LMFDB - Newform orbit 3648.1.cn.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3648,1,Mod(161,3648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3648, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 9, 9, 8]))

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

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

chi := DirichletCharacter("3648.161");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3648 = 2^{6} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3648.cn (of order $18$, degree $6$, 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:	$1.82058916609$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	$\Q(\zeta_{36})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{6} + 1 $ x^12 - x^6 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{18}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{18} - \cdots)$

$q$-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_{36}^{7} q^{3} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} + \zeta_{36}^{14} q^{9}+O(q^{10}) $ q + z^7 * q^3 + (z^17 - z^13) * q^7 + z^14 * q^9	$ q + \zeta_{36}^{7} q^{3} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} + \zeta_{36}^{14} q^{9} + ( - \zeta_{36}^{12} - \zeta_{36}^{10}) q^{13} - \zeta_{36}^{15} q^{19} + ( - \zeta_{36}^{6} + \zeta_{36}^{2}) q^{21} + \zeta_{36}^{2} q^{25} - \zeta_{36}^{3} q^{27} + (\zeta_{36}^{11} + \zeta_{36}) q^{31} + ( - \zeta_{36}^{14} - \zeta_{36}^{4}) q^{37} + ( - \zeta_{36}^{17} + \zeta_{36}) q^{39} + (\zeta_{36}^{15} - \zeta_{36}^{5}) q^{43} + ( - \zeta_{36}^{16} + \cdots - \zeta_{36}^{8}) q^{49} + \cdots + \zeta_{36}^{4} q^{97} +O(q^{100}) $ q + z^7 * q^3 + (z^17 - z^13) * q^7 + z^14 * q^9 + (-z^12 - z^10) * q^13 - z^15 * q^19 + (-z^6 + z^2) * q^21 + z^2 * q^25 - z^3 * q^27 + (z^11 + z) * q^31 + (-z^14 - z^4) * q^37 + (-z^17 + z) * q^39 + (z^15 - z^5) * q^43 + (-z^16 + z^12 - z^8) * q^49 + z^4 * q^57 + (-z^16 + 1) * q^61 + (-z^13 + z^9) * q^63 + (z^7 + z^3) * q^67 + (z^8 - z^6) * q^73 + z^9 * q^75 + (-z^9 + z^5) * q^79 - z^10 * q^81 + (z^11 + z^9 - z^7 - z^5) * q^91 + (z^8 - 1) * q^93 + z^4 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q + 6 q^{13} - 6 q^{21} - 6 q^{49} + 12 q^{61} - 6 q^{73} - 12 q^{93}+O(q^{100}) $ 12 * q + 6 * q^13 - 6 * q^21 - 6 * q^49 + 12 * q^61 - 6 * q^73 - 12 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3648\mathbb{Z}\right)^\times$.

