LMFDB - Newform orbit 320.3.h.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,3,Mod(319,320)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("320.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	320.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.71936845953$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 20)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $\beta = 4i$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta - 3) q^{5} - 9 q^{9}+O(q^{10}) $ q + (b - 3) * q^5 - 9 * q^9	$ q + (\beta - 3) q^{5} - 9 q^{9} - 6 \beta q^{13} - 4 \beta q^{17} + ( - 6 \beta - 7) q^{25} - 42 q^{29} - 6 \beta q^{37} - 18 q^{41} + ( - 9 \beta + 27) q^{45} - 49 q^{49} + 14 \beta q^{53} - 22 q^{61} + (18 \beta + 96) q^{65} - 24 \beta q^{73} + 81 q^{81} + (12 \beta + 64) q^{85} - 78 q^{89} + 36 \beta q^{97} +O(q^{100}) $ q + (b - 3) * q^5 - 9 * q^9 - 6b q^13 - 4b q^17 + (-6b - 7) q^25 - 42 * q^29 - 6b q^37 - 18 * q^41 + (-9b + 27) q^45 - 49 * q^49 + 14b q^53 - 22 * q^61 + (18b + 96) q^65 - 24b q^73 + 81 * q^81 + (12b + 64) q^85 - 78 * q^89 + 36b q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{5} - 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^5 - 18 * q^9	$ 2 q - 6 q^{5} - 18 q^{9} - 14 q^{25} - 84 q^{29} - 36 q^{41} + 54 q^{45} - 98 q^{49} - 44 q^{61} + 192 q^{65} + 162 q^{81} + 128 q^{85} - 156 q^{89}+O(q^{100}) $ 2 * q - 6 * q^5 - 18 * q^9 - 14 * q^25 - 84 * q^29 - 36 * q^41 + 54 * q^45 - 98 * q^49 - 44 * q^61 + 192 * q^65 + 162 * q^81 + 128 * q^85 - 156 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$-1$	$-1$	$1$

