LMFDB - Newform orbit 880.2.cd.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,2,Mod(49,880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 5, 4]))

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

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

chi := DirichletCharacter("880.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	880.cd (of order $10$, degree $4$, not 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:	$7.02683537787$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$-\zeta_{20}^{2}$	$1$

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} $

49.1

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

−1.90211

2.61803i

0.333023

2.21113i

0.951057

1.30902i

−2.30902

−

7.10642i

49.2

1.90211

−

2.61803i

−1.56909

1.59310i

−0.951057

−

1.30902i

−2.30902

−

7.10642i

289.1

−1.17557

0.381966i

2.20582

0.366554i

0.587785

0.190983i

−1.19098

0.865300i

289.2

1.17557

−

0.381966i

1.03025

1.98459i

−0.587785

−

0.190983i

−1.19098

0.865300i

449.1

−1.90211

−

2.61803i

0.333023

−

2.21113i

0.951057

−

1.30902i

−2.30902

7.10642i

449.2

1.90211

2.61803i

−1.56909

−

1.59310i

−0.951057

1.30902i

−2.30902

7.10642i

609.1

−1.17557

−

0.381966i

2.20582

−

0.366554i

0.587785

−

0.190983i

−1.19098

−

0.865300i

609.2

1.17557

0.381966i

1.03025

−

1.98459i

−0.587785

0.190983i

−1.19098

−

0.865300i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
11.c	even	5	1	inner
55.j	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	880.2.cd.a		8
4.b	odd	2	1		110.2.j.a	✓	8
5.b	even	2	1	inner	880.2.cd.a		8
11.c	even	5	1	inner	880.2.cd.a		8
12.b	even	2	1		990.2.ba.b		8
20.d	odd	2	1		110.2.j.a	✓	8
20.e	even	4	1		550.2.h.d		4
20.e	even	4	1		550.2.h.e		4
44.g	even	10	1		1210.2.b.g		4
44.h	odd	10	1		110.2.j.a	✓	8
44.h	odd	10	1		1210.2.b.f		4
55.j	even	10	1	inner	880.2.cd.a		8
60.h	even	2	1		990.2.ba.b		8
132.o	even	10	1		990.2.ba.b		8
220.n	odd	10	1		110.2.j.a	✓	8
220.n	odd	10	1		1210.2.b.f		4
220.o	even	10	1		1210.2.b.g		4
220.v	even	20	1		550.2.h.d		4
220.v	even	20	1		550.2.h.e		4
220.v	even	20	1		6050.2.a.ce		2
220.v	even	20	1		6050.2.a.cl		2
220.w	odd	20	1		6050.2.a.bv		2
220.w	odd	20	1		6050.2.a.ct		2
660.bd	even	10	1		990.2.ba.b		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
110.2.j.a	✓	8	4.b	odd	2	1
110.2.j.a	✓	8	20.d	odd	2	1
110.2.j.a	✓	8	44.h	odd	10	1
110.2.j.a	✓	8	220.n	odd	10	1
550.2.h.d		4	20.e	even	4	1
550.2.h.d		4	220.v	even	20	1
550.2.h.e		4	20.e	even	4	1
550.2.h.e		4	220.v	even	20	1
880.2.cd.a		8	1.a	even	1	1	trivial
880.2.cd.a		8	5.b	even	2	1	inner
880.2.cd.a		8	11.c	even	5	1	inner
880.2.cd.a		8	55.j	even	10	1	inner
990.2.ba.b		8	12.b	even	2	1
