LMFDB - Newform orbit 2772.2.s.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2772,2,Mod(793,2772)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2772, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2772.793");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2772 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2772.s (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:	$22.1345314403$
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 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 308)
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} + \cdots + \beta_1) q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{4} - 1) q^{11} + ( - \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{5} + 2 \beta_{4} - 2) q^{17}+ \cdots + ( - \beta_{3} + 6 \beta_1 + 10) q^{97}+O(q^{100}) $ q + (-b5 + b3 + b2 + b1) * q^5 + (-b5 - b2 + b1 + 1) * q^7 + (b4 - 1) * q^11 + (-b3 - 2b1) q^13 + (-b5 + 2b4 - 2) q^17 + (b5 - 4b4 - b1) q^19 + (-b5 - b4 + b1) * q^23 + (-2b5 + 4b4 - b2 - 4) * q^25 + (-b3 - 3b1 - 1) q^29 + (-3b5 + 7b4 - b2 - 7) * q^31 + (-b5 + 4b4 - b3 + 2b2 + 3b1 + 1) q^35 + (-2b5 + 4b4 + 2b3 + 2b2 + 2b1) q^37 + (b3 - 2b1 - 6) q^41 + (-3b3 + 2b1 + 3) * q^43 + (b5 + 4b4 - b1) q^47 + (-b4 + 2b3 + b2 + 3b1) * q^49 + (-4b5 - 2b4 - b2 + 2) * q^53 + (-b3 - b1) * q^55 + (-3b4 - 3b2 + 3) * q^59 + (-2b5 - 2b4 - 4b3 - 4b2 + 2b1) q^61 + (2b5 - 13b4 - 2b1) q^65 + (4b5 + 2b4 + 2b2 - 2) q^67 + (-b1 + 4) * q^71 + (7b4 - b2 - 7) q^73 + (b4 + b3 + b2 - b1 - 1) * q^77 + (-5b5 - b4 - 3b3 - 3b2 + 5b1) * q^79 + (-4b3 + b1 - 2) q^83 + (-3b3 - 2b1 - 4) * q^85 + (3b5 - 3b4 - 3b3 - 3b2 - 3b1) q^89 + (-4b5 - b4 - 3b3 - 4b2 - b1 - 6) q^91 + (4b5 - 4b4 - 5b2 + 4) q^95 + (-b3 + 6b1 + 10) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} + 4 q^{7} - 3 q^{11} + 6 q^{13} - 5 q^{17} - 11 q^{19} - 4 q^{23} - 11 q^{25} + 2 q^{29} - 19 q^{31} + 17 q^{35} + 8 q^{37} - 34 q^{41} + 20 q^{43} + 13 q^{47} - 12 q^{49} + 9 q^{53} + 4 q^{55}+ \cdots + 50 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 + 4 * q^7 - 3 * q^11 + 6 * q^13 - 5 * q^17 - 11 * q^19 - 4 * q^23 - 11 * q^25 + 2 * q^29 - 19 * q^31 + 17 * q^35 + 8 * q^37 - 34 * q^41 + 20 * q^43 + 13 * q^47 - 12 * q^49 + 9 * q^53 + 4 * q^55 + 6 * q^59 - 4 * q^61 - 37 * q^65 - 8 * q^67 + 26 * q^71 - 22 * q^73 - 2 * q^77 - 5 * q^79 - 6 * q^83 - 14 * q^85 - 3 * q^89 - 31 * q^91 + 3 * q^95 + 50 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 :

$\beta_{1}$	$=$	$ \nu^{2} - \nu + 2 $ v^2 - v + 2
$\beta_{2}$	$=$	$ ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3 $ (-v^5 + v^4 - 8v^3 + 5v^2 - 18*v + 6) / 3
$\beta_{3}$	$=$	$ \nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3 $ v^4 - 2v^3 + 6v^2 - 5*v + 3
$\beta_{4}$	$=$	$ ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3 $ (-2v^5 + 5v^4 - 16v^3 + 19v^2 - 21*v + 9) / 3
$\beta_{5}$	$=$	$ ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3 $ (2v^5 - 5v^4 + 19v^3 - 22v^2 + 30*v - 9) / 3

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

$\nu$	$=$	$ ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3 $ (-2b5 - b4 - b3 - 2b2 + b1 + 2) / 3
$\nu^{2}$	$=$	$ ( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3 $ (-2b5 - b4 - b3 - 2b2 + 4*b1 - 4) / 3
$\nu^{3}$	$=$	$ ( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3 $ (7b5 + 5b4 + 2b3 + 4b2 + b1 - 10) / 3
$\nu^{4}$	$=$	$ ( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3 $ (16b5 + 11b4 + 8b3 + 10b2 - 17*b1 + 5) / 3
$\nu^{5}$	$=$	$ ( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3 $ (-14b5 - 16b4 + 5b3 - 5b2 - 23*b1 + 47) / 3

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2772\mathbb{Z}\right)^\times$.

