LMFDB - Newform orbit 1776.1.em.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1776,1,Mod(143,1776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1776, base_ring=CyclotomicField(36))

chi = DirichletCharacter(H, H._module([18, 0, 18, 5]))

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

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

chi := DirichletCharacter("1776.143");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1776 = 2^{4} \cdot 3 \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1776.em (of order $36$, degree $12$, 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:	$0.886339462436$
Analytic rank:	$0$
Dimension:	$12$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{36}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{36} - \cdots)$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times$.

$n$	$223$	$593$	$1297$	$1333$
$\chi(n)$	$-1$	$-1$	$\zeta_{36}^{13}$	$1$

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} $

143.1

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

−0.939693

−

0.342020i

1.50881

−

0.266044i

0.766044

0.642788i

239.1

0.173648

−

0.984808i

1.20805

−

1.43969i

−0.939693

−

0.342020i

335.1

0.766044

0.642788i

0.118782

0.326352i

0.173648

0.984808i

383.1

0.173648

0.984808i

−1.20805

−

1.43969i

−0.939693

0.342020i

431.1

0.173648

0.984808i

1.20805

1.43969i

−0.939693

0.342020i

479.1

0.766044

0.642788i

−0.118782

−

0.326352i

0.173648

0.984808i

575.1

0.173648

−

0.984808i

−1.20805

1.43969i

−0.939693

−

0.342020i

671.1

−0.939693

−

0.342020i

−1.50881

0.266044i

0.766044

0.642788i

1055.1

0.766044

−

0.642788i

0.118782

−

0.326352i

0.173648

−

0.984808i

1199.1

−0.939693

0.342020i

−1.50881

−

0.266044i

0.766044

−

0.642788i

1391.1

−0.939693

0.342020i

1.50881

0.266044i

0.766044

−

0.642788i

1535.1

0.766044

−

0.642788i

−0.118782

0.326352i

0.173648

−

0.984808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
148.q	even	36	1	inner
444.bi	odd	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1776.1.em.a	✓	12
3.b	odd	2	1	CM	1776.1.em.a	✓	12
4.b	odd	2	1		1776.1.em.b	yes	12
12.b	even	2	1		1776.1.em.b	yes	12
37.i	odd	36	1		1776.1.em.b	yes	12
111.q	even	36	1		1776.1.em.b	yes	12
148.q	even	36	1	inner	1776.1.em.a	✓	12
444.bi	odd	36	1	inner	1776.1.em.a	✓	12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1776.1.em.a	✓	12	1.a	even	1	1	trivial
1776.1.em.a	✓	12	3.b	odd	2	1	CM
1776.1.em.a	✓	12	148.q	even	36	1	inner
1776.1.em.a	✓	12	444.bi	odd	36	1	inner
1776.1.em.b	yes	12	4.b	odd	2	1
1776.1.em.b	yes	12	12.b	even	2	1
1776.1.em.b	yes	12	37.i	odd	36	1
1776.1.em.b	yes	12	111.q	even	36	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{19}^{12} - 4T_{19}^{9} + 53T_{19}^{6} - 14T_{19}^{3} + 1 $ T19^12 - 4*T19^9 + 53*T19^6 - 14*T19^3 + 1 acting on $S_{1}^{\mathrm{new}}(1776, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ (T^{6} + T^{3} + 1)^{2} $ (T^6 + T^3 + 1)^2
$5$	$ T^{12} $ T^12
$7$	$ T^{12} - 3 T^{10} + \cdots + 1 $ T^12 - 3T^10 + 12T^8 - 46T^6 + 60T^4 + 12*T^2 + 1
$11$	$ T^{12} $ T^12
$13$	$ T^{12} + 6 T^{10} + \cdots + 1 $ T^12 + 6T^10 - 2T^9 + 15T^8 + 23T^6 + 12T^5 - 12T^4 + 14T^3 + 39T^2 + 12*T + 1
$17$	$ T^{12} $ T^12
$19$	$ T^{12} - 4 T^{9} + \cdots + 1 $ T^12 - 4T^9 + 53T^6 - 14*T^3 + 1
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} + 2 T^{9} + \cdots + 1 $ T^12 + 2T^9 + 18T^8 + 6T^7 + 2T^6 + 18T^5 + 69T^4 + 52T^3 + 18T^2 - 6*T + 1
$37$	$ (T^{2} - T + 1)^{6} $ (T^2 - T + 1)^6
$41$	$ T^{12} $ T^12
$43$	$ T^{12} - 2 T^{9} + \cdots + 1 $ T^12 - 2T^9 + 18T^8 - 6T^7 + 2T^6 - 18T^5 + 69T^4 - 52T^3 + 18T^2 + 6*T + 1
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} $ T^12
$61$	$ T^{12} - 4 T^{9} + \cdots + 64 $ T^12 - 4T^9 + 8T^6 - 32*T^3 + 64
$67$	$ T^{12} - 3 T^{10} + \cdots + 9 $ T^12 - 3T^10 + 18T^8 - 24T^6 + 9T^4 + 27*T^2 + 9
$71$	$ T^{12} $ T^12
$73$	$ (T^{6} + 6 T^{4} + 9 T^{2} + 1)^{2} $ (T^6 + 6T^4 + 9T^2 + 1)^2
$79$	$ T^{12} - 3 T^{10} + \cdots + 1 $ T^12 - 3T^10 + 2T^9 + 6T^8 - 18T^7 - 4T^6 - 12T^5 + 15T^4 + 40T^3 + 39T^2 + 6T + 1
$83$	$ T^{12} $ T^12
$89$	$ T^{12} $ T^12
$97$	$ T^{12} - 2 T^{9} + \cdots + 1 $ T^12 - 2T^9 - 9T^8 + 12T^7 + 2T^6 - 18T^5 + 69T^4 - 52T^3 + 45T^2 - 12*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:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1776.em (of order \(36\), degree \(12\), minimal)