990.2.ba.b		8	60.h	even	2	1
990.2.ba.b		8	132.o	even	10	1
990.2.ba.b		8	660.bd	even	10	1
1210.2.b.f		4	44.h	odd	10	1
1210.2.b.f		4	220.n	odd	10	1
1210.2.b.g		4	44.g	even	10	1
1210.2.b.g		4	220.o	even	10	1
6050.2.a.bv		2	220.w	odd	20	1
6050.2.a.ce		2	220.v	even	20	1
6050.2.a.cl		2	220.v	even	20	1
6050.2.a.ct		2	220.w	odd	20	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} + 4T_{3}^{6} + 96T_{3}^{4} - 256T_{3}^{2} + 256 $ T3^8 + 4*T3^6 + 96*T3^4 - 256*T3^2 + 256 acting on $S_{2}^{\mathrm{new}}(880, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 4 T^{6} + \cdots + 256 $ T^8 + 4T^6 + 96T^4 - 256*T^2 + 256
$5$	$ T^{8} - 4 T^{7} + \cdots + 625 $ T^8 - 4T^7 + 11T^6 - 24T^5 + 41T^4 - 120T^3 + 275T^2 - 500*T + 625
$7$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$11$	$ (T^{4} + 9 T^{3} + \cdots + 121)^{2} $ (T^4 + 9T^3 + 41T^2 + 99*T + 121)^2
$13$	$ T^{8} - 31 T^{6} + \cdots + 14641 $ T^8 - 31T^6 + 366T^4 + 484*T^2 + 14641
$17$	$ T^{8} + 16 T^{6} + \cdots + 65536 $ T^8 + 16T^6 + 1536T^4 - 16384*T^2 + 65536
$19$	$ (T^{4} + 10 T^{2} + \cdots + 25)^{2} $ (T^4 + 10T^2 + 25T + 25)^2
$23$	$ (T^{4} + 27 T^{2} + 121)^{2} $ (T^4 + 27*T^2 + 121)^2
$29$	$ (T^{4} + 10 T^{3} + \cdots + 400)^{2} $ (T^4 + 10T^3 + 60T^2 + 200*T + 400)^2
$31$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} $ (T^4 - 2T^3 + 4T^2 - 8*T + 16)^2
$37$	$ T^{8} - 44 T^{6} + \cdots + 130321 $ T^8 - 44T^6 + 766T^4 + 361*T^2 + 130321
$41$	$ (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$43$	$ (T^{4} + 72 T^{2} + 16)^{2} $ (T^4 + 72*T^2 + 16)^2
$47$	$ T^{8} - 164 T^{6} + \cdots + 14641 $ T^8 - 164T^6 + 10246T^4 + 7381*T^2 + 14641
$53$	$ T^{8} + 4 T^{6} + \cdots + 707281 $ T^8 + 4T^6 + 1966T^4 - 59711*T^2 + 707281
$59$	$ (T^{4} + 90 T^{2} + \cdots + 2025)^{2} $ (T^4 + 90T^2 - 675T + 2025)^2
$61$	$ (T^{4} - 18 T^{3} + \cdots + 5776)^{2} $ (T^4 - 18T^3 + 244T^2 - 1672*T + 5776)^2
$67$	$ (T^{4} + 108 T^{2} + 1936)^{2} $ (T^4 + 108*T^2 + 1936)^2
$71$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} $ (T^4 - 2T^3 + 4T^2 - 8*T + 16)^2
$73$	$ T^{8} + 64 T^{6} + \cdots + 65536 $ T^8 + 64T^6 + 11776T^4 - 45056*T^2 + 65536
$79$	$ (T^{4} - 20 T^{3} + \cdots + 10000)^{2} $ (T^4 - 20T^3 + 400T^2 - 3000*T + 10000)^2
$83$	$ T^{8} - 176 T^{6} + \cdots + 236421376 $ T^8 - 176T^6 + 17376T^4 - 1168576*T^2 + 236421376
$89$	$ (T^{2} + 25 T + 155)^{4} $ (T^2 + 25*T + 155)^4
$97$	$ T^{8} + 16 T^{6} + \cdots + 3748096 $ T^8 + 16T^6 + 5856T^4 - 240064*T^2 + 3748096
