LMFDB - Newform orbit 1215.1.h.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1215,1,Mod(404,1215)] 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("1215.404"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1215, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 1215 = 3^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1215.h (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,-3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$0.606363990349$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{18})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.242137805625.3

$q$-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_{18}^{2} + \zeta_{18}) q^{2} + (\zeta_{18}^{4} + \cdots + \zeta_{18}^{2}) q^{4} + \zeta_{18}^{3} q^{5} + ( - \zeta_{18}^{6} + \cdots + \zeta_{18}^{3}) q^{8} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{10}+ \cdots + (\zeta_{18}^{5} - \zeta_{18}^{4}) q^{98}+O(q^{100}) $ q + (-z^2 + z) * q^2 + (z^4 - z^3 + z^2) * q^4 + z^3 * q^5 + (-z^6 + z^5 - z^4 + z^3) * q^8 + (-z^5 + z^4) * q^10 + (z^8 - z^7 + z^6 - z^5 + z^4) * q^16 + (z^7 - z^2) * q^17 + (-z^7 + z^2) * q^19 + (z^7 - z^6 + z^5) * q^20 + (-z^4 - z^2) * q^23 + z^6 * q^25 + (z^8 - z^7) * q^31 + (-z^8 + z^7 - z^6 + z^5 + z - 1) * q^32 + (z^8 + z^4 - z^3 + 1) * q^34 + (-z^8 - z^4 + z^3 - 1) * q^38 + (z^8 - z^7 + z^6 + 1) * q^40 + (z^6 - z^5 + z^4 - z^3) * q^46 + z^6 * q^47 - z^3 * q^49 + (-z^8 + z^7) * q^50 + (-z^8 + z) * q^53 + (z^2 - z) * q^61 + (-z^8 + z - 2) * q^62 + (z^8 - z^7 + z^6 - z^3 + z^2 - z + 1) * q^64 + (-z^6 + z^5 - z^4 - z^2 + z - 1) * q^68 + (z^6 - z^5 + z^4 + z^2 - z + 1) * q^76 + (-z^7 - z^5) * q^79 + (-z^8 + z^7 - z^2 + z - 1) * q^80 + (-z^8 - z^4) * q^83 + (-z^5 - z) * q^85 + (-z^8 + z^7 - 2z^6 + z^5 - z^4) q^92 + (-z^8 + z^7) * q^94 + (z^5 + z) * q^95 + (z^5 - z^4) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{4} + 3 q^{5} + 6 q^{8} - 3 q^{16} + 3 q^{20} - 3 q^{25} - 3 q^{32} + 3 q^{34} - 3 q^{38} + 3 q^{40} - 6 q^{46} - 3 q^{47} - 3 q^{49} - 12 q^{62} - 3 q^{68} + 3 q^{76} - 6 q^{80} + 6 q^{92}+O(q^{100}) $ 6 * q - 3 * q^4 + 3 * q^5 + 6 * q^8 - 3 * q^16 + 3 * q^20 - 3 * q^25 - 3 * q^32 + 3 * q^34 - 3 * q^38 + 3 * q^40 - 6 * q^46 - 3 * q^47 - 3 * q^49 - 12 * q^62 - 3 * q^68 + 3 * q^76 - 6 * q^80 + 6 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1215\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$\zeta_{18}^{3}$

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} $

404.1

−0.766044	−	0.642788i
0.939693	−	0.342020i
−0.173648	+	0.984808i
−0.766044	+	0.642788i
0.939693	+	0.342020i
−0.173648	−	0.984808i

