LMFDB - Newform orbit 736.1.p.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [736,1,Mod(45,736)] 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("736.45"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 7, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	736.p (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$0.367311849298$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{8})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{24}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{24} + \cdots)$

$q$-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_{24}^{5} q^{2} + ( - \zeta_{24}^{8} + \zeta_{24}) q^{3} + \zeta_{24}^{10} q^{4} + ( - \zeta_{24}^{6} - \zeta_{24}) q^{6} + \zeta_{24}^{3} q^{8} + ( - \zeta_{24}^{9} + \cdots + \zeta_{24}^{2}) q^{9} + \cdots - \zeta_{24}^{11} q^{98} +O(q^{100}) $ q - z^5 * q^2 + (-z^8 + z) * q^3 + z^10 * q^4 + (-z^6 - z) * q^6 + z^3 * q^8 + (-z^9 - z^4 + z^2) * q^9 + (z^11 + z^6) * q^12 + (z^11 - z^10) * q^13 - z^8 * q^16 + (z^9 - z^7 - z^2) * q^18 + z^9 * q^23 + (-z^11 + z^4) * q^24 - z^3 * q^25 + (z^4 - z^3) * q^26 + (-z^10 - z^5 + z^3 - 1) * q^27 + (z^7 - z^2) * q^29 + (-z^7 + z^5) * q^31 - z * q^32 + (z^7 + z^2 - 1) * q^36 + (-z^11 + z^7 - z^6 - 1) * q^39 - z^9 * q^41 + z^2 * q^46 + (z^8 + z^4) * q^47 + (-z^9 - z^4) * q^48 + z^6 * q^49 + z^8 * q^50 + (-z^9 + z^8) * q^52 + (z^10 - z^8 + z^5 - z^3) * q^54 + (z^7 + 1) * q^58 + (z^9 + z^6) * q^59 + (-z^10 - 1) * q^62 + z^6 * q^64 + (z^10 + z^5) * q^69 + (-z^4 - z^2) * q^71 + (-z^7 + z^5 + 1) * q^72 + (-z^11 - z^7) * q^73 + (z^11 - z^4) * q^75 + (z^11 + z^5 - z^4 + 1) * q^78 + (-z^11 + z^8 - z^6 + z^4 - z) * q^81 - z^2 * q^82 + (z^10 + z^8) * q^87 - z^7 * q^92 + (-z^8 + z^6 - z^3 + z) * q^93 + (-z^9 + z) * q^94 + (z^9 - z^2) * q^96 - z^11 * q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 4 q^{9} + 4 q^{16} + 4 q^{24} + 4 q^{26} - 8 q^{27} - 8 q^{36} - 8 q^{39} - 4 q^{48} - 4 q^{50} - 4 q^{52} + 4 q^{54} + 8 q^{58} - 8 q^{62} - 4 q^{71} + 8 q^{72} - 4 q^{75} + 4 q^{78} - 4 q^{87}+ \cdots + 4 q^{93}+O(q^{100}) $ 8 * q + 4 * q^3 - 4 * q^9 + 4 * q^16 + 4 * q^24 + 4 * q^26 - 8 * q^27 - 8 * q^36 - 8 * q^39 - 4 * q^48 - 4 * q^50 - 4 * q^52 + 4 * q^54 + 8 * q^58 - 8 * q^62 - 4 * q^71 + 8 * q^72 - 4 * q^75 + 4 * q^78 - 4 * q^87 + 4 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/736\mathbb{Z}\right)^\times$.

$n$	$97$	$415$	$645$
$\chi(n)$	$-1$	$1$	$-\zeta_{24}^{3}$

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} $

45.1

0.258819	−	0.965926i
−0.965926	+	0.258819i
0.258819	+	0.965926i
−0.965926	−	0.258819i
0.965926	−	0.258819i
−0.258819	+	0.965926i
0.965926	+	0.258819i
−0.258819	−	0.965926i

