LMFDB - Newform orbit 3025.1.bq.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3025,1,Mod(3,3025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3025, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([7, 16])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3025.3"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3025.bq (of order $20$, degree $8$, not minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.50967166321$
Analytic rank:	$0$
Dimension:	$8$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{20}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{20} - \cdots)$

$q$-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_{20}^{5} - \zeta_{20}^{2}) q^{3} + \zeta_{20}^{3} q^{4} - \zeta_{20}^{7} q^{5} + (\zeta_{20}^{7} + \zeta_{20}^{4} - 1) q^{9} + ( - \zeta_{20}^{8} - \zeta_{20}^{5}) q^{12} + (\zeta_{20}^{9} - \zeta_{20}^{2}) q^{15} + \cdots + ( - \zeta_{20}^{9} + \zeta_{20}^{6}) q^{97} +O(q^{100}) $ q + (-z^5 - z^2) * q^3 + z^3 * q^4 - z^7 * q^5 + (z^7 + z^4 - 1) * q^9 + (-z^8 - z^5) * q^12 + (z^9 - z^2) * q^15 + z^6 * q^16 + q^20 + (-z^4 - z^3) * q^23 - z^4 * q^25 + (-z^9 - z^6 + z^5 + z^2) * q^27 + (-z^7 - z^5) * q^31 + (z^7 - z^3 - 1) * q^36 + (-z^5 - 1) * q^37 + (z^7 + z^4 + z) * q^45 + (-z^6 + z) * q^47 + (-z^8 + z) * q^48 + z^7 * q^49 + (-z^4 + z) * q^53 + (-z^6 - z^4) * q^59 + (-z^5 - z^2) * q^60 + z^9 * q^64 + (-z^6 + z^5) * q^67 + (z^9 + z^8 + z^6 + z^5) * q^69 + (z^8 + 1) * q^71 + (z^9 + z^6) * q^75 + z^3 * q^80 + (z^8 - z^7 - z^4 - z + 1) * q^81 + (-z^9 + z^3) * q^89 + (-z^7 - z^6) * q^92 + (z^9 + z^7 - z^2 - 1) * q^93 + (-z^9 + z^6) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 10 q^{9} + 2 q^{12} - 2 q^{15} + 2 q^{16} + 8 q^{20} + 2 q^{23} + 2 q^{25} - 8 q^{36} - 8 q^{37} - 2 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{53} - 2 q^{60} - 2 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{81}+ \cdots + 2 q^{97}+O(q^{100}) $ 8 * q - 2 * q^3 - 10 * q^9 + 2 * q^12 - 2 * q^15 + 2 * q^16 + 8 * q^20 + 2 * q^23 + 2 * q^25 - 8 * q^36 - 8 * q^37 - 2 * q^45 - 2 * q^47 + 2 * q^48 + 2 * q^53 - 2 * q^60 - 2 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^81 - 2 * q^92 - 10 * q^93 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3025\mathbb{Z}\right)^\times$.

