LMFDB - Newform orbit 1288.1.bi.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1288,1,Mod(13,1288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1288, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([0, 11, 11, 14]))

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

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

chi := DirichletCharacter("1288.13");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1288 = 2^{3} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1288.bi (of order $22$, degree $10$, 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:	$0.642795736271$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + 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:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} + \cdots)$

$q$-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_{22}^{9} q^{2} - \zeta_{22}^{7} q^{4} + \zeta_{22} q^{7} + \zeta_{22}^{5} q^{8} - \zeta_{22}^{4} q^{9} +O(q^{10}) $ q + z^9 * q^2 - z^7 * q^4 + z * q^7 + z^5 * q^8 - z^4 * q^9	\( q + \zeta_{22}^{9} q^{2} - \zeta_{22}^{7} q^{4} + \zeta_{22} q^{7} + \zeta_{22}^{5} q^{8} - \zeta_{22}^{4} q^{9} + ( - \zeta_{22}^{10} + \zeta_{22}^{6}) q^{11} + \zeta_{22}^{10} q^{14} - \zeta_{22}^{3} q^{16} + \zeta_{22}^{2} q^{18} + (\zeta_{22}^{8} - \zeta_{22}^{4}) q^{22} - \zeta_{22}^{7} q^{23} + \zeta_{22}^{3} q^{25} - \zeta_{22}^{8} q^{28} + (\zeta_{22}^{8} - \zeta_{22}^{2}) q^{29} + \zeta_{22} q^{32} - q^{36} + ( - \zeta_{22}^{5} + \zeta_{22}^{3}) q^{37} + ( - \zeta_{22}^{9} - \zeta_{22}^{6}) q^{43} + ( - \zeta_{22}^{6} + \zeta_{22}^{2}) q^{44} + \zeta_{22}^{5} q^{46} + \zeta_{22}^{2} q^{49} - \zeta_{22} q^{50} + (\zeta_{22}^{9} + \zeta_{22}^{4}) q^{53} + \zeta_{22}^{6} q^{56} + ( - \zeta_{22}^{6} + 1) q^{58} - \zeta_{22}^{5} q^{63} + \zeta_{22}^{10} q^{64} + ( - \zeta_{22}^{9} - \zeta_{22}^{8}) q^{67} + (\zeta_{22}^{5} + \zeta_{22}) q^{71} - \zeta_{22}^{9} q^{72} + (\zeta_{22}^{3} - \zeta_{22}) q^{74} + (\zeta_{22}^{7} + 1) q^{77} + (\zeta_{22}^{7} - \zeta_{22}^{2}) q^{79} + \zeta_{22}^{8} q^{81} + (\zeta_{22}^{7} + \zeta_{22}^{4}) q^{86} + (\zeta_{22}^{4} - 1) q^{88} - \zeta_{22}^{3} q^{92} - q^{98} + ( - \zeta_{22}^{10} - \zeta_{22}^{3}) q^{99} +O(q^{100}) \) q + z^9 * q^2 - z^7 * q^4 + z * q^7 + z^5 * q^8 - z^4 * q^9 + (-z^10 + z^6) * q^11 + z^10 * q^14 - z^3 * q^16 + z^2 * q^18 + (z^8 - z^4) * q^22 - z^7 * q^23 + z^3 * q^25 - z^8 * q^28 + (z^8 - z^2) * q^29 + z * q^32 - q^36 + (-z^5 + z^3) * q^37 + (-z^9 - z^6) * q^43 + (-z^6 + z^2) * q^44 + z^5 * q^46 + z^2 * q^49 - z * q^50 + (z^9 + z^4) * q^53 + z^6 * q^56 + (-z^6 + 1) * q^58 - z^5 * q^63 + z^10 * q^64 + (-z^9 - z^8) * q^67 + (z^5 + z) * q^71 - z^9 * q^72 + (z^3 - z) * q^74 + (z^7 + 1) * q^77 + (z^7 - z^2) * q^79 + z^8 * q^81 + (z^7 + z^4) * q^86 + (z^4 - 1) * q^88 - z^3 * q^92 - q^98 + (-z^10 - z^3) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{2} - q^{4} + q^{7} + q^{8} + q^{9}+O(q^{10}) $ 10 * q + q^2 - q^4 + q^7 + q^8 + q^9	$ 10 q + q^{2} - q^{4} + q^{7} + q^{8} + q^{9} - q^{14} - q^{16} - q^{18} - q^{23} + q^{25} + q^{28} + q^{32} - 10 q^{36} + q^{46} - q^{49} - q^{50} - q^{56} + 11 q^{58} - q^{63} - q^{64} + 2 q^{71} - q^{72} + 11 q^{77} + 2 q^{79} - q^{81} - 11 q^{88} - q^{92} - 10 q^{98}+O(q^{100}) $ 10 * q + q^2 - q^4 + q^7 + q^8 + q^9 - q^14 - q^16 - q^18 - q^23 + q^25 + q^28 + q^32 - 10 * q^36 + q^46 - q^49 - q^50 - q^56 + 11 * q^58 - q^63 - q^64 + 2 * q^71 - q^72 + 11 * q^77 + 2 * q^79 - q^81 - 11 * q^88 - q^92 - 10 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1288\mathbb{Z}\right)^\times$.