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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:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	880.cd (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + (2 \zeta_{20}^{5} + 2 \zeta_{20}) q^{3} + (\zeta_{20}^{7} - \zeta_{20}^{5} + \cdots - \zeta_{20}) q^{5}+ \cdots + (5 \zeta_{20}^{6} + 4 \zeta_{20}^{2} - 4) q^{9}+O(q^{10}) \) q + (2z^5 + 2z) * q^3 + (z^7 - z^5 - 2z^4 + z^3 - z) q^5 + (z^7 + z^3 - z) * q^7 + (5z^6 + 4z^2 - 4) * q^9	\( q + (2 \zeta_{20}^{5} + 2 \zeta_{20}) q^{3} + (\zeta_{20}^{7} - \zeta_{20}^{5} + \cdots - \zeta_{20}) q^{5}+ \cdots + ( - 13 \zeta_{20}^{6} - 3 \zeta_{20}^{4} + \cdots + 3) q^{99}+O(q^{100}) \) q + (2z^5 + 2z) * q^3 + (z^7 - z^5 - 2z^4 + z^3 - z) q^5 + (z^7 + z^3 - z) * q^7 + (5z^6 + 4z^2 - 4) * q^9 + (2z^6 - 2z^4 + 3z^2 - 4) q^11 + (-z^7 + z^5 + 4z) q^13 + (-4z^7 - 2z^4 - 4z^3 + 4z - 2) * q^15 + (4z^5 - 4z^3) * q^17 + (2z^6 - 3z^4 + 3z^2 - 2) q^19 + (2z^6 - 2z^4 - 4) * q^21 + (z^7 - 4z^5 + z^3) q^23 + (3z^6 - 3z^4 + 4z^3 + 3z^2 - 3) * q^25 + (12z^7 - 8z^5 + 8z^3 - 12z) * q^27 + (-4z^4 + 2z^2 - 4) * q^29 + 2z^6 q^31 + (6z^7 - 8z^5 + 2z^3 - 8z) * q^33 + (-2z^7 - z^6 + 2z^5 - z^2 + 2z + 1) q^35 + (4z^7 - z^3 + z) q^37 + (8z^6 + 2z^4 + 8z^2) q^39 + (-z^6 + 1) * q^41 + (-4z^7 + 6z^5 - 4z^3) q^43 + (-4z^7 - 8z^6 - z^5 + 8z^4 - 4z^3 + 10) * q^45 + (5z^5 - 7z^3 + 5z) q^47 + (-z^6 + 6z^4 - z^2) q^49 - 8z^2 q^51 + (-5z^7 + 5z^5 + z) * q^53 + (-4z^7 - 2z^6 + 2z^5 + 4z^4 - 2z^3 + 4z^2 + z) * q^55 + (4z^7 - 4z^5 - 2z) q^57 + (-3z^4 + 9z^2 - 3) * q^59 + (8z^6 - 2z^4 + 8z^2) q^61 + (-5z^5 - 4z^3 - 5z) q^63 + (-2z^7 + z^6 - 6z^5 - z^4 - 2z^3 - 4) q^65 + (-2z^7 + 8z^5 - 2z^3) q^67 + (-4z^6 - 2z^4 + 2z^2 + 4) q^69 - 2z^4 q^71 + (-8z^7 - 4z^3 + 4z) q^73 + (6z^7 + 8z^6 - 6z^5 + 8z^2 - 6z - 8) q^75 + (-3z^7 - 4z^3 + z) * q^77 + (-10z^6 - 10z^2 + 10) * q^79 + (-12z^4 - 5z^2 - 12) * q^81 + (-6z^7 - 4z^5 + 4z^3 + 6z) * q^83 + (8z^5 - 4z^4 - 8z^3 + 4z^2 + 8z) q^85 + (-4z^7 - 8z^5 - 4z^3) q^87 + (z^6 - z^4 - 13) * q^89 + (5z^6 - z^4 + z^2 - 5) q^91 + (4z^7 - 4z) * q^93 + (-2z^7 - 2z^4 + z^3 + 6z^2 - z - 2) q^95 + (-2z^7 + 2z^5 - 6z) q^97 + (-13z^6 - 3z^4 - 15z^2 + 3) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} - 14 q^{9}+O(q^{10}) \) 8 * q + 4 * q^5 - 14 * q^9	\( 8 q + 4 q^{5} - 14 q^{9} - 18 q^{11} - 12 q^{15} - 24 q^{21} - 6 q^{25} - 20 q^{29} + 4 q^{31} + 4 q^{35} + 28 q^{39} + 6 q^{41} + 48 q^{45} - 16 q^{49} - 16 q^{51} - 4 q^{55} + 36 q^{61} - 28 q^{65} + 32 q^{69} + 4 q^{71} - 32 q^{75} + 40 q^{79} - 82 q^{81} + 16 q^{85} - 100 q^{89} - 26 q^{91} - 26 q^{99}+O(q^{100}) \) 8 * q + 4 * q^5 - 14 * q^9 - 18 * q^11 - 12 * q^15 - 24 * q^21 - 6 * q^25 - 20 * q^29 + 4 * q^31 + 4 * q^35 + 28 * q^39 + 6 * q^41 + 48 * q^45 - 16 * q^49 - 16 * q^51 - 4 * q^55 + 36 * q^61 - 28 * q^65 + 32 * q^69 + 4 * q^71 - 32 * q^75 + 40 * q^79 - 82 * q^81 + 16 * q^85 - 100 * q^89 - 26 * q^91 - 26 * q^99