$n$	$1217$	$1921$	$2053$	$2623$
$\chi(n)$	$-1$	$\zeta_{36}^{8}$	$-1$	$1$

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} $

161.1

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

−0.642788

−

0.766044i

0.642788

−

1.11334i

−0.173648

0.984808i

161.2

0.642788

0.766044i

−0.642788

1.11334i

−0.173648

0.984808i

929.1

−0.642788

0.766044i

0.642788

1.11334i

−0.173648

−

0.984808i

929.2

0.642788

−

0.766044i

−0.642788

−

1.11334i

−0.173648

−

0.984808i

1505.1

−0.984808

0.173648i

0.984808

−

1.70574i

0.939693

−

0.342020i

1505.2

0.984808

−

0.173648i

−0.984808

1.70574i

0.939693

−

0.342020i

1697.1

−0.342020

0.939693i

0.342020

−

0.592396i

−0.766044

−

0.642788i

1697.2

0.342020

−

0.939693i

−0.342020

0.592396i

−0.766044

−

0.642788i

2657.1

−0.342020

−

0.939693i

0.342020

0.592396i

−0.766044

0.642788i

2657.2

0.342020

0.939693i

−0.342020

−

0.592396i

−0.766044

0.642788i

3425.1

−0.984808

−

0.173648i

0.984808

1.70574i

0.939693

0.342020i

3425.2

0.984808

0.173648i

−0.984808

−

1.70574i

0.939693

0.342020i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
4.b	odd	2	1	inner
12.b	even	2	1	inner
152.t	even	18	1	inner
152.u	odd	18	1	inner
456.bh	odd	18	1	inner
456.bu	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3648.1.cn.c	yes	12
3.b	odd	2	1	CM	3648.1.cn.c	yes	12
4.b	odd	2	1	inner	3648.1.cn.c	yes	12
8.b	even	2	1		3648.1.cn.a	✓	12
8.d	odd	2	1		3648.1.cn.a	✓	12
12.b	even	2	1	inner	3648.1.cn.c	yes	12
19.e	even	9	1		3648.1.cn.a	✓	12
24.f	even	2	1		3648.1.cn.a	✓	12
24.h	odd	2	1		3648.1.cn.a	✓	12
57.l	odd	18	1		3648.1.cn.a	✓	12
76.l	odd	18	1		3648.1.cn.a	✓	12
152.t	even	18	1	inner	3648.1.cn.c	yes	12
152.u	odd	18	1	inner	3648.1.cn.c	yes	12
228.v	even	18	1		3648.1.cn.a	✓	12
456.bh	odd	18	1	inner	3648.1.cn.c	yes	12
456.bu	even	18	1	inner	3648.1.cn.c	yes	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3648.1.cn.a	✓	12	8.b	even	2	1
3648.1.cn.a	✓	12	8.d	odd	2	1
3648.1.cn.a	✓	12	19.e	even	9	1
3648.1.cn.a	✓	12	24.f	even	2	1
3648.1.cn.a	✓	12	24.h	odd	2	1
3648.1.cn.a	✓	12	57.l	odd	18	1
3648.1.cn.a	✓	12	76.l	odd	18	1
3648.1.cn.a	✓	12	228.v	even	18	1
3648.1.cn.c	yes	12	1.a	even	1	1	trivial
3648.1.cn.c	yes	12	3.b	odd	2	1	CM
3648.1.cn.c	yes	12	4.b	odd	2	1	inner
3648.1.cn.c	yes	12	12.b	even	2	1	inner
3648.1.cn.c	yes	12	152.t	even	18	1	inner
3648.1.cn.c	yes	12	152.u	odd	18	1	inner
3648.1.cn.c	yes	12	456.bh	odd	18	1	inner
3648.1.cn.c	yes	12	456.bu	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(3648, [\chi])$:

$ T_{7}^{12} + 6T_{7}^{10} + 27T_{7}^{8} + 48T_{7}^{6} + 63T_{7}^{4} + 27T_{7}^{2} + 9 $

$ T_{13}^{6} - 3T_{13}^{5} + 6T_{13}^{4} - 6T_{13}^{3} + 3 $

$ T_{17} $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - T^{6} + 1 $ T^12 - T^6 + 1
$5$	$ T^{12} $ T^12
$7$	$ T^{12} + 6 T^{10} + \cdots + 9 $ T^12 + 6T^10 + 27T^8 + 48T^6 + 63T^4 + 27*T^2 + 9
$11$	$ T^{12} $ T^12
$13$	$ (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 3)^{2} $ (T^6 - 3T^5 + 6T^4 - 6*T^3 + 3)^2
$17$	$ T^{12} $ T^12
$19$	$ (T^{4} - T^{2} + 1)^{3} $ (T^4 - T^2 + 1)^3
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} + 6 T^{10} + \cdots + 9 $ T^12 + 6T^10 + 27T^8 + 48T^6 + 63T^4 + 27*T^2 + 9
$37$	$ (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{2} $ (T^6 + 6T^4 + 9T^2 + 3)^2
$41$	$ T^{12} $ T^12
$43$	$ T^{12} - 3 T^{10} + \cdots + 1 $ T^12 - 3T^10 + 12T^8 - 46T^6 + 60T^4 + 12*T^2 + 1
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ (T^{6} - 6 T^{5} + 15 T^{4} + \cdots + 3)^{2} $ (T^6 - 6T^5 + 15T^4 - 21T^3 + 18T^2 - 9*T + 3)^2
$67$	$ T^{12} - 3 T^{10} + \cdots + 1 $ T^12 - 3T^10 + 12T^8 - 46T^6 + 60T^4 + 12*T^2 + 1
$71$	$ T^{12} $ T^12
$73$	$ (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} $ (T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1)^2
$79$	$ T^{12} + 6 T^{10} + \cdots + 9 $ T^12 + 6T^10 + 9T^8 + 3T^6 + 36T^4 - 27*T^2 + 9
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ (T^{6} + T^{3} + 1)^{2} $ (T^6 + T^3 + 1)^2
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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:	\( N \)	\(=\)	\( 3648 = 2^{6} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3648.cn (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{36}^{7} q^{3} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} + \zeta_{36}^{14} q^{9}+O(q^{10}) \) q + z^7 * q^3 + (z^17 - z^13) * q^7 + z^14 * q^9	\( q + \zeta_{36}^{7} q^{3} + (\zeta_{36}^{17} - \zeta_{36}^{13}) q^{7} + \zeta_{36}^{14} q^{9} + ( - \zeta_{36}^{12} - \zeta_{36}^{10}) q^{13} - \zeta_{36}^{15} q^{19} + ( - \zeta_{36}^{6} + \zeta_{36}^{2}) q^{21} + \zeta_{36}^{2} q^{25} - \zeta_{36}^{3} q^{27} + (\zeta_{36}^{11} + \zeta_{36}) q^{31} + ( - \zeta_{36}^{14} - \zeta_{36}^{4}) q^{37} + ( - \zeta_{36}^{17} + \zeta_{36}) q^{39} + (\zeta_{36}^{15} - \zeta_{36}^{5}) q^{43} + ( - \zeta_{36}^{16} + \cdots - \zeta_{36}^{8}) q^{49} + \cdots + \zeta_{36}^{4} q^{97} +O(q^{100}) \) q + z^7 * q^3 + (z^17 - z^13) * q^7 + z^14 * q^9 + (-z^12 - z^10) * q^13 - z^15 * q^19 + (-z^6 + z^2) * q^21 + z^2 * q^25 - z^3 * q^27 + (z^11 + z) * q^31 + (-z^14 - z^4) * q^37 + (-z^17 + z) * q^39 + (z^15 - z^5) * q^43 + (-z^16 + z^12 - z^8) * q^49 + z^4 * q^57 + (-z^16 + 1) * q^61 + (-z^13 + z^9) * q^63 + (z^7 + z^3) * q^67 + (z^8 - z^6) * q^73 + z^9 * q^75 + (-z^9 + z^5) * q^79 - z^10 * q^81 + (z^11 + z^9 - z^7 - z^5) * q^91 + (z^8 - 1) * q^93 + z^4 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q + 6 q^{13} - 6 q^{21} - 6 q^{49} + 12 q^{61} - 6 q^{73} - 12 q^{93}+O(q^{100}) \) 12 * q + 6 * q^13 - 6 * q^21 - 6 * q^49 + 12 * q^61 - 6 * q^73 - 12 * 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	3648.1.cn.c

