LMFDB - Newform orbit 448.1.bf.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,1,Mod(13,448)] 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("448.13"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 15, 8])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	448.bf (of order $16$, degree $8$, 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:	$0.223581125660$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

$q$-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_{16} q^{2} + \zeta_{16}^{2} q^{4} + \zeta_{16}^{5} q^{7} - \zeta_{16}^{3} q^{8} + \zeta_{16}^{3} q^{9} + ( - \zeta_{16}^{7} + \zeta_{16}^{6}) q^{11} - \zeta_{16}^{6} q^{14} + \zeta_{16}^{4} q^{16} + \cdots + (\zeta_{16}^{2} - \zeta_{16}) q^{99} +O(q^{100}) $ q - z * q^2 + z^2 * q^4 + z^5 * q^7 - z^3 * q^8 + z^3 * q^9 + (-z^7 + z^6) * q^11 - z^6 * q^14 + z^4 * q^16 - z^4 * q^18 + (-z^7 - 1) * q^22 + (z^4 - z^2) * q^23 + z * q^25 + z^7 * q^28 + (-z^6 - z^5) * q^29 - z^5 * q^32 + z^5 * q^36 + (z^7 + z^2) * q^37 + (-z^4 + z) * q^43 + (z - 1) * q^44 + (-z^5 + z^3) * q^46 - z^2 * q^49 - z^2 * q^50 + (-z^4 - z) * q^53 + q^56 + (z^7 + z^6) * q^58 - q^63 + z^6 * q^64 + (-z^3 - 1) * q^67 + (z^7 + z^3) * q^71 - z^6 * q^72 + (-z^3 + 1) * q^74 + (z^4 - z^3) * q^77 + (-z^3 - z) * q^79 + z^6 * q^81 + (z^5 - z^2) * q^86 + (-z^2 + z) * q^88 + (z^6 - z^4) * q^92 + z^3 * q^98 + (z^2 - z) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{22} - 8 q^{44} + 8 q^{56} - 8 q^{63} - 8 q^{67} + 8 q^{74}+O(q^{100}) $ 8 * q - 8 * q^22 - 8 * q^44 + 8 * q^56 - 8 * q^63 - 8 * q^67 + 8 * q^74

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$\zeta_{16}$

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} $

13.1

0.923880	−	0.382683i
0.923880	+	0.382683i
0.382683	+	0.923880i
−0.382683	+	0.923880i
−0.923880	+	0.382683i
−0.923880	−	0.382683i
−0.382683	−	0.923880i
0.382683	−	0.923880i

