LMFDB - Newform orbit 3312.1.bm.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,1,Mod(1057,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))

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

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

chi := DirichletCharacter("3312.1057");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3312.bm (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:	$1.65290332184$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 207)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.148718980881.2

$q$-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_{18}^{7} q^{3} - \zeta_{18}^{5} q^{9} +O(q^{10}) $ q + z^7 * q^3 - z^5 * q^9	$ q + \zeta_{18}^{7} q^{3} - \zeta_{18}^{5} q^{9} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{13} - \zeta_{18}^{6} q^{23} - \zeta_{18}^{3} q^{25} + \zeta_{18}^{3} q^{27} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{29} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{31} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{39} + (\zeta_{18}^{2} - \zeta_{18}) q^{41} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{47} + \zeta_{18}^{6} q^{49} + \zeta_{18}^{6} q^{59} + \zeta_{18}^{4} q^{69} + ( - \zeta_{18}^{8} + \zeta_{18}) q^{71} + (\zeta_{18}^{8} - \zeta_{18}) q^{73} + \zeta_{18} q^{75} - \zeta_{18} q^{81} + ( - \zeta_{18}^{2} - 1) q^{87} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{93} +O(q^{100}) $ q + z^7 * q^3 - z^5 * q^9 + (z^8 + z^4) * q^13 - z^6 * q^23 - z^3 * q^25 + z^3 * q^27 + (z^4 + z^2) * q^29 + (z^7 + z^5) * q^31 + (-z^6 - z^2) * q^39 + (z^2 - z) * q^41 + (-z^8 + z^7) * q^47 + z^6 * q^49 + z^6 * q^59 + z^4 * q^69 + (-z^8 + z) * q^71 + (z^8 - z) * q^73 + z * q^75 - z * q^81 + (-z^2 - 1) * q^87 + (-z^5 - z^3) * q^93
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q+O(q^{10}) $ 6 * q	$ 6 q + 3 q^{23} - 3 q^{25} + 3 q^{27} + 3 q^{39} - 3 q^{49} - 3 q^{59} - 6 q^{87} - 3 q^{93}+O(q^{100}) $ 6 * q + 3 * q^23 - 3 * q^25 + 3 * q^27 + 3 * q^39 - 3 * q^49 - 3 * q^59 - 6 * q^87 - 3 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3312\mathbb{Z}\right)^\times$.

$n$	$415$	$2305$	$2485$	$2945$
$\chi(n)$	$1$	$-1$	$1$	$\zeta_{18}^{6}$

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} $

1057.1

0.939693	+	0.342020i
−0.766044	+	0.642788i
−0.173648	−	0.984808i
0.939693	−	0.342020i
−0.766044	−	0.642788i
−0.173648	+	0.984808i

−0.766044

0.642788i

0.173648

−

0.984808i

1057.2

−0.173648

−

0.984808i

−0.939693

0.342020i

1057.3

0.939693

0.342020i

0.766044

0.642788i

3265.1

−0.766044

−

0.642788i

0.173648

0.984808i

3265.2

−0.173648

0.984808i

−0.939693

−

0.342020i

3265.3

0.939693

−

0.342020i

0.766044

−

0.642788i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
9.c	even	3	1	inner
207.f	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3312.1.bm.a		6
4.b	odd	2	1		207.1.f.a	✓	6
9.c	even	3	1	inner	3312.1.bm.a		6
12.b	even	2	1		621.1.f.a		6
23.b	odd	2	1	CM	3312.1.bm.a		6
36.f	odd	6	1		207.1.f.a	✓	6
36.f	odd	6	1		1863.1.d.a		3
36.h	even	6	1		621.1.f.a		6
36.h	even	6	1		1863.1.d.b		3
92.b	even	2	1		207.1.f.a	✓	6
207.f	odd	6	1	inner	3312.1.bm.a		6
276.h	odd	2	1		621.1.f.a		6
828.j	odd	6	1		621.1.f.a		6
828.j	odd	6	1		1863.1.d.b		3
828.m	even	6	1		207.1.f.a	✓	6
828.m	even	6	1		1863.1.d.a		3

