LMFDB - Newform orbit 1224.1.cx.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1224,1,Mod(11,1224)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1224, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 24, 8, 21]))

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

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

chi := DirichletCharacter("1224.11");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1224.cx (of order $48$, degree $16$, 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:	$0.610855575463$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} - \cdots)$

$q$-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_{48}^{5} q^{2} + \zeta_{48}^{17} q^{3} + \zeta_{48}^{10} q^{4} - \zeta_{48}^{22} q^{6} - \zeta_{48}^{15} q^{8} - \zeta_{48}^{10} q^{9} + (\zeta_{48}^{21} - \zeta_{48}^{4}) q^{11} - \zeta_{48}^{3} q^{12} + \cdots + (\zeta_{48}^{14} + \zeta_{48}^{7}) q^{99} +O(q^{100}) $ q - z^5 * q^2 + z^17 * q^3 + z^10 * q^4 - z^22 * q^6 - z^15 * q^8 - z^10 * q^9 + (z^21 - z^4) * q^11 - z^3 * q^12 + z^20 * q^16 - z * q^17 + z^15 * q^18 + (z^23 - z^19) * q^19 + (z^9 + z^2) * q^22 + z^8 * q^24 - z^11 * q^25 + z^3 * q^27 + z * q^32 + (-z^21 - z^14) * q^33 + z^6 * q^34 - z^20 * q^36 + (z^4 - 1) * q^38 + (-z^23 - z^6) * q^41 + (z^22 - 1) * q^43 + (-z^14 - z^7) * q^44 - z^13 * q^48 - z^13 * q^49 + z^16 * q^50 - z^18 * q^51 - z^8 * q^54 + (-z^16 + z^12) * q^57 + (-z^12 - z^2) * q^59 - z^6 * q^64 + (z^19 - z^2) * q^66 + (-z^17 + z^15) * q^67 - z^11 * q^68 - z * q^72 + (z^14 + z^13) * q^73 + z^4 * q^75 + (-z^9 + z^5) * q^76 + z^20 * q^81 + (z^11 - z^4) * q^82 + (-z^8 + z^2) * q^83 + (z^5 + z^3) * q^86 + (z^19 + z^12) * q^88 + (-z^9 - z^3) * q^89 + z^18 * q^96 + (-z^12 - z^7) * q^97 + z^18 * q^98 + (z^14 + z^7) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{24} - 16 q^{38} - 16 q^{43} - 8 q^{50} - 8 q^{54} + 8 q^{57} - 8 q^{83}+O(q^{100}) $ 16 * q + 8 * q^24 - 16 * q^38 - 16 * q^43 - 8 * q^50 - 8 * q^54 + 8 * q^57 - 8 * q^83

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$-\zeta_{48}^{16}$	$-1$	$\zeta_{48}^{15}$	$-1$

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} $

11.1

0.130526	+	0.991445i
−0.991445	−	0.130526i
−0.130526	−	0.991445i
−0.130526	+	0.991445i
−0.991445	+	0.130526i
−0.608761	−	0.793353i
0.793353	+	0.608761i
0.793353	−	0.608761i
−0.608761	+	0.793353i
−0.793353	+	0.608761i
0.608761	+	0.793353i
0.130526	−	0.991445i
0.991445	+	0.130526i
0.608761	−	0.793353i
−0.793353	−	0.608761i
0.991445	−	0.130526i

