LMFDB - Newform orbit 1475.1.d.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1475,1,Mod(1474,1475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1475.1474"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1475 = 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1475.d (of order $2$, degree $1$, 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:	$0.736120893634$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 59)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.59.1
Artin image:	$C_4\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

$q$-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 - i q^{3} - q^{4} + i q^{7} + i q^{12} + q^{16} - 2 i q^{17} + q^{19} + q^{21} - i q^{27} - i q^{28} + q^{29} - q^{41} - i q^{48} - 2 q^{51} - i q^{53} - i q^{57} - q^{59} - q^{64} + 2 i q^{68} + \cdots - i q^{87} +O(q^{100}) $ q - z * q^3 - q^4 + z * q^7 + z * q^12 + q^16 - 2z q^17 + q^19 + q^21 - z * q^27 - z * q^28 + q^29 - q^41 - z * q^48 - 2 * q^51 - z * q^53 - z * q^57 - q^59 - q^64 + 2z q^68 + 2 * q^71 - q^76 + q^79 - q^81 - q^84 - z * q^87
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 2 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{29} - 2 q^{41} - 4 q^{51} - 2 q^{59} - 2 q^{64} + 4 q^{71} - 2 q^{76} + 2 q^{79} - 2 q^{81} - 2 q^{84}+O(q^{100}) $ 2 * q - 2 * q^4 + 2 * q^16 + 2 * q^19 + 2 * q^21 + 2 * q^29 - 2 * q^41 - 4 * q^51 - 2 * q^59 - 2 * q^64 + 4 * q^71 - 2 * q^76 + 2 * q^79 - 2 * q^81 - 2 * q^84

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1475\mathbb{Z}\right)^\times$.

