LMFDB - Newform orbit 1296.1.x.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,1,Mod(53,1296)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 3, 10]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1296.53");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1296.x (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.646788256372$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
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:	no (minimal twist has level 432)
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.55296.2

$q$-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} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} - \zeta_{24}^{10} q^{7} - \zeta_{24}^{3} q^{8} +O(q^{10}) $ q - z * q^2 + z^2 * q^4 + z^5 * q^5 - z^10 * q^7 - z^3 * q^8	\( q - \zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} - \zeta_{24}^{10} q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{6} q^{10} + \zeta_{24} q^{11} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{13} + \zeta_{24}^{11} q^{14} + \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{17} + \zeta_{24}^{7} q^{20} - \zeta_{24}^{2} q^{22} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{26} + q^{28} + \zeta_{24}^{8} q^{31} - \zeta_{24}^{5} q^{32} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{34} + \zeta_{24}^{3} q^{35} - \zeta_{24}^{8} q^{40} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{43} + \zeta_{24}^{3} q^{44} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{47} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{52} + \zeta_{24}^{9} q^{53} + \zeta_{24}^{6} q^{55} - \zeta_{24} q^{56} - \zeta_{24}^{9} q^{62} + \zeta_{24}^{6} q^{64} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{65} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{67} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{68} - \zeta_{24}^{4} q^{70} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{71} - \zeta_{24}^{6} q^{73} - \zeta_{24}^{11} q^{77} + \zeta_{24}^{9} q^{80} + \zeta_{24}^{7} q^{83} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{85} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{86} - \zeta_{24}^{4} q^{88} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{89} + ( - \zeta_{24}^{6} - 1) q^{91} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{94} - \zeta_{24}^{4} q^{97} +O(q^{100}) \) q - z * q^2 + z^2 * q^4 + z^5 * q^5 - z^10 * q^7 - z^3 * q^8 - z^6 * q^10 + z * q^11 + (-z^8 - z^2) * q^13 + z^11 * q^14 + z^4 * q^16 + (-z^9 - z^3) * q^17 + z^7 * q^20 - z^2 * q^22 + (z^9 + z^3) * q^26 + q^28 + z^8 * q^31 - z^5 * q^32 + (z^10 + z^4) * q^34 + z^3 * q^35 - z^8 * q^40 + (z^10 + z^4) * q^43 + z^3 * q^44 + (-z^7 + z) * q^47 + (-z^10 - z^4) * q^52 + z^9 * q^53 + z^6 * q^55 - z * q^56 - z^9 * q^62 + z^6 * q^64 + (-z^7 + z) * q^65 + (z^8 + z^2) * q^67 + (-z^11 - z^5) * q^68 - z^4 * q^70 + (z^9 - z^3) * q^71 - z^6 * q^73 - z^11 * q^77 + z^9 * q^80 + z^7 * q^83 + (-z^8 + z^2) * q^85 + (-z^11 - z^5) * q^86 - z^4 * q^88 + (-z^9 + z^3) * q^89 + (-z^6 - 1) * q^91 + (z^8 - z^2) * q^94 - z^4 * q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97}+O(q^{100}) $ 8 * q + 4 * q^13 + 4 * q^16 + 8 * q^28 - 4 * q^31 + 4 * q^34 + 4 * q^40 + 4 * q^43 - 4 * q^52 - 4 * q^67 - 4 * q^70 + 4 * q^85 - 4 * q^88 - 8 * q^91 - 4 * q^94 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$\zeta_{24}^{6}$	$1$	$-\zeta_{24}^{8}$

