 + 2b2 - 2b1 + 2) q^87 - 11 * q^89 + (3b5 - 3b4 - 3b1 + 12) q^91 + (-2b4 - 4b3 - 5b2 - b1 - 2) q^93 + (-b5 + b4 + 2b3 + b2) q^95 + (2b5 + 2b4 - 8b3 + 2b2 - 2b1 - 8) q^97 + (5b5 + b4 + 2b3 - 3b2 + 2b1 + 4) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{3} - 3 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{21} - 7 q^{23} - 3 q^{25} - 2 q^{27} + 7 q^{29} - 16 q^{31} - 20 q^{33} + 10 q^{35} + 4 q^{37} + 18 q^{39} + q^{41}+ \cdots + 14 q^{99}+O(q^{100}) $ 6 * q + q^3 - 3 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + q^15 + 4 * q^17 + 8 * q^19 + 12 * q^21 - 7 * q^23 - 3 * q^25 - 2 * q^27 + 7 * q^29 - 16 * q^31 - 20 * q^33 + 10 * q^35 + 4 * q^37 + 18 * q^39 + q^41 + 2 * q^43 + 2 * q^45 - 13 * q^47 - 10 * q^49 - 20 * q^53 + 4 * q^55 - 14 * q^57 - 6 * q^59 + 11 * q^61 - 27 * q^63 + q^67 - 33 * q^69 + 28 * q^71 + 32 * q^73 - 2 * q^75 + 12 * q^77 - 6 * q^79 + 29 * q^81 - 21 * q^83 - 2 * q^85 - 2 * q^87 - 66 * q^89 + 60 * q^91 + 2 * q^93 - 4 * q^95 - 30 * q^97 + 14 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2*x^4 + 3*x^3 - 6*x^2 - 9*x + 27 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 45 ) / 27 $ (v^5 + 5v^4 + v^3 + 9v^2 - 6*v - 45) / 27
$\beta_{3}$	$=$	$ ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 $ (-2v^5 - v^4 - 2v^3 + 12*v + 9) / 27
$\beta_{4}$	$=$	$ ( \nu^{5} + 2\nu^{4} - 2\nu^{3} - 6\nu - 18 ) / 9 $ (v^5 + 2v^4 - 2v^3 - 6*v - 18) / 9
$\beta_{5}$	$=$	$ ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} - 24\nu - 72 ) / 27 $ (4v^5 + 2v^4 - 5v^3 + 18v^2 - 24*v - 72) / 27

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 $ b5 - b4 + b3 + b2 + 2
$\nu^{3}$	$=$	$ -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2 $ -b5 - 2b4 - 4b3 + 2*b2 - 2
$\nu^{4}$	$=$	$ -2\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} + 5 $ -2b5 + 2b4 + b3 + 4*b2 + 5
$\nu^{5}$	$=$	$ 2\beta_{5} + \beta_{4} - 10\beta_{3} - 4\beta_{2} + 6\beta _1 + 4 $ 2b5 + b4 - 10b3 - 4b2 + 6b1 + 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$1$	$\beta_{3}$

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} $

241.1

−1.62241	+	0.606458i
0.403374	−	1.68443i
1.71903	+	0.211943i
−1.62241	−	0.606458i
0.403374	+	1.68443i
1.71903	−	0.211943i