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
207.1.f.a	✓	6	4.b	odd	2	1
207.1.f.a	✓	6	36.f	odd	6	1
207.1.f.a	✓	6	92.b	even	2	1
207.1.f.a	✓	6	828.m	even	6	1
621.1.f.a		6	12.b	even	2	1
621.1.f.a		6	36.h	even	6	1
621.1.f.a		6	276.h	odd	2	1
621.1.f.a		6	828.j	odd	6	1
1863.1.d.a		3	36.f	odd	6	1
1863.1.d.a		3	828.m	even	6	1
1863.1.d.b		3	36.h	even	6	1
1863.1.d.b		3	828.j	odd	6	1
3312.1.bm.a		6	1.a	even	1	1	trivial
3312.1.bm.a		6	9.c	even	3	1	inner
3312.1.bm.a		6	23.b	odd	2	1	CM
3312.1.bm.a		6	207.f	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3312, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - T^{3} + 1 $ T^6 - T^3 + 1
$5$	$ T^{6} $ T^6
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$17$	$ T^{6} $ T^6
$19$	$ T^{6} $ T^6
$23$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$29$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$31$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$37$	$ T^{6} $ T^6
$41$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$43$	$ T^{6} $ T^6
$47$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$53$	$ T^{6} $ T^6
$59$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$61$	$ T^{6} $ T^6
$67$	$ T^{6} $ T^6
$71$	$ (T^{3} - 3 T - 1)^{2} $ (T^3 - 3*T - 1)^2
$73$	$ (T^{3} - 3 T + 1)^{2} $ (T^3 - 3*T + 1)^2
$79$	$ T^{6} $ T^6
$83$	$ T^{6} $ T^6
$89$	$ T^{6} $ T^6
$97$	$ T^{6} $ T^6
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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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3312.bm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{18}^{7} q^{3} - \zeta_{18}^{5} q^{9} +O(q^{10}) \) q + z^7 * q^3 - z^5 * q^9	\( q + \zeta_{18}^{7} q^{3} - \zeta_{18}^{5} q^{9} + (\zeta_{18}^{8} + \zeta_{18}^{4}) q^{13} - \zeta_{18}^{6} q^{23} - \zeta_{18}^{3} q^{25} + \zeta_{18}^{3} q^{27} + (\zeta_{18}^{4} + \zeta_{18}^{2}) q^{29} + (\zeta_{18}^{7} + \zeta_{18}^{5}) q^{31} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{39} + (\zeta_{18}^{2} - \zeta_{18}) q^{41} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{47} + \zeta_{18}^{6} q^{49} + \zeta_{18}^{6} q^{59} + \zeta_{18}^{4} q^{69} + ( - \zeta_{18}^{8} + \zeta_{18}) q^{71} + (\zeta_{18}^{8} - \zeta_{18}) q^{73} + \zeta_{18} q^{75} - \zeta_{18} q^{81} + ( - \zeta_{18}^{2} - 1) q^{87} + ( - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{93} +O(q^{100}) \) q + z^7 * q^3 - z^5 * q^9 + (z^8 + z^4) * q^13 - z^6 * q^23 - z^3 * q^25 + z^3 * q^27 + (z^4 + z^2) * q^29 + (z^7 + z^5) * q^31 + (-z^6 - z^2) * q^39 + (z^2 - z) * q^41 + (-z^8 + z^7) * q^47 + z^6 * q^49 + z^6 * q^59 + z^4 * q^69 + (-z^8 + z) * q^71 + (z^8 - z) * q^73 + z * q^75 - z * q^81 + (-z^2 - 1) * q^87 + (-z^5 - z^3) * q^93
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q+O(q^{10}) \) 6 * q	\( 6 q + 3 q^{23} - 3 q^{25} + 3 q^{27} + 3 q^{39} - 3 q^{49} - 3 q^{59} - 6 q^{87} - 3 q^{93}+O(q^{100}) \) 6 * q + 3 * q^23 - 3 * q^25 + 3 * q^27 + 3 * q^39 - 3 * q^49 - 3 * q^59 - 6 * q^87 - 3 * q^93

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	3312.1.bm.a

Level	$3312$
Weight	$1$
Character orbit	3312.bm
Analytic conductor	$1.653$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-23
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.65290332184\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{18})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 207)
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.148718980881.2

\(n\)	\(415\)	\(2305\)	\(2485\)	\(2945\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)	\(\zeta_{18}^{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 1057.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} \) T^6
$3$	\( T^{6} - T^{3} + 1 \) T^6 - T^3 + 1
$5$	\( T^{6} \) T^6
$7$	\( T^{6} \) T^6
$11$	\( T^{6} \) T^6
$13$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$17$	\( T^{6} \) T^6
$19$	\( T^{6} \) T^6
$23$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$29$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$31$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$37$	\( T^{6} \) T^6
$41$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$43$	\( T^{6} \) T^6
$47$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$53$	\( T^{6} \) T^6
$59$	\( (T^{2} + T + 1)^{3} \) (T^2 + T + 1)^3
$61$	\( T^{6} \) T^6
$67$	\( T^{6} \) T^6
$71$	\( (T^{3} - 3 T - 1)^{2} \) (T^3 - 3*T - 1)^2
$73$	\( (T^{3} - 3 T + 1)^{2} \) (T^3 - 3*T + 1)^2
$79$	\( T^{6} \) T^6
$83$	\( T^{6} \) T^6
$89$	\( T^{6} \) T^6
$97$	\( T^{6} \) T^6
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