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} $

53.1

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

−0.965926

−

0.258819i

0.866025

0.500000i

0.258819

0.965926i

0.866025

−

0.500000i

−0.707107

−

0.707107i

−

1.00000i

53.2

0.965926

0.258819i

0.866025

0.500000i

−0.258819

−

0.965926i

0.866025

−

0.500000i

0.707107

0.707107i

−

1.00000i

269.1

−0.965926

0.258819i

0.866025

−

0.500000i

0.258819

−

0.965926i

0.866025

0.500000i

−0.707107

0.707107i

1.00000i

269.2

0.965926

−

0.258819i

0.866025

−

0.500000i

−0.258819

0.965926i

0.866025

0.500000i

0.707107

−

0.707107i

1.00000i

701.1

−0.258819

0.965926i

−0.866025

−

0.500000i

0.965926

−

0.258819i

−0.866025

0.500000i

0.707107

−

0.707107i

1.00000i

701.2

0.258819

−

0.965926i

−0.866025

−

0.500000i

−0.965926

0.258819i

−0.866025

0.500000i

−0.707107

0.707107i

1.00000i

917.1

−0.258819

−

0.965926i

−0.866025

0.500000i

0.965926

0.258819i

−0.866025

−

0.500000i

0.707107

0.707107i

−

1.00000i

917.2

0.258819

0.965926i

−0.866025

0.500000i

−0.965926

−

0.258819i

−0.866025

−

0.500000i

−0.707107

−

0.707107i

−

1.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
9.c	even	3	1	inner
9.d	odd	6	1	inner
16.e	even	4	1	inner
48.i	odd	4	1	inner
144.w	odd	12	1	inner
144.x	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1296.1.x.a		8
3.b	odd	2	1	inner	1296.1.x.a		8
9.c	even	3	1		432.1.j.a	✓	4
9.c	even	3	1	inner	1296.1.x.a		8
9.d	odd	6	1		432.1.j.a	✓	4
9.d	odd	6	1	inner	1296.1.x.a		8
16.e	even	4	1	inner	1296.1.x.a		8
36.f	odd	6	1		1728.1.j.a		4
36.h	even	6	1		1728.1.j.a		4
48.i	odd	4	1	inner	1296.1.x.a		8
72.j	odd	6	1		3456.1.j.b		4
72.l	even	6	1		3456.1.j.a		4
72.n	even	6	1		3456.1.j.b		4
72.p	odd	6	1		3456.1.j.a		4
144.u	even	12	1		1728.1.j.a		4
144.u	even	12	1		3456.1.j.a		4
144.v	odd	12	1		1728.1.j.a		4
144.v	odd	12	1		3456.1.j.a		4
144.w	odd	12	1		432.1.j.a	✓	4
144.w	odd	12	1	inner	1296.1.x.a		8
144.w	odd	12	1		3456.1.j.b		4
144.x	even	12	1		432.1.j.a	✓	4
144.x	even	12	1	inner	1296.1.x.a		8
144.x	even	12	1		3456.1.j.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
432.1.j.a	✓	4	9.c	even	3	1
432.1.j.a	✓	4	9.d	odd	6	1
432.1.j.a	✓	4	144.w	odd	12	1
432.1.j.a	✓	4	144.x	even	12	1
1296.1.x.a		8	1.a	even	1	1	trivial
1296.1.x.a		8	3.b	odd	2	1	inner
1296.1.x.a		8	9.c	even	3	1	inner
1296.1.x.a		8	9.d	odd	6	1	inner
1296.1.x.a		8	16.e	even	4	1	inner
1296.1.x.a		8	48.i	odd	4	1	inner
1296.1.x.a		8	144.w	odd	12	1	inner
1296.1.x.a		8	144.x	even	12	1	inner
1728.1.j.a		4	36.f	odd	6	1
1728.1.j.a		4	36.h	even	6	1
1728.1.j.a		4	144.u	even	12	1
1728.1.j.a		4	144.v	odd	12	1
3456.1.j.a		4	72.l	even	6	1
3456.1.j.a		4	72.p	odd	6	1
3456.1.j.a		4	144.u	even	12	1
3456.1.j.a		4	144.v	odd	12	1
3456.1.j.b		4	72.j	odd	6	1
3456.1.j.b		4	72.n	even	6	1
3456.1.j.b		4	144.w	odd	12	1
3456.1.j.b		4	144.x	even	12	1

Hecke kernels

This newform subspace is the entire newspace $S_{1}^{\mathrm{new}}(1296, [\chi])$.

