LMFDB - Newform orbit 324.1.f.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,1,Mod(55,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("324.55");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	324.f (of order $6$, degree $2$, 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:	$0.161697064093$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.324.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.419904.3

$q$-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_{6}^{2} q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{8} +O(q^{10}) $ q + z^2 * q^2 - z * q^4 + z * q^5 + q^8	$ q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{8} - q^{10} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - q^{17} - \zeta_{6}^{2} q^{20} - q^{26} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{32} - \zeta_{6}^{2} q^{34} - q^{37} + \zeta_{6} q^{40} - \zeta_{6} q^{41} - \zeta_{6} q^{49} - \zeta_{6}^{2} q^{52} + q^{53} + \zeta_{6} q^{58} - \zeta_{6}^{2} q^{61} + q^{64} + \zeta_{6}^{2} q^{65} + \zeta_{6} q^{68} - q^{73} - \zeta_{6}^{2} q^{74} - q^{80} + 2 q^{82} - \zeta_{6} q^{85} - q^{89} + \zeta_{6}^{2} q^{97} + q^{98} +O(q^{100}) $ q + z^2 * q^2 - z * q^4 + z * q^5 + q^8 - q^10 + z * q^13 + z^2 * q^16 - q^17 - z^2 * q^20 - q^26 - z^2 * q^29 - z * q^32 - z^2 * q^34 - q^37 + z * q^40 - z * q^41 - z * q^49 - z^2 * q^52 + q^53 + z * q^58 - z^2 * q^61 + q^64 + z^2 * q^65 + z * q^68 - q^73 - z^2 * q^74 - q^80 + 2 * q^82 - z * q^85 - q^89 + z^2 * q^97 + q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - q^4 + q^5 + 2 * q^8	$ 2 q - q^{2} - q^{4} + q^{5} + 2 q^{8} - 2 q^{10} + q^{13} - q^{16} - 2 q^{17} + q^{20} - 2 q^{26} + q^{29} - q^{32} + q^{34} - 2 q^{37} + q^{40} - 2 q^{41} - q^{49} + q^{52} + 4 q^{53} + q^{58} + q^{61} + 2 q^{64} - q^{65} + q^{68} - 2 q^{73} + q^{74} - 2 q^{80} + 4 q^{82} - q^{85} - 2 q^{89} - 2 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + q^5 + 2 * q^8 - 2 * q^10 + q^13 - q^16 - 2 * q^17 + q^20 - 2 * q^26 + q^29 - q^32 + q^34 - 2 * q^37 + q^40 - 2 * q^41 - q^49 + q^52 + 4 * q^53 + q^58 + q^61 + 2 * q^64 - q^65 + q^68 - 2 * q^73 + q^74 - 2 * q^80 + 4 * q^82 - q^85 - 2 * q^89 - 2 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$-1$	$-\zeta_{6}$

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} $

55.1

0.500000	+	0.866025i
0.500000	−	0.866025i

−0.500000

0.866025i

−0.500000

−

0.866025i

0.500000

0.866025i

1.00000

−1.00000

271.1

−0.500000

−

0.866025i

−0.500000

0.866025i

0.500000

−

0.866025i

1.00000

−1.00000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1}) $
9.c	even	3	1	inner
36.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	324.1.f.a		2
3.b	odd	2	1		324.1.f.b		2
4.b	odd	2	1	CM	324.1.f.a		2
9.c	even	3	1		324.1.d.b	yes	1
9.c	even	3	1	inner	324.1.f.a		2
9.d	odd	6	1		324.1.d.a	✓	1
9.d	odd	6	1		324.1.f.b		2
12.b	even	2	1		324.1.f.b		2
27.e	even	9	6		2916.1.j.d		6
27.f	odd	18	6		2916.1.j.e		6
36.f	odd	6	1		324.1.d.b	yes	1
36.f	odd	6	1	inner	324.1.f.a		2
36.h	even	6	1		324.1.d.a	✓	1
36.h	even	6	1		324.1.f.b		2
108.j	odd	18	6		2916.1.j.d		6
108.l	even	18	6		2916.1.j.e		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
324.1.d.a	✓	1	9.d	odd	6	1
324.1.d.a	✓	1	36.h	even	6	1
324.1.d.b	yes	1	9.c	even	3	1
324.1.d.b	yes	1	36.f	odd	6	1
324.1.f.a		2	1.a	even	1	1	trivial
324.1.f.a		2	4.b	odd	2	1	CM
324.1.f.a		2	9.c	even	3	1	inner
324.1.f.a		2	36.f	odd	6	1	inner
324.1.f.b		2	3.b	odd	2	1
324.1.f.b		2	9.d	odd	6	1
324.1.f.b		2	12.b	even	2	1
324.1.f.b		2	36.h	even	6	1
2916.1.j.d		6	27.e	even	9	6
2916.1.j.d		6	108.j	odd	18	6
2916.1.j.e		6	27.f	odd	18	6
2916.1.j.e		6	108.l	even	18	6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} - T_{5} + 1 $ T5^2 - T5 + 1 acting on $S_{1}^{\mathrm{new}}(324, [\chi])$.