−1.62241

−

0.606458i

−0.500000

−

0.866025i

−2.62241

4.54214i

2.26442

1.96784i

241.2

0.403374

1.68443i

−0.500000

−

0.866025i

−0.596626

1.03339i

−2.67458

1.35891i

241.3

1.71903

−

0.211943i

−0.500000

−

0.866025i

0.719035

−

1.24540i

2.91016

−

0.728674i

481.1

−1.62241

0.606458i

−0.500000

0.866025i

−2.62241

−

4.54214i

2.26442

−

1.96784i

481.2

0.403374

−

1.68443i

−0.500000

0.866025i

−0.596626

−

1.03339i

−2.67458

−

1.35891i

481.3

1.71903

0.211943i

−0.500000

0.866025i

0.719035

1.24540i

2.91016

0.728674i

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	720.2.q.j		6
3.b	odd	2	1		2160.2.q.j		6
4.b	odd	2	1		360.2.q.d	✓	6
9.c	even	3	1	inner	720.2.q.j		6
9.c	even	3	1		6480.2.a.bx		3
9.d	odd	6	1		2160.2.q.j		6
9.d	odd	6	1		6480.2.a.bu		3
12.b	even	2	1		1080.2.q.d		6
36.f	odd	6	1		360.2.q.d	✓	6
36.f	odd	6	1		3240.2.a.r		3
36.h	even	6	1		1080.2.q.d		6
36.h	even	6	1		3240.2.a.q		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
360.2.q.d	✓	6	4.b	odd	2	1
360.2.q.d	✓	6	36.f	odd	6	1
720.2.q.j		6	1.a	even	1	1	trivial
720.2.q.j		6	9.c	even	3	1	inner
1080.2.q.d		6	12.b	even	2	1
1080.2.q.d		6	36.h	even	6	1
2160.2.q.j		6	3.b	odd	2	1
2160.2.q.j		6	9.d	odd	6	1
3240.2.a.q		3	36.h	even	6	1
3240.2.a.r		3	36.f	odd	6	1
6480.2.a.bu		3	9.d	odd	6	1
6480.2.a.bx		3	9.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(720, [\chi])$:

