LMFDB - Newform orbit 528.2.ba.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,2,Mod(79,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([5, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("528.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	528.ba (of order $10$, degree $4$, 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:	$4.21610122672$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{10})$
Coefficient field:	$\Q(\zeta_{20})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-\zeta_{20}^{4}$	$1$	$-1$

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} $

79.1

−0.587785	−	0.809017i
0.587785	+	0.809017i
−0.587785	+	0.809017i
0.587785	−	0.809017i
−0.951057	−	0.309017i
0.951057	+	0.309017i
−0.951057	+	0.309017i
0.951057	−	0.309017i

−0.587785

−

0.809017i

1.19098

3.66547i

−2.48990

−

1.80902i

−0.309017

0.951057i

79.2

0.587785

0.809017i

1.19098

3.66547i

2.48990

1.80902i

−0.309017

0.951057i

127.1

−0.587785

0.809017i

1.19098

−

3.66547i

−2.48990

1.80902i

−0.309017

−

0.951057i

127.2

0.587785

−

0.809017i

1.19098

−

3.66547i

2.48990

−

1.80902i

−0.309017

−

0.951057i

271.1

−0.951057

−

0.309017i

2.30902

−

1.67760i

0.224514

0.690983i

0.809017

0.587785i

271.2

0.951057

0.309017i

2.30902

−

1.67760i

−0.224514

−

0.690983i

0.809017

0.587785i

415.1

−0.951057

0.309017i

2.30902

1.67760i

0.224514

−

0.690983i

0.809017

−

0.587785i

415.2

0.951057

−

0.309017i

2.30902

1.67760i

−0.224514

0.690983i

0.809017

−

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
11.d	odd	10	1	inner
44.g	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	528.2.ba.b	✓	8
4.b	odd	2	1	inner	528.2.ba.b	✓	8
11.d	odd	10	1	inner	528.2.ba.b	✓	8
44.g	even	10	1	inner	528.2.ba.b	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
528.2.ba.b	✓	8	1.a	even	1	1	trivial
528.2.ba.b	✓	8	4.b	odd	2	1	inner
528.2.ba.b	✓	8	11.d	odd	10	1	inner
528.2.ba.b	✓	8	44.g	even	10	1	inner