$n$	$727$	$2301$
$\chi(n)$	$-\zeta_{20}^{7}$	$-\zeta_{20}^{6}$

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} $

3.1

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

−0.809017

0.412215i

−0.587785

−

0.809017i

−0.587785

0.809017i

−0.103198

0.142040i

27.1

0.309017

−

1.95106i

0.951057

−

0.309017i

0.951057

0.309017i

−2.76007

−

0.896802i

148.1

0.309017

0.0489435i

−0.951057

0.309017i

−0.951057

−

0.309017i

−0.857960

−

0.278768i

1533.1

−0.809017

1.58779i

0.587785

−

0.809017i

0.587785

0.809017i

−1.27877

−

1.76007i

1697.1

−0.809017

−

1.58779i

0.587785

0.809017i

0.587785

−

0.809017i

−1.27877

1.76007i

2017.1

−0.809017

−

0.412215i

−0.587785

0.809017i

−0.587785

−

0.809017i

−0.103198

−

0.142040i

2187.1

0.309017

−

0.0489435i

−0.951057

−

0.309017i

−0.951057

0.309017i

−0.857960

0.278768i

2913.1

0.309017

1.95106i

0.951057

0.309017i

0.951057

−

0.309017i

−2.76007

0.896802i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
275.bg	even	20	1	inner
275.bp	odd	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3025.1.bq.a		8
11.b	odd	2	1	CM	3025.1.bq.a		8
11.c	even	5	1		3025.1.bf.a		8
11.c	even	5	1		3025.1.bi.a		8
11.c	even	5	1		3025.1.bj.a	✓	8
11.c	even	5	1		3025.1.bk.a		8
11.d	odd	10	1		3025.1.bf.a		8
11.d	odd	10	1		3025.1.bi.a		8
11.d	odd	10	1		3025.1.bj.a	✓	8
11.d	odd	10	1		3025.1.bk.a		8
25.f	odd	20	1		3025.1.bk.a		8
275.be	odd	20	1		3025.1.bj.a	✓	8
275.bf	even	20	1		3025.1.bj.a	✓	8
275.bg	even	20	1	inner	3025.1.bq.a		8
275.bh	odd	20	1		3025.1.bf.a		8
275.bj	odd	20	1		3025.1.bi.a		8
275.bl	even	20	1		3025.1.bf.a		8
275.bn	even	20	1		3025.1.bi.a		8
275.bo	even	20	1		3025.1.bk.a		8
275.bp	odd	20	1	inner	3025.1.bq.a		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3025.1.bf.a		8	11.c	even	5	1
3025.1.bf.a		8	11.d	odd	10	1
3025.1.bf.a		8	275.bh	odd	20	1
3025.1.bf.a		8	275.bl	even	20	1
3025.1.bi.a		8	11.c	even	5	1
3025.1.bi.a		8	11.d	odd	10	1
3025.1.bi.a		8	275.bj	odd	20	1
3025.1.bi.a		8	275.bn	even	20	1
3025.1.bj.a	✓	8	11.c	even	5	1
3025.1.bj.a	✓	8	11.d	odd	10	1
3025.1.bj.a	✓	8	275.be	odd	20	1
3025.1.bj.a	✓	8	275.bf	even	20	1
3025.1.bk.a		8	11.c	even	5	1
3025.1.bk.a		8	11.d	odd	10	1
3025.1.bk.a		8	25.f	odd	20	1
3025.1.bk.a		8	275.bo	even	20	1
3025.1.bq.a		8	1.a	even	1	1	trivial
3025.1.bq.a		8	11.b	odd	2	1	CM
3025.1.bq.a		8	275.bg	even	20	1	inner
3025.1.bq.a		8	275.bp	odd	20	1	inner