−0.939693

−

1.62760i

−1.26604

2.19285i

0.500000

−

0.866025i

2.87939

−1.87939

404.2

0.173648

0.300767i

0.439693

−

0.761570i

0.500000

−

0.866025i

0.652704

0.347296

404.3

0.766044

1.32683i

−0.673648

1.16679i

0.500000

−

0.866025i

−0.532089

1.53209

809.1

−0.939693

1.62760i

−1.26604

−

2.19285i

0.500000

0.866025i

2.87939

−1.87939

809.2

0.173648

−

0.300767i

0.439693

0.761570i

0.500000

0.866025i

0.652704

0.347296

809.3

0.766044

−

1.32683i

−0.673648

−

1.16679i

0.500000

0.866025i

−0.532089

1.53209

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
15.d	odd	2	1	CM by $\Q(\sqrt{-15}) $
9.c	even	3	1	inner
45.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1215.1.h.b		6
3.b	odd	2	1		1215.1.h.a		6
5.b	even	2	1		1215.1.h.a		6
9.c	even	3	1		1215.1.d.a	✓	3
9.c	even	3	1	inner	1215.1.h.b		6
9.d	odd	6	1		1215.1.d.b	yes	3
9.d	odd	6	1		1215.1.h.a		6
15.d	odd	2	1	CM	1215.1.h.b		6
27.e	even	9	2		3645.1.n.a		6
27.e	even	9	2		3645.1.n.f		6
27.e	even	9	2		3645.1.n.g		6
27.f	odd	18	2		3645.1.n.b		6
27.f	odd	18	2		3645.1.n.c		6
27.f	odd	18	2		3645.1.n.h		6
45.h	odd	6	1		1215.1.d.a	✓	3
45.h	odd	6	1	inner	1215.1.h.b		6
45.j	even	6	1		1215.1.d.b	yes	3
45.j	even	6	1		1215.1.h.a		6
135.n	odd	18	2		3645.1.n.a		6
135.n	odd	18	2		3645.1.n.f		6
135.n	odd	18	2		3645.1.n.g		6
135.p	even	18	2		3645.1.n.b		6
135.p	even	18	2		3645.1.n.c		6
135.p	even	18	2		3645.1.n.h		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1215.1.d.a	✓	3	9.c	even	3	1
1215.1.d.a	✓	3	45.h	odd	6	1
1215.1.d.b	yes	3	9.d	odd	6	1
1215.1.d.b	yes	3	45.j	even	6	1
1215.1.h.a		6	3.b	odd	2	1
1215.1.h.a		6	5.b	even	2	1
1215.1.h.a		6	9.d	odd	6	1
1215.1.h.a		6	45.j	even	6	1
1215.1.h.b		6	1.a	even	1	1	trivial
1215.1.h.b		6	9.c	even	3	1	inner
1215.1.h.b		6	15.d	odd	2	1	CM
1215.1.h.b		6	45.h	odd	6	1	inner
3645.1.n.a		6	27.e	even	9	2
3645.1.n.a		6	135.n	odd	18	2
3645.1.n.b		6	27.f	odd	18	2
3645.1.n.b		6	135.p	even	18	2
3645.1.n.c		6	27.f	odd	18	2
3645.1.n.c		6	135.p	even	18	2
3645.1.n.f		6	27.e	even	9	2
3645.1.n.f		6	135.n	odd	18	2
3645.1.n.g		6	27.e	even	9	2
3645.1.n.g		6	135.n	odd	18	2
3645.1.n.h		6	27.f	odd	18	2
3645.1.n.h		6	135.p	even	18	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{6} + 3T_{2}^{4} - 2T_{2}^{3} + 9T_{2}^{2} - 3T_{2} + 1 $ T2^6 + 3*T2^4 - 2*T2^3 + 9*T2^2 - 3*T2 + 1 acting on $S_{1}^{\mathrm{new}}(1215, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$3$	$ T^{6} $ T^6
$5$	$ (T^{2} - T + 1)^{3} $ (T^2 - T + 1)^3
$7$	$ T^{6} $ T^6
$11$	$ T^{6} $ T^6
$13$	$ T^{6} $ T^6
$17$	$ (T^{3} - 3 T - 1)^{2} $ (T^3 - 3*T - 1)^2
$19$	$ (T^{3} - 3 T + 1)^{2} $ (T^3 - 3*T + 1)^2
$23$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$29$	$ T^{6} $ T^6
$31$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$37$	$ T^{6} $ T^6
$41$	$ T^{6} $ T^6
$43$	$ T^{6} $ T^6
$47$	$ (T^{2} + T + 1)^{3} $ (T^2 + T + 1)^3
$53$	$ (T^{3} - 3 T - 1)^{2} $ (T^3 - 3*T - 1)^2
$59$	$ T^{6} $ T^6
$61$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$67$	$ T^{6} $ T^6
$71$	$ T^{6} $ T^6
$73$	$ T^{6} $ T^6
$79$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$83$	$ T^{6} + 3 T^{4} + \cdots + 1 $ T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$89$	$ T^{6} $ T^6
$97$	$ T^{6} $ T^6
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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:	\( N \)	\(=\)	\( 1215 = 3^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1215.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{18}^{2} + \zeta_{18}) q^{2} + (\zeta_{18}^{4} + \cdots + \zeta_{18}^{2}) q^{4} + \zeta_{18}^{3} q^{5} + ( - \zeta_{18}^{6} + \cdots + \zeta_{18}^{3}) q^{8} + ( - \zeta_{18}^{5} + \zeta_{18}^{4}) q^{10}+ \cdots + (\zeta_{18}^{5} - \zeta_{18}^{4}) q^{98}+O(q^{100}) \) q + (-z^2 + z) * q^2 + (z^4 - z^3 + z^2) * q^4 + z^3 * q^5 + (-z^6 + z^5 - z^4 + z^3) * q^8 + (-z^5 + z^4) * q^10 + (z^8 - z^7 + z^6 - z^5 + z^4) * q^16 + (z^7 - z^2) * q^17 + (-z^7 + z^2) * q^19 + (z^7 - z^6 + z^5) * q^20 + (-z^4 - z^2) * q^23 + z^6 * q^25 + (z^8 - z^7) * q^31 + (-z^8 + z^7 - z^6 + z^5 + z - 1) * q^32 + (z^8 + z^4 - z^3 + 1) * q^34 + (-z^8 - z^4 + z^3 - 1) * q^38 + (z^8 - z^7 + z^6 + 1) * q^40 + (z^6 - z^5 + z^4 - z^3) * q^46 + z^6 * q^47 - z^3 * q^49 + (-z^8 + z^7) * q^50 + (-z^8 + z) * q^53 + (z^2 - z) * q^61 + (-z^8 + z - 2) * q^62 + (z^8 - z^7 + z^6 - z^3 + z^2 - z + 1) * q^64 + (-z^6 + z^5 - z^4 - z^2 + z - 1) * q^68 + (z^6 - z^5 + z^4 + z^2 - z + 1) * q^76 + (-z^7 - z^5) * q^79 + (-z^8 + z^7 - z^2 + z - 1) * q^80 + (-z^8 - z^4) * q^83 + (-z^5 - z) * q^85 + (-z^8 + z^7 - 2z^6 + z^5 - z^4) q^92 + (-z^8 + z^7) * q^94 + (z^5 + z) * q^95 + (z^5 - z^4) * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{4} + 3 q^{5} + 6 q^{8} - 3 q^{16} + 3 q^{20} - 3 q^{25} - 3 q^{32} + 3 q^{34} - 3 q^{38} + 3 q^{40} - 6 q^{46} - 3 q^{47} - 3 q^{49} - 12 q^{62} - 3 q^{68} + 3 q^{76} - 6 q^{80} + 6 q^{92}+O(q^{100}) \) 6 * q - 3 * q^4 + 3 * q^5 + 6 * q^8 - 3 * q^16 + 3 * q^20 - 3 * q^25 - 3 * q^32 + 3 * q^34 - 3 * q^38 + 3 * q^40 - 6 * q^46 - 3 * q^47 - 3 * q^49 - 12 * q^62 - 3 * q^68 + 3 * q^76 - 6 * q^80 + 6 * q^92