$ T_{7}^{6} + 5T_{7}^{5} + 28T_{7}^{4} + 3T_{7}^{3} + 54T_{7}^{2} + 27T_{7} + 81 $

T7^6 + 5*T7^5 + 28*T7^4 + 3*T7^3 + 54*T7^2 + 27*T7 + 81

$ T_{11}^{6} + 2T_{11}^{5} + 12T_{11}^{4} + 8T_{11}^{3} + 88T_{11}^{2} + 96T_{11} + 144 $

T11^6 + 2*T11^5 + 12*T11^4 + 8*T11^3 + 88*T11^2 + 96*T11 + 144

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - T^{5} + \cdots + 27 $ T^6 - T^5 - 2T^4 + 3T^3 - 6T^2 - 9T + 27
$5$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$7$	$ T^{6} + 5 T^{5} + \cdots + 81 $ T^6 + 5T^5 + 28T^4 + 3T^3 + 54T^2 + 27*T + 81
$11$	$ T^{6} + 2 T^{5} + \cdots + 144 $ T^6 + 2T^5 + 12T^4 + 8T^3 + 88T^2 + 96*T + 144
$13$	$ T^{6} + 24 T^{4} + \cdots + 1296 $ T^6 + 24T^4 + 72T^3 + 576T^2 + 864T + 1296
$17$	$ (T^{3} - 2 T^{2} + \cdots + 108)^{2} $ (T^3 - 2T^2 - 36T + 108)^2
$19$	$ (T^{3} - 4 T^{2} - 4 T + 4)^{2} $ (T^3 - 4T^2 - 4T + 4)^2
$23$	$ T^{6} + 7 T^{5} + \cdots + 91809 $ T^6 + 7T^5 + 100T^4 + 249T^3 + 4722T^2 + 15453*T + 91809
$29$	$ T^{6} - 7 T^{5} + \cdots + 729 $ T^6 - 7T^5 + 54T^4 - 19T^3 + 214T^2 - 135*T + 729
$31$	$ T^{6} + 16 T^{5} + \cdots + 11664 $ T^6 + 16T^5 + 180T^4 + 1000T^3 + 4048T^2 + 8208*T + 11664
$37$	$ (T^{3} - 2 T^{2} - 8 T + 12)^{2} $ (T^3 - 2T^2 - 8T + 12)^2
$41$	$ T^{6} - T^{5} + 10 T^{4} + \cdots + 9 $ T^6 - T^5 + 10T^4 + 15T^3 + 78T^2 + 27T + 9
$43$	$ T^{6} - 2 T^{5} + \cdots + 11664 $ T^6 - 2T^5 + 40T^4 - 144T^3 + 1512T^2 - 3888*T + 11664
$47$	$ T^{6} + 13 T^{5} + \cdots + 1261129 $ T^6 + 13T^5 + 254T^4 + 1141T^3 + 21824T^2 + 95455*T + 1261129
$53$	$ (T^{3} + 10 T^{2} + \cdots - 24)^{2} $ (T^3 + 10T^2 + 12T - 24)^2
$59$	$ T^{6} + 6 T^{5} + \cdots + 5184 $ T^6 + 6T^5 + 96T^4 - 504T^3 + 3168T^2 - 4320*T + 5184
$61$	$ T^{6} - 11 T^{5} + \cdots + 269361 $ T^6 - 11T^5 + 162T^4 - 587T^3 + 7390T^2 - 21279*T + 269361
$67$	$ T^{6} - T^{5} + \cdots + 12769 $ T^6 - T^5 + 134T^4 + 359T^3 + 17576T^2 + 15029T + 12769
$71$	$ (T^{3} - 14 T^{2} + \cdots + 36)^{2} $ (T^3 - 14T^2 - 20T + 36)^2
$73$	$ (T^{3} - 16 T^{2} + 384)^{2} $ (T^3 - 16*T^2 + 384)^2
$79$	$ T^{6} + 6 T^{5} + \cdots + 10816 $ T^6 + 6T^5 + 120T^4 - 712T^3 + 6432T^2 - 8736*T + 10816
$83$	$ T^{6} + 21 T^{5} + \cdots + 59049 $ T^6 + 21T^5 + 312T^4 + 2223T^3 + 11538T^2 + 31347*T + 59049
$89$	$ (T + 11)^{6} $ (T + 11)^6
$97$	$ T^{6} + 30 T^{5} + \cdots + 1201216 $ T^6 + 30T^5 + 744T^4 + 6872T^3 + 57216T^2 - 170976*T + 1201216
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Level:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	720.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} + \beta_{2}) q^{3} + \beta_{3} q^{5} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - 2) q^{7} + ( - 2 \beta_{4} - \beta_{3} + \beta_{2} + \cdots + 1) q^{9} + (\beta_{2} - \beta_1) q^{11}+ \cdots + (5 \beta_{5} + \beta_{4} + 2 \beta_{3} + \cdots + 4) q^{99}+O(q^{100}) \) q + (-b5 + b2) * q^3 + b3 * q^5 + (-b5 - b4 - 2b3 - 2) q^7 + (-2b4 - b3 + b2 - b1 + 1) q^9 + (b2 - b1) * q^11 + (b5 + b4 + b2 + 2b1) q^13 - b2 * q^15 + (b5 - 3b4 + 2b2 - b1 + 2) * q^17 + (b5 - b4 - b1 + 2) * q^19 + (b5 - b4 + 4b3 + b2 - 2b1 + 5) * q^21 + (-4b5 + b4 + 4b3 + b2 - 3b1) q^23 + (-b3 - 1) * q^25 + (-b4 + 4b3 + 2b2 + b1 + 2) * q^27 + (-2b5 - 2b4 + 3b3 + 2b2 - 2b1 + 3) q^29 + (-b5 + b4 + 6b3 + b2) q^31 + (b5 - b4 - 2b3 + b1 - 4) q^33 + (b5 - b2 - b1 + 2) * q^35 + (-b5 + b4 + b1) * q^37 + (2b5 - 6b3 - b2 + 3b1) q^39 + (b5 - b4 - b3 - b2) * q^41 + (-2b5 - 2b4 + 2b3 + 3b2 - 3b1 + 2) q^43 + (-b5 + b4 + 2b3 - b1 + 1) q^45 + (b5 + b4 - 2b3 + 3b2 - 3b1 - 2) q^47 + (-3b5 + 5b4 + 6b3 + 5b2 + 2b1) q^49 + (b5 - b4 + 4b3 + b2 + 4b1 + 2) * q^51 + (2b4 - 2b2 - 4) * q^53 + (-b5 + b4 + b1) * q^55 + (-b5 + b4 + 2b3 + b2 + 2b1 - 2) * q^57 + (4b5 - 2b4 - 2b2 + 2b1) * q^59 + (4b5 + 4b4 + 3b3 - 3b2 + 3b1 + 3) q^61 + (-3b5 + 3b1 - 6) * q^63 + (-2b5 - 2b4 + b2 - b1) * q^65 + (5b5 - 2b3 + 5b1) q^67 + (-3b5 - 5b4 - b3 - 7b1 - 5) q^69 + (3b5 - b4 - 2b2 - 3b1 + 6) q^71 + (4b4 - 4b2 + 4) * q^73 + b5 * q^75 + (3b5 - 3b4 - 6b3 - 3b2) * q^77 + (4b5 + 4b4 - 2b3 - 2b2 + 2b1 - 2) q^79 + (-2b4 - b3 - 2b2 + 2b1 + 4) q^81 + (-b5 - b4 - 6b3 + 2b2 - 2b1 - 6) q^83 + (-3b5 + b4 + 2b3 + b2 - 2b1) q^85 + (-3b5 - 4b4 + 4b3 + 2b2 - 2b1 + 2) q^87 - 11 * q^89 + (3b5 - 3b4 - 3b1 + 12) q^91 + (-2b4 - 4b3 - 5b2 - b1 - 2) q^93 + (-b5 + b4 + 2b3 + b2) q^95 + (2b5 + 2b4 - 8b3 + 2b2 - 2b1 - 8) q^97 + (5b5 + b4 + 2b3 - 3b2 + 2b1 + 4) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{3} - 3 q^{5} - 5 q^{7} + 5 q^{9} - 2 q^{11} + q^{15} + 4 q^{17} + 8 q^{19} + 12 q^{21} - 7 q^{23} - 3 q^{25} - 2 q^{27} + 7 q^{29} - 16 q^{31} - 20 q^{33} + 10 q^{35} + 4 q^{37} + 18 q^{39} + q^{41}+ \cdots + 14 q^{99}+O(q^{100}) \) 6 * q + q^3 - 3 * q^5 - 5 * q^7 + 5 * q^9 - 2 * q^11 + q^15 + 4 * q^17 + 8 * q^19 + 12 * q^21 - 7 * q^23 - 3 * q^25 - 2 * q^27 + 7 * q^29 - 16 * q^31 - 20 * q^33 + 10 * q^35 + 4 * q^37 + 18 * q^39 + q^41 + 2 * q^43 + 2 * q^45 - 13 * q^47 - 10 * q^49 - 20 * q^53 + 4 * q^55 - 14 * q^57 - 6 * q^59 + 11 * q^61 - 27 * q^63 + q^67 - 33 * q^69 + 28 * q^71 + 32 * q^73 - 2 * q^75 + 12 * q^77 - 6 * q^79 + 29 * q^81 - 21 * q^83 - 2 * q^85 - 2 * q^87 - 66 * q^89 + 60 * q^91 + 2 * q^93 - 4 * q^95 - 30 * q^97 + 14 * q^99