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

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1776.1.em.a

Level	$1776$
Weight	$1$
Character orbit	1776.em
Analytic conductor	$0.886$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{36}$
CM discriminant	-3
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.886339462436\)
Analytic rank:	\(0\)
Dimension:	\(12\)
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:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{36}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

\(n\)	\(223\)	\(593\)	\(1297\)	\(1333\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(\zeta_{36}^{13}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 143.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
148.q	even	36	1	inner
444.bi	odd	36	1	inner

$p$	$F_p(T)$
$2$	\( T^{12} \) T^12
$3$	\( (T^{6} + T^{3} + 1)^{2} \) (T^6 + T^3 + 1)^2
$5$	\( T^{12} \) T^12
$7$	\( T^{12} - 3 T^{10} + \cdots + 1 \) T^12 - 3T^10 + 12T^8 - 46T^6 + 60T^4 + 12*T^2 + 1
$11$	\( T^{12} \) T^12
$13$	\( T^{12} + 6 T^{10} + \cdots + 1 \) T^12 + 6T^10 - 2T^9 + 15T^8 + 23T^6 + 12T^5 - 12T^4 + 14T^3 + 39T^2 + 12*T + 1
$17$	\( T^{12} \) T^12
$19$	\( T^{12} - 4 T^{9} + \cdots + 1 \) T^12 - 4T^9 + 53T^6 - 14*T^3 + 1
$23$	\( T^{12} \) T^12
$29$	\( T^{12} \) T^12
$31$	\( T^{12} + 2 T^{9} + \cdots + 1 \) T^12 + 2T^9 + 18T^8 + 6T^7 + 2T^6 + 18T^5 + 69T^4 + 52T^3 + 18T^2 - 6*T + 1
$37$	\( (T^{2} - T + 1)^{6} \) (T^2 - T + 1)^6
$41$	\( T^{12} \) T^12
$43$	\( T^{12} - 2 T^{9} + \cdots + 1 \) T^12 - 2T^9 + 18T^8 - 6T^7 + 2T^6 - 18T^5 + 69T^4 - 52T^3 + 18T^2 + 6*T + 1
$47$	\( T^{12} \) T^12
$53$	\( T^{12} \) T^12
$59$	\( T^{12} \) T^12
$61$	\( T^{12} - 4 T^{9} + \cdots + 64 \) T^12 - 4T^9 + 8T^6 - 32*T^3 + 64
$67$	\( T^{12} - 3 T^{10} + \cdots + 9 \) T^12 - 3T^10 + 18T^8 - 24T^6 + 9T^4 + 27*T^2 + 9
$71$	\( T^{12} \) T^12
$73$	\( (T^{6} + 6 T^{4} + 9 T^{2} + 1)^{2} \) (T^6 + 6T^4 + 9T^2 + 1)^2
$79$	\( T^{12} - 3 T^{10} + \cdots + 1 \) T^12 - 3T^10 + 2T^9 + 6T^8 - 18T^7 - 4T^6 - 12T^5 + 15T^4 + 40T^3 + 39T^2 + 6T + 1
$83$	\( T^{12} \) T^12
$89$	\( T^{12} \) T^12
$97$	\( T^{12} - 2 T^{9} + \cdots + 1 \) T^12 - 2T^9 - 9T^8 + 12T^7 + 2T^6 - 18T^5 + 69T^4 - 52T^3 + 45T^2 - 12*T + 1
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