LMFDB - Newform orbit 1386.2.k.w

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1386,2,Mod(793,1386)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1386.793"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1386, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,3,0,-3,0,0,0,-6,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.1156923.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 $ x^6 - 3x^5 + 12x^4 - 19x^3 + 27x^2 - 18*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 462)
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_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + \beta_{4} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} - q^{8} + (\beta_{4} - \beta_1) q^{10} + (\beta_{2} + 1) q^{11} + 2 \beta_1 q^{13} + ( - \beta_{4} + \beta_{3} - 1) q^{14}+ \cdots + (\beta_{5} - 2 \beta_{4} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) $ q - b2 * q^2 + (-b2 - 1) * q^4 + b4 * q^5 + (b5 - b4 + b3) * q^7 - q^8 + (b4 - b1) * q^10 + (b2 + 1) * q^11 + 2b1 q^13 + (-b4 + b3 - 1) * q^14 + b2 * q^16 + (b2 + 1) * q^17 + (2b5 + b3 + b1 + 1) q^19 - b1 * q^20 + q^22 + (-2b5 + b4 - b3 - 4b2 - b1 - 1) * q^23 + (b5 - 2b4 + 2b3 + b1 - 1) * q^25 + 2b4 q^26 + (-b5 - 1) * q^28 + (-b5 + b3 - 4) * q^29 + (2b5 - 4b4 + 4b3 + 2b1 - 2) * q^31 + (b2 + 1) * q^32 + q^34 + (-b5 + 2b4 + 3b2 - 3b1) q^35 + (2b5 + b3 + 4b2 + b1 + 1) * q^37 + (b5 + 2b3 - b1 - 1) q^38 - b4 * q^40 + (-b5 + b3 - 5) * q^41 + (-b5 + b3 - 2b1 + 2) q^43 - b2 * q^44 + (-b5 + b4 - 2b3 - 4b2 - 3) * q^46 + (-2b5 + b4 - b3 + 4b2 - b1 - 1) * q^47 + (-2b4 - b3 - b2 + b1 - 4) q^49 + (-b5 + b3 - 2) * q^50 + (2b4 - 2b1) * q^52 + (-2b4 + 2b1) * q^53 + b1 * q^55 + (-b5 + b4 - b3) * q^56 + (-2b5 + 2b4 - b3 + 2b2 - b1 - 1) q^58 + (b5 + 2b4 + 2b3 + 4b2 - 3b1 + 3) * q^59 + (-4b5 + 5b4 - 2b3 - 4b2 - 2b1 - 2) q^61 + (-2b5 + 2b3 - 4) * q^62 + q^64 + (4b5 - 4b4 + 2b3 - 10b2 + 2b1 + 2) q^65 + (b5 - 2b4 + 2b3 - b2 + b1 - 2) * q^67 - b2 * q^68 + (-b5 - b3 + 2b2 - 2b1 + 3) * q^70 + (-b5 + b3 - 2) * q^71 + (-2b5 + 2b4 - 4b3 - 2b2) * q^73 + (b5 + 2b3 + 4b2 - b1 + 3) * q^74 + (-b5 + b3 - 2b1 - 2) q^76 + (b5 + 1) * q^77 + (-3b4 + 4b2) * q^79 + (-b4 + b1) * q^80 + (-2b5 + 2b4 - b3 + 3b2 - b1 - 1) q^82 + (b5 - b3 + 4b1 + 3) q^83 + b1 * q^85 + (-2b5 - b3 - 4b2 - b1 - 1) * q^86 + (-b2 - 1) * q^88 + (2b4 + 12b2) * q^89 + (4b4 + 2b3 + 2b2 - 2b1 - 6) * q^91 + (b5 - b3 + b1 - 2) * q^92 + (-b5 + b4 - 2b3 + 4b2 + 5) * q^94 + (2b5 + 4b3 - 6b2 - 2b1 - 8) * q^95 + (-2b5 + 2b3 + 1) * q^97 + (b5 - 2b4 + 4b2 + 3b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} + 3 q^{11} - 3 q^{14} - 3 q^{16} + 3 q^{17} + 3 q^{19} + 6 q^{22} + 9 q^{23} - 3 q^{25} - 3 q^{28} - 18 q^{29} - 6 q^{31} + 3 q^{32} + 6 q^{34} - 6 q^{35} - 9 q^{37} - 3 q^{38}+ \cdots - 15 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 + 3 * q^11 - 3 * q^14 - 3 * q^16 + 3 * q^17 + 3 * q^19 + 6 * q^22 + 9 * q^23 - 3 * q^25 - 3 * q^28 - 18 * q^29 - 6 * q^31 + 3 * q^32 + 6 * q^34 - 6 * q^35 - 9 * q^37 - 3 * q^38 - 24 * q^41 + 18 * q^43 + 3 * q^44 - 9 * q^46 - 15 * q^47 - 24 * q^49 - 6 * q^50 - 9 * q^58 + 9 * q^59 + 6 * q^61 - 12 * q^62 + 6 * q^64 + 36 * q^65 - 6 * q^67 + 3 * q^68 + 12 * q^70 - 6 * q^71 + 9 * q^74 - 6 * q^76 + 3 * q^77 - 12 * q^79 - 12 * q^82 + 12 * q^83 + 9 * q^86 - 3 * q^88 - 36 * q^89 - 36 * q^91 - 18 * q^92 + 15 * q^94 - 24 * q^95 + 18 * q^97 - 15 * q^98

