LMFDB - Newform orbit 3060.1.f.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3060,1,Mod(2719,3060)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3060, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3060.2719"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3060.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,-1,0,0,1,0,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

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

Self dual:	yes
Analytic conductor:	$1.52713893866$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{2}$
Projective field:	Galois closure of $\Q(\sqrt{-15}, \sqrt{51})$
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.45900.2
Stark unit:	Root of $x^{4} - 4228636x^{3} - 7891770x^{2} - 4228636x + 1$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3060\mathbb{Z}\right)^\times$.

$n$	$1261$	$1361$	$1531$	$1837$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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} $

2719.1

1.00000

−1.00000

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
204.h	even	2	1	RM by $\Q(\sqrt{51}) $
340.d	odd	2	1	CM by $\Q(\sqrt{-85}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3060.1.f.c	yes	1
3.b	odd	2	1		3060.1.f.b	yes	1
4.b	odd	2	1		3060.1.f.a	✓	1
5.b	even	2	1		3060.1.f.b	yes	1
12.b	even	2	1		3060.1.f.d	yes	1
15.d	odd	2	1	CM	3060.1.f.c	yes	1
17.b	even	2	1		3060.1.f.d	yes	1
20.d	odd	2	1		3060.1.f.d	yes	1
51.c	odd	2	1		3060.1.f.a	✓	1
60.h	even	2	1		3060.1.f.a	✓	1
68.d	odd	2	1		3060.1.f.b	yes	1
85.c	even	2	1		3060.1.f.a	✓	1
204.h	even	2	1	RM	3060.1.f.c	yes	1
255.h	odd	2	1		3060.1.f.d	yes	1
340.d	odd	2	1	CM	3060.1.f.c	yes	1
1020.b	even	2	1		3060.1.f.b	yes	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3060.1.f.a	✓	1	4.b	odd	2	1
3060.1.f.a	✓	1	51.c	odd	2	1
3060.1.f.a	✓	1	60.h	even	2	1
3060.1.f.a	✓	1	85.c	even	2	1
3060.1.f.b	yes	1	3.b	odd	2	1
3060.1.f.b	yes	1	5.b	even	2	1
3060.1.f.b	yes	1	68.d	odd	2	1
3060.1.f.b	yes	1	1020.b	even	2	1
3060.1.f.c	yes	1	1.a	even	1	1	trivial
3060.1.f.c	yes	1	15.d	odd	2	1	CM
3060.1.f.c	yes	1	204.h	even	2	1	RM
3060.1.f.c	yes	1	340.d	odd	2	1	CM
3060.1.f.d	yes	1	12.b	even	2	1
3060.1.f.d	yes	1	17.b	even	2	1
3060.1.f.d	yes	1	20.d	odd	2	1
3060.1.f.d	yes	1	255.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3060, [\chi])$:

$ T_{7} $

T7

$ T_{29} $

T29

$ T_{31} - 2 $

T31 - 2

$ T_{37} $

T37

$ T_{47} + 2 $

T47 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T + 1 $ T + 1
$7$	$ T $ T
$11$	$ T $ T
$13$	$ T $ T
$17$	$ T - 1 $ T - 1
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T - 2 $ T - 2
$37$	$ T $ T
$41$	$ T $ T
$43$	$ T $ T
$47$	$ T + 2 $ T + 2
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T $ T
$73$	$ T $ T
$79$	$ T + 2 $ T + 2
$83$	$ T + 2 $ T + 2
$89$	$ T $ T
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 3060 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3060.f (of order \(2\), degree \(1\), minimal)

\(n\)	\(1261\)	\(1361\)	\(1531\)	\(1837\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-1\)	\(-1\)

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