LMFDB - Newform orbit 3872.1.r.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3872,1,Mod(161,3872)] 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("3872.161"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3872, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 7])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3872.r (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8] 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:	$1.93237972891$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	8.0.64000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.21296.1
Artin image:	$C_5\times \SD_{16}$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{40} - \cdots)$

$q$-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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{6} q^{9} - \beta_1 q^{13} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{17} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{25} - \beta_{7} q^{29} + 2 \beta_{2} q^{37} + \beta_{3} q^{41}+ \cdots + 2 q^{89}+O(q^{100}) $ q + b6 * q^9 - b1 * q^13 + (-b7 + b5 - b3 + b1) * q^17 + (-b6 + b4 - b2 + 1) * q^25 - b7 * q^29 + 2b2 q^37 + b3 * q^41 + b4 * q^49 + 2b6 q^53 + (-b7 + b5 - b3 + b1) * q^61 + b7 * q^73 - b2 * q^81 + 2 * q^89
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9} + 2 q^{25} + 4 q^{37} - 2 q^{49} + 4 q^{53} - 2 q^{81} + 16 q^{89}+O(q^{100}) $ 8 * q + 2 * q^9 + 2 * q^25 + 4 * q^37 - 2 * q^49 + 4 * q^53 - 2 * q^81 + 16 * q^89

Basis of coefficient ring in terms of a root $\nu$ of $ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2*x^6 + 4*x^4 - 8*x^2 + 16 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3872\mathbb{Z}\right)^\times$.

$n$	$485$	$1695$	$2785$
$\chi(n)$	$1$	$1$	$\beta_{2}$

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} $

161.1

0.831254	−	1.14412i
−0.831254	+	1.14412i
0.831254	+	1.14412i
−0.831254	−	1.14412i
1.34500	+	0.437016i
−1.34500	−	0.437016i
1.34500	−	0.437016i
−1.34500	+	0.437016i

0.809017

0.587785i

161.2

0.809017

0.587785i

481.1

0.809017

−

0.587785i

481.2

0.809017

−

0.587785i

3137.1

−0.309017

0.951057i

3137.2

−0.309017

0.951057i

3361.1

−0.309017

−

0.951057i

3361.2

−0.309017

−

0.951057i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
44.c	even	2	1	inner
44.g	even	10	3	inner
44.h	odd	10	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3872.1.r.a		8
4.b	odd	2	1	CM	3872.1.r.a		8
11.b	odd	2	1	inner	3872.1.r.a		8
11.c	even	5	1		3872.1.h.a	✓	2
11.c	even	5	3	inner	3872.1.r.a		8
11.d	odd	10	1		3872.1.h.a	✓	2
11.d	odd	10	3	inner	3872.1.r.a		8
44.c	even	2	1	inner	3872.1.r.a		8
44.g	even	10	1		3872.1.h.a	✓	2
44.g	even	10	3	inner	3872.1.r.a		8
44.h	odd	10	1		3872.1.h.a	✓	2
44.h	odd	10	3	inner	3872.1.r.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3872.1.h.a	✓	2	11.c	even	5	1
3872.1.h.a	✓	2	11.d	odd	10	1
3872.1.h.a	✓	2	44.g	even	10	1
3872.1.h.a	✓	2	44.h	odd	10	1
3872.1.r.a		8	1.a	even	1	1	trivial
3872.1.r.a		8	4.b	odd	2	1	CM
3872.1.r.a		8	11.b	odd	2	1	inner
3872.1.r.a		8	11.c	even	5	3	inner
3872.1.r.a		8	11.d	odd	10	3	inner
3872.1.r.a		8	44.c	even	2	1	inner
3872.1.r.a		8	44.g	even	10	3	inner
3872.1.r.a		8	44.h	odd	10	3	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{1}^{\mathrm{new}}(3872, [\chi])$.

Hecke characteristic polynomials

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

Classical	Maass
Hilbert	Bianchi

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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3872.r (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{6} q^{9} - \beta_1 q^{13} + ( - \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{17} + ( - \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{25} - \beta_{7} q^{29} + 2 \beta_{2} q^{37} + \beta_{3} q^{41}+ \cdots + 2 q^{89}+O(q^{100}) \) q + b6 * q^9 - b1 * q^13 + (-b7 + b5 - b3 + b1) * q^17 + (-b6 + b4 - b2 + 1) * q^25 - b7 * q^29 + 2b2 q^37 + b3 * q^41 + b4 * q^49 + 2b6 q^53 + (-b7 + b5 - b3 + b1) * q^61 + b7 * q^73 - b2 * q^81 + 2 * q^89
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9} + 2 q^{25} + 4 q^{37} - 2 q^{49} + 4 q^{53} - 2 q^{81} + 16 q^{89}+O(q^{100}) \) 8 * q + 2 * q^9 + 2 * q^25 + 4 * q^37 - 2 * q^49 + 4 * q^53 - 2 * q^81 + 16 * q^89

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	3872.1.r.a

Level	$3872$
Weight	$1$
Character orbit	3872.r
Analytic conductor	$1.932$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{4}$
CM discriminant	-4
Inner twists	$16$

Self dual:	no
Analytic conductor:	\(1.93237972891\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	8.0.64000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	\(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{4}\)
Projective field:	Galois closure of 4.0.21296.1
Artin image:	$C_5\times \SD_{16}$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

\(\beta_{1}\)	\(=\)	\( \nu \) v
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 2 \) (v^2) / 2
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 2 \) (v^3) / 2
\(\beta_{4}\)	\(=\)	\( ( \nu^{4} ) / 4 \) (v^4) / 4
\(\beta_{5}\)	\(=\)	\( ( \nu^{5} ) / 4 \) (v^5) / 4
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} ) / 8 \) (v^6) / 8
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} ) / 8 \) (v^7) / 8

\(\nu\)	\(=\)	\( \beta_1 \) b1
\(\nu^{2}\)	\(=\)	\( 2\beta_{2} \) 2*b2
\(\nu^{3}\)	\(=\)	\( 2\beta_{3} \) 2*b3
\(\nu^{4}\)	\(=\)	\( 4\beta_{4} \) 4*b4
\(\nu^{5}\)	\(=\)	\( 4\beta_{5} \) 4*b5
\(\nu^{6}\)	\(=\)	\( 8\beta_{6} \) 8*b6
\(\nu^{7}\)	\(=\)	\( 8\beta_{7} \) 8*b7

\(n\)	\(485\)	\(1695\)	\(2785\)
\(\chi(n)\)	\(1\)	\(1\)	\(\beta_{2}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 161.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
11.b	odd	2	1	inner
11.c	even	5	3	inner
11.d	odd	10	3	inner
44.c	even	2	1	inner
44.g	even	10	3	inner
44.h	odd	10	3	inner

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