LMFDB - Newform orbit 3969.1.bz.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,1,Mod(55,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(42))

chi = DirichletCharacter(H, H._module([28, 27]))

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

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

chi := DirichletCharacter("3969.55");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3969.bz (of order $42$, degree $12$, 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:	$1.98078903514$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{21})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 $ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 441)
Projective image:	$D_{14}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{14} - \cdots)$

$q$-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_{42}^{16} q^{4} - \zeta_{42}^{8} q^{7} +O(q^{10}) $ q - z^16 * q^4 - z^8 * q^7	$ q - \zeta_{42}^{16} q^{4} - \zeta_{42}^{8} q^{7} + ( - \zeta_{42}^{19} - \zeta_{42}^{4}) q^{13} - \zeta_{42}^{11} q^{16} + ( - \zeta_{42}^{15} - \zeta_{42}^{6}) q^{19} + \zeta_{42}^{20} q^{25} - \zeta_{42}^{3} q^{28} + (\zeta_{42}^{19} + \zeta_{42}^{16}) q^{31} + (\zeta_{42}^{3} - 1) q^{37} + (\zeta_{42}^{17} + \zeta_{42}^{5}) q^{43} + \zeta_{42}^{16} q^{49} + (\zeta_{42}^{20} - \zeta_{42}^{14}) q^{52} + ( - \zeta_{42}^{17} - \zeta_{42}^{14}) q^{61} - \zeta_{42}^{6} q^{64} + (\zeta_{42}^{10} + \zeta_{42}^{4}) q^{67} + ( - \zeta_{42}^{18} - \zeta_{42}^{15}) q^{73} + ( - \zeta_{42}^{10} - \zeta_{42}) q^{76} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{79} + (\zeta_{42}^{12} - \zeta_{42}^{6}) q^{91} + (\zeta_{42}^{20} - \zeta_{42}^{8}) q^{97} +O(q^{100}) $ q - z^16 * q^4 - z^8 * q^7 + (-z^19 - z^4) * q^13 - z^11 * q^16 + (-z^15 - z^6) * q^19 + z^20 * q^25 - z^3 * q^28 + (z^19 + z^16) * q^31 + (z^3 - 1) * q^37 + (z^17 + z^5) * q^43 + z^16 * q^49 + (z^20 - z^14) * q^52 + (-z^17 - z^14) * q^61 - z^6 * q^64 + (z^10 + z^4) * q^67 + (-z^18 - z^15) * q^73 + (-z^10 - z) * q^76 + (-z^5 + z^2) * q^79 + (z^12 - z^6) * q^91 + (z^20 - z^8) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{4} - q^{7}+O(q^{10}) $ 12 * q - q^4 - q^7	$ 12 q - q^{4} - q^{7} + q^{16} + q^{25} - 2 q^{28} - 10 q^{37} - 2 q^{43} + q^{49} + 7 q^{52} + 7 q^{61} + 2 q^{64} + 2 q^{67} + 2 q^{79}+O(q^{100}) $ 12 * q - q^4 - q^7 + q^16 + q^25 - 2 * q^28 - 10 * q^37 - 2 * q^43 + q^49 + 7 * q^52 + 7 * q^61 + 2 * q^64 + 2 * q^67 + 2 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3969\mathbb{Z}\right)^\times$.

$n$	$2108$	$3727$
$\chi(n)$	$-\zeta_{42}^{7}$	$-\zeta_{42}^{6}$

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

−0.988831	−	0.149042i
−0.988831	+	0.149042i
0.955573	−	0.294755i
0.826239	−	0.563320i
−0.733052	+	0.680173i
0.365341	−	0.930874i
0.0747301	−	0.997204i
0.0747301	+	0.997204i
0.365341	+	0.930874i
−0.733052	−	0.680173i
0.826239	+	0.563320i
0.955573	+	0.294755i

