LMFDB - Newform orbit 1176.1.bj.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,1,Mod(29,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(14))

chi = DirichletCharacter(H, H._module([0, 7, 7, 6]))

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

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

chi := DirichletCharacter("1176.29");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1176.bj (of order $14$, 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:	$0.586900454856$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{14})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{7}$
Projective field:	Galois closure of 7.1.191341954266624.2

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$-1$	$-1$	$\zeta_{14}^{4}$

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} $

29.1

−0.623490	+	0.781831i
−0.623490	−	0.781831i
0.222521	−	0.974928i
0.900969	−	0.433884i
0.900969	+	0.433884i
0.222521	+	0.974928i

0.900969

0.433884i

0.222521

0.974928i

0.623490

0.781831i

−0.400969

−

1.75676i

−0.222521

0.974928i

−0.222521

0.974928i

0.222521

0.974928i

−0.900969

0.433884i

0.400969

−

1.75676i

365.1

0.900969

−

0.433884i

0.222521

−

0.974928i

0.623490

−

0.781831i

−0.400969

1.75676i

−0.222521

−

0.974928i

−0.222521

−

0.974928i

0.222521

−

0.974928i

−0.900969

−

0.433884i

0.400969

1.75676i

533.1

−0.623490

0.781831i

0.900969

0.433884i

−0.222521

−

0.974928i

1.12349

0.541044i

−0.900969

0.433884i

−0.900969

0.433884i

0.900969

0.433884i

0.623490

0.781831i

−1.12349

0.541044i

701.1

0.222521

−

0.974928i

−0.623490

0.781831i

−0.900969

−

0.433884i

0.277479

−

0.347948i

0.623490

0.781831i

0.623490

0.781831i

−0.623490

0.781831i

−0.222521

−

0.974928i

−0.277479

−

0.347948i

869.1

0.222521

0.974928i

−0.623490

−

0.781831i

−0.900969

0.433884i

0.277479

0.347948i

0.623490

−

0.781831i

0.623490

−

0.781831i

−0.623490

−

0.781831i

−0.222521

0.974928i

−0.277479

0.347948i

1037.1

−0.623490

−

0.781831i

0.900969

−

0.433884i

−0.222521

0.974928i

1.12349

−

0.541044i

−0.900969

−

0.433884i

−0.900969

−

0.433884i

0.900969

−

0.433884i

0.623490

−

0.781831i

−1.12349

−

0.541044i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
24.h	odd	2	1	CM by $\Q(\sqrt{-6}) $
49.e	even	7	1	inner
1176.bj	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1176.1.bj.b	yes	6
3.b	odd	2	1		1176.1.bj.a	✓	6
8.b	even	2	1		1176.1.bj.a	✓	6
24.h	odd	2	1	CM	1176.1.bj.b	yes	6
49.e	even	7	1	inner	1176.1.bj.b	yes	6
147.l	odd	14	1		1176.1.bj.a	✓	6
392.x	even	14	1		1176.1.bj.a	✓	6
1176.bj	odd	14	1	inner	1176.1.bj.b	yes	6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1176.1.bj.a	✓	6	3.b	odd	2	1
1176.1.bj.a	✓	6	8.b	even	2	1
1176.1.bj.a	✓	6	147.l	odd	14	1
1176.1.bj.a	✓	6	392.x	even	14	1
1176.1.bj.b	yes	6	1.a	even	1	1	trivial
1176.1.bj.b	yes	6	24.h	odd	2	1	CM
1176.1.bj.b	yes	6	49.e	even	7	1	inner
1176.1.bj.b	yes	6	1176.bj	odd	14	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{6} - 2T_{5}^{5} + 4T_{5}^{4} - 8T_{5}^{3} + 9T_{5}^{2} - 4T_{5} + 1 $ T5^6 - 2*T5^5 + 4*T5^4 - 8*T5^3 + 9*T5^2 - 4*T5 + 1 acting on $S_{1}^{\mathrm{new}}(1176, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} - T^{5} + T^{4} + \cdots + 1 $ T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$ T^{6} - T^{5} + T^{4} + \cdots + 1 $ T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$7$	$ T^{6} + T^{5} + T^{4} + \cdots + 1 $ T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$13$	$ T^{6} $ T^6
$17$	$ T^{6} $ T^6
$19$	$ T^{6} $ T^6
$23$	$ T^{6} $ T^6
$29$	$ T^{6} - 2 T^{5} + \cdots + 1 $ T^6 - 2T^5 + 4T^4 - T^3 + 2T^2 + 3T + 1
$31$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$37$	$ T^{6} $ T^6
$41$	$ T^{6} $ T^6
$43$	$ T^{6} $ T^6
$47$	$ T^{6} $ T^6
$53$	$ T^{6} - 2 T^{5} + \cdots + 64 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64
$59$	$ T^{6} - 2 T^{5} + \cdots + 64 $ T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64
$61$	$ T^{6} $ T^6
