LMFDB - Newform orbit 1875.1.h.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1875,1,Mod(374,1875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 3])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1875.374"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1875.h (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,2,0,0,0,0,2,0,0,0,0,0,0,-2,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$0.935746898687$
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
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.3515625.1

$q$-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}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{7} - \zeta_{20}^{4} q^{9} + \zeta_{20}^{9} q^{12} + ( - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{13} + \zeta_{20}^{4} q^{16} + \cdots + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{97} +O(q^{100}) $ q + z^7 * q^3 + z^2 * q^4 + (-z^7 - z^3) * q^7 - z^4 * q^9 + z^9 * q^12 + (-z^5 + z^3) * q^13 + z^4 * q^16 + (-z^4 + z^2) * q^19 + (z^4 + 1) * q^21 + z * q^27 + (-z^9 - z^5) * q^28 + (-z^6 + 1) * q^31 - z^6 * q^36 + (z^5 - z^3) * q^37 + (z^2 - 1) * q^39 + (-z^9 - z) * q^43 - z * q^48 + (z^6 - z^4 - 1) * q^49 + (-z^7 + z^5) * q^52 + (z^9 + z) * q^57 + (z^8 + z^4) * q^61 + (z^7 - z) * q^63 + z^6 * q^64 + (z^5 + z) * q^67 + (-z^9 + z^3) * q^73 + (-z^6 + z^4) * q^76 + (-z^4 - 1) * q^79 + z^8 * q^81 + (z^6 + z^2) * q^84 + (z^8 - z^6 - z^2 + 1) * q^91 + (z^7 + z^3) * q^93 + (z^9 + z^5) * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 4 q^{19} + 6 q^{21} + 6 q^{31} - 2 q^{36} - 6 q^{39} - 4 q^{49} - 4 q^{61} + 2 q^{64} - 4 q^{76} - 6 q^{79} - 2 q^{81} + 4 q^{84} + 2 q^{91}+O(q^{100}) $ 8 * q + 2 * q^4 + 2 * q^9 - 2 * q^16 + 4 * q^19 + 6 * q^21 + 6 * q^31 - 2 * q^36 - 6 * q^39 - 4 * q^49 - 4 * q^61 + 2 * q^64 - 4 * q^76 - 6 * q^79 - 2 * q^81 + 4 * q^84 + 2 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$-1$	$\zeta_{20}^{2}$