−0.923880

0.382683i

0.707107

−

0.707107i

−0.382683

−

0.923880i

−0.382683

0.923880i

0.382683

−

0.923880i

69.1

−0.923880

−

0.382683i

0.707107

0.707107i

−0.382683

0.923880i

−0.382683

−

0.923880i

0.382683

0.923880i

125.1

−0.382683

−

0.923880i

−0.707107

0.707107i

0.923880

−

0.382683i

0.923880

0.382683i

−0.923880

−

0.382683i

181.1

0.382683

−

0.923880i

−0.707107

−

0.707107i

−0.923880

−

0.382683i

−0.923880

0.382683i

0.923880

−

0.382683i

237.1

0.923880

−

0.382683i

0.707107

−

0.707107i

0.382683

0.923880i

0.382683

−

0.923880i

−0.382683

0.923880i

293.1

0.923880

0.382683i

0.707107

0.707107i

0.382683

−

0.923880i

0.382683

0.923880i

−0.382683

−

0.923880i

349.1

0.382683

0.923880i

−0.707107

0.707107i

−0.923880

0.382683i

−0.923880

−

0.382683i

0.923880

0.382683i

405.1

−0.382683

0.923880i

−0.707107

−

0.707107i

0.923880

0.382683i

0.923880

−

0.382683i

−0.923880

0.382683i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
64.i	even	16	1	inner
448.bf	odd	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	448.1.bf.a	✓	8
4.b	odd	2	1		1792.1.bf.a		8
7.b	odd	2	1	CM	448.1.bf.a	✓	8
7.c	even	3	2		3136.1.ce.a		16
7.d	odd	6	2		3136.1.ce.a		16
8.b	even	2	1		3584.1.bf.a		8
8.d	odd	2	1		3584.1.bf.b		8
28.d	even	2	1		1792.1.bf.a		8
56.e	even	2	1		3584.1.bf.b		8
56.h	odd	2	1		3584.1.bf.a		8
64.i	even	16	1	inner	448.1.bf.a	✓	8
64.i	even	16	1		3584.1.bf.a		8
64.j	odd	16	1		1792.1.bf.a		8
64.j	odd	16	1		3584.1.bf.b		8
448.bd	even	16	1		1792.1.bf.a		8
448.bd	even	16	1		3584.1.bf.b		8
448.bf	odd	16	1	inner	448.1.bf.a	✓	8
448.bf	odd	16	1		3584.1.bf.a		8
448.bk	odd	48	2		3136.1.ce.a		16
448.bn	even	48	2		3136.1.ce.a		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
448.1.bf.a	✓	8	1.a	even	1	1	trivial
448.1.bf.a	✓	8	7.b	odd	2	1	CM
448.1.bf.a	✓	8	64.i	even	16	1	inner
448.1.bf.a	✓	8	448.bf	odd	16	1	inner
1792.1.bf.a		8	4.b	odd	2	1
1792.1.bf.a		8	28.d	even	2	1
1792.1.bf.a		8	64.j	odd	16	1
1792.1.bf.a		8	448.bd	even	16	1
3136.1.ce.a		16	7.c	even	3	2
3136.1.ce.a		16	7.d	odd	6	2
3136.1.ce.a		16	448.bk	odd	48	2
3136.1.ce.a		16	448.bn	even	48	2
3584.1.bf.a		8	8.b	even	2	1
3584.1.bf.a		8	56.h	odd	2	1
3584.1.bf.a		8	64.i	even	16	1
3584.1.bf.a		8	448.bf	odd	16	1
3584.1.bf.b		8	8.d	odd	2	1
3584.1.bf.b		8	56.e	even	2	1
3584.1.bf.b		8	64.j	odd	16	1
3584.1.bf.b		8	448.bd	even	16	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(448, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} + 1 $ T^8 + 1
$3$	$ T^{8} $ T^8
$5$	$ T^{8} $ T^8
$7$	$ T^{8} + 1 $ T^8 + 1
$11$	$ T^{8} + 2 T^{4} + \cdots + 2 $ T^8 + 2T^4 - 16T^3 + 20T^2 - 8T + 2
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ (T^{4} + 2 T^{2} + 4 T + 2)^{2} $ (T^4 + 2T^2 + 4T + 2)^2
$29$	$ T^{8} + 8 T^{5} + \cdots + 2 $ T^8 + 8T^5 + 2T^4 + 12T^2 - 8T + 2
$31$	$ T^{8} $ T^8
$37$	$ T^{8} - 8 T^{5} + \cdots + 2 $ T^8 - 8T^5 + 2T^4 + 12T^2 + 8T + 2
$41$	$ T^{8} $ T^8
$43$	$ T^{8} + 4 T^{6} + \cdots + 2 $ T^8 + 4T^6 + 6T^4 - 8T^3 + 4T^2 + 8*T + 2
$47$	$ T^{8} $ T^8
$53$	$ T^{8} + 4 T^{6} + \cdots + 2 $ T^8 + 4T^6 + 6T^4 - 8T^3 + 4T^2 + 8*T + 2
$59$	$ T^{8} $ T^8
$61$	$ T^{8} $ T^8
$67$	$ T^{8} + 8 T^{7} + \cdots + 2 $ T^8 + 8T^7 + 28T^6 + 56T^5 + 70T^4 + 56T^3 + 28T^2 + 8*T + 2
$71$	$ T^{8} + 16 $ T^8 + 16
$73$	$ T^{8} $ T^8
$79$	$ T^{8} + 12T^{4} + 4 $ T^8 + 12*T^4 + 4
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	448.bf (of order \(16\), degree \(8\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{16} q^{2} + \zeta_{16}^{2} q^{4} + \zeta_{16}^{5} q^{7} - \zeta_{16}^{3} q^{8} + \zeta_{16}^{3} q^{9} + ( - \zeta_{16}^{7} + \zeta_{16}^{6}) q^{11} - \zeta_{16}^{6} q^{14} + \zeta_{16}^{4} q^{16} + \cdots + (\zeta_{16}^{2} - \zeta_{16}) q^{99} +O(q^{100}) \) q - z * q^2 + z^2 * q^4 + z^5 * q^7 - z^3 * q^8 + z^3 * q^9 + (-z^7 + z^6) * q^11 - z^6 * q^14 + z^4 * q^16 - z^4 * q^18 + (-z^7 - 1) * q^22 + (z^4 - z^2) * q^23 + z * q^25 + z^7 * q^28 + (-z^6 - z^5) * q^29 - z^5 * q^32 + z^5 * q^36 + (z^7 + z^2) * q^37 + (-z^4 + z) * q^43 + (z - 1) * q^44 + (-z^5 + z^3) * q^46 - z^2 * q^49 - z^2 * q^50 + (-z^4 - z) * q^53 + q^56 + (z^7 + z^6) * q^58 - q^63 + z^6 * q^64 + (-z^3 - 1) * q^67 + (z^7 + z^3) * q^71 - z^6 * q^72 + (-z^3 + 1) * q^74 + (z^4 - z^3) * q^77 + (-z^3 - z) * q^79 + z^6 * q^81 + (z^5 - z^2) * q^86 + (-z^2 + z) * q^88 + (z^6 - z^4) * q^92 + z^3 * q^98 + (z^2 - z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{22} - 8 q^{44} + 8 q^{56} - 8 q^{63} - 8 q^{67} + 8 q^{74}+O(q^{100}) \) 8 * q - 8 * q^22 - 8 * q^44 + 8 * q^56 - 8 * q^63 - 8 * q^67 + 8 * q^74