$67$	$ T^{6} $ T^6
$71$	$ T^{6} $ T^6
$73$	$ T^{6} + 2 T^{5} + \cdots + 1 $ T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$79$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 1)^2
$83$	$ T^{6} + 5 T^{5} + \cdots + 1 $ T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1
$89$	$ T^{6} $ T^6
$97$	$ (T^{3} + T^{2} - 2 T - 1)^{2} $ (T^3 + T^2 - 2*T - 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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1176.bj (of order \(14\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{14}^{3} q^{2} - \zeta_{14}^{2} q^{3} + \zeta_{14}^{6} q^{4} + ( - \zeta_{14}^{6} + \zeta_{14}^{5}) q^{5} - \zeta_{14}^{5} q^{6} - \zeta_{14}^{5} q^{7} - \zeta_{14}^{2} q^{8} + \zeta_{14}^{4} q^{9} +O(q^{10}) \) q + z^3 * q^2 - z^2 * q^3 + z^6 * q^4 + (-z^6 + z^5) * q^5 - z^5 * q^6 - z^5 * q^7 - z^2 * q^8 + z^4 * q^9	\( q + \zeta_{14}^{3} q^{2} - \zeta_{14}^{2} q^{3} + \zeta_{14}^{6} q^{4} + ( - \zeta_{14}^{6} + \zeta_{14}^{5}) q^{5} - \zeta_{14}^{5} q^{6} - \zeta_{14}^{5} q^{7} - \zeta_{14}^{2} q^{8} + \zeta_{14}^{4} q^{9} + (\zeta_{14}^{2} - \zeta_{14}) q^{10} + ( - \zeta_{14}^{4} - \zeta_{14}^{2}) q^{11} + \zeta_{14} q^{12} + \zeta_{14} q^{14} + ( - \zeta_{14} + 1) q^{15} - \zeta_{14}^{5} q^{16} - q^{18} + (\zeta_{14}^{5} - \zeta_{14}^{4}) q^{20} - q^{21} + ( - \zeta_{14}^{5} + 1) q^{22} + \zeta_{14}^{4} q^{24} + ( - \zeta_{14}^{5} + \cdots - \zeta_{14}^{3}) q^{25} + \cdots + ( - \zeta_{14}^{6} + \zeta_{14}) q^{99} +O(q^{100}) \) q + z^3 * q^2 - z^2 * q^3 + z^6 * q^4 + (-z^6 + z^5) * q^5 - z^5 * q^6 - z^5 * q^7 - z^2 * q^8 + z^4 * q^9 + (z^2 - z) * q^10 + (-z^4 - z^2) * q^11 + z * q^12 + z * q^14 + (-z + 1) * q^15 - z^5 * q^16 - q^18 + (z^5 - z^4) * q^20 - q^21 + (-z^5 + 1) * q^22 + z^4 * q^24 + (-z^5 + z^4 - z^3) * q^25 - z^6 * q^27 + z^4 * q^28 + (z^5 - z^4) * q^29 + (-z^4 + z^3) * q^30 + (-z^5 + z^2) * q^31 + z * q^32 + (z^6 + z^4) * q^33 + (-z^4 + z^3) * q^35 - z^3 * q^36 + (-z + 1) * q^40 - z^3 * q^42 + (z^3 + z) * q^44 + (z^3 - z^2) * q^45 - q^48 - z^3 * q^49 + (-z^6 + z - 1) * q^50 - z^6 * q^53 + z^2 * q^54 + (-z^3 + z^2 - z + 1) * q^55 - q^56 + (-z + 1) * q^58 + z^5 * q^59 + (z^6 + 1) * q^60 + (z^5 + z) * q^62 + z^2 * q^63 + z^4 * q^64 + (-z^2 - 1) * q^66 + (z^6 + 1) * q^70 - z^6 * q^72 + (z^6 + z^2) * q^73 + (-z^6 + z^5 - 1) * q^75 + (-z^2 - 1) * q^77 + (-z^5 + z^2) * q^79 + (-z^4 + z^3) * q^80 - z * q^81 + (z - 1) * q^83 - z^6 * q^84 + (z^6 + 1) * q^87 + (z^6 + z^4) * q^88 + (z^6 - z^5) * q^90 + (-z^4 - 1) * q^93 - z^3 * q^96 + (z^6 - z) * q^97 - z^6 * q^98 + (-z^6 + z) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} - q^{7} + q^{8} - q^{9}+O(q^{10}) \) 6 * q + q^2 + q^3 - q^4 + 2 * q^5 - q^6 - q^7 + q^8 - q^9	\( 6 q + q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} - q^{7} + q^{8} - q^{9} - 2 q^{10} + 2 q^{11} + q^{12} + q^{14} + 5 q^{15} - q^{16} - 6 q^{18} + 2 q^{20} - 6 q^{21} + 5 q^{22} - q^{24} - 3 q^{25} + q^{27} - q^{28} + 2 q^{29} + 2 q^{30} - 2 q^{31} + q^{32} - 2 q^{33} + 2 q^{35} - q^{36} + 5 q^{40} - q^{42} + 2 q^{44} + 2 q^{45} - 6 q^{48} - q^{49} - 4 q^{50} + 2 q^{53} - q^{54} + 3 q^{55} - 6 q^{56} + 5 q^{58} + 2 q^{59} + 5 q^{60} + 2 q^{62} - q^{63} - q^{64} - 5 q^{66} + 5 q^{70} + q^{72} - 2 q^{73} - 4 q^{75} - 5 q^{77} - 2 q^{79} + 2 q^{80} - q^{81} - 5 q^{83} + q^{84} + 5 q^{87} - 2 q^{88} - 2 q^{90} - 5 q^{93} - q^{96} - 2 q^{97} + q^{98} + 2 q^{99}+O(q^{100}) \) 6 * q + q^2 + q^3 - q^4 + 2 * q^5 - q^6 - q^7 + q^8 - q^9 - 2 * q^10 + 2 * q^11 + q^12 + q^14 + 5 * q^15 - q^16 - 6 * q^18 + 2 * q^20 - 6 * q^21 + 5 * q^22 - q^24 - 3 * q^25 + q^27 - q^28 + 2 * q^29 + 2 * q^30 - 2 * q^31 + q^32 - 2 * q^33 + 2 * q^35 - q^36 + 5 * q^40 - q^42 + 2 * q^44 + 2 * q^45 - 6 * q^48 - q^49 - 4 * q^50 + 2 * q^53 - q^54 + 3 * q^55 - 6 * q^56 + 5 * q^58 + 2 * q^59 + 5 * q^60 + 2 * q^62 - q^63 - q^64 - 5 * q^66 + 5 * q^70 + q^72 - 2 * q^73 - 4 * q^75 - 5 * q^77 - 2 * q^79 + 2 * q^80 - q^81 - 5 * q^83 + q^84 + 5 * q^87 - 2 * q^88 - 2 * q^90 - 5 * q^93 - q^96 - 2 * q^97 + q^98 + 2 * q^99