−0.965926

0.258819i

0.758819

−

1.83195i

0.866025

−

0.500000i

−0.258819

1.96593i

−0.707107

0.707107i

−2.07313

−

2.07313i

45.2

0.258819

−

0.965926i

−0.465926

1.12484i

−0.866025

−

0.500000i

0.965926

0.741181i

−0.707107

0.707107i

−0.341081

−

0.341081i

229.1

−0.965926

−

0.258819i

0.758819

1.83195i

0.866025

0.500000i

−0.258819

−

1.96593i

−0.707107

−

0.707107i

−2.07313

2.07313i

229.2

0.258819

0.965926i

−0.465926

−

1.12484i

−0.866025

0.500000i

0.965926

−

0.741181i

−0.707107

−

0.707107i

−0.341081

0.341081i

413.1

−0.258819

0.965926i

1.46593

0.607206i

−0.866025

−

0.500000i

−0.965926

1.25882i

0.707107

−

0.707107i

1.07313

1.07313i

413.2

0.965926

−

0.258819i

0.241181

0.0999004i

0.866025

−

0.500000i

0.258819

0.0340742i

0.707107

−

0.707107i

−0.658919

−

0.658919i

597.1

−0.258819

−

0.965926i

1.46593

−

0.607206i

−0.866025

0.500000i

−0.965926

−

1.25882i

0.707107

0.707107i

1.07313

−

1.07313i

597.2

0.965926

0.258819i

0.241181

−

0.0999004i

0.866025

0.500000i

0.258819

−

0.0340742i

0.707107

0.707107i

−0.658919

0.658919i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.b	odd	2	1	CM by $\Q(\sqrt{-23}) $
32.g	even	8	1	inner
736.p	odd	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	736.1.p.b	✓	8
4.b	odd	2	1		2944.1.p.b		8
23.b	odd	2	1	CM	736.1.p.b	✓	8
32.g	even	8	1	inner	736.1.p.b	✓	8
32.h	odd	8	1		2944.1.p.b		8
92.b	even	2	1		2944.1.p.b		8
736.n	even	8	1		2944.1.p.b		8
736.p	odd	8	1	inner	736.1.p.b	✓	8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
736.1.p.b	✓	8	1.a	even	1	1	trivial
736.1.p.b	✓	8	23.b	odd	2	1	CM
736.1.p.b	✓	8	32.g	even	8	1	inner
736.1.p.b	✓	8	736.p	odd	8	1	inner
2944.1.p.b		8	4.b	odd	2	1
2944.1.p.b		8	32.h	odd	8	1
2944.1.p.b		8	92.b	even	2	1
2944.1.p.b		8	736.n	even	8	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{8} - 4T_{3}^{7} + 10T_{3}^{6} - 16T_{3}^{5} + 18T_{3}^{4} - 20T_{3}^{3} + 22T_{3}^{2} - 8T_{3} + 1 $ T3^8 - 4*T3^7 + 10*T3^6 - 16*T3^5 + 18*T3^4 - 20*T3^3 + 22*T3^2 - 8*T3 + 1 acting on $S_{1}^{\mathrm{new}}(736, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{8} - T^{4} + 1 $ T^8 - T^4 + 1
$3$	$ T^{8} - 4 T^{7} + \cdots + 1 $ T^8 - 4T^7 + 10T^6 - 16T^5 + 18T^4 - 20T^3 + 22T^2 - 8*T + 1
$5$	$ T^{8} $ T^8
$7$	$ T^{8} $ T^8
$11$	$ T^{8} $ T^8
$13$	$ T^{8} - 2 T^{6} + \cdots + 1 $ T^8 - 2T^6 - 4T^5 + 2T^4 + 16T^3 + 18T^2 + 4T + 1
$17$	$ T^{8} $ T^8
$19$	$ T^{8} $ T^8
$23$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$29$	$ T^{8} - 2 T^{6} + \cdots + 1 $ T^8 - 2T^6 - 4T^5 + 2T^4 + 16T^3 + 18T^2 + 4T + 1
$31$	$ (T^{4} - 4 T^{2} + 1)^{2} $ (T^4 - 4*T^2 + 1)^2
$37$	$ T^{8} $ T^8
$41$	$ (T^{4} + 1)^{2} $ (T^4 + 1)^2
$43$	$ T^{8} $ T^8
$47$	$ (T^{2} + 3)^{4} $ (T^2 + 3)^4
$53$	$ T^{8} $ T^8
$59$	$ (T^{4} + 2 T^{2} - 4 T + 2)^{2} $ (T^4 + 2T^2 - 4T + 2)^2
$61$	$ T^{8} $ T^8
$67$	$ T^{8} $ T^8
$71$	$ (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} $ (T^4 + 2T^3 + 2T^2 - 2*T + 1)^2
$73$	$ (T^{4} + 9)^{2} $ (T^4 + 9)^2
$79$	$ T^{8} $ T^8
$83$	$ T^{8} $ T^8
$89$	$ T^{8} $ T^8
$97$	$ T^{8} $ T^8
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	736.p (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{24}^{5} q^{2} + ( - \zeta_{24}^{8} + \zeta_{24}) q^{3} + \zeta_{24}^{10} q^{4} + ( - \zeta_{24}^{6} - \zeta_{24}) q^{6} + \zeta_{24}^{3} q^{8} + ( - \zeta_{24}^{9} + \cdots + \zeta_{24}^{2}) q^{9} + \cdots - \zeta_{24}^{11} q^{98} +O(q^{100}) \) q - z^5 * q^2 + (-z^8 + z) * q^3 + z^10 * q^4 + (-z^6 - z) * q^6 + z^3 * q^8 + (-z^9 - z^4 + z^2) * q^9 + (z^11 + z^6) * q^12 + (z^11 - z^10) * q^13 - z^8 * q^16 + (z^9 - z^7 - z^2) * q^18 + z^9 * q^23 + (-z^11 + z^4) * q^24 - z^3 * q^25 + (z^4 - z^3) * q^26 + (-z^10 - z^5 + z^3 - 1) * q^27 + (z^7 - z^2) * q^29 + (-z^7 + z^5) * q^31 - z * q^32 + (z^7 + z^2 - 1) * q^36 + (-z^11 + z^7 - z^6 - 1) * q^39 - z^9 * q^41 + z^2 * q^46 + (z^8 + z^4) * q^47 + (-z^9 - z^4) * q^48 + z^6 * q^49 + z^8 * q^50 + (-z^9 + z^8) * q^52 + (z^10 - z^8 + z^5 - z^3) * q^54 + (z^7 + 1) * q^58 + (z^9 + z^6) * q^59 + (-z^10 - 1) * q^62 + z^6 * q^64 + (z^10 + z^5) * q^69 + (-z^4 - z^2) * q^71 + (-z^7 + z^5 + 1) * q^72 + (-z^11 - z^7) * q^73 + (z^11 - z^4) * q^75 + (z^11 + z^5 - z^4 + 1) * q^78 + (-z^11 + z^8 - z^6 + z^4 - z) * q^81 - z^2 * q^82 + (z^10 + z^8) * q^87 - z^7 * q^92 + (-z^8 + z^6 - z^3 + z) * q^93 + (-z^9 + z) * q^94 + (z^9 - z^2) * q^96 - z^11 * q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 4 q^{9} + 4 q^{16} + 4 q^{24} + 4 q^{26} - 8 q^{27} - 8 q^{36} - 8 q^{39} - 4 q^{48} - 4 q^{50} - 4 q^{52} + 4 q^{54} + 8 q^{58} - 8 q^{62} - 4 q^{71} + 8 q^{72} - 4 q^{75} + 4 q^{78} - 4 q^{87}+ \cdots + 4 q^{93}+O(q^{100}) \) 8 * q + 4 * q^3 - 4 * q^9 + 4 * q^16 + 4 * q^24 + 4 * q^26 - 8 * q^27 - 8 * q^36 - 8 * q^39 - 4 * q^48 - 4 * q^50 - 4 * q^52 + 4 * q^54 + 8 * q^58 - 8 * q^62 - 4 * q^71 + 8 * q^72 - 4 * q^75 + 4 * q^78 - 4 * q^87 + 4 * q^93