0.733052

−

0.680173i

−0.365341

−

0.930874i

433.1

0.733052

0.680173i

−0.365341

0.930874i

622.1

−0.0747301

−

0.997204i

0.733052

0.680173i

1000.1

0.988831

−

0.149042i

−0.0747301

−

0.997204i

1189.1

−0.826239

−

0.563320i

−0.955573

−

0.294755i

1756.1

−0.955573

0.294755i

0.988831

−

0.149042i

2134.1

−0.365341

−

0.930874i

−0.826239

−

0.563320i

2323.1

−0.365341

0.930874i

−0.826239

0.563320i

2701.1

−0.955573

−

0.294755i

0.988831

0.149042i

3268.1

−0.826239

0.563320i

−0.955573

0.294755i

3457.1

0.988831

0.149042i

−0.0747301

0.997204i

3835.1

−0.0747301

0.997204i

0.733052

−

0.680173i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
9.c	even	3	1	inner
9.d	odd	6	1	inner
49.f	odd	14	1	inner
147.k	even	14	1	inner
441.bh	even	42	1	inner
441.bk	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3969.1.bz.a		12
3.b	odd	2	1	CM	3969.1.bz.a		12
9.c	even	3	1		441.1.v.a	✓	6
9.c	even	3	1	inner	3969.1.bz.a		12
9.d	odd	6	1		441.1.v.a	✓	6
9.d	odd	6	1	inner	3969.1.bz.a		12
49.f	odd	14	1	inner	3969.1.bz.a		12
63.g	even	3	1		3087.1.bj.b		12
63.h	even	3	1		3087.1.bj.b		12
63.i	even	6	1		3087.1.bj.a		12
63.j	odd	6	1		3087.1.bj.b		12
63.k	odd	6	1		3087.1.bj.a		12
63.l	odd	6	1		3087.1.v.a		6
63.n	odd	6	1		3087.1.bj.b		12
63.o	even	6	1		3087.1.v.a		6
63.s	even	6	1		3087.1.bj.a		12
63.t	odd	6	1		3087.1.bj.a		12
147.k	even	14	1	inner	3969.1.bz.a		12
441.y	even	21	1		3087.1.bj.a		12
441.z	even	21	1		3087.1.bj.a		12
441.ba	even	21	1		3087.1.v.a		6
441.bc	odd	42	1		3087.1.bj.b		12
441.bd	even	42	1		3087.1.bj.b		12
441.be	odd	42	1		3087.1.v.a		6
441.bh	even	42	1		441.1.v.a	✓	6
441.bh	even	42	1	inner	3969.1.bz.a		12
441.bi	odd	42	1		3087.1.bj.a		12
441.bk	odd	42	1		441.1.v.a	✓	6
441.bk	odd	42	1	inner	3969.1.bz.a		12
441.bl	odd	42	1		3087.1.bj.b		12
441.bm	odd	42	1		3087.1.bj.a		12
441.bn	even	42	1		3087.1.bj.b		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
441.1.v.a	✓	6	9.c	even	3	1
441.1.v.a	✓	6	9.d	odd	6	1
441.1.v.a	✓	6	441.bh	even	42	1
441.1.v.a	✓	6	441.bk	odd	42	1
3087.1.v.a		6	63.l	odd	6	1
3087.1.v.a		6	63.o	even	6	1
3087.1.v.a		6	441.ba	even	21	1
3087.1.v.a		6	441.be	odd	42	1
3087.1.bj.a		12	63.i	even	6	1
3087.1.bj.a		12	63.k	odd	6	1
3087.1.bj.a		12	63.s	even	6	1
3087.1.bj.a		12	63.t	odd	6	1
3087.1.bj.a		12	441.y	even	21	1
3087.1.bj.a		12	441.z	even	21	1
3087.1.bj.a		12	441.bi	odd	42	1
3087.1.bj.a		12	441.bm	odd	42	1
3087.1.bj.b		12	63.g	even	3	1
3087.1.bj.b		12	63.h	even	3	1
3087.1.bj.b		12	63.j	odd	6	1
3087.1.bj.b		12	63.n	odd	6	1
3087.1.bj.b		12	441.bc	odd	42	1
3087.1.bj.b		12	441.bd	even	42	1
3087.1.bj.b		12	441.bl	odd	42	1
3087.1.bj.b		12	441.bn	even	42	1
3969.1.bz.a		12	1.a	even	1	1	trivial
3969.1.bz.a		12	3.b	odd	2	1	CM
3969.1.bz.a		12	9.c	even	3	1	inner
3969.1.bz.a		12	9.d	odd	6	1	inner
3969.1.bz.a		12	49.f	odd	14	1	inner
3969.1.bz.a		12	147.k	even	14	1	inner
3969.1.bz.a		12	441.bh	even	42	1	inner
3969.1.bz.a		12	441.bk	odd	42	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3969, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} $ T^12
$5$	$ T^{12} $ T^12
