LMFDB - Newform orbit 1568.1.x.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1568,1,Mod(99,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([4, 3, 0])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1568.99"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$1568 = 2^{5} \cdot 7^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1568.x (of order $8$ , degree $4$ , minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$0.782533939809$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.5156108238848.2

$q$ -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.

$f(q)$	$=$	$q - \zeta_{8}^{3} q^{2} - \zeta_{8}^{2} q^{4} - \zeta_{8} q^{8} + \zeta_{8} q^{9} + ( - \zeta_{8}^{2} - \zeta_{8}) q^{11} - q^{16} + q^{18} + ( - \zeta_{8} - 1) q^{22} + ( - \zeta_{8}^{2} + 1) q^{23} + \cdots + ( - \zeta_{8}^{3} - \zeta_{8}^{2}) q^{99} +O(q^{100})$ q - z^3 * q^2 - z^2 * q^4 - z * q^8 + z * q^9 + (-z^2 - z) * q^11 - q^16 + q^18 + (-z - 1) * q^22 + (-z^2 + 1) * q^23 - z^3 * q^25 + (z^3 - z^2) * q^29 + z^3 * q^32 - z^3 * q^36 + (z^2 + z) * q^37 + (z^3 + 1) * q^43 + (z^3 - 1) * q^44 + (-z^3 - z) * q^46 - z^2 * q^50 + (z^3 - 1) * q^53 + (z^2 - z) * q^58 + z^2 * q^64 + (-z + 1) * q^67 + 2z q^71 - z^2 * q^72 + (z + 1) * q^74 + (z^3 - z) * q^79 + z^2 * q^81 + (-z^3 + z^2) * q^86 + (z^3 + z^2) * q^88 + (-z^2 - 1) * q^92 + (-z^3 - z^2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{16} + 4 q^{18} - 4 q^{22} + 4 q^{23} + 4 q^{43} - 4 q^{44} - 4 q^{53} + 4 q^{67} + 4 q^{74} - 4 q^{92}+O(q^{100})$ 4 * q - 4 * q^16 + 4 * q^18 - 4 * q^22 + 4 * q^23 + 4 * q^43 - 4 * q^44 - 4 * q^53 + 4 * q^67 + 4 * q^74 - 4 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$ .

$n$	$197$	$1471$	$1473$
$\chi(n)$	$\zeta_{8}^{3}$	$-1$	$1$

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}

99.1

0.707107	+	0.707107i
0.707107	−	0.707107i
−0.707107	−	0.707107i
−0.707107	+	0.707107i

0.707107

−

0.707107i

−

1.00000i

−0.707107

−

0.707107i

0.707107

0.707107i

491.1

0.707107

0.707107i

1.00000i

−0.707107

0.707107i

0.707107

−

0.707107i

883.1

−0.707107

0.707107i

−

1.00000i

0.707107

0.707107i

−0.707107

−

0.707107i

1275.1

−0.707107

−

0.707107i

1.00000i

0.707107

−

0.707107i

−0.707107

0.707107i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7})$
32.h	odd	8	1	inner
224.x	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1568.1.x.a	✓	4
7.b	odd	2	1	CM	1568.1.x.a	✓	4
7.c	even	3	2		1568.1.bo.a		8
7.d	odd	6	2		1568.1.bo.a		8
32.h	odd	8	1	inner	1568.1.x.a	✓	4
224.x	even	8	1	inner	1568.1.x.a	✓	4
224.be	even	24	2		1568.1.bo.a		8
224.bf	odd	24	2		1568.1.bo.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1568.1.x.a	✓	4	1.a	even	1	1	trivial
1568.1.x.a	✓	4	7.b	odd	2	1	CM
1568.1.x.a	✓	4	32.h	odd	8	1	inner
1568.1.x.a	✓	4	224.x	even	8	1	inner
1568.1.bo.a		8	7.c	even	3	2
1568.1.bo.a		8	7.d	odd	6	2
1568.1.bo.a		8	224.be	even	24	2
1568.1.bo.a		8	224.bf	odd	24	2

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1568, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{4} + 1$ T^4 + 1
$3$	$T^{4}$ T^4
$5$	$T^{4}$ T^4
$7$	$T^{4}$ T^4
$11$	$T^{4} + 2 T^{2} + \cdots + 2$ T^4 + 2T^2 - 4T + 2
$13$	$T^{4}$ T^4
$17$	$T^{4}$ T^4
$19$	$T^{4}$ T^4
$23$	$(T^{2} - 2 T + 2)^{2}$ (T^2 - 2*T + 2)^2
$29$	$T^{4} + 2 T^{2} + \cdots + 2$ T^4 + 2T^2 + 4T + 2
$31$	$T^{4}$ T^4
$37$	$T^{4} + 2 T^{2} + \cdots + 2$ T^4 + 2T^2 + 4T + 2
$41$	$T^{4}$ T^4
$43$	$T^{4} - 4 T^{3} + \cdots + 2$ T^4 - 4T^3 + 6T^2 - 4*T + 2
$47$	$T^{4}$ T^4
$53$	$T^{4} + 4 T^{3} + \cdots + 2$ T^4 + 4T^3 + 6T^2 + 4*T + 2
$59$	$T^{4}$ T^4
$61$	$T^{4}$ T^4
$67$	$T^{4} - 4 T^{3} + \cdots + 2$ T^4 - 4T^3 + 6T^2 - 4*T + 2
$71$	$T^{4} + 16$ T^4 + 16
$73$	$T^{4}$ T^4
$79$	$(T^{2} - 2)^{2}$ (T^2 - 2)^2
$83$	$T^{4}$ T^4
$89$	$T^{4}$ T^4
$97$	$T^{4}$ T^4
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Newspace parameters

Newform invariants

$q$ -expansion

Character values

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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	1568.1.x.a

Level	$1568$
Weight	$1$
Character orbit	1568.x
Analytic conductor	$0.783$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{8}$
CM discriminant	-7
Inner twists	$4$