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

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	448.1.bf.a

Level	$448$
Weight	$1$
Character orbit	448.bf
Analytic conductor	$0.224$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{16}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.223581125660\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	\(\Q(\zeta_{16})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} + 1 \) x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{16}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{16} - \cdots)\)

\(n\)	\(127\)	\(129\)	\(197\)
\(\chi(n)\)	\(1\)	\(-1\)	\(\zeta_{16}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
64.i	even	16	1	inner
448.bf	odd	16	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} + 1 \) T^8 + 1
$3$	\( T^{8} \) T^8
$5$	\( T^{8} \) T^8
$7$	\( T^{8} + 1 \) T^8 + 1
$11$	\( T^{8} + 2 T^{4} + \cdots + 2 \) T^8 + 2T^4 - 16T^3 + 20T^2 - 8T + 2
$13$	\( T^{8} \) T^8
$17$	\( T^{8} \) T^8
$19$	\( T^{8} \) T^8
$23$	\( (T^{4} + 2 T^{2} + 4 T + 2)^{2} \) (T^4 + 2T^2 + 4T + 2)^2
$29$	\( T^{8} + 8 T^{5} + \cdots + 2 \) T^8 + 8T^5 + 2T^4 + 12T^2 - 8T + 2
$31$	\( T^{8} \) T^8
$37$	\( T^{8} - 8 T^{5} + \cdots + 2 \) T^8 - 8T^5 + 2T^4 + 12T^2 + 8T + 2
$41$	\( T^{8} \) T^8
$43$	\( T^{8} + 4 T^{6} + \cdots + 2 \) T^8 + 4T^6 + 6T^4 - 8T^3 + 4T^2 + 8*T + 2
$47$	\( T^{8} \) T^8
$53$	\( T^{8} + 4 T^{6} + \cdots + 2 \) T^8 + 4T^6 + 6T^4 - 8T^3 + 4T^2 + 8*T + 2
$59$	\( T^{8} \) T^8
$61$	\( T^{8} \) T^8
$67$	\( T^{8} + 8 T^{7} + \cdots + 2 \) T^8 + 8T^7 + 28T^6 + 56T^5 + 70T^4 + 56T^3 + 28T^2 + 8*T + 2
$71$	\( T^{8} + 16 \) T^8 + 16
$73$	\( T^{8} \) T^8
$79$	\( T^{8} + 12T^{4} + 4 \) T^8 + 12*T^4 + 4
$83$	\( T^{8} \) T^8
$89$	\( T^{8} \) T^8
$97$	\( T^{8} \) T^8
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