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Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	720.2.q.j

Level	$720$
Weight	$2$
Character orbit	720.q
Analytic conductor	$5.749$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(5.74922894553\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.954288.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{5} + 5\nu^{4} + \nu^{3} + 9\nu^{2} - 6\nu - 45 ) / 27 \) (v^5 + 5v^4 + v^3 + 9v^2 - 6*v - 45) / 27
\(\beta_{3}\)	\(=\)	\( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) (-2v^5 - v^4 - 2v^3 + 12*v + 9) / 27
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} + 2\nu^{4} - 2\nu^{3} - 6\nu - 18 ) / 9 \) (v^5 + 2v^4 - 2v^3 - 6*v - 18) / 9
\(\beta_{5}\)	\(=\)	\( ( 4\nu^{5} + 2\nu^{4} - 5\nu^{3} + 18\nu^{2} - 24\nu - 72 ) / 27 \) (4v^5 + 2v^4 - 5v^3 + 18v^2 - 24*v - 72) / 27

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 2 \) b5 - b4 + b3 + b2 + 2
\(\nu^{3}\)	\(=\)	\( -\beta_{5} - 2\beta_{4} - 4\beta_{3} + 2\beta_{2} - 2 \) -b5 - 2b4 - 4b3 + 2*b2 - 2
\(\nu^{4}\)	\(=\)	\( -2\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} + 5 \) -2b5 + 2b4 + b3 + 4*b2 + 5
\(\nu^{5}\)	\(=\)	\( 2\beta_{5} + \beta_{4} - 10\beta_{3} - 4\beta_{2} + 6\beta _1 + 4 \) 2b5 + b4 - 10b3 - 4b2 + 6b1 + 4