$7$	$ T^{12} + T^{11} + \cdots + 1 $ T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$11$	$ T^{12} $ T^12
$13$	$ T^{12} + 14 T^{9} + \cdots + 49 $ T^12 + 14T^9 + 7T^7 + 63T^6 + 49T^4 + 98T^3 + 49T^2 + 49*T + 49
$17$	$ T^{12} $ T^12
$19$	$ (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} $ (T^6 + 7T^4 + 14T^2 + 7)^2
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} - 7 T^{10} + \cdots + 49 $ T^12 - 7T^10 + 35T^8 - 84T^6 + 147T^4 - 98*T^2 + 49
$37$	$ (T^{6} + 5 T^{5} + 11 T^{4} + \cdots + 1)^{2} $ (T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1)^2
$41$	$ T^{12} $ T^12
$43$	$ T^{12} + 2 T^{11} + \cdots + 1 $ T^12 + 2T^11 - 8T^9 - 9T^8 + 22T^6 + 42T^5 + 45T^4 + 20T^3 + 7T^2 + 4*T + 1
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ T^{12} - 7 T^{11} + \cdots + 49 $ T^12 - 7T^11 + 28T^10 - 77T^9 + 161T^8 - 266T^7 + 357T^6 - 392T^5 + 343T^4 - 245T^3 + 196T^2 - 147*T + 49
$67$	$ (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$71$	$ T^{12} $ T^12
$73$	$ (T^{6} + 7 T^{2} + 14 T + 7)^{2} $ (T^6 + 7T^2 + 14T + 7)^2
$79$	$ (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} $ (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ T^{12} - 7 T^{10} + \cdots + 49 $ T^12 - 7T^10 + 35T^8 - 84T^6 + 147T^4 - 98*T^2 + 49
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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3969.bz (of order \(42\), degree \(12\), not minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{42}^{16} q^{4} - \zeta_{42}^{8} q^{7} +O(q^{10}) \) q - z^16 * q^4 - z^8 * q^7	\( q - \zeta_{42}^{16} q^{4} - \zeta_{42}^{8} q^{7} + ( - \zeta_{42}^{19} - \zeta_{42}^{4}) q^{13} - \zeta_{42}^{11} q^{16} + ( - \zeta_{42}^{15} - \zeta_{42}^{6}) q^{19} + \zeta_{42}^{20} q^{25} - \zeta_{42}^{3} q^{28} + (\zeta_{42}^{19} + \zeta_{42}^{16}) q^{31} + (\zeta_{42}^{3} - 1) q^{37} + (\zeta_{42}^{17} + \zeta_{42}^{5}) q^{43} + \zeta_{42}^{16} q^{49} + (\zeta_{42}^{20} - \zeta_{42}^{14}) q^{52} + ( - \zeta_{42}^{17} - \zeta_{42}^{14}) q^{61} - \zeta_{42}^{6} q^{64} + (\zeta_{42}^{10} + \zeta_{42}^{4}) q^{67} + ( - \zeta_{42}^{18} - \zeta_{42}^{15}) q^{73} + ( - \zeta_{42}^{10} - \zeta_{42}) q^{76} + ( - \zeta_{42}^{5} + \zeta_{42}^{2}) q^{79} + (\zeta_{42}^{12} - \zeta_{42}^{6}) q^{91} + (\zeta_{42}^{20} - \zeta_{42}^{8}) q^{97} +O(q^{100}) \) q - z^16 * q^4 - z^8 * q^7 + (-z^19 - z^4) * q^13 - z^11 * q^16 + (-z^15 - z^6) * q^19 + z^20 * q^25 - z^3 * q^28 + (z^19 + z^16) * q^31 + (z^3 - 1) * q^37 + (z^17 + z^5) * q^43 + z^16 * q^49 + (z^20 - z^14) * q^52 + (-z^17 - z^14) * q^61 - z^6 * q^64 + (z^10 + z^4) * q^67 + (-z^18 - z^15) * q^73 + (-z^10 - z) * q^76 + (-z^5 + z^2) * q^79 + (z^12 - z^6) * q^91 + (z^20 - z^8) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{4} - q^{7}+O(q^{10}) \) 12 * q - q^4 - q^7	\( 12 q - q^{4} - q^{7} + q^{16} + q^{25} - 2 q^{28} - 10 q^{37} - 2 q^{43} + q^{49} + 7 q^{52} + 7 q^{61} + 2 q^{64} + 2 q^{67} + 2 q^{79}+O(q^{100}) \) 12 * q - q^4 - q^7 + q^16 + q^25 - 2 * q^28 - 10 * q^37 - 2 * q^43 + q^49 + 7 * q^52 + 7 * q^61 + 2 * q^64 + 2 * q^67 + 2 * q^79