$n$	$651$	$827$
$\chi(n)$	$-1$	$-1$

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} $

1474.1

		1.00000i
	−	1.00000i

−

1.00000i

−1.00000

1.00000i

1474.2

1.00000i

−1.00000

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
59.b	odd	2	1	CM by $\Q(\sqrt{-59}) $
5.b	even	2	1	inner
295.d	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1475.1.d.a		2
5.b	even	2	1	inner	1475.1.d.a		2
5.c	odd	4	1		59.1.b.a	✓	1
5.c	odd	4	1		1475.1.c.b		1
15.e	even	4	1		531.1.c.a		1
20.e	even	4	1		944.1.h.a		1
35.f	even	4	1		2891.1.c.e		1
35.k	even	12	2		2891.1.g.b		2
35.l	odd	12	2		2891.1.g.d		2
40.i	odd	4	1		3776.1.h.b		1
40.k	even	4	1		3776.1.h.a		1
59.b	odd	2	1	CM	1475.1.d.a		2
295.d	odd	2	1	inner	1475.1.d.a		2
295.e	even	4	1		59.1.b.a	✓	1
295.e	even	4	1		1475.1.c.b		1
295.k	odd	116	28		3481.1.d.a		28
295.l	even	116	28		3481.1.d.a		28
885.k	odd	4	1		531.1.c.a		1
1180.l	odd	4	1		944.1.h.a		1
2065.l	odd	4	1		2891.1.c.e		1
2065.v	even	12	2		2891.1.g.d		2
2065.x	odd	12	2		2891.1.g.b		2
2360.q	odd	4	1		3776.1.h.a		1
2360.w	even	4	1		3776.1.h.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
59.1.b.a	✓	1	5.c	odd	4	1
59.1.b.a	✓	1	295.e	even	4	1
531.1.c.a		1	15.e	even	4	1
531.1.c.a		1	885.k	odd	4	1
944.1.h.a		1	20.e	even	4	1
944.1.h.a		1	1180.l	odd	4	1
1475.1.c.b		1	5.c	odd	4	1
1475.1.c.b		1	295.e	even	4	1
1475.1.d.a		2	1.a	even	1	1	trivial
1475.1.d.a		2	5.b	even	2	1	inner
1475.1.d.a		2	59.b	odd	2	1	CM
1475.1.d.a		2	295.d	odd	2	1	inner
2891.1.c.e		1	35.f	even	4	1
2891.1.c.e		1	2065.l	odd	4	1
2891.1.g.b		2	35.k	even	12	2
2891.1.g.b		2	2065.x	odd	12	2
2891.1.g.d		2	35.l	odd	12	2
2891.1.g.d		2	2065.v	even	12	2
3481.1.d.a		28	295.k	odd	116	28
3481.1.d.a		28	295.l	even	116	28
3776.1.h.a		1	40.k	even	4	1
3776.1.h.a		1	2360.q	odd	4	1
3776.1.h.b		1	40.i	odd	4	1
3776.1.h.b		1	2360.w	even	4	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1475, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} $ T^2
$3$	$ T^{2} + 1 $ T^2 + 1
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 1 $ T^2 + 1
$11$	$ T^{2} $ T^2
$13$	$ T^{2} $ T^2
$17$	$ T^{2} + 4 $ T^2 + 4
$19$	$ (T - 1)^{2} $ (T - 1)^2
$23$	$ T^{2} $ T^2
$29$	$ (T - 1)^{2} $ (T - 1)^2
$31$	$ T^{2} $ T^2
$37$	$ T^{2} $ T^2
$41$	$ (T + 1)^{2} $ (T + 1)^2
$43$	$ T^{2} $ T^2
$47$	$ T^{2} $ T^2
$53$	$ T^{2} + 1 $ T^2 + 1
$59$	$ (T + 1)^{2} $ (T + 1)^2
$61$	$ T^{2} $ T^2
$67$	$ T^{2} $ T^2
$71$	$ (T - 2)^{2} $ (T - 2)^2
$73$	$ T^{2} $ T^2
$79$	$ (T - 1)^{2} $ (T - 1)^2
$83$	$ T^{2} $ T^2
$89$	$ T^{2} $ T^2
$97$	$ T^{2} $ T^2
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Overview	Random
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Classical	Maass
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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:	\( N \)	\(=\)	\( 1475 = 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1475.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q - i q^{3} - q^{4} + i q^{7} + i q^{12} + q^{16} - 2 i q^{17} + q^{19} + q^{21} - i q^{27} - i q^{28} + q^{29} - q^{41} - i q^{48} - 2 q^{51} - i q^{53} - i q^{57} - q^{59} - q^{64} + 2 i q^{68} + \cdots - i q^{87} +O(q^{100}) \) q - z * q^3 - q^4 + z * q^7 + z * q^12 + q^16 - 2z q^17 + q^19 + q^21 - z * q^27 - z * q^28 + q^29 - q^41 - z * q^48 - 2 * q^51 - z * q^53 - z * q^57 - q^59 - q^64 + 2z q^68 + 2 * q^71 - q^76 + q^79 - q^81 - q^84 - z * q^87
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 2 q^{16} + 2 q^{19} + 2 q^{21} + 2 q^{29} - 2 q^{41} - 4 q^{51} - 2 q^{59} - 2 q^{64} + 4 q^{71} - 2 q^{76} + 2 q^{79} - 2 q^{81} - 2 q^{84}+O(q^{100}) \) 2 * q - 2 * q^4 + 2 * q^16 + 2 * q^19 + 2 * q^21 + 2 * q^29 - 2 * q^41 - 4 * q^51 - 2 * q^59 - 2 * q^64 + 4 * q^71 - 2 * q^76 + 2 * q^79 - 2 * q^81 - 2 * q^84

Introduction

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Properties

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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	1475.1.d.a

Level	$1475$
Weight	$1$
Character orbit	1475.d
Analytic conductor	$0.736$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-59
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.736120893634\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(i)\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{2} + 1 \) x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 59)
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.59.1
Artin image:	$C_4\times S_3$
Artin field:	Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$p$	$F_p(T)$
$2$	\( T^{2} \) T^2
$3$	\( T^{2} + 1 \) T^2 + 1
$5$	\( T^{2} \) T^2
$7$	\( T^{2} + 1 \) T^2 + 1
$11$	\( T^{2} \) T^2
$13$	\( T^{2} \) T^2
$17$	\( T^{2} + 4 \) T^2 + 4
$19$	\( (T - 1)^{2} \) (T - 1)^2
$23$	\( T^{2} \) T^2
$29$	\( (T - 1)^{2} \) (T - 1)^2
$31$	\( T^{2} \) T^2
$37$	\( T^{2} \) T^2
$41$	\( (T + 1)^{2} \) (T + 1)^2
$43$	\( T^{2} \) T^2
$47$	\( T^{2} \) T^2
$53$	\( T^{2} + 1 \) T^2 + 1
$59$	\( (T + 1)^{2} \) (T + 1)^2
$61$	\( T^{2} \) T^2
$67$	\( T^{2} \) T^2
$71$	\( (T - 2)^{2} \) (T - 2)^2
$73$	\( T^{2} \) T^2
$79$	\( (T - 1)^{2} \) (T - 1)^2
$83$	\( T^{2} \) T^2
$89$	\( T^{2} \) T^2
$97$	\( T^{2} \) T^2
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\(n\)	\(651\)	\(827\)
\(\chi(n)\)	\(-1\)	\(-1\)

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