$n$	$1387$	$1541$	$1585$	$2521$
$\chi(n)$	$1$	$1$	$-\beta_{4}$	$1$

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	−	1.41036i
0.500000	+	2.05195i
0.500000	+	0.224437i
0.500000	+	1.41036i
0.500000	−	2.05195i
0.500000	−	0.224437i

−1.71053

2.96273i

−0.710533

−

2.54856i

793.2

−0.933463

1.61680i

0.0665372

2.64491i

793.3

1.64400

−

2.84748i

2.64400

−

0.0963576i

2377.1

−1.71053

−

2.96273i

−0.710533

2.54856i

2377.2

−0.933463

−

1.61680i

0.0665372

−

2.64491i

2377.3

1.64400

2.84748i

2.64400

0.0963576i

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	2772.2.s.f		6
3.b	odd	2	1		308.2.i.a	✓	6
7.c	even	3	1	inner	2772.2.s.f		6
12.b	even	2	1		1232.2.q.l		6
21.c	even	2	1		2156.2.i.l		6
21.g	even	6	1		2156.2.a.h		3
21.g	even	6	1		2156.2.i.l		6
21.h	odd	6	1		308.2.i.a	✓	6
21.h	odd	6	1		2156.2.a.i		3
84.j	odd	6	1		8624.2.a.cn		3
84.n	even	6	1		1232.2.q.l		6
84.n	even	6	1		8624.2.a.ci		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
308.2.i.a	✓	6	3.b	odd	2	1
308.2.i.a	✓	6	21.h	odd	6	1
1232.2.q.l		6	12.b	even	2	1
1232.2.q.l		6	84.n	even	6	1
2156.2.a.h		3	21.g	even	6	1
2156.2.a.i		3	21.h	odd	6	1
2156.2.i.l		6	21.c	even	2	1
2156.2.i.l		6	21.g	even	6	1
2772.2.s.f		6	1.a	even	1	1	trivial
2772.2.s.f		6	7.c	even	3	1	inner
8624.2.a.ci		3	84.n	even	6	1
8624.2.a.cn		3	84.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2772, [\chi])$:

$ T_{5}^{6} + 2T_{5}^{5} + 15T_{5}^{4} + 20T_{5}^{3} + 163T_{5}^{2} + 231T_{5} + 441 $

$ T_{13}^{3} - 3T_{13}^{2} - 24T_{13} + 79 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} $ T^6
$5$	$ T^{6} + 2 T^{5} + \cdots + 441 $ T^6 + 2T^5 + 15T^4 + 20T^3 + 163T^2 + 231*T + 441
$7$	$ T^{6} - 4 T^{5} + \cdots + 343 $ T^6 - 4T^5 + 14T^4 - 55T^3 + 98T^2 - 196*T + 343
$11$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$13$	$ (T^{3} - 3 T^{2} - 24 T + 79)^{2} $ (T^3 - 3T^2 - 24T + 79)^2
$17$	$ T^{6} + 5 T^{5} + \cdots + 9 $ T^6 + 5T^5 + 21T^4 + 26T^3 + 31T^2 - 12*T + 9
$19$	$ T^{6} + 11 T^{5} + \cdots + 1089 $ T^6 + 11T^5 + 85T^4 + 330T^3 + 933T^2 + 1188*T + 1089
$23$	$ T^{6} + 4 T^{5} + \cdots + 9 $ T^6 + 4T^5 + 15T^4 + 10T^3 + 13T^2 - 3*T + 9
$29$	$ (T^{3} - T^{2} - 50 T + 129)^{2} $ (T^3 - T^2 - 50*T + 129)^2
$31$	$ T^{6} + 19 T^{5} + \cdots + 9409 $ T^6 + 19T^5 + 281T^4 + 1714T^3 + 8243T^2 - 7760*T + 9409
$37$	$ T^{6} - 8 T^{5} + \cdots + 64 $ T^6 - 8T^5 + 92T^4 + 208T^3 + 848T^2 - 224*T + 64
$41$	$ (T^{3} + 17 T^{2} + \cdots + 33)^{2} $ (T^3 + 17T^2 + 76T + 33)^2
$43$	$ (T^{3} - 10 T^{2} + \cdots + 73)^{2} $ (T^3 - 10T^2 - 31T + 73)^2
$47$	$ T^{6} - 13 T^{5} + \cdots + 3969 $ T^6 - 13T^5 + 117T^4 - 550T^3 + 1885T^2 - 3276*T + 3969
$53$	$ T^{6} - 9 T^{5} + \cdots + 81 $ T^6 - 9T^5 + 123T^4 + 396T^3 + 1683T^2 + 378*T + 81
$59$	$ T^{6} - 6 T^{5} + \cdots + 59049 $ T^6 - 6T^5 + 81T^4 - 216T^3 + 3483T^2 - 10935*T + 59049
$61$	$ T^{6} + 4 T^{5} + \cdots + 222784 $ T^6 + 4T^5 + 116T^4 + 544T^3 + 11888T^2 + 47200*T + 222784
$67$	$ T^{6} + 8 T^{5} + \cdots + 5184 $ T^6 + 8T^5 + 124T^4 - 624T^3 + 3024T^2 - 4320*T + 5184
$71$	$ (T^{3} - 13 T^{2} + \cdots - 63)^{2} $ (T^3 - 13T^2 + 52T - 63)^2
$73$	$ T^{6} + 22 T^{5} + \cdots + 124609 $ T^6 + 22T^5 + 329T^4 + 2704T^3 + 16259T^2 + 54715*T + 124609
$79$	$ T^{6} + 5 T^{5} + \cdots + 91809 $ T^6 + 5T^5 + 157T^4 - 1266T^3 + 15909T^2 - 39996*T + 91809
$83$	$ (T^{3} + 3 T^{2} + \cdots - 477)^{2} $ (T^3 + 3T^2 - 96T - 477)^2
$89$	$ T^{6} + 3 T^{5} + \cdots + 59049 $ T^6 + 3T^5 + 117T^4 - 810T^3 + 10935T^2 - 26244*T + 59049
$97$	$ (T^{3} - 25 T^{2} + \cdots + 1171)^{2} $ (T^3 - 25T^2 + 56T + 1171)^2
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2772 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2772.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{5} + ( - \beta_{5} - \beta_{2} + \beta_1 + 1) q^{7} + (\beta_{4} - 1) q^{11} + ( - \beta_{3} - 2 \beta_1) q^{13} + ( - \beta_{5} + 2 \beta_{4} - 2) q^{17}+ \cdots + ( - \beta_{3} + 6 \beta_1 + 10) q^{97}+O(q^{100}) \) q + (-b5 + b3 + b2 + b1) * q^5 + (-b5 - b2 + b1 + 1) * q^7 + (b4 - 1) * q^11 + (-b3 - 2b1) q^13 + (-b5 + 2b4 - 2) q^17 + (b5 - 4b4 - b1) q^19 + (-b5 - b4 + b1) * q^23 + (-2b5 + 4b4 - b2 - 4) * q^25 + (-b3 - 3b1 - 1) q^29 + (-3b5 + 7b4 - b2 - 7) * q^31 + (-b5 + 4b4 - b3 + 2b2 + 3b1 + 1) q^35 + (-2b5 + 4b4 + 2b3 + 2b2 + 2b1) q^37 + (b3 - 2b1 - 6) q^41 + (-3b3 + 2b1 + 3) * q^43 + (b5 + 4b4 - b1) q^47 + (-b4 + 2b3 + b2 + 3b1) * q^49 + (-4b5 - 2b4 - b2 + 2) * q^53 + (-b3 - b1) * q^55 + (-3b4 - 3b2 + 3) * q^59 + (-2b5 - 2b4 - 4b3 - 4b2 + 2b1) q^61 + (2b5 - 13b4 - 2b1) q^65 + (4b5 + 2b4 + 2b2 - 2) q^67 + (-b1 + 4) * q^71 + (7b4 - b2 - 7) q^73 + (b4 + b3 + b2 - b1 - 1) * q^77 + (-5b5 - b4 - 3b3 - 3b2 + 5b1) * q^79 + (-4b3 + b1 - 2) q^83 + (-3b3 - 2b1 - 4) * q^85 + (3b5 - 3b4 - 3b3 - 3b2 - 3b1) q^89 + (-4b5 - b4 - 3b3 - 4b2 - b1 - 6) q^91 + (4b5 - 4b4 - 5b2 + 4) q^95 + (-b3 + 6b1 + 10) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} + 4 q^{7} - 3 q^{11} + 6 q^{13} - 5 q^{17} - 11 q^{19} - 4 q^{23} - 11 q^{25} + 2 q^{29} - 19 q^{31} + 17 q^{35} + 8 q^{37} - 34 q^{41} + 20 q^{43} + 13 q^{47} - 12 q^{49} + 9 q^{53} + 4 q^{55}+ \cdots + 50 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 + 4 * q^7 - 3 * q^11 + 6 * q^13 - 5 * q^17 - 11 * q^19 - 4 * q^23 - 11 * q^25 + 2 * q^29 - 19 * q^31 + 17 * q^35 + 8 * q^37 - 34 * q^41 + 20 * q^43 + 13 * q^47 - 12 * q^49 + 9 * q^53 + 4 * q^55 + 6 * q^59 - 4 * q^61 - 37 * q^65 - 8 * q^67 + 26 * q^71 - 22 * q^73 - 2 * q^77 - 5 * q^79 - 6 * q^83 - 14 * q^85 - 3 * q^89 - 31 * q^91 + 3 * q^95 + 50 * q^97

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