−0.608761

−

0.793353i

0.793353

−

0.608761i

−0.258819

0.965926i

−0.965926

−

0.258819i

0.923880

−

0.382683i

0.258819

−

0.965926i

131.1

0.793353

0.608761i

0.608761

−

0.793353i

0.258819

0.965926i

0.965926

−

0.258819i

−0.382683

0.923880i

−0.258819

−

0.965926i

227.1

0.608761

0.793353i

−0.793353

0.608761i

−0.258819

0.965926i

−0.965926

−

0.258819i

−0.923880

0.382683i

0.258819

−

0.965926i

275.1

0.608761

−

0.793353i

−0.793353

−

0.608761i

−0.258819

−

0.965926i

−0.965926

0.258819i

−0.923880

−

0.382683i

0.258819

0.965926i

299.1

0.793353

−

0.608761i

0.608761

0.793353i

0.258819

−

0.965926i

0.965926

0.258819i

−0.382683

−

0.923880i

−0.258819

0.965926i

347.1

−0.130526

−

0.991445i

0.991445

−

0.130526i

−0.965926

0.258819i

−0.258819

−

0.965926i

0.382683

0.923880i

0.965926

−

0.258819i

371.1

0.991445

0.130526i

0.130526

−

0.991445i

0.965926

0.258819i

0.258819

−

0.965926i

0.923880

0.382683i

−0.965926

−

0.258819i

419.1

0.991445

−

0.130526i

0.130526

0.991445i

0.965926

−

0.258819i

0.258819

0.965926i

0.923880

−

0.382683i

−0.965926

0.258819i

515.1

−0.130526

0.991445i

0.991445

0.130526i

−0.965926

−

0.258819i

−0.258819

0.965926i

0.382683

−

0.923880i

0.965926

0.258819i

635.1

−0.991445

0.130526i

−0.130526

−

0.991445i

0.965926

−

0.258819i

0.258819

0.965926i

−0.923880

0.382683i

−0.965926

0.258819i

707.1

0.130526

0.991445i

−0.991445

0.130526i

−0.965926

0.258819i

−0.258819

−

0.965926i

−0.382683

−

0.923880i

0.965926

−

0.258819i

779.1

−0.608761

0.793353i

0.793353

0.608761i

−0.258819

−

0.965926i

−0.965926

0.258819i

0.923880

0.382683i

0.258819

0.965926i

923.1

−0.793353

−

0.608761i

−0.608761

0.793353i

0.258819

0.965926i

0.965926

−

0.258819i

0.382683

−

0.923880i

−0.258819

−

0.965926i

947.1

0.130526

−

0.991445i

−0.991445

−

0.130526i

−0.965926

−

0.258819i

−0.258819

0.965926i

−0.382683

0.923880i

0.965926

0.258819i

1091.1

−0.991445

−

0.130526i

−0.130526

0.991445i

0.965926

0.258819i

0.258819

−

0.965926i

−0.923880

−

0.382683i

−0.965926

−

0.258819i

1163.1

−0.793353

0.608761i

−0.608761

−

0.793353i

0.258819

−

0.965926i

0.965926

0.258819i

0.382683

0.923880i

−0.258819

0.965926i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
153.s	even	48	1	inner
1224.cx	odd	48	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1224.1.cx.b	yes	16
3.b	odd	2	1		3672.1.dv.b		16
8.d	odd	2	1	CM	1224.1.cx.b	yes	16
9.c	even	3	1		3672.1.dv.a		16
9.d	odd	6	1		1224.1.cx.a	✓	16
17.e	odd	16	1		1224.1.cx.a	✓	16
24.f	even	2	1		3672.1.dv.b		16
51.i	even	16	1		3672.1.dv.a		16
72.l	even	6	1		1224.1.cx.a	✓	16
72.p	odd	6	1		3672.1.dv.a		16
136.s	even	16	1		1224.1.cx.a	✓	16
153.s	even	48	1	inner	1224.1.cx.b	yes	16
153.t	odd	48	1		3672.1.dv.b		16
408.bg	odd	16	1		3672.1.dv.a		16
1224.cx	odd	48	1	inner	1224.1.cx.b	yes	16
1224.da	even	48	1		3672.1.dv.b		16