$n$	$185$	$281$	$645$	$967$
$\chi(n)$	$-1$	$-\zeta_{22}^{3}$	$-1$	$1$

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} $

13.1

0.959493	+	0.281733i
0.142315	+	0.989821i
−0.841254	−	0.540641i
−0.415415	−	0.909632i
0.654861	−	0.755750i
0.654861	+	0.755750i
−0.415415	+	0.909632i
0.142315	−	0.989821i
0.959493	−	0.281733i
−0.841254	+	0.540641i

−0.841254

0.540641i

0.415415

−

0.909632i

0.959493

0.281733i

0.142315

0.989821i

−0.415415

−

0.909632i

349.1

0.959493

0.281733i

0.841254

0.540641i

0.142315

0.989821i

0.654861

0.755750i

−0.841254

0.540641i

629.1

−0.415415

0.909632i

−0.654861

−

0.755750i

−0.841254

−

0.540641i

0.959493

−

0.281733i

0.654861

−

0.755750i

685.1

0.654861

0.755750i

−0.142315

0.989821i

−0.415415

−

0.909632i

−0.841254

0.540641i

0.142315

0.989821i

853.1

0.142315

−

0.989821i

−0.959493

−

0.281733i

0.654861

−

0.755750i

−0.415415

0.909632i

0.959493

−

0.281733i

909.1

0.142315

0.989821i

−0.959493

0.281733i

0.654861

0.755750i

−0.415415

−

0.909632i

0.959493

0.281733i

1021.1

0.654861

−

0.755750i

−0.142315

−

0.989821i

−0.415415

0.909632i

−0.841254

−

0.540641i

0.142315

−

0.989821i

1133.1

0.959493

−

0.281733i

0.841254

−

0.540641i

0.142315

−

0.989821i

0.654861

−

0.755750i

−0.841254

−

0.540641i

1189.1

−0.841254

−

0.540641i

0.415415

0.909632i

0.959493

−

0.281733i

0.142315

−

0.989821i

−0.415415

0.909632i

1245.1

−0.415415

−

0.909632i

−0.654861

0.755750i

−0.841254

0.540641i

0.959493

0.281733i

0.654861

0.755750i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by $\Q(\sqrt{-7}) $
184.p	even	22	1	inner
1288.bi	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1288.1.bi.c	✓	10
7.b	odd	2	1	CM	1288.1.bi.c	✓	10
8.b	even	2	1		1288.1.bi.d	yes	10
23.c	even	11	1		1288.1.bi.d	yes	10
56.h	odd	2	1		1288.1.bi.d	yes	10
161.l	odd	22	1		1288.1.bi.d	yes	10
184.p	even	22	1	inner	1288.1.bi.c	✓	10
1288.bi	odd	22	1	inner	1288.1.bi.c	✓	10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1288.1.bi.c	✓	10	1.a	even	1	1	trivial
1288.1.bi.c	✓	10	7.b	odd	2	1	CM
1288.1.bi.c	✓	10	184.p	even	22	1	inner
1288.1.bi.c	✓	10	1288.bi	odd	22	1	inner
1288.1.bi.d	yes	10	8.b	even	2	1
1288.1.bi.d	yes	10	23.c	even	11	1
1288.1.bi.d	yes	10	56.h	odd	2	1
1288.1.bi.d	yes	10	161.l	odd	22	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1288, [\chi])$:

$ T_{3} $

$ T_{11}^{10} + 11T_{11}^{7} + 33T_{11}^{4} - 11T_{11}^{3} + 22T_{11} + 11 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	$ T^{10} $ T^10
$5$	$ T^{10} $ T^10
$7$	$ T^{10} - T^{9} + \cdots + 1 $ T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$11$	$ T^{10} + 11 T^{7} + \cdots + 11 $ T^10 + 11T^7 + 33T^4 - 11T^3 + 22T + 11
$13$	$ T^{10} $ T^10
$17$	$ T^{10} $ T^10
$19$	$ T^{10} $ T^10
$23$	$ T^{10} + T^{9} + \cdots + 1 $ T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$29$	$ T^{10} + 22 T^{5} + \cdots + 11 $ T^10 + 22T^5 - 33T^3 + 11T^2 + 11T + 11
$31$	$ T^{10} $ T^10
$37$	$ T^{10} - 11 T^{7} + \cdots + 11 $ T^10 - 11T^7 + 33T^4 + 11T^3 - 22T + 11
$41$	$ T^{10} $ T^10
$43$	$ T^{10} + 11 T^{6} + \cdots + 11 $ T^10 + 11T^6 - 11T^5 + 22T^2 + 33T + 11
$47$	$ T^{10} $ T^10
$53$	$ T^{10} - 11 T^{7} + \cdots + 11 $ T^10 - 11T^7 + 33T^4 + 11T^3 - 22T + 11
$59$	$ T^{10} $ T^10
$61$	$ T^{10} $ T^10
$67$	$ T^{10} + 11 T^{6} + \cdots + 11 $ T^10 + 11T^6 + 11T^5 + 22T^2 - 33T + 11
$71$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$73$	$ T^{10} $ T^10
$79$	$ T^{10} - 2 T^{9} + \cdots + 1 $ T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$83$	$ T^{10} $ T^10
$89$	$ T^{10} $ T^10
$97$	$ T^{10} $ T^10
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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:	\( N \)	\(=\)	\( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1288.bi (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{22}^{9} q^{2} - \zeta_{22}^{7} q^{4} + \zeta_{22} q^{7} + \zeta_{22}^{5} q^{8} - \zeta_{22}^{4} q^{9} +O(q^{10}) \) q + z^9 * q^2 - z^7 * q^4 + z * q^7 + z^5 * q^8 - z^4 * q^9	\( q + \zeta_{22}^{9} q^{2} - \zeta_{22}^{7} q^{4} + \zeta_{22} q^{7} + \zeta_{22}^{5} q^{8} - \zeta_{22}^{4} q^{9} + ( - \zeta_{22}^{10} + \zeta_{22}^{6}) q^{11} + \zeta_{22}^{10} q^{14} - \zeta_{22}^{3} q^{16} + \zeta_{22}^{2} q^{18} + (\zeta_{22}^{8} - \zeta_{22}^{4}) q^{22} - \zeta_{22}^{7} q^{23} + \zeta_{22}^{3} q^{25} - \zeta_{22}^{8} q^{28} + (\zeta_{22}^{8} - \zeta_{22}^{2}) q^{29} + \zeta_{22} q^{32} - q^{36} + ( - \zeta_{22}^{5} + \zeta_{22}^{3}) q^{37} + ( - \zeta_{22}^{9} - \zeta_{22}^{6}) q^{43} + ( - \zeta_{22}^{6} + \zeta_{22}^{2}) q^{44} + \zeta_{22}^{5} q^{46} + \zeta_{22}^{2} q^{49} - \zeta_{22} q^{50} + (\zeta_{22}^{9} + \zeta_{22}^{4}) q^{53} + \zeta_{22}^{6} q^{56} + ( - \zeta_{22}^{6} + 1) q^{58} - \zeta_{22}^{5} q^{63} + \zeta_{22}^{10} q^{64} + ( - \zeta_{22}^{9} - \zeta_{22}^{8}) q^{67} + (\zeta_{22}^{5} + \zeta_{22}) q^{71} - \zeta_{22}^{9} q^{72} + (\zeta_{22}^{3} - \zeta_{22}) q^{74} + (\zeta_{22}^{7} + 1) q^{77} + (\zeta_{22}^{7} - \zeta_{22}^{2}) q^{79} + \zeta_{22}^{8} q^{81} + (\zeta_{22}^{7} + \zeta_{22}^{4}) q^{86} + (\zeta_{22}^{4} - 1) q^{88} - \zeta_{22}^{3} q^{92} - q^{98} + ( - \zeta_{22}^{10} - \zeta_{22}^{3}) q^{99} +O(q^{100}) \) q + z^9 * q^2 - z^7 * q^4 + z * q^7 + z^5 * q^8 - z^4 * q^9 + (-z^10 + z^6) * q^11 + z^10 * q^14 - z^3 * q^16 + z^2 * q^18 + (z^8 - z^4) * q^22 - z^7 * q^23 + z^3 * q^25 - z^8 * q^28 + (z^8 - z^2) * q^29 + z * q^32 - q^36 + (-z^5 + z^3) * q^37 + (-z^9 - z^6) * q^43 + (-z^6 + z^2) * q^44 + z^5 * q^46 + z^2 * q^49 - z * q^50 + (z^9 + z^4) * q^53 + z^6 * q^56 + (-z^6 + 1) * q^58 - z^5 * q^63 + z^10 * q^64 + (-z^9 - z^8) * q^67 + (z^5 + z) * q^71 - z^9 * q^72 + (z^3 - z) * q^74 + (z^7 + 1) * q^77 + (z^7 - z^2) * q^79 + z^8 * q^81 + (z^7 + z^4) * q^86 + (z^4 - 1) * q^88 - z^3 * q^92 - q^98 + (-z^10 - z^3) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - q^{4} + q^{7} + q^{8} + q^{9}+O(q^{10}) \) 10 * q + q^2 - q^4 + q^7 + q^8 + q^9	\( 10 q + q^{2} - q^{4} + q^{7} + q^{8} + q^{9} - q^{14} - q^{16} - q^{18} - q^{23} + q^{25} + q^{28} + q^{32} - 10 q^{36} + q^{46} - q^{49} - q^{50} - q^{56} + 11 q^{58} - q^{63} - q^{64} + 2 q^{71} - q^{72} + 11 q^{77} + 2 q^{79} - q^{81} - 11 q^{88} - q^{92} - 10 q^{98}+O(q^{100}) \) 10 * q + q^2 - q^4 + q^7 + q^8 + q^9 - q^14 - q^16 - q^18 - q^23 + q^25 + q^28 + q^32 - 10 * q^36 + q^46 - q^49 - q^50 - q^56 + 11 * q^58 - q^63 - q^64 + 2 * q^71 - q^72 + 11 * q^77 + 2 * q^79 - q^81 - 11 * q^88 - q^92 - 10 * q^98