Level	$3648$
Weight	$1$
Character orbit	3648.cn
Analytic conductor	$1.821$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{18}$
CM discriminant	-3
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(1.82058916609\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{18})\)
Coefficient field:	\(\Q(\zeta_{36})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{12} - x^{6} + 1 \) x^12 - x^6 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{18}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{18} - \cdots)\)

\(n\)	\(1217\)	\(1921\)	\(2053\)	\(2623\)
\(\chi(n)\)	\(-1\)	\(\zeta_{36}^{8}\)	\(-1\)	\(1\)

\(n\):		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
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
4.b	odd	2	1	inner
12.b	even	2	1	inner
152.t	even	18	1	inner
152.u	odd	18	1	inner
456.bh	odd	18	1	inner
456.bu	even	18	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( T^{12} - T^{6} + 1 \) T^12 - T^6 + 1
$5$	\( T^{12} \) T^12
$7$	\( T^{12} + 6 T^{10} + \cdots + 9 \) T^12 + 6T^10 + 27T^8 + 48T^6 + 63T^4 + 27*T^2 + 9
$11$	\( T^{12} \) T^12
$13$	\( (T^{6} - 3 T^{5} + 6 T^{4} + \cdots + 3)^{2} \) (T^6 - 3T^5 + 6T^4 - 6*T^3 + 3)^2
$17$	\( T^{12} \) T^12
$19$	\( (T^{4} - T^{2} + 1)^{3} \) (T^4 - T^2 + 1)^3
$23$	\( T^{12} \) T^12
$29$	\( T^{12} \) T^12
$31$	\( T^{12} + 6 T^{10} + \cdots + 9 \) T^12 + 6T^10 + 27T^8 + 48T^6 + 63T^4 + 27*T^2 + 9
$37$	\( (T^{6} + 6 T^{4} + 9 T^{2} + 3)^{2} \) (T^6 + 6T^4 + 9T^2 + 3)^2
$41$	\( T^{12} \) T^12
$43$	\( T^{12} - 3 T^{10} + \cdots + 1 \) T^12 - 3T^10 + 12T^8 - 46T^6 + 60T^4 + 12*T^2 + 1
$47$	\( T^{12} \) T^12
$53$	\( T^{12} \) T^12
$59$	\( T^{12} \) T^12
$61$	\( (T^{6} - 6 T^{5} + 15 T^{4} + \cdots + 3)^{2} \) (T^6 - 6T^5 + 15T^4 - 21T^3 + 18T^2 - 9*T + 3)^2
$67$	\( T^{12} - 3 T^{10} + \cdots + 1 \) T^12 - 3T^10 + 12T^8 - 46T^6 + 60T^4 + 12*T^2 + 1
$71$	\( T^{12} \) T^12
$73$	\( (T^{6} + 3 T^{5} + 6 T^{4} + \cdots + 1)^{2} \) (T^6 + 3T^5 + 6T^4 + 8T^3 + 12T^2 + 6*T + 1)^2
$79$	\( T^{12} + 6 T^{10} + \cdots + 9 \) T^12 + 6T^10 + 9T^8 + 3T^6 + 36T^4 - 27*T^2 + 9
$83$	\( T^{12} \) T^12
$89$	\( T^{12} \) T^12
$97$	\( (T^{6} + T^{3} + 1)^{2} \) (T^6 + T^3 + 1)^2
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