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 $ x^6 - 3*x^5 + 12*x^4 - 19*x^3 + 27*x^2 - 18*x + 4 :

$\beta_{1}$	$=$	$ \nu^{2} - \nu + 3 $ v^2 - v + 3
$\beta_{2}$	$=$	$ ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 18 ) / 2 $ (2v^5 - 5v^4 + 22v^3 - 28v^2 + 43*v - 18) / 2
$\beta_{3}$	$=$	$ -3\nu^{5} + 7\nu^{4} - 31\nu^{3} + 37\nu^{2} - 56\nu + 22 $ -3v^5 + 7v^4 - 31v^3 + 37v^2 - 56*v + 22
$\beta_{4}$	$=$	$ ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 $ (-6v^5 + 15v^4 - 64v^3 + 82v^2 - 121*v + 50) / 2
$\beta_{5}$	$=$	$ -3\nu^{5} + 8\nu^{4} - 33\nu^{3} + 44\nu^{2} - 62\nu + 24 $ -3v^5 + 8v^4 - 33v^3 + 44v^2 - 62*v + 24

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

$\nu$	$=$	$ ( \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 $ (b5 - 2*b4 + b3 + b1 + 1) / 2
$\nu^{2}$	$=$	$ ( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta _1 - 5 ) / 2 $ (b5 - 2b4 + b3 + 3b1 - 5) / 2
$\nu^{3}$	$=$	$ ( -3\beta_{5} + 8\beta_{4} - 3\beta_{3} + 6\beta_{2} - \beta _1 - 5 ) / 2 $ (-3b5 + 8b4 - 3b3 + 6b2 - b1 - 5) / 2
$\nu^{4}$	$=$	$ ( -5\beta_{5} + 18\beta_{4} - 9\beta_{3} + 12\beta_{2} - 17\beta _1 + 27 ) / 2 $ (-5b5 + 18b4 - 9b3 + 12b2 - 17*b1 + 27) / 2
$\nu^{5}$	$=$	$ ( 13\beta_{5} - 28\beta_{4} + 3\beta_{3} - 34\beta_{2} - 11\beta _1 + 49 ) / 2 $ (13b5 - 28b4 + 3b3 - 34b2 - 11*b1 + 49) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$1$	$\beta_{2}$	$1$

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} $

793.1

0.500000	−	2.43956i
0.500000	+	1.51496i
0.500000	+	0.0585812i
0.500000	+	2.43956i
0.500000	−	1.51496i
0.500000	−	0.0585812i