\(n\)	\(181\)	\(271\)	\(577\)	\(641\)
\(\chi(n)\)	\(1\)	\(1\)	\(1\)	\(\beta_{3}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 241.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} \) T^6
$3$	\( T^{6} - T^{5} + \cdots + 27 \) T^6 - T^5 - 2T^4 + 3T^3 - 6T^2 - 9T + 27
$5$	\( (T^{2} + T + 1)^{3} \) (T^2 + T + 1)^3
$7$	\( T^{6} + 5 T^{5} + \cdots + 81 \) T^6 + 5T^5 + 28T^4 + 3T^3 + 54T^2 + 27*T + 81
$11$	\( T^{6} + 2 T^{5} + \cdots + 144 \) T^6 + 2T^5 + 12T^4 + 8T^3 + 88T^2 + 96*T + 144
$13$	\( T^{6} + 24 T^{4} + \cdots + 1296 \) T^6 + 24T^4 + 72T^3 + 576T^2 + 864T + 1296
$17$	\( (T^{3} - 2 T^{2} + \cdots + 108)^{2} \) (T^3 - 2T^2 - 36T + 108)^2
$19$	\( (T^{3} - 4 T^{2} - 4 T + 4)^{2} \) (T^3 - 4T^2 - 4T + 4)^2
$23$	\( T^{6} + 7 T^{5} + \cdots + 91809 \) T^6 + 7T^5 + 100T^4 + 249T^3 + 4722T^2 + 15453*T + 91809
$29$	\( T^{6} - 7 T^{5} + \cdots + 729 \) T^6 - 7T^5 + 54T^4 - 19T^3 + 214T^2 - 135*T + 729
$31$	\( T^{6} + 16 T^{5} + \cdots + 11664 \) T^6 + 16T^5 + 180T^4 + 1000T^3 + 4048T^2 + 8208*T + 11664
$37$	\( (T^{3} - 2 T^{2} - 8 T + 12)^{2} \) (T^3 - 2T^2 - 8T + 12)^2
$41$	\( T^{6} - T^{5} + 10 T^{4} + \cdots + 9 \) T^6 - T^5 + 10T^4 + 15T^3 + 78T^2 + 27T + 9
$43$	\( T^{6} - 2 T^{5} + \cdots + 11664 \) T^6 - 2T^5 + 40T^4 - 144T^3 + 1512T^2 - 3888*T + 11664
$47$	\( T^{6} + 13 T^{5} + \cdots + 1261129 \) T^6 + 13T^5 + 254T^4 + 1141T^3 + 21824T^2 + 95455*T + 1261129
$53$	\( (T^{3} + 10 T^{2} + \cdots - 24)^{2} \) (T^3 + 10T^2 + 12T - 24)^2
$59$	\( T^{6} + 6 T^{5} + \cdots + 5184 \) T^6 + 6T^5 + 96T^4 - 504T^3 + 3168T^2 - 4320*T + 5184
$61$	\( T^{6} - 11 T^{5} + \cdots + 269361 \) T^6 - 11T^5 + 162T^4 - 587T^3 + 7390T^2 - 21279*T + 269361
$67$	\( T^{6} - T^{5} + \cdots + 12769 \) T^6 - T^5 + 134T^4 + 359T^3 + 17576T^2 + 15029T + 12769
$71$	\( (T^{3} - 14 T^{2} + \cdots + 36)^{2} \) (T^3 - 14T^2 - 20T + 36)^2
$73$	\( (T^{3} - 16 T^{2} + 384)^{2} \) (T^3 - 16*T^2 + 384)^2
$79$	\( T^{6} + 6 T^{5} + \cdots + 10816 \) T^6 + 6T^5 + 120T^4 - 712T^3 + 6432T^2 - 8736*T + 10816
$83$	\( T^{6} + 21 T^{5} + \cdots + 59049 \) T^6 + 21T^5 + 312T^4 + 2223T^3 + 11538T^2 + 31347*T + 59049
$89$	\( (T + 11)^{6} \) (T + 11)^6
$97$	\( T^{6} + 30 T^{5} + \cdots + 1201216 \) T^6 + 30T^5 + 744T^4 + 6872T^3 + 57216T^2 - 170976*T + 1201216
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