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	1288.1.bi.c

Level	$1288$
Weight	$1$
Character orbit	1288.bi
Analytic conductor	$0.643$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{22}$
CM discriminant	-7
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.642795736271\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\Q(\zeta_{22})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) x^10 - x^9 + x^8 - x^7 + 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:	\(D_{22}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{22} + \cdots)\)

\(n\)	\(185\)	\(281\)	\(645\)	\(967\)
\(\chi(n)\)	\(-1\)	\(-\zeta_{22}^{3}\)	\(-1\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 13.1
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	CM by \(\Q(\sqrt{-7}) \)
184.p	even	22	1	inner
1288.bi	odd	22	1	inner

$p$	$F_p(T)$
$2$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$3$	\( T^{10} \) T^10
$5$	\( T^{10} \) T^10
$7$	\( T^{10} - T^{9} + \cdots + 1 \) T^10 - T^9 + T^8 - T^7 + T^6 - T^5 + T^4 - T^3 + T^2 - T + 1
$11$	\( T^{10} + 11 T^{7} + \cdots + 11 \) T^10 + 11T^7 + 33T^4 - 11T^3 + 22T + 11
$13$	\( T^{10} \) T^10
$17$	\( T^{10} \) T^10
$19$	\( T^{10} \) T^10
$23$	\( T^{10} + T^{9} + \cdots + 1 \) T^10 + T^9 + T^8 + T^7 + T^6 + T^5 + T^4 + T^3 + T^2 + T + 1
$29$	\( T^{10} + 22 T^{5} + \cdots + 11 \) T^10 + 22T^5 - 33T^3 + 11T^2 + 11T + 11
$31$	\( T^{10} \) T^10
$37$	\( T^{10} - 11 T^{7} + \cdots + 11 \) T^10 - 11T^7 + 33T^4 + 11T^3 - 22T + 11
$41$	\( T^{10} \) T^10
$43$	\( T^{10} + 11 T^{6} + \cdots + 11 \) T^10 + 11T^6 - 11T^5 + 22T^2 + 33T + 11
$47$	\( T^{10} \) T^10
$53$	\( T^{10} - 11 T^{7} + \cdots + 11 \) T^10 - 11T^7 + 33T^4 + 11T^3 - 22T + 11
$59$	\( T^{10} \) T^10
$61$	\( T^{10} \) T^10
$67$	\( T^{10} + 11 T^{6} + \cdots + 11 \) T^10 + 11T^6 + 11T^5 + 22T^2 - 33T + 11
$71$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$73$	\( T^{10} \) T^10
$79$	\( T^{10} - 2 T^{9} + \cdots + 1 \) T^10 - 2T^9 + 4T^8 + 3T^7 - 6T^6 + 12T^5 + 9T^4 - 7T^3 + 14T^2 - 6*T + 1
$83$	\( T^{10} \) T^10
$89$	\( T^{10} \) T^10
$97$	\( T^{10} \) T^10
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