0.500000

−

0.866025i

−0.500000

−

0.866025i

−1.60074

2.77256i

1.60074

−

2.10657i

−1.00000

1.60074

2.77256i

793.2

0.500000

−

0.866025i

−0.500000

−

0.866025i

0.227452

−

0.393958i

−0.227452

2.63596i

−1.00000

−0.227452

−

0.393958i

793.3

0.500000

−

0.866025i

−0.500000

−

0.866025i

1.37328

−

2.37860i

−1.37328

−

2.26144i

−1.00000

−1.37328

−

2.37860i

991.1

0.500000

0.866025i

−0.500000

0.866025i

−1.60074

−

2.77256i

1.60074

2.10657i

−1.00000

1.60074

−

2.77256i

991.2

0.500000

0.866025i

−0.500000

0.866025i

0.227452

0.393958i

−0.227452

−

2.63596i

−1.00000

−0.227452

0.393958i

991.3

0.500000

0.866025i

−0.500000

0.866025i

1.37328

2.37860i

−1.37328

2.26144i

−1.00000

−1.37328

2.37860i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1386.2.k.w		6
3.b	odd	2	1		462.2.i.f	✓	6
7.c	even	3	1	inner	1386.2.k.w		6
7.c	even	3	1		9702.2.a.dt		3
7.d	odd	6	1		9702.2.a.du		3
21.g	even	6	1		3234.2.a.bg		3
21.h	odd	6	1		462.2.i.f	✓	6
21.h	odd	6	1		3234.2.a.bi		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
462.2.i.f	✓	6	3.b	odd	2	1
462.2.i.f	✓	6	21.h	odd	6	1
1386.2.k.w		6	1.a	even	1	1	trivial
1386.2.k.w		6	7.c	even	3	1	inner
3234.2.a.bg		3	21.g	even	6	1
3234.2.a.bi		3	21.h	odd	6	1
9702.2.a.dt		3	7.c	even	3	1
9702.2.a.du		3	7.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 $S_{2}^{\mathrm{new}}(1386, [\chi])$:

$ T_{5}^{6} + 9T_{5}^{4} - 8T_{5}^{3} + 81T_{5}^{2} - 36T_{5} + 16 $

T5^6 + 9*T5^4 - 8*T5^3 + 81*T5^2 - 36*T5 + 16

$ T_{13}^{3} - 36T_{13} + 32 $

T13^3 - 36*T13 + 32

$ T_{17}^{2} - T_{17} + 1 $

T17^2 - T17 + 1

$ T_{23}^{6} - 9T_{23}^{5} + 78T_{23}^{4} - 69T_{23}^{3} + 198T_{23}^{2} + 63T_{23} + 441 $

T23^6 - 9*T23^5 + 78*T23^4 - 69*T23^3 + 198*T23^2 + 63*T23 + 441

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 9 T^{4} + \cdots + 16 $ T^6 + 9T^4 - 8T^3 + 81T^2 - 36T + 16
$7$	$ T^{6} + 12 T^{4} + \cdots + 343 $ T^6 + 12T^4 - 4T^3 + 84*T^2 + 343
$11$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$13$	$ (T^{3} - 36 T + 32)^{2} $ (T^3 - 36*T + 32)^2
$17$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$19$	$ T^{6} - 3 T^{5} + \cdots + 1296 $ T^6 - 3T^5 + 45T^4 + 180T^3 + 1188T^2 + 1296*T + 1296
$23$	$ T^{6} - 9 T^{5} + \cdots + 441 $ T^6 - 9T^5 + 78T^4 - 69T^3 + 198T^2 + 63*T + 441
$29$	$ (T^{3} + 9 T^{2} - 92)^{2} $ (T^3 + 9*T^2 - 92)^2
$31$	$ T^{6} + 6 T^{5} + \cdots + 262144 $ T^6 + 6T^5 + 132T^4 + 448T^3 + 12288T^2 + 49152*T + 262144
$37$	$ T^{6} + 9 T^{5} + \cdots + 26896 $ T^6 + 9T^5 + 93T^4 + 220T^3 + 1620T^2 + 1968*T + 26896
$41$	$ (T^{3} + 12 T^{2} + \cdots - 82)^{2} $ (T^3 + 12T^2 + 21T - 82)^2
$43$	$ (T^{3} - 9 T^{2} + \cdots + 164)^{2} $ (T^3 - 9T^2 - 12T + 164)^2
$47$	$ T^{6} + 15 T^{5} + \cdots + 361 $ T^6 + 15T^5 + 174T^4 + 803T^3 + 2886T^2 - 969*T + 361
$53$	$ T^{6} + 36 T^{4} + \cdots + 1024 $ T^6 + 36T^4 - 64T^3 + 1296T^2 - 1152T + 1024
$59$	$ T^{6} - 9 T^{5} + \cdots + 589824 $ T^6 - 9T^5 + 177T^4 - 672T^3 + 16128T^2 - 73728*T + 589824
$61$	$ T^{6} - 6 T^{5} + \cdots + 51076 $ T^6 - 6T^5 + 165T^4 + 322T^3 + 17997T^2 - 29154*T + 51076
$67$	$ T^{6} + 6 T^{5} + \cdots + 7056 $ T^6 + 6T^5 + 51T^4 + 78T^3 + 729T^2 + 1260*T + 7056
$71$	$ (T^{3} + 3 T^{2} - 24 T - 64)^{2} $ (T^3 + 3T^2 - 24T - 64)^2
$73$	$ T^{6} + 96 T^{4} + \cdots + 36864 $ T^6 + 96T^4 - 384T^3 + 9216T^2 - 18432T + 36864
$79$	$ T^{6} + 12 T^{5} + \cdots + 135424 $ T^6 + 12T^5 + 177T^4 + 340T^3 + 5505T^2 + 12144*T + 135424
$83$	$ (T^{3} - 6 T^{2} + \cdots - 188)^{2} $ (T^3 - 6T^2 - 111T - 188)^2
$89$	$ T^{6} + 36 T^{5} + \cdots + 1763584 $ T^6 + 36T^5 + 900T^4 + 11600T^3 + 109008T^2 + 525888*T + 1763584
$97$	$ (T^{3} - 9 T^{2} - 81 T - 7)^{2} $ (T^3 - 9T^2 - 81T - 7)^2
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Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_{2} q^{2} + ( - \beta_{2} - 1) q^{4} + \beta_{4} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} - q^{8} + (\beta_{4} - \beta_1) q^{10} + (\beta_{2} + 1) q^{11} + 2 \beta_1 q^{13} + ( - \beta_{4} + \beta_{3} - 1) q^{14}+ \cdots + (\beta_{5} - 2 \beta_{4} + \cdots + 3 \beta_1) q^{98}+O(q^{100}) \) q - b2 * q^2 + (-b2 - 1) * q^4 + b4 * q^5 + (b5 - b4 + b3) * q^7 - q^8 + (b4 - b1) * q^10 + (b2 + 1) * q^11 + 2b1 q^13 + (-b4 + b3 - 1) * q^14 + b2 * q^16 + (b2 + 1) * q^17 + (2b5 + b3 + b1 + 1) q^19 - b1 * q^20 + q^22 + (-2b5 + b4 - b3 - 4b2 - b1 - 1) * q^23 + (b5 - 2b4 + 2b3 + b1 - 1) * q^25 + 2b4 q^26 + (-b5 - 1) * q^28 + (-b5 + b3 - 4) * q^29 + (2b5 - 4b4 + 4b3 + 2b1 - 2) * q^31 + (b2 + 1) * q^32 + q^34 + (-b5 + 2b4 + 3b2 - 3b1) q^35 + (2b5 + b3 + 4b2 + b1 + 1) * q^37 + (b5 + 2b3 - b1 - 1) q^38 - b4 * q^40 + (-b5 + b3 - 5) * q^41 + (-b5 + b3 - 2b1 + 2) q^43 - b2 * q^44 + (-b5 + b4 - 2b3 - 4b2 - 3) * q^46 + (-2b5 + b4 - b3 + 4b2 - b1 - 1) * q^47 + (-2b4 - b3 - b2 + b1 - 4) q^49 + (-b5 + b3 - 2) * q^50 + (2b4 - 2b1) * q^52 + (-2b4 + 2b1) * q^53 + b1 * q^55 + (-b5 + b4 - b3) * q^56 + (-2b5 + 2b4 - b3 + 2b2 - b1 - 1) q^58 + (b5 + 2b4 + 2b3 + 4b2 - 3b1 + 3) * q^59 + (-4b5 + 5b4 - 2b3 - 4b2 - 2b1 - 2) q^61 + (-2b5 + 2b3 - 4) * q^62 + q^64 + (4b5 - 4b4 + 2b3 - 10b2 + 2b1 + 2) q^65 + (b5 - 2b4 + 2b3 - b2 + b1 - 2) * q^67 - b2 * q^68 + (-b5 - b3 + 2b2 - 2b1 + 3) * q^70 + (-b5 + b3 - 2) * q^71 + (-2b5 + 2b4 - 4b3 - 2b2) * q^73 + (b5 + 2b3 + 4b2 - b1 + 3) * q^74 + (-b5 + b3 - 2b1 - 2) q^76 + (b5 + 1) * q^77 + (-3b4 + 4b2) * q^79 + (-b4 + b1) * q^80 + (-2b5 + 2b4 - b3 + 3b2 - b1 - 1) q^82 + (b5 - b3 + 4b1 + 3) q^83 + b1 * q^85 + (-2b5 - b3 - 4b2 - b1 - 1) * q^86 + (-b2 - 1) * q^88 + (2b4 + 12b2) * q^89 + (4b4 + 2b3 + 2b2 - 2b1 - 6) * q^91 + (b5 - b3 + b1 - 2) * q^92 + (-b5 + b4 - 2b3 + 4b2 + 5) * q^94 + (2b5 + 4b3 - 6b2 - 2b1 - 8) * q^95 + (-2b5 + 2b3 + 1) * q^97 + (b5 - 2b4 + 4b2 + 3b1) q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} + 3 q^{11} - 3 q^{14} - 3 q^{16} + 3 q^{17} + 3 q^{19} + 6 q^{22} + 9 q^{23} - 3 q^{25} - 3 q^{28} - 18 q^{29} - 6 q^{31} + 3 q^{32} + 6 q^{34} - 6 q^{35} - 9 q^{37} - 3 q^{38}+ \cdots - 15 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 + 3 * q^11 - 3 * q^14 - 3 * q^16 + 3 * q^17 + 3 * q^19 + 6 * q^22 + 9 * q^23 - 3 * q^25 - 3 * q^28 - 18 * q^29 - 6 * q^31 + 3 * q^32 + 6 * q^34 - 6 * q^35 - 9 * q^37 - 3 * q^38 - 24 * q^41 + 18 * q^43 + 3 * q^44 - 9 * q^46 - 15 * q^47 - 24 * q^49 - 6 * q^50 - 9 * q^58 + 9 * q^59 + 6 * q^61 - 12 * q^62 + 6 * q^64 + 36 * q^65 - 6 * q^67 + 3 * q^68 + 12 * q^70 - 6 * q^71 + 9 * q^74 - 6 * q^76 + 3 * q^77 - 12 * q^79 - 12 * q^82 + 12 * q^83 + 9 * q^86 - 3 * q^88 - 36 * q^89 - 36 * q^91 - 18 * q^92 + 15 * q^94 - 24 * q^95 + 18 * q^97 - 15 * q^98