Introduction

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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	1176.1.bj.b

Level	$1176$
Weight	$1$
Character orbit	1176.bj
Analytic conductor	$0.587$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{7}$
CM discriminant	-24
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.586900454856\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Coefficient field:	\(\Q(\zeta_{14})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{7}\)
Projective field:	Galois closure of 7.1.191341954266624.2

\(n\)	\(295\)	\(589\)	\(785\)	\(1081\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-1\)	\(\zeta_{14}^{4}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 29.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} - T^{5} + T^{4} + \cdots + 1 \) T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{6} - T^{5} + T^{4} + \cdots + 1 \) T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$5$	\( T^{6} - 2 T^{5} + \cdots + 1 \) T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$7$	\( T^{6} + T^{5} + T^{4} + \cdots + 1 \) T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$11$	\( T^{6} - 2 T^{5} + \cdots + 1 \) T^6 - 2T^5 + 4T^4 - 8T^3 + 9T^2 - 4*T + 1
$13$	\( T^{6} \) T^6
$17$	\( T^{6} \) T^6
$19$	\( T^{6} \) T^6
$23$	\( T^{6} \) T^6
$29$	\( T^{6} - 2 T^{5} + \cdots + 1 \) T^6 - 2T^5 + 4T^4 - T^3 + 2T^2 + 3T + 1
$31$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$37$	\( T^{6} \) T^6
$41$	\( T^{6} \) T^6
$43$	\( T^{6} \) T^6
$47$	\( T^{6} \) T^6
$53$	\( T^{6} - 2 T^{5} + \cdots + 64 \) T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64
$59$	\( T^{6} - 2 T^{5} + \cdots + 64 \) T^6 - 2T^5 + 4T^4 - 8T^3 + 16T^2 - 32*T + 64
$61$	\( T^{6} \) T^6
$67$	\( T^{6} \) T^6
$71$	\( T^{6} \) T^6
$73$	\( T^{6} + 2 T^{5} + \cdots + 1 \) T^6 + 2T^5 + 4T^4 + T^3 + 2T^2 - 3T + 1
$79$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
$83$	\( T^{6} + 5 T^{5} + \cdots + 1 \) T^6 + 5T^5 + 11T^4 + 13T^3 + 9T^2 + 3*T + 1
$89$	\( T^{6} \) T^6
$97$	\( (T^{3} + T^{2} - 2 T - 1)^{2} \) (T^3 + T^2 - 2*T - 1)^2
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