LMFDB - Newform orbit 2704.1.bb.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,1,Mod(191,2704)] 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("2704.191"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2704.bb (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.34947179416$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 208)
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(i, \sqrt{13})$
Artin image:	$C_3\times D_4$
Artin field:	Galois closure of 12.6.7340688973975552.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

The $q$-expansion and trace form are shown below.

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times$.

$n$	$677$	$1185$	$2367$
$\chi(n)$	$1$	$-\zeta_{6}$	$-1$

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} $

191.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

−

0.866025i

991.1

−0.500000

0.866025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
13.b	even	2	1	RM by $\Q(\sqrt{13}) $
52.b	odd	2	1	CM by $\Q(\sqrt{-13}) $
13.c	even	3	1	inner
13.e	even	6	1	inner
52.i	odd	6	1	inner
52.j	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2704.1.bb.a		2
4.b	odd	2	1	CM	2704.1.bb.a		2
13.b	even	2	1	RM	2704.1.bb.a		2
13.c	even	3	1		2704.1.d.a		1
13.c	even	3	1	inner	2704.1.bb.a		2
13.d	odd	4	2		2704.1.y.b		2
13.e	even	6	1		2704.1.d.a		1
13.e	even	6	1	inner	2704.1.bb.a		2
13.f	odd	12	2		208.1.c.a	✓	1
13.f	odd	12	2		2704.1.y.b		2
39.k	even	12	2		1872.1.i.b		1
52.b	odd	2	1	CM	2704.1.bb.a		2
52.f	even	4	2		2704.1.y.b		2
52.i	odd	6	1		2704.1.d.a		1
52.i	odd	6	1	inner	2704.1.bb.a		2
52.j	odd	6	1		2704.1.d.a		1
52.j	odd	6	1	inner	2704.1.bb.a		2
52.l	even	12	2		208.1.c.a	✓	1
52.l	even	12	2		2704.1.y.b		2
104.u	even	12	2		832.1.c.a		1
104.x	odd	12	2		832.1.c.a		1
156.v	odd	12	2		1872.1.i.b		1
208.be	odd	12	2		3328.1.h.a		2
208.bf	even	12	2		3328.1.h.a		2
208.bk	even	12	2		3328.1.h.a		2
208.bl	odd	12	2		3328.1.h.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
208.1.c.a	✓	1	13.f	odd	12	2
208.1.c.a	✓	1	52.l	even	12	2
832.1.c.a		1	104.u	even	12	2
832.1.c.a		1	104.x	odd	12	2
1872.1.i.b		1	39.k	even	12	2
1872.1.i.b		1	156.v	odd	12	2
2704.1.d.a		1	13.c	even	3	1
2704.1.d.a		1	13.e	even	6	1
2704.1.d.a		1	52.i	odd	6	1
2704.1.d.a		1	52.j	odd	6	1
2704.1.y.b		2	13.d	odd	4	2
2704.1.y.b		2	13.f	odd	12	2
2704.1.y.b		2	52.f	even	4	2
2704.1.y.b		2	52.l	even	12	2
2704.1.bb.a		2	1.a	even	1	1	trivial
2704.1.bb.a		2	4.b	odd	2	1	CM
2704.1.bb.a		2	13.b	even	2	1	RM
2704.1.bb.a		2	13.c	even	3	1	inner
2704.1.bb.a		2	13.e	even	6	1	inner
2704.1.bb.a		2	52.b	odd	2	1	CM
2704.1.bb.a		2	52.i	odd	6	1	inner
2704.1.bb.a		2	52.j	odd	6	1	inner
3328.1.h.a		2	208.be	odd	12	2
3328.1.h.a		2	208.bf	even	12	2
3328.1.h.a		2	208.bk	even	12	2
3328.1.h.a		2	208.bl	odd	12	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{1}^{\mathrm{new}}(2704, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ T^{2} - 2T + 4 $ T^2 - 2*T + 4
$31$	$ T^{2} $ T^2
$37$	$ T^{2} $ T^2
$41$	$ T^{2} $ T^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ (T - 2)^{2} $ (T - 2)^2
$59$	$ T^{2} $ T^2
$61$	$ T^{2} + 2T + 4 $ T^2 + 2*T + 4
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} $ T^2
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ T^{2} $ T^2
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Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2704.bb (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{6} q^{9} - 2 \zeta_{6} q^{17} - q^{25} - 2 \zeta_{6}^{2} q^{29} + \zeta_{6}^{2} q^{49} + 2 q^{53} - 2 \zeta_{6} q^{61} + \zeta_{6}^{2} q^{81} +O(q^{100}) \) q - z * q^9 - 2z q^17 - q^25 - 2z^2 q^29 + z^2 * q^49 + 2 * q^53 - 2z q^61 + z^2 * q^81
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{9} - 2 q^{17} - 2 q^{25} + 2 q^{29} - q^{49} + 4 q^{53} - 2 q^{61} - q^{81}+O(q^{100}) \) 2 * q - q^9 - 2 * q^17 - 2 * q^25 + 2 * q^29 - q^49 + 4 * q^53 - 2 * q^61 - q^81

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