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Newspace parameters

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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	1386.2.k.w

Level	$1386$
Weight	$2$
Character orbit	1386.k
Analytic conductor	$11.067$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$

Self dual:	no
Analytic conductor:	\(11.0672657201\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.1156923.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4 \) x^6 - 3x^5 + 12x^4 - 19x^3 + 27x^2 - 18*x + 4
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	no (minimal twist has level 462)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

\(\beta_{1}\)	\(=\)	\( \nu^{2} - \nu + 3 \) v^2 - v + 3
\(\beta_{2}\)	\(=\)	\( ( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 18 ) / 2 \) (2v^5 - 5v^4 + 22v^3 - 28v^2 + 43*v - 18) / 2
\(\beta_{3}\)	\(=\)	\( -3\nu^{5} + 7\nu^{4} - 31\nu^{3} + 37\nu^{2} - 56\nu + 22 \) -3v^5 + 7v^4 - 31v^3 + 37v^2 - 56*v + 22
\(\beta_{4}\)	\(=\)	\( ( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2 \) (-6v^5 + 15v^4 - 64v^3 + 82v^2 - 121*v + 50) / 2
\(\beta_{5}\)	\(=\)	\( -3\nu^{5} + 8\nu^{4} - 33\nu^{3} + 44\nu^{2} - 62\nu + 24 \) -3v^5 + 8v^4 - 33v^3 + 44v^2 - 62*v + 24

\(\nu\)	\(=\)	\( ( \beta_{5} - 2\beta_{4} + \beta_{3} + \beta _1 + 1 ) / 2 \) (b5 - 2*b4 + b3 + b1 + 1) / 2
\(\nu^{2}\)	\(=\)	\( ( \beta_{5} - 2\beta_{4} + \beta_{3} + 3\beta _1 - 5 ) / 2 \) (b5 - 2b4 + b3 + 3b1 - 5) / 2
\(\nu^{3}\)	\(=\)	\( ( -3\beta_{5} + 8\beta_{4} - 3\beta_{3} + 6\beta_{2} - \beta _1 - 5 ) / 2 \) (-3b5 + 8b4 - 3b3 + 6b2 - b1 - 5) / 2
\(\nu^{4}\)	\(=\)	\( ( -5\beta_{5} + 18\beta_{4} - 9\beta_{3} + 12\beta_{2} - 17\beta _1 + 27 ) / 2 \) (-5b5 + 18b4 - 9b3 + 12b2 - 17*b1 + 27) / 2
\(\nu^{5}\)	\(=\)	\( ( 13\beta_{5} - 28\beta_{4} + 3\beta_{3} - 34\beta_{2} - 11\beta _1 + 49 ) / 2 \) (13b5 - 28b4 + 3b3 - 34b2 - 11*b1 + 49) / 2