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} $

374.1

−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
0.587785	−	0.809017i

−0.951057

−

0.309017i

−0.309017

0.951057i

0.618034i

0.809017

0.587785i

374.2

0.951057

0.309017i

−0.309017

0.951057i

−

0.618034i

0.809017

0.587785i

749.1

−0.587785

0.809017i

0.809017

0.587785i

−

1.61803i

−0.309017

−

0.951057i

749.2

0.587785

−

0.809017i

0.809017

0.587785i

1.61803i

−0.309017

−

0.951057i

1124.1

−0.587785

−

0.809017i

0.809017

−

0.587785i

1.61803i

−0.309017

0.951057i

1124.2

0.587785

0.809017i

0.809017

−

0.587785i

−

1.61803i

−0.309017

0.951057i

1499.1

−0.951057

0.309017i

−0.309017

−

0.951057i

−

0.618034i

0.809017

−

0.587785i

1499.2

0.951057

−

0.309017i

−0.309017

−

0.951057i

0.618034i

0.809017

−

0.587785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
5.b	even	2	1	inner
15.d	odd	2	1	inner
25.d	even	5	1	inner
25.e	even	10	1	inner
75.h	odd	10	1	inner
75.j	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1875.1.h.f		8
3.b	odd	2	1	CM	1875.1.h.f		8
5.b	even	2	1	inner	1875.1.h.f		8
5.c	odd	4	1		1875.1.j.b		4
5.c	odd	4	1		1875.1.j.c		4
15.d	odd	2	1	inner	1875.1.h.f		8
15.e	even	4	1		1875.1.j.b		4
15.e	even	4	1		1875.1.j.c		4
25.d	even	5	1		1875.1.d.a		4
25.d	even	5	2		1875.1.h.e		8
25.d	even	5	1	inner	1875.1.h.f		8
25.e	even	10	1		1875.1.d.a		4
25.e	even	10	2		1875.1.h.e		8
25.e	even	10	1	inner	1875.1.h.f		8
25.f	odd	20	1		1875.1.c.a	✓	2
25.f	odd	20	1		1875.1.c.b	yes	2
25.f	odd	20	2		1875.1.j.a		4
25.f	odd	20	1		1875.1.j.b		4
25.f	odd	20	1		1875.1.j.c		4
25.f	odd	20	2		1875.1.j.d		4
75.h	odd	10	1		1875.1.d.a		4
75.h	odd	10	2		1875.1.h.e		8
75.h	odd	10	1	inner	1875.1.h.f		8
75.j	odd	10	1		1875.1.d.a		4
75.j	odd	10	2		1875.1.h.e		8
75.j	odd	10	1	inner	1875.1.h.f		8
75.l	even	20	1		1875.1.c.a	✓	2
75.l	even	20	1		1875.1.c.b	yes	2
75.l	even	20	2		1875.1.j.a		4
75.l	even	20	1		1875.1.j.b		4
75.l	even	20	1		1875.1.j.c		4
75.l	even	20	2		1875.1.j.d		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1875.1.c.a	✓	2	25.f	odd	20	1
1875.1.c.a	✓	2	75.l	even	20	1
1875.1.c.b	yes	2	25.f	odd	20	1
1875.1.c.b	yes	2	75.l	even	20	1
1875.1.d.a		4	25.d	even	5	1
1875.1.d.a		4	25.e	even	10	1
1875.1.d.a		4	75.h	odd	10	1
1875.1.d.a		4	75.j	odd	10	1
1875.1.h.e		8	25.d	even	5	2
1875.1.h.e		8	25.e	even	10	2
1875.1.h.e		8	75.h	odd	10	2
1875.1.h.e		8	75.j	odd	10	2
1875.1.h.f		8	1.a	even	1	1	trivial
1875.1.h.f		8	3.b	odd	2	1	CM
1875.1.h.f		8	5.b	even	2	1	inner
1875.1.h.f		8	15.d	odd	2	1	inner
1875.1.h.f		8	25.d	even	5	1	inner
1875.1.h.f		8	25.e	even	10	1	inner
1875.1.h.f		8	75.h	odd	10	1	inner
1875.1.h.f		8	75.j	odd	10	1	inner
1875.1.j.a		4	25.f	odd	20	2
1875.1.j.a		4	75.l	even	20	2
1875.1.j.b		4	5.c	odd	4	1
1875.1.j.b		4	15.e	even	4	1
1875.1.j.b		4	25.f	odd	20	1
1875.1.j.b		4	75.l	even	20	1
1875.1.j.c		4	5.c	odd	4	1
1875.1.j.c		4	15.e	even	4	1
1875.1.j.c		4	25.f	odd	20	1
1875.1.j.c		4	75.l	even	20	1
1875.1.j.d		4	25.f	odd	20	2
1875.1.j.d		4	75.l	even	20	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{1}^{\mathrm{new}}(1875, [\chi])$:

$ T_{2} $

T2

$ T_{13}^{8} + T_{13}^{6} + 6T_{13}^{4} - 4T_{13}^{2} + 1 $

T13^8 + T13^6 + 6*T13^4 - 4*T13^2 + 1

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^{8} $ T^8
$7$	$ (T^{4} + 3 T^{2} + 1)^{2} $ (T^4 + 3*T^2 + 1)^2
$11$	$ T^{8} $ T^8
$13$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$17$	$ T^{8} $ T^8
$19$	$ (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$23$	$ T^{8} $ T^8
$29$	$ T^{8} $ T^8
$31$	$ (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$37$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$41$	$ T^{8} $ T^8
$43$	$ (T^{4} + 3 T^{2} + 1)^{2} $ (T^4 + 3*T^2 + 1)^2
$47$	$ T^{8} $ T^8
$53$	$ T^{8} $ T^8
$59$	$ T^{8} $ T^8
$61$	$ (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$67$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
$71$	$ T^{8} $ T^8
$73$	$ T^{8} - 4 T^{6} + \cdots + 1 $ T^8 - 4T^6 + 6T^4 + T^2 + 1
$79$	$ (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} $ (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} + T^{6} + 6 T^{4} + \cdots + 1 $ T^8 + T^6 + 6T^4 - 4T^2 + 1
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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1875.h (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q + \zeta_{20}^{7} q^{3} + \zeta_{20}^{2} q^{4} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{7} - \zeta_{20}^{4} q^{9} + \zeta_{20}^{9} q^{12} + ( - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{13} + \zeta_{20}^{4} q^{16} + \cdots + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{97} +O(q^{100}) \) q + z^7 * q^3 + z^2 * q^4 + (-z^7 - z^3) * q^7 - z^4 * q^9 + z^9 * q^12 + (-z^5 + z^3) * q^13 + z^4 * q^16 + (-z^4 + z^2) * q^19 + (z^4 + 1) * q^21 + z * q^27 + (-z^9 - z^5) * q^28 + (-z^6 + 1) * q^31 - z^6 * q^36 + (z^5 - z^3) * q^37 + (z^2 - 1) * q^39 + (-z^9 - z) * q^43 - z * q^48 + (z^6 - z^4 - 1) * q^49 + (-z^7 + z^5) * q^52 + (z^9 + z) * q^57 + (z^8 + z^4) * q^61 + (z^7 - z) * q^63 + z^6 * q^64 + (z^5 + z) * q^67 + (-z^9 + z^3) * q^73 + (-z^6 + z^4) * q^76 + (-z^4 - 1) * q^79 + z^8 * q^81 + (z^6 + z^2) * q^84 + (z^8 - z^6 - z^2 + 1) * q^91 + (z^7 + z^3) * q^93 + (z^9 + z^5) * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 4 q^{19} + 6 q^{21} + 6 q^{31} - 2 q^{36} - 6 q^{39} - 4 q^{49} - 4 q^{61} + 2 q^{64} - 4 q^{76} - 6 q^{79} - 2 q^{81} + 4 q^{84} + 2 q^{91}+O(q^{100}) \) 8 * q + 2 * q^4 + 2 * q^9 - 2 * q^16 + 4 * q^19 + 6 * q^21 + 6 * q^31 - 2 * q^36 - 6 * q^39 - 4 * q^49 - 4 * q^61 + 2 * q^64 - 4 * q^76 - 6 * q^79 - 2 * q^81 + 4 * q^84 + 2 * q^91

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	1875.1.h.f

Level	$1875$
Weight	$1$
Character orbit	1875.h
Analytic conductor	$0.936$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{5}$
CM discriminant	-3
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.935746898687\)
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
Projective image:	\(D_{5}\)
Projective field:	Galois closure of 5.1.3515625.1

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 374.2
Significant digits:
Format:

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
5.b	even	2	1	inner
15.d	odd	2	1	inner
25.d	even	5	1	inner
25.e	even	10	1	inner
75.h	odd	10	1	inner
75.j	odd	10	1	inner

$p$	$F_p(T)$
$2$	\( T^{8} \) T^8
$3$	\( T^{8} - T^{6} + T^{4} + \cdots + 1 \) T^8 - T^6 + T^4 - T^2 + 1
$5$	\( T^{8} \) T^8
$7$	\( (T^{4} + 3 T^{2} + 1)^{2} \) (T^4 + 3*T^2 + 1)^2
$11$	\( T^{8} \) T^8
$13$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$17$	\( T^{8} \) T^8
$19$	\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 2T^3 + 4T^2 - 3*T + 1)^2
$23$	\( T^{8} \) T^8
$29$	\( T^{8} \) T^8
$31$	\( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 - 3T^3 + 4T^2 - 2*T + 1)^2
$37$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$41$	\( T^{8} \) T^8
$43$	\( (T^{4} + 3 T^{2} + 1)^{2} \) (T^4 + 3*T^2 + 1)^2
$47$	\( T^{8} \) T^8
$53$	\( T^{8} \) T^8
$59$	\( T^{8} \) T^8
$61$	\( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 + 2T^3 + 4T^2 + 3*T + 1)^2
$67$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
$71$	\( T^{8} \) T^8
$73$	\( T^{8} - 4 T^{6} + \cdots + 1 \) T^8 - 4T^6 + 6T^4 + T^2 + 1
$79$	\( (T^{4} + 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) (T^4 + 3T^3 + 4T^2 + 2*T + 1)^2
$83$	\( T^{8} \) T^8
$89$	\( T^{8} \) T^8
$97$	\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \) T^8 + T^6 + 6T^4 - 4T^2 + 1
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\(n\)	\(626\)	\(1252\)
\(\chi(n)\)	\(-1\)	\(\zeta_{20}^{2}\)