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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	736.1.p.b

Level	$736$
Weight	$1$
Character orbit	736.p
Analytic conductor	$0.367$
Analytic rank	$0$
Dimension	$8$
Projective image	$D_{24}$
CM discriminant	-23
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(0.367311849298\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{8})\)
Coefficient field:	\(\Q(\zeta_{24})\)
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{8} - x^{4} + 1 \) x^8 - x^4 + 1
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{24}\)
Projective field:	Galois closure of \(\mathbb{Q}[x]/(x^{24} + \cdots)\)

\(n\)	\(97\)	\(415\)	\(645\)
\(\chi(n)\)	\(-1\)	\(1\)	\(-\zeta_{24}^{3}\)

\(n\):		e.g. 2-40 or 80-90
Embeddings:		e.g. 1-3 or 45.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{8} - T^{4} + 1 \) T^8 - T^4 + 1
$3$	\( T^{8} - 4 T^{7} + \cdots + 1 \) T^8 - 4T^7 + 10T^6 - 16T^5 + 18T^4 - 20T^3 + 22T^2 - 8*T + 1
$5$	\( T^{8} \) T^8
$7$	\( T^{8} \) T^8
$11$	\( T^{8} \) T^8
$13$	\( T^{8} - 2 T^{6} + \cdots + 1 \) T^8 - 2T^6 - 4T^5 + 2T^4 + 16T^3 + 18T^2 + 4T + 1
$17$	\( T^{8} \) T^8
$19$	\( T^{8} \) T^8
$23$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$29$	\( T^{8} - 2 T^{6} + \cdots + 1 \) T^8 - 2T^6 - 4T^5 + 2T^4 + 16T^3 + 18T^2 + 4T + 1
$31$	\( (T^{4} - 4 T^{2} + 1)^{2} \) (T^4 - 4*T^2 + 1)^2
$37$	\( T^{8} \) T^8
$41$	\( (T^{4} + 1)^{2} \) (T^4 + 1)^2
$43$	\( T^{8} \) T^8
$47$	\( (T^{2} + 3)^{4} \) (T^2 + 3)^4
$53$	\( T^{8} \) T^8
$59$	\( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) (T^4 + 2T^2 - 4T + 2)^2
$61$	\( T^{8} \) T^8
$67$	\( T^{8} \) T^8
$71$	\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) (T^4 + 2T^3 + 2T^2 - 2*T + 1)^2
$73$	\( (T^{4} + 9)^{2} \) (T^4 + 9)^2
$79$	\( T^{8} \) T^8
$83$	\( T^{8} \) T^8
$89$	\( T^{8} \) T^8
$97$	\( T^{8} \) T^8
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