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} - 7T_{5}^{3} + 34T_{5}^{2} - 88T_{5} + 121 $ T5^4 - 7*T5^3 + 34*T5^2 - 88*T5 + 121 acting on $S_{2}^{\mathrm{new}}(528, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$5$	$ (T^{4} - 7 T^{3} + \cdots + 121)^{2} $ (T^4 - 7T^3 + 34T^2 - 88*T + 121)^2
$7$	$ T^{8} - 5 T^{6} + \cdots + 25 $ T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$11$	$ T^{8} + 19 T^{6} + \cdots + 14641 $ T^8 + 19T^6 + 301T^4 + 2299*T^2 + 14641
$13$	$ (T^{4} - 5 T^{3} + 15 T^{2} + \cdots + 5)^{2} $ (T^4 - 5T^3 + 15T^2 - 15*T + 5)^2
$17$	$ (T^{4} + 10 T^{3} + 30 T^{2} + \cdots + 5)^{2} $ (T^4 + 10T^3 + 30T^2 + 15*T + 5)^2
$19$	$ T^{8} + 250 T^{4} + \cdots + 15625 $ T^8 + 250T^4 + 3125T^2 + 15625
$23$	$ (T^{4} + 42 T^{2} + 121)^{2} $ (T^4 + 42*T^2 + 121)^2
$29$	$ (T^{4} - 10 T^{3} + \cdots + 80)^{2} $ (T^4 - 10T^3 + 40T^2 - 80*T + 80)^2
$31$	$ T^{8} - 44 T^{6} + \cdots + 923521 $ T^8 - 44T^6 + 1086T^4 - 18259*T^2 + 923521
$37$	$ (T^{4} - 13 T^{3} + \cdots + 121)^{2} $ (T^4 - 13T^3 + 69T^2 - 77*T + 121)^2
$41$	$ (T^{4} - 5 T^{3} + \cdots + 125)^{2} $ (T^4 - 5T^3 + 25T^2 + 125*T + 125)^2
$43$	$ (T^{4} - 250 T^{2} + 15125)^{2} $ (T^4 - 250*T^2 + 15125)^2
$47$	$ T^{8} - 199 T^{6} + \cdots + 1 $ T^8 - 199T^6 + 15126T^4 + 76*T^2 + 1
$53$	$ (T^{4} - 14 T^{3} + \cdots + 841)^{2} $ (T^4 - 14T^3 + 96T^2 - 319*T + 841)^2
$59$	$ T^{8} - 36 T^{6} + \cdots + 96059601 $ T^8 - 36T^6 + 13446T^4 - 1675971*T^2 + 96059601
$61$	$ (T^{4} - 25 T^{3} + \cdots + 4805)^{2} $ (T^4 - 25T^3 + 300T^2 - 1860*T + 4805)^2
$67$	$ (T^{4} + 223 T^{2} + 11881)^{2} $ (T^4 + 223*T^2 + 11881)^2
$71$	$ T^{8} + 5 T^{6} + \cdots + 625 $ T^8 + 5T^6 + 150T^4 - 500*T^2 + 625
$73$	$ (T^{4} + 40 T^{3} + \cdots + 20480)^{2} $ (T^4 + 40T^3 + 640T^2 + 5120*T + 20480)^2
$79$	$ T^{8} - 165 T^{6} + \cdots + 3258025 $ T^8 - 165T^6 + 78685T^4 + 821275*T^2 + 3258025
$83$	$ T^{8} + 75 T^{6} + \cdots + 17682025 $ T^8 + 75T^6 + 4285T^4 + 231275*T^2 + 17682025
$89$	$ (T^{2} + 18 T + 61)^{4} $ (T^2 + 18*T + 61)^4
$97$	$ (T^{4} + 13 T^{3} + \cdots + 57121)^{2} $ (T^4 + 13T^3 + 244T^2 + 4302*T + 57121)^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:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	528.ba (of order \(10\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{20} q^{3} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{4} + \cdots + 1) q^{5}+ \cdots + \zeta_{20}^{2} q^{9}+O(q^{10}) \) q + z * q^3 + (-z^6 - 2z^4 + 2z^2 + 1) * q^5 + (z^7 - z^5 - z^3 + z) * q^7 + z^2 * q^9	\( q + \zeta_{20} q^{3} + ( - \zeta_{20}^{6} - 2 \zeta_{20}^{4} + \cdots + 1) q^{5}+ \cdots + ( - 2 \zeta_{20}^{7} + \cdots + 2 \zeta_{20}) q^{99}+O(q^{100}) \) q + z * q^3 + (-z^6 - 2z^4 + 2z^2 + 1) * q^5 + (z^7 - z^5 - z^3 + z) * q^7 + z^2 * q^9 + (-2z^7 - z^3 - 2z) * q^11 + (-2z^6 - z^2 + 2) q^13 + (-z^7 - 2z^5 + 2z^3 + z) * q^15 + (-2z^6 + z^4 + z^2 - 2) q^17 + (z^7 + 3z^5 - 2z^3 + z) * q^19 + (-2z^4 + 2z^2 - 1) * q^21 + (-2z^7 + 5z^5 - 2z^3) q^23 + (-5z^6 + 3z^2 - 3) * q^25 + z^3 * q^27 + (-2z^6 + 2z^4 - 2z^2 + 4) q^29 + (2z^7 + 5z^3 - 5z) q^31 + (-2z^6 + z^4 - 4z^2 + 2) * q^33 + (5z^7 - 10z^5 + 7z^3 - 3z) * q^35 + (4z^6 - 5z^4 + 4z^2) q^37 + (-2z^7 - z^3 + 2z) * q^39 + (3z^6 - 6z^4 + 4z^2 - 2) q^41 + (8z^7 - z^5 - 6z^3 - 2z) q^43 + (-3z^6 + 3z^4 + 1) * q^45 + (-5z^7 + 5z^5 + 8z) q^47 + (-4z^6 + 4) q^49 + (-2z^7 + z^5 + z^3 - 2z) * q^51 + (5z^4 - z^2 + 5) q^53 + (-z^7 + 5z^5 - 10z^3 - 3z) q^55 + (4z^6 - 3z^4 + 2z^2 - 1) q^57 + (-3z^7 - 6z^5 + 6z^3 + 3z) * q^59 + (5z^6 + 2z^4 + 2z^2 + 5) q^61 + (-2z^5 + 2z^3 - z) * q^63 + (-5z^6 - 3z^4 - 2z^2 + 1) q^65 + (z^7 + 10z^5 + z^3) q^67 + (3z^6 - 2z^2 + 2) * q^69 + (z^5 + 2z^3 + z) q^71 + (8z^6 - 8z^4 + 8z^2 - 16) q^73 + (-5z^7 + 3z^3 - 3z) q^75 + (6z^4 - 7z^2 + 2) * q^77 + (3z^7 - 6z^5 - 4z^3 - 10z) * q^79 + z^4 * q^81 + (-2z^7 - 4z^5 - z^3 + 6z) q^83 + (-7z^6 + 14z^4 - 12z^2 + 2) q^85 + (-2z^7 + 2z^5 - 2z^3 + 4z) * q^87 + (4z^6 - 4z^4 - 11) * q^89 + (2z^7 - 2z^5 - z) * q^91 + (2z^6 + 3z^4 - 3z^2 - 2) q^93 + (8z^7 - z^5 - z^3 + 8z) * q^95 + (z^4 - 16z^2 + 1) q^97 + (-2z^7 + z^5 - 4z^3 + 2z) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 2 q^{9}+O(q^{10}) \) 8 * q + 14 * q^5 + 2 * q^9	\( 8 q + 14 q^{5} + 2 q^{9} + 10 q^{13} - 20 q^{17} - 28 q^{25} + 20 q^{29} + 2 q^{33} + 26 q^{37} + 10 q^{41} - 4 q^{45} + 24 q^{49} + 28 q^{53} + 10 q^{57} + 50 q^{61} + 18 q^{69} - 80 q^{73} - 10 q^{77} - 2 q^{81} - 50 q^{85} - 72 q^{89} - 24 q^{93} - 26 q^{97}+O(q^{100}) \) 8 * q + 14 * q^5 + 2 * q^9 + 10 * q^13 - 20 * q^17 - 28 * q^25 + 20 * q^29 + 2 * q^33 + 26 * q^37 + 10 * q^41 - 4 * q^45 + 24 * q^49 + 28 * q^53 + 10 * q^57 + 50 * q^61 + 18 * q^69 - 80 * q^73 - 10 * q^77 - 2 * q^81 - 50 * q^85 - 72 * q^89 - 24 * q^93 - 26 * q^97