Hecke characteristic polynomials

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

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	324.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{8} +O(q^{10}) \) q + z^2 * q^2 - z * q^4 + z * q^5 + q^8	\( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{8} - q^{10} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - q^{17} - \zeta_{6}^{2} q^{20} - q^{26} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{32} - \zeta_{6}^{2} q^{34} - q^{37} + \zeta_{6} q^{40} - \zeta_{6} q^{41} - \zeta_{6} q^{49} - \zeta_{6}^{2} q^{52} + q^{53} + \zeta_{6} q^{58} - \zeta_{6}^{2} q^{61} + q^{64} + \zeta_{6}^{2} q^{65} + \zeta_{6} q^{68} - q^{73} - \zeta_{6}^{2} q^{74} - q^{80} + 2 q^{82} - \zeta_{6} q^{85} - q^{89} + \zeta_{6}^{2} q^{97} + q^{98} +O(q^{100}) \) q + z^2 * q^2 - z * q^4 + z * q^5 + q^8 - q^10 + z * q^13 + z^2 * q^16 - q^17 - z^2 * q^20 - q^26 - z^2 * q^29 - z * q^32 - z^2 * q^34 - q^37 + z * q^40 - z * q^41 - z * q^49 - z^2 * q^52 + q^53 + z * q^58 - z^2 * q^61 + q^64 + z^2 * q^65 + z * q^68 - q^73 - z^2 * q^74 - q^80 + 2 * q^82 - z * q^85 - q^89 + z^2 * q^97 + q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - q^4 + q^5 + 2 * q^8	\( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{8} - 2 q^{10} + q^{13} - q^{16} - 2 q^{17} + q^{20} - 2 q^{26} + q^{29} - q^{32} + q^{34} - 2 q^{37} + q^{40} - 2 q^{41} - q^{49} + q^{52} + 4 q^{53} + q^{58} + q^{61} + 2 q^{64} - q^{65} + q^{68} - 2 q^{73} + q^{74} - 2 q^{80} + 4 q^{82} - q^{85} - 2 q^{89} - 2 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + q^5 + 2 * q^8 - 2 * q^10 + q^13 - q^16 - 2 * q^17 + q^20 - 2 * q^26 + q^29 - q^32 + q^34 - 2 * q^37 + q^40 - 2 * q^41 - q^49 + q^52 + 4 * q^53 + q^58 + q^61 + 2 * q^64 - q^65 + q^68 - 2 * q^73 + q^74 - 2 * q^80 + 4 * q^82 - q^85 - 2 * q^89 - 2 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	324.1.f.a

Level	$324$
Weight	$1$
Character orbit	324.f
Analytic conductor	$0.162$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-4
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.161697064093\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{2} - x + 1 \) x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.324.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.419904.3

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by \(\Q(\sqrt{-1}) \)
9.c	even	3	1	inner
36.f	odd	6	1	inner

$p$	$F_p(T)$
$2$	\( T^{2} + T + 1 \) T^2 + T + 1
$3$	\( T^{2} \) T^2
$5$	\( T^{2} - T + 1 \) T^2 - T + 1
$7$	\( T^{2} \) T^2
$11$	\( T^{2} \) T^2
$13$	\( T^{2} - T + 1 \) T^2 - T + 1
$17$	\( (T + 1)^{2} \) (T + 1)^2
$19$	\( T^{2} \) T^2
$23$	\( T^{2} \) T^2
$29$	\( T^{2} - T + 1 \) T^2 - T + 1
$31$	\( T^{2} \) T^2
$37$	\( (T + 1)^{2} \) (T + 1)^2
$41$	\( T^{2} + 2T + 4 \) T^2 + 2*T + 4
$43$	\( T^{2} \) T^2
$47$	\( T^{2} \) T^2
$53$	\( (T - 2)^{2} \) (T - 2)^2
$59$	\( T^{2} \) T^2
$61$	\( T^{2} - T + 1 \) T^2 - T + 1
$67$	\( T^{2} \) T^2
$71$	\( T^{2} \) T^2
$73$	\( (T + 1)^{2} \) (T + 1)^2
$79$	\( T^{2} \) T^2
$83$	\( T^{2} \) T^2
$89$	\( (T + 1)^{2} \) (T + 1)^2
$97$	\( T^{2} + 2T + 4 \) T^2 + 2*T + 4
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\(n\)	\(163\)	\(245\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{6}\)

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