Hecke characteristic polynomials

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

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1296.x (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\( q - \zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} - \zeta_{24}^{10} q^{7} - \zeta_{24}^{3} q^{8} +O(q^{10}) \) q - z * q^2 + z^2 * q^4 + z^5 * q^5 - z^10 * q^7 - z^3 * q^8	\( q - \zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} - \zeta_{24}^{10} q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{6} q^{10} + \zeta_{24} q^{11} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{13} + \zeta_{24}^{11} q^{14} + \zeta_{24}^{4} q^{16} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{17} + \zeta_{24}^{7} q^{20} - \zeta_{24}^{2} q^{22} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{26} + q^{28} + \zeta_{24}^{8} q^{31} - \zeta_{24}^{5} q^{32} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{34} + \zeta_{24}^{3} q^{35} - \zeta_{24}^{8} q^{40} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{43} + \zeta_{24}^{3} q^{44} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{47} + ( - \zeta_{24}^{10} - \zeta_{24}^{4}) q^{52} + \zeta_{24}^{9} q^{53} + \zeta_{24}^{6} q^{55} - \zeta_{24} q^{56} - \zeta_{24}^{9} q^{62} + \zeta_{24}^{6} q^{64} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{65} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{67} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{68} - \zeta_{24}^{4} q^{70} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{71} - \zeta_{24}^{6} q^{73} - \zeta_{24}^{11} q^{77} + \zeta_{24}^{9} q^{80} + \zeta_{24}^{7} q^{83} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{85} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{86} - \zeta_{24}^{4} q^{88} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{89} + ( - \zeta_{24}^{6} - 1) q^{91} + (\zeta_{24}^{8} - \zeta_{24}^{2}) q^{94} - \zeta_{24}^{4} q^{97} +O(q^{100}) \) q - z * q^2 + z^2 * q^4 + z^5 * q^5 - z^10 * q^7 - z^3 * q^8 - z^6 * q^10 + z * q^11 + (-z^8 - z^2) * q^13 + z^11 * q^14 + z^4 * q^16 + (-z^9 - z^3) * q^17 + z^7 * q^20 - z^2 * q^22 + (z^9 + z^3) * q^26 + q^28 + z^8 * q^31 - z^5 * q^32 + (z^10 + z^4) * q^34 + z^3 * q^35 - z^8 * q^40 + (z^10 + z^4) * q^43 + z^3 * q^44 + (-z^7 + z) * q^47 + (-z^10 - z^4) * q^52 + z^9 * q^53 + z^6 * q^55 - z * q^56 - z^9 * q^62 + z^6 * q^64 + (-z^7 + z) * q^65 + (z^8 + z^2) * q^67 + (-z^11 - z^5) * q^68 - z^4 * q^70 + (z^9 - z^3) * q^71 - z^6 * q^73 - z^11 * q^77 + z^9 * q^80 + z^7 * q^83 + (-z^8 + z^2) * q^85 + (-z^11 - z^5) * q^86 - z^4 * q^88 + (-z^9 + z^3) * q^89 + (-z^6 - 1) * q^91 + (z^8 - z^2) * q^94 - z^4 * q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97}+O(q^{100}) \) 8 * q + 4 * q^13 + 4 * q^16 + 8 * q^28 - 4 * q^31 + 4 * q^34 + 4 * q^40 + 4 * q^43 - 4 * q^52 - 4 * q^67 - 4 * q^70 + 4 * q^85 - 4 * q^88 - 8 * q^91 - 4 * q^94 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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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	1296.1.x.a

Level	$1296$
Weight	$1$
Character orbit	1296.x
Analytic conductor	$0.647$
Analytic rank	$0$
Dimension	$8$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$8$

Self dual:	no
Analytic conductor:	\(0.646788256372\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{12})\)
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:	no (minimal twist has level 432)
Projective image:	\(S_{4}\)
Projective field:	Galois closure of 4.2.55296.2

\(n\)	\(325\)	\(1135\)	\(1217\)
\(\chi(n)\)	\(\zeta_{24}^{6}\)	\(1\)	\(-\zeta_{24}^{8}\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 53.2
Significant digits:
Format:

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