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} $

319.1

	−	1.00000i
		1.00000i

−3.00000

−

4.00000i

−9.00000

319.2

−3.00000

4.00000i

−9.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
5.b	even	2	1	inner
20.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	320.3.h.d		2
4.b	odd	2	1	CM	320.3.h.d		2
5.b	even	2	1	inner	320.3.h.d		2
5.c	odd	4	1		1600.3.b.a		1
5.c	odd	4	1		1600.3.b.c		1
8.b	even	2	1		20.3.d.c	✓	2
8.d	odd	2	1		20.3.d.c	✓	2
16.e	even	4	1		1280.3.e.a		2
16.e	even	4	1		1280.3.e.d		2
16.f	odd	4	1		1280.3.e.a		2
16.f	odd	4	1		1280.3.e.d		2
20.d	odd	2	1	inner	320.3.h.d		2
20.e	even	4	1		1600.3.b.a		1
20.e	even	4	1		1600.3.b.c		1
24.f	even	2	1		180.3.f.c		2
24.h	odd	2	1		180.3.f.c		2
40.e	odd	2	1		20.3.d.c	✓	2
40.f	even	2	1		20.3.d.c	✓	2
40.i	odd	4	1		100.3.b.a		1
40.i	odd	4	1		100.3.b.b		1
40.k	even	4	1		100.3.b.a		1
40.k	even	4	1		100.3.b.b		1
80.k	odd	4	1		1280.3.e.a		2
80.k	odd	4	1		1280.3.e.d		2
80.q	even	4	1		1280.3.e.a		2
80.q	even	4	1		1280.3.e.d		2
120.i	odd	2	1		180.3.f.c		2
120.m	even	2	1		180.3.f.c		2
120.q	odd	4	1		900.3.c.a		1
120.q	odd	4	1		900.3.c.d		1
120.w	even	4	1		900.3.c.a		1
120.w	even	4	1		900.3.c.d		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
20.3.d.c	✓	2	8.b	even	2	1
20.3.d.c	✓	2	8.d	odd	2	1
20.3.d.c	✓	2	40.e	odd	2	1
20.3.d.c	✓	2	40.f	even	2	1
100.3.b.a		1	40.i	odd	4	1
100.3.b.a		1	40.k	even	4	1
100.3.b.b		1	40.i	odd	4	1
100.3.b.b		1	40.k	even	4	1
180.3.f.c		2	24.f	even	2	1
180.3.f.c		2	24.h	odd	2	1
180.3.f.c		2	120.i	odd	2	1
180.3.f.c		2	120.m	even	2	1
320.3.h.d		2	1.a	even	1	1	trivial
320.3.h.d		2	4.b	odd	2	1	CM
320.3.h.d		2	5.b	even	2	1	inner
320.3.h.d		2	20.d	odd	2	1	inner
900.3.c.a		1	120.q	odd	4	1
900.3.c.a		1	120.w	even	4	1
900.3.c.d		1	120.q	odd	4	1
900.3.c.d		1	120.w	even	4	1
1280.3.e.a		2	16.e	even	4	1
1280.3.e.a		2	16.f	odd	4	1
1280.3.e.a		2	80.k	odd	4	1
1280.3.e.a		2	80.q	even	4	1
1280.3.e.d		2	16.e	even	4	1
1280.3.e.d		2	16.f	odd	4	1
1280.3.e.d		2	80.k	odd	4	1
1280.3.e.d		2	80.q	even	4	1
1600.3.b.a		1	5.c	odd	4	1
1600.3.b.a		1	20.e	even	4	1
1600.3.b.c		1	5.c	odd	4	1
1600.3.b.c		1	20.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} $ T3 acting on $S_{3}^{\mathrm{new}}(320, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 6T + 25 $ T^2 + 6*T + 25
$7$	$ T^{2} $ T^2
$11$	$ T^{2} $ T^2
$13$	$ T^{2} + 576 $ T^2 + 576
$17$	$ T^{2} + 256 $ T^2 + 256
$19$	$ T^{2} $ T^2
$23$	$ T^{2} $ T^2
$29$	$ (T + 42)^{2} $ (T + 42)^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} + 576 $ T^2 + 576
$41$	$ (T + 18)^{2} $ (T + 18)^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ T^{2} + 3136 $ T^2 + 3136
$59$	$ T^{2} $ T^2
$61$	$ (T + 22)^{2} $ (T + 22)^2
$67$	$ T^{2} $ T^2
$71$	$ T^{2} $ T^2
$73$	$ T^{2} + 9216 $ T^2 + 9216
$79$	$ T^{2} $ T^2
$83$	$ T^{2} $ T^2
$89$	$ (T + 78)^{2} $ (T + 78)^2
$97$	$ T^{2} + 20736 $ T^2 + 20736
show more
show less

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	320.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + (\beta - 3) q^{5} - 9 q^{9}+O(q^{10}) \) q + (b - 3) * q^5 - 9 * q^9	\( q + (\beta - 3) q^{5} - 9 q^{9} - 6 \beta q^{13} - 4 \beta q^{17} + ( - 6 \beta - 7) q^{25} - 42 q^{29} - 6 \beta q^{37} - 18 q^{41} + ( - 9 \beta + 27) q^{45} - 49 q^{49} + 14 \beta q^{53} - 22 q^{61} + (18 \beta + 96) q^{65} - 24 \beta q^{73} + 81 q^{81} + (12 \beta + 64) q^{85} - 78 q^{89} + 36 \beta q^{97} +O(q^{100}) \) q + (b - 3) * q^5 - 9 * q^9 - 6b q^13 - 4b q^17 + (-6b - 7) q^25 - 42 * q^29 - 6b q^37 - 18 * q^41 + (-9b + 27) q^45 - 49 * q^49 + 14b q^53 - 22 * q^61 + (18b + 96) q^65 - 24b q^73 + 81 * q^81 + (12b + 64) q^85 - 78 * q^89 + 36b q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{5} - 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^5 - 18 * q^9	\( 2 q - 6 q^{5} - 18 q^{9} - 14 q^{25} - 84 q^{29} - 36 q^{41} + 54 q^{45} - 98 q^{49} - 44 q^{61} + 192 q^{65} + 162 q^{81} + 128 q^{85} - 156 q^{89}+O(q^{100}) \) 2 * q - 6 * q^5 - 18 * q^9 - 14 * q^25 - 84 * q^29 - 36 * q^41 + 54 * q^45 - 98 * q^49 - 44 * q^61 + 192 * q^65 + 162 * q^81 + 128 * q^85 - 156 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more