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1224.1.cx.a	✓	16	9.d	odd	6	1
1224.1.cx.a	✓	16	17.e	odd	16	1
1224.1.cx.a	✓	16	72.l	even	6	1
1224.1.cx.a	✓	16	136.s	even	16	1
1224.1.cx.b	yes	16	1.a	even	1	1	trivial
1224.1.cx.b	yes	16	8.d	odd	2	1	CM
1224.1.cx.b	yes	16	153.s	even	48	1	inner
1224.1.cx.b	yes	16	1224.cx	odd	48	1	inner
3672.1.dv.a		16	9.c	even	3	1
3672.1.dv.a		16	51.i	even	16	1
3672.1.dv.a		16	72.p	odd	6	1
3672.1.dv.a		16	408.bg	odd	16	1
3672.1.dv.b		16	3.b	odd	2	1
3672.1.dv.b		16	24.f	even	2	1
3672.1.dv.b		16	153.t	odd	48	1
3672.1.dv.b		16	1224.da	even	48	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{11}^{16} - 4 T_{11}^{14} + 10 T_{11}^{12} + 8 T_{11}^{11} - 16 T_{11}^{10} + 48 T_{11}^{9} + 21 T_{11}^{8} + \cdots + 1 $ T11^16 - 4*T11^14 + 10*T11^12 + 8*T11^11 - 16*T11^10 + 48*T11^9 + 21*T11^8 - 128*T11^7 + 44*T11^6 - 16*T11^5 + 68*T11^4 + 80*T11^3 + 68*T11^2 + 8*T11 + 1 acting on $S_{1}^{\mathrm{new}}(1224, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$3$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$5$	$ T^{16} $ T^16
$7$	$ T^{16} $ T^16
$11$	$ T^{16} - 4 T^{14} + \cdots + 1 $ T^16 - 4T^14 + 10T^12 + 8T^11 - 16T^10 + 48T^9 + 21T^8 - 128T^7 + 44T^6 - 16T^5 + 68T^4 + 80T^3 + 68T^2 + 8*T + 1
$13$	$ T^{16} $ T^16
$17$	$ T^{16} - T^{8} + 1 $ T^16 - T^8 + 1
$19$	$ T^{16} + 194T^{8} + 1 $ T^16 + 194*T^8 + 1
$23$	$ T^{16} $ T^16
$29$	$ T^{16} $ T^16
$31$	$ T^{16} $ T^16
$37$	$ T^{16} $ T^16
$41$	$ T^{16} + 4 T^{12} + \cdots + 1 $ T^16 + 4T^12 - 16T^11 - 20T^10 + 16T^9 + 5T^8 - 32T^7 + 216T^6 - 288T^5 + 274T^4 - 144T^3 + 24T^2 + 8T + 1
$43$	$ (T^{8} + 8 T^{7} + 28 T^{6} + \cdots + 1)^{2} $ (T^8 + 8T^7 + 28T^6 + 56T^5 + 69T^4 + 52T^3 + 22T^2 + 4*T + 1)^2
$47$	$ T^{16} $ T^16
$53$	$ T^{16} $ T^16
$59$	$ (T^{8} + 4 T^{6} - 4 T^{5} + \cdots + 1)^{2} $ (T^8 + 4T^6 - 4T^5 + 5T^4 - 8T^3 + 18T^2 - 8T + 1)^2
$61$	$ T^{16} $ T^16
$67$	$ T^{16} - 8 T^{14} + \cdots + 1 $ T^16 - 8T^14 + 44T^12 - 128T^10 + 271T^8 - 304T^6 + 236T^4 - 16*T^2 + 1
$71$	$ T^{16} $ T^16
$73$	$ T^{16} - 8 T^{13} + \cdots + 1 $ T^16 - 8T^13 - 2T^12 + 52T^10 - 32T^9 + 2T^8 - 96T^7 + 112T^6 + 24T^5 + 146T^4 + 96T^3 + 76T^2 + 16T + 1
$79$	$ T^{16} $ T^16
$83$	$ (T^{8} + 4 T^{7} + 10 T^{6} + \cdots + 4)^{2} $ (T^8 + 4T^7 + 10T^6 + 16T^5 + 18T^4 + 8T^3 + 4T^2 + 8*T + 4)^2
$89$	$ (T^{8} + 12 T^{4} + 4)^{2} $ (T^8 + 12*T^4 + 4)^2
$97$	$ T^{16} + 8 T^{14} + \cdots + 1 $ T^16 + 8T^14 + 28T^12 + 8T^11 + 56T^10 + 24T^9 + 69T^8 + 16T^7 + 116T^6 - 16T^5 - 106T^4 - 16T^3 + 68T^2 - 16*T + 1
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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:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1224.cx (of order \(48\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{48}^{5} q^{2} + \zeta_{48}^{17} q^{3} + \zeta_{48}^{10} q^{4} - \zeta_{48}^{22} q^{6} - \zeta_{48}^{15} q^{8} - \zeta_{48}^{10} q^{9} + (\zeta_{48}^{21} - \zeta_{48}^{4}) q^{11} - \zeta_{48}^{3} q^{12} + \cdots + (\zeta_{48}^{14} + \zeta_{48}^{7}) q^{99} +O(q^{100}) \) q - z^5 * q^2 + z^17 * q^3 + z^10 * q^4 - z^22 * q^6 - z^15 * q^8 - z^10 * q^9 + (z^21 - z^4) * q^11 - z^3 * q^12 + z^20 * q^16 - z * q^17 + z^15 * q^18 + (z^23 - z^19) * q^19 + (z^9 + z^2) * q^22 + z^8 * q^24 - z^11 * q^25 + z^3 * q^27 + z * q^32 + (-z^21 - z^14) * q^33 + z^6 * q^34 - z^20 * q^36 + (z^4 - 1) * q^38 + (-z^23 - z^6) * q^41 + (z^22 - 1) * q^43 + (-z^14 - z^7) * q^44 - z^13 * q^48 - z^13 * q^49 + z^16 * q^50 - z^18 * q^51 - z^8 * q^54 + (-z^16 + z^12) * q^57 + (-z^12 - z^2) * q^59 - z^6 * q^64 + (z^19 - z^2) * q^66 + (-z^17 + z^15) * q^67 - z^11 * q^68 - z * q^72 + (z^14 + z^13) * q^73 + z^4 * q^75 + (-z^9 + z^5) * q^76 + z^20 * q^81 + (z^11 - z^4) * q^82 + (-z^8 + z^2) * q^83 + (z^5 + z^3) * q^86 + (z^19 + z^12) * q^88 + (-z^9 - z^3) * q^89 + z^18 * q^96 + (-z^12 - z^7) * q^97 + z^18 * q^98 + (z^14 + z^7) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{24} - 16 q^{38} - 16 q^{43} - 8 q^{50} - 8 q^{54} + 8 q^{57} - 8 q^{83}+O(q^{100}) \) 16 * q + 8 * q^24 - 16 * q^38 - 16 * q^43 - 8 * q^50 - 8 * q^54 + 8 * q^57 - 8 * q^83