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	3969.1.bz.a

Level	$3969$
Weight	$1$
Character orbit	3969.bz
Analytic conductor	$1.981$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{14}$
CM discriminant	-3
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.98078903514\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Coefficient field:	\(\Q(\zeta_{21})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1 \) x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 441)
Projective image:	\(D_{14}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{14} - \cdots)\)

\(n\)	\(2108\)	\(3727\)
\(\chi(n)\)	\(-\zeta_{42}^{7}\)	\(-\zeta_{42}^{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 55.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
9.c	even	3	1	inner
9.d	odd	6	1	inner
49.f	odd	14	1	inner
147.k	even	14	1	inner
441.bh	even	42	1	inner
441.bk	odd	42	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} \) T^12
$5$	\( T^{12} \) T^12
$7$	\( T^{12} + T^{11} + \cdots + 1 \) T^12 + T^11 - T^9 - T^8 + T^6 - T^4 - T^3 + T + 1
$11$	\( T^{12} \) T^12
$13$	\( T^{12} + 14 T^{9} + \cdots + 49 \) T^12 + 14T^9 + 7T^7 + 63T^6 + 49T^4 + 98T^3 + 49T^2 + 49*T + 49
$17$	\( T^{12} \) T^12
$19$	\( (T^{6} + 7 T^{4} + 14 T^{2} + 7)^{2} \) (T^6 + 7T^4 + 14T^2 + 7)^2
$23$	\( T^{12} \) T^12
$29$	\( T^{12} \) T^12
$31$	\( T^{12} - 7 T^{10} + \cdots + 49 \) T^12 - 7T^10 + 35T^8 - 84T^6 + 147T^4 - 98*T^2 + 49
$37$	\( (T^{6} + 5 T^{5} + 11 T^{4} + \cdots + 1)^{2} \) (T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1)^2
$41$	\( T^{12} \) T^12
$43$	\( T^{12} + 2 T^{11} + \cdots + 1 \) T^12 + 2T^11 - 8T^9 - 9T^8 + 22T^6 + 42T^5 + 45T^4 + 20T^3 + 7T^2 + 4*T + 1
$47$	\( T^{12} \) T^12
$53$	\( T^{12} \) T^12
$59$	\( T^{12} \) T^12
$61$	\( T^{12} - 7 T^{11} + \cdots + 49 \) T^12 - 7T^11 + 28T^10 - 77T^9 + 161T^8 - 266T^7 + 357T^6 - 392T^5 + 343T^4 - 245T^3 + 196T^2 - 147*T + 49
$67$	\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$71$	\( T^{12} \) T^12
$73$	\( (T^{6} + 7 T^{2} + 14 T + 7)^{2} \) (T^6 + 7T^2 + 14T + 7)^2
$79$	\( (T^{6} - T^{5} + 3 T^{4} + \cdots + 1)^{2} \) (T^6 - T^5 + 3T^4 + 5T^2 - 2*T + 1)^2
$83$	\( T^{12} \) T^12
$89$	\( T^{12} \) T^12
$97$	\( T^{12} - 7 T^{10} + \cdots + 49 \) T^12 - 7T^10 + 35T^8 - 84T^6 + 147T^4 - 98*T^2 + 49
show more
show less