Introduction

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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	880.2.cd.a

Level	$880$
Weight	$2$
Character orbit	880.cd
Analytic conductor	$7.027$
Analytic rank	$0$
Dimension	$8$
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(7.02683537787\)
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:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

\(n\)	\(111\)	\(177\)	\(321\)	\(661\)
\(\chi(n)\)	\(1\)	\(-1\)	\(-\zeta_{20}^{2}\)	\(1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + 4 T^{6} + \cdots + 256 \) T^8 + 4T^6 + 96T^4 - 256*T^2 + 256
$5$	\( T^{8} - 4 T^{7} + \cdots + 625 \) T^8 - 4T^7 + 11T^6 - 24T^5 + 41T^4 - 120T^3 + 275T^2 - 500*T + 625
$7$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$11$	\( (T^{4} + 9 T^{3} + \cdots + 121)^{2} \) (T^4 + 9T^3 + 41T^2 + 99*T + 121)^2
$13$	\( T^{8} - 31 T^{6} + \cdots + 14641 \) T^8 - 31T^6 + 366T^4 + 484*T^2 + 14641
$17$	\( T^{8} + 16 T^{6} + \cdots + 65536 \) T^8 + 16T^6 + 1536T^4 - 16384*T^2 + 65536
$19$	\( (T^{4} + 10 T^{2} + \cdots + 25)^{2} \) (T^4 + 10T^2 + 25T + 25)^2
$23$	\( (T^{4} + 27 T^{2} + 121)^{2} \) (T^4 + 27*T^2 + 121)^2
$29$	\( (T^{4} + 10 T^{3} + \cdots + 400)^{2} \) (T^4 + 10T^3 + 60T^2 + 200*T + 400)^2
$31$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) (T^4 - 2T^3 + 4T^2 - 8*T + 16)^2
$37$	\( T^{8} - 44 T^{6} + \cdots + 130321 \) T^8 - 44T^6 + 766T^4 + 361*T^2 + 130321
$41$	\( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$43$	\( (T^{4} + 72 T^{2} + 16)^{2} \) (T^4 + 72*T^2 + 16)^2
$47$	\( T^{8} - 164 T^{6} + \cdots + 14641 \) T^8 - 164T^6 + 10246T^4 + 7381*T^2 + 14641
$53$	\( T^{8} + 4 T^{6} + \cdots + 707281 \) T^8 + 4T^6 + 1966T^4 - 59711*T^2 + 707281
$59$	\( (T^{4} + 90 T^{2} + \cdots + 2025)^{2} \) (T^4 + 90T^2 - 675T + 2025)^2
$61$	\( (T^{4} - 18 T^{3} + \cdots + 5776)^{2} \) (T^4 - 18T^3 + 244T^2 - 1672*T + 5776)^2
$67$	\( (T^{4} + 108 T^{2} + 1936)^{2} \) (T^4 + 108*T^2 + 1936)^2
$71$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) (T^4 - 2T^3 + 4T^2 - 8*T + 16)^2
$73$	\( T^{8} + 64 T^{6} + \cdots + 65536 \) T^8 + 64T^6 + 11776T^4 - 45056*T^2 + 65536
$79$	\( (T^{4} - 20 T^{3} + \cdots + 10000)^{2} \) (T^4 - 20T^3 + 400T^2 - 3000*T + 10000)^2
$83$	\( T^{8} - 176 T^{6} + \cdots + 236421376 \) T^8 - 176T^6 + 17376T^4 - 1168576*T^2 + 236421376
$89$	\( (T^{2} + 25 T + 155)^{4} \) (T^2 + 25*T + 155)^4
$97$	\( T^{8} + 16 T^{6} + \cdots + 3748096 \) T^8 + 16T^6 + 5856T^4 - 240064*T^2 + 3748096
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