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	1224.1.cx.b

Level	$1224$
Weight	$1$
Character orbit	1224.cx
Analytic conductor	$0.611$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{48}$
CM discriminant	-8
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.610855575463\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Coefficient field:	\(\Q(\zeta_{48})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{16} - x^{8} + 1 \) x^16 - x^8 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{48}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{48} - \cdots)\)

\(n\)	\(137\)	\(613\)	\(649\)	\(919\)
\(\chi(n)\)	\(-\zeta_{48}^{16}\)	\(-1\)	\(\zeta_{48}^{15}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 11.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by \(\Q(\sqrt{-2}) \)
153.s	even	48	1	inner
1224.cx	odd	48	1	inner

$p$	$F_p(T)$
$2$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$3$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$5$	\( T^{16} \) T^16
$7$	\( T^{16} \) T^16
$11$	\( T^{16} - 4 T^{14} + \cdots + 1 \) T^16 - 4T^14 + 10T^12 + 8T^11 - 16T^10 + 48T^9 + 21T^8 - 128T^7 + 44T^6 - 16T^5 + 68T^4 + 80T^3 + 68T^2 + 8*T + 1
$13$	\( T^{16} \) T^16
$17$	\( T^{16} - T^{8} + 1 \) T^16 - T^8 + 1
$19$	\( T^{16} + 194T^{8} + 1 \) T^16 + 194*T^8 + 1
$23$	\( T^{16} \) T^16
$29$	\( T^{16} \) T^16
$31$	\( T^{16} \) T^16
$37$	\( T^{16} \) T^16
$41$	\( T^{16} + 4 T^{12} + \cdots + 1 \) T^16 + 4T^12 - 16T^11 - 20T^10 + 16T^9 + 5T^8 - 32T^7 + 216T^6 - 288T^5 + 274T^4 - 144T^3 + 24T^2 + 8T + 1
$43$	\( (T^{8} + 8 T^{7} + 28 T^{6} + \cdots + 1)^{2} \) (T^8 + 8T^7 + 28T^6 + 56T^5 + 69T^4 + 52T^3 + 22T^2 + 4*T + 1)^2
$47$	\( T^{16} \) T^16
$53$	\( T^{16} \) T^16
$59$	\( (T^{8} + 4 T^{6} - 4 T^{5} + \cdots + 1)^{2} \) (T^8 + 4T^6 - 4T^5 + 5T^4 - 8T^3 + 18T^2 - 8T + 1)^2
$61$	\( T^{16} \) T^16
$67$	\( T^{16} - 8 T^{14} + \cdots + 1 \) T^16 - 8T^14 + 44T^12 - 128T^10 + 271T^8 - 304T^6 + 236T^4 - 16*T^2 + 1
$71$	\( T^{16} \) T^16
$73$	\( T^{16} - 8 T^{13} + \cdots + 1 \) T^16 - 8T^13 - 2T^12 + 52T^10 - 32T^9 + 2T^8 - 96T^7 + 112T^6 + 24T^5 + 146T^4 + 96T^3 + 76T^2 + 16T + 1
$79$	\( T^{16} \) T^16
$83$	\( (T^{8} + 4 T^{7} + 10 T^{6} + \cdots + 4)^{2} \) (T^8 + 4T^7 + 10T^6 + 16T^5 + 18T^4 + 8T^3 + 4T^2 + 8*T + 4)^2
$89$	\( (T^{8} + 12 T^{4} + 4)^{2} \) (T^8 + 12*T^4 + 4)^2
$97$	\( T^{16} + 8 T^{14} + \cdots + 1 \) T^16 + 8T^14 + 28T^12 + 8T^11 + 56T^10 + 24T^9 + 69T^8 + 16T^7 + 116T^6 - 16T^5 - 106T^4 - 16T^3 + 68T^2 - 16*T + 1
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