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

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	1215.1.h.b

Level	$1215$
Weight	$1$
Character orbit	1215.h
Analytic conductor	$0.606$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-15
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.606363990349\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{6})\)
Coefficient field:	\(\Q(\zeta_{18})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{6} - x^{3} + 1 \) x^6 - x^3 + 1
Coefficient ring:	\(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index:	\( 3 \)
Twist minimal:	yes
Projective image:	\(D_{9}\)
Projective field:	Galois closure of 9.1.242137805625.3

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 404.3
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$3$	\( T^{6} \) T^6
$5$	\( (T^{2} - T + 1)^{3} \) (T^2 - T + 1)^3
$7$	\( T^{6} \) T^6
$11$	\( T^{6} \) T^6
$13$	\( T^{6} \) T^6
$17$	\( (T^{3} - 3 T - 1)^{2} \) (T^3 - 3*T - 1)^2
$19$	\( (T^{3} - 3 T + 1)^{2} \) (T^3 - 3*T + 1)^2
$23$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$29$	\( T^{6} \) T^6
$31$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$37$	\( T^{6} \) T^6
$41$	\( T^{6} \) T^6
$43$	\( T^{6} \) T^6
$47$	\( (T^{2} + T + 1)^{3} \) (T^2 + T + 1)^3
$53$	\( (T^{3} - 3 T - 1)^{2} \) (T^3 - 3*T - 1)^2
$59$	\( T^{6} \) T^6
$61$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$67$	\( T^{6} \) T^6
$71$	\( T^{6} \) T^6
$73$	\( T^{6} \) T^6
$79$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 + 2T^3 + 9T^2 + 3T + 1
$83$	\( T^{6} + 3 T^{4} + \cdots + 1 \) T^6 + 3T^4 - 2T^3 + 9T^2 - 3T + 1
$89$	\( T^{6} \) T^6
$97$	\( T^{6} \) T^6
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\(n\)	\(487\)	\(731\)
\(\chi(n)\)	\(-1\)	\(\zeta_{18}^{3}\)