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(3025, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} $ T^8
$3$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1
$5$	$ T^{8} - T^{6} + T^{4} + \cdots + 1 $ T^8 - T^6 + T^4 - T^2 + 1
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} $ T^8
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 + 2T^6 - 10T^5 + 16T^4 - 10T^3 + 13T^2 - 6*T + 1
$29$	$ T^{8} $ T^8
$31$	$ T^{8} + 5 T^{6} + \cdots + 25 $ T^8 + 5T^6 + 10T^4 + 25
$37$	$ (T^{2} + 2 T + 2)^{4} $ (T^2 + 2*T + 2)^4
$41$	$ T^{8} $ T^8
$43$	$ T^{8} $ T^8
$47$	$ T^{8} + 2 T^{7} + \cdots + 16 $ T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$53$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 + 2T^6 + 11T^4 - 20T^3 + 18T^2 - 6T + 1
$59$	$ (T^{4} + 5 T^{2} + 5)^{2} $ (T^4 + 5*T^2 + 5)^2
$61$	$ T^{8} $ T^8
$67$	$ T^{8} + 2 T^{7} + \cdots + 1 $ T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1
$71$	$ (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$73$	$ T^{8} $ T^8
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 6T^4 + T^2 + 1
$97$	$ T^{8} - 2 T^{7} + \cdots + 1 $ T^8 - 2T^7 + 2T^6 + 11T^4 - 20T^3 + 18T^2 - 6T + 1
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3025.bq (of order \(20\), degree \(8\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{20}^{5} - \zeta_{20}^{2}) q^{3} + \zeta_{20}^{3} q^{4} - \zeta_{20}^{7} q^{5} + (\zeta_{20}^{7} + \zeta_{20}^{4} - 1) q^{9} + ( - \zeta_{20}^{8} - \zeta_{20}^{5}) q^{12} + (\zeta_{20}^{9} - \zeta_{20}^{2}) q^{15} + \cdots + ( - \zeta_{20}^{9} + \zeta_{20}^{6}) q^{97} +O(q^{100}) \) q + (-z^5 - z^2) * q^3 + z^3 * q^4 - z^7 * q^5 + (z^7 + z^4 - 1) * q^9 + (-z^8 - z^5) * q^12 + (z^9 - z^2) * q^15 + z^6 * q^16 + q^20 + (-z^4 - z^3) * q^23 - z^4 * q^25 + (-z^9 - z^6 + z^5 + z^2) * q^27 + (-z^7 - z^5) * q^31 + (z^7 - z^3 - 1) * q^36 + (-z^5 - 1) * q^37 + (z^7 + z^4 + z) * q^45 + (-z^6 + z) * q^47 + (-z^8 + z) * q^48 + z^7 * q^49 + (-z^4 + z) * q^53 + (-z^6 - z^4) * q^59 + (-z^5 - z^2) * q^60 + z^9 * q^64 + (-z^6 + z^5) * q^67 + (z^9 + z^8 + z^6 + z^5) * q^69 + (z^8 + 1) * q^71 + (z^9 + z^6) * q^75 + z^3 * q^80 + (z^8 - z^7 - z^4 - z + 1) * q^81 + (-z^9 + z^3) * q^89 + (-z^7 - z^6) * q^92 + (z^9 + z^7 - z^2 - 1) * q^93 + (-z^9 + z^6) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 10 q^{9} + 2 q^{12} - 2 q^{15} + 2 q^{16} + 8 q^{20} + 2 q^{23} + 2 q^{25} - 8 q^{36} - 8 q^{37} - 2 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{53} - 2 q^{60} - 2 q^{67} + 6 q^{71} + 2 q^{75} + 8 q^{81}+ \cdots + 2 q^{97}+O(q^{100}) \) 8 * q - 2 * q^3 - 10 * q^9 + 2 * q^12 - 2 * q^15 + 2 * q^16 + 8 * q^20 + 2 * q^23 + 2 * q^25 - 8 * q^36 - 8 * q^37 - 2 * q^45 - 2 * q^47 + 2 * q^48 + 2 * q^53 - 2 * q^60 - 2 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^81 - 2 * q^92 - 10 * q^93 + 2 * 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	3025.1.bq.a

Level	$3025$
Weight	$1$
Character orbit	3025.bq
Analytic conductor	$1.510$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{20}$
CM discriminant	-11
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(1.50967166321\)
Analytic rank:	\(0\)
Dimension:	\(8\)
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:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{20}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

\(n\)	\(727\)	\(2301\)
\(\chi(n)\)	\(-\zeta_{20}^{7}\)	\(-\zeta_{20}^{6}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 3.1
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1
$5$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} \) T^8
$17$	\( T^{8} \) T^8
$19$	\( T^{8} \) T^8
$23$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 + 2T^6 - 10T^5 + 16T^4 - 10T^3 + 13T^2 - 6*T + 1
$29$	\( T^{8} \) T^8
$31$	\( T^{8} + 5 T^{6} + \cdots + 25 \) T^8 + 5T^6 + 10T^4 + 25
$37$	\( (T^{2} + 2 T + 2)^{4} \) (T^2 + 2*T + 2)^4
$41$	\( T^{8} \) T^8
$43$	\( T^{8} \) T^8
$47$	\( T^{8} + 2 T^{7} + \cdots + 16 \) T^8 + 2T^7 + 2T^6 - 4T^4 + 8T^2 + 16*T + 16
$53$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 + 2T^6 + 11T^4 - 20T^3 + 18T^2 - 6T + 1
$59$	\( (T^{4} + 5 T^{2} + 5)^{2} \) (T^4 + 5*T^2 + 5)^2
$61$	\( T^{8} \) T^8
$67$	\( T^{8} + 2 T^{7} + \cdots + 1 \) T^8 + 2T^7 + 7T^6 + 10T^5 + 16T^4 + 10T^3 - 2T^2 - 4*T + 1
$71$	\( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$73$	\( T^{8} \) T^8
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 6T^4 + T^2 + 1
$97$	\( T^{8} - 2 T^{7} + \cdots + 1 \) T^8 - 2T^7 + 2T^6 + 11T^4 - 20T^3 + 18T^2 - 6T + 1
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