\(n\)	\(155\)	\(199\)	\(1135\)
\(\chi(n)\)	\(1\)	\(\beta_{2}\)	\(1\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 793.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$3$	\( T^{6} \) T^6
$5$	\( T^{6} + 9 T^{4} + \cdots + 16 \) T^6 + 9T^4 - 8T^3 + 81T^2 - 36T + 16
$7$	\( T^{6} + 12 T^{4} + \cdots + 343 \) T^6 + 12T^4 - 4T^3 + 84*T^2 + 343
$11$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$13$	\( (T^{3} - 36 T + 32)^{2} \) (T^3 - 36*T + 32)^2
$17$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$19$	\( T^{6} - 3 T^{5} + \cdots + 1296 \) T^6 - 3T^5 + 45T^4 + 180T^3 + 1188T^2 + 1296*T + 1296
$23$	\( T^{6} - 9 T^{5} + \cdots + 441 \) T^6 - 9T^5 + 78T^4 - 69T^3 + 198T^2 + 63*T + 441
$29$	\( (T^{3} + 9 T^{2} - 92)^{2} \) (T^3 + 9*T^2 - 92)^2
$31$	\( T^{6} + 6 T^{5} + \cdots + 262144 \) T^6 + 6T^5 + 132T^4 + 448T^3 + 12288T^2 + 49152*T + 262144
$37$	\( T^{6} + 9 T^{5} + \cdots + 26896 \) T^6 + 9T^5 + 93T^4 + 220T^3 + 1620T^2 + 1968*T + 26896
$41$	\( (T^{3} + 12 T^{2} + \cdots - 82)^{2} \) (T^3 + 12T^2 + 21T - 82)^2
$43$	\( (T^{3} - 9 T^{2} + \cdots + 164)^{2} \) (T^3 - 9T^2 - 12T + 164)^2
$47$	\( T^{6} + 15 T^{5} + \cdots + 361 \) T^6 + 15T^5 + 174T^4 + 803T^3 + 2886T^2 - 969*T + 361
$53$	\( T^{6} + 36 T^{4} + \cdots + 1024 \) T^6 + 36T^4 - 64T^3 + 1296T^2 - 1152T + 1024
$59$	\( T^{6} - 9 T^{5} + \cdots + 589824 \) T^6 - 9T^5 + 177T^4 - 672T^3 + 16128T^2 - 73728*T + 589824
$61$	\( T^{6} - 6 T^{5} + \cdots + 51076 \) T^6 - 6T^5 + 165T^4 + 322T^3 + 17997T^2 - 29154*T + 51076
$67$	\( T^{6} + 6 T^{5} + \cdots + 7056 \) T^6 + 6T^5 + 51T^4 + 78T^3 + 729T^2 + 1260*T + 7056
$71$	\( (T^{3} + 3 T^{2} - 24 T - 64)^{2} \) (T^3 + 3T^2 - 24T - 64)^2
$73$	\( T^{6} + 96 T^{4} + \cdots + 36864 \) T^6 + 96T^4 - 384T^3 + 9216T^2 - 18432T + 36864
$79$	\( T^{6} + 12 T^{5} + \cdots + 135424 \) T^6 + 12T^5 + 177T^4 + 340T^3 + 5505T^2 + 12144*T + 135424
$83$	\( (T^{3} - 6 T^{2} + \cdots - 188)^{2} \) (T^3 - 6T^2 - 111T - 188)^2
$89$	\( T^{6} + 36 T^{5} + \cdots + 1763584 \) T^6 + 36T^5 + 900T^4 + 11600T^3 + 109008T^2 + 525888*T + 1763584
$97$	\( (T^{3} - 9 T^{2} - 81 T - 7)^{2} \) (T^3 - 9T^2 - 81T - 7)^2
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