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	528.2.ba.b

Level	$528$
Weight	$2$
Character orbit	528.ba
Analytic conductor	$4.216$
Analytic rank	$0$
Dimension	$8$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.21610122672\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{10})\)
Coefficient field:	\(\Q(\zeta_{20})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\)	\(133\)	\(145\)	\(353\)	\(463\)
\(\chi(n)\)	\(1\)	\(-\zeta_{20}^{4}\)	\(1\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 79.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$5$	\( (T^{4} - 7 T^{3} + \cdots + 121)^{2} \) (T^4 - 7T^3 + 34T^2 - 88*T + 121)^2
$7$	\( T^{8} - 5 T^{6} + \cdots + 25 \) T^8 - 5T^6 + 85T^4 + 75*T^2 + 25
$11$	\( T^{8} + 19 T^{6} + \cdots + 14641 \) T^8 + 19T^6 + 301T^4 + 2299*T^2 + 14641
$13$	\( (T^{4} - 5 T^{3} + 15 T^{2} + \cdots + 5)^{2} \) (T^4 - 5T^3 + 15T^2 - 15*T + 5)^2
$17$	\( (T^{4} + 10 T^{3} + 30 T^{2} + \cdots + 5)^{2} \) (T^4 + 10T^3 + 30T^2 + 15*T + 5)^2
$19$	\( T^{8} + 250 T^{4} + \cdots + 15625 \) T^8 + 250T^4 + 3125T^2 + 15625
$23$	\( (T^{4} + 42 T^{2} + 121)^{2} \) (T^4 + 42*T^2 + 121)^2
$29$	\( (T^{4} - 10 T^{3} + \cdots + 80)^{2} \) (T^4 - 10T^3 + 40T^2 - 80*T + 80)^2
$31$	\( T^{8} - 44 T^{6} + \cdots + 923521 \) T^8 - 44T^6 + 1086T^4 - 18259*T^2 + 923521
$37$	\( (T^{4} - 13 T^{3} + \cdots + 121)^{2} \) (T^4 - 13T^3 + 69T^2 - 77*T + 121)^2
$41$	\( (T^{4} - 5 T^{3} + \cdots + 125)^{2} \) (T^4 - 5T^3 + 25T^2 + 125*T + 125)^2
$43$	\( (T^{4} - 250 T^{2} + 15125)^{2} \) (T^4 - 250*T^2 + 15125)^2
$47$	\( T^{8} - 199 T^{6} + \cdots + 1 \) T^8 - 199T^6 + 15126T^4 + 76*T^2 + 1
$53$	\( (T^{4} - 14 T^{3} + \cdots + 841)^{2} \) (T^4 - 14T^3 + 96T^2 - 319*T + 841)^2
$59$	\( T^{8} - 36 T^{6} + \cdots + 96059601 \) T^8 - 36T^6 + 13446T^4 - 1675971*T^2 + 96059601
$61$	\( (T^{4} - 25 T^{3} + \cdots + 4805)^{2} \) (T^4 - 25T^3 + 300T^2 - 1860*T + 4805)^2
$67$	\( (T^{4} + 223 T^{2} + 11881)^{2} \) (T^4 + 223*T^2 + 11881)^2
$71$	\( T^{8} + 5 T^{6} + \cdots + 625 \) T^8 + 5T^6 + 150T^4 - 500*T^2 + 625
$73$	\( (T^{4} + 40 T^{3} + \cdots + 20480)^{2} \) (T^4 + 40T^3 + 640T^2 + 5120*T + 20480)^2
$79$	\( T^{8} - 165 T^{6} + \cdots + 3258025 \) T^8 - 165T^6 + 78685T^4 + 821275*T^2 + 3258025
$83$	\( T^{8} + 75 T^{6} + \cdots + 17682025 \) T^8 + 75T^6 + 4285T^4 + 231275*T^2 + 17682025
$89$	\( (T^{2} + 18 T + 61)^{4} \) (T^2 + 18*T + 61)^4
$97$	\( (T^{4} + 13 T^{3} + \cdots + 57121)^{2} \) (T^4 + 13T^3 + 244T^2 + 4302*T + 57121)^2
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