LMFDB - Newform orbit 2592.2.r.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(433,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2592.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.r (of order $6$, degree $2$, 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:	$20.6972242039$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 648)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a primitive root of unity $\zeta_{12}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{7}+O(q^{10}) $ q + (-z^2 + 2z - 1) q^5 + (z^3 + z^2 + z) * q^7	$ q + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{7} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{11}+ \cdots + 8 \zeta_{12}^{2} q^{97}+O(q^{100}) $ q + (-z^2 + 2z - 1) q^5 + (z^3 + z^2 + z) * q^7 + (-3z^3 + z^2 + 3z - 2) * q^11 + (2z^2 + z + 2) q^13 + (z^3 - 2z + 2) q^17 + (5z^3 + 2z^2 - 1) * q^19 + (-2z^3 + 7z^2 + z - 7) * q^23 + (-4z^3 + 2z^2 - 4z) q^25 + (3z^2 - 6) q^29 + (-2z^2 + 2) q^31 + (-z^3 + 2z^2 - 1) q^35 + 5z^3 q^37 + (-4z^3 - 8z^2 + 2z + 8) q^41 + (-z^3 + 5z^2 + z - 10) q^43 + (2z^3 + 2z) * q^47 + (4z^3 - 3z^2 - 2z + 3) q^49 + (-6z^3 - 8z^2 + 4) * q^53 + (5z^3 - 10z + 9) * q^55 + (6z^2 + 2z + 6) * q^59 + (9z^3 - 9z) * q^61 + (3z^3 - 4z^2 + 3z) q^65 + (z^2 - 3z + 1) q^67 + (5z^3 - 10z - 3) * q^71 + 9 * q^73 + (2z^2 + 2) q^77 + (z^3 + 7z^2 + z) q^79 + (14z^3 - 14z) * q^83 + (-4z^2 + 7z - 4) * q^85 + (z^3 - 2z + 10) q^89 + (7z^3 + 6z^2 - 3) * q^91 + (-6z^3 + 7z^2 + 3z - 7) q^95 + 8z^2 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q - 6 * q^5 + 2 * q^7	$ 4 q - 6 q^{5} + 2 q^{7} - 6 q^{11} + 12 q^{13} + 8 q^{17} - 14 q^{23} + 4 q^{25} - 18 q^{29} + 4 q^{31} + 16 q^{41} - 30 q^{43} + 6 q^{49} + 36 q^{55} + 36 q^{59} - 8 q^{65} + 6 q^{67} - 12 q^{71} + 36 q^{73} + 12 q^{77} + 14 q^{79} - 24 q^{85} + 40 q^{89} - 14 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 + 2 * q^7 - 6 * q^11 + 12 * q^13 + 8 * q^17 - 14 * q^23 + 4 * q^25 - 18 * q^29 + 4 * q^31 + 16 * q^41 - 30 * q^43 + 6 * q^49 + 36 * q^55 + 36 * q^59 - 8 * q^65 + 6 * q^67 - 12 * q^71 + 36 * q^73 + 12 * q^77 + 14 * q^79 - 24 * q^85 + 40 * q^89 - 14 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$-1 + \zeta_{12}^{2}$	$1$

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} $

433.1

−0.866025	−	0.500000i
0.866025	+	0.500000i
−0.866025	+	0.500000i
0.866025	−	0.500000i

−3.23205

−

1.86603i

−0.366025

−

0.633975i

433.2

0.232051

0.133975i

1.36603

2.36603i

2161.1

−3.23205

1.86603i

−0.366025

0.633975i

2161.2

0.232051

−

0.133975i

1.36603

−

2.36603i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
72.n	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2592.2.r.b		4
3.b	odd	2	1		2592.2.r.k		4
4.b	odd	2	1		648.2.n.b		4
8.b	even	2	1		2592.2.r.j		4
8.d	odd	2	1		648.2.n.m		4
9.c	even	3	1		2592.2.d.h		4
9.c	even	3	1		2592.2.r.j		4
9.d	odd	6	1		2592.2.d.g		4
9.d	odd	6	1		2592.2.r.a		4
12.b	even	2	1		648.2.n.l		4
24.f	even	2	1		648.2.n.a		4
24.h	odd	2	1		2592.2.r.a		4
36.f	odd	6	1		648.2.d.e	✓	4
36.f	odd	6	1		648.2.n.m		4
36.h	even	6	1		648.2.d.i	yes	4
36.h	even	6	1		648.2.n.a		4
72.j	odd	6	1		2592.2.d.g		4
72.j	odd	6	1		2592.2.r.k		4
72.l	even	6	1		648.2.d.i	yes	4
72.l	even	6	1		648.2.n.l		4
72.n	even	6	1		2592.2.d.h		4
72.n	even	6	1	inner	2592.2.r.b		4
72.p	odd	6	1		648.2.d.e	✓	4
72.p	odd	6	1		648.2.n.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
648.2.d.e	✓	4	36.f	odd	6	1
648.2.d.e	✓	4	72.p	odd	6	1
648.2.d.i	yes	4	36.h	even	6	1
648.2.d.i	yes	4	72.l	even	6	1
648.2.n.a		4	24.f	even	2	1
648.2.n.a		4	36.h	even	6	1
648.2.n.b		4	4.b	odd	2	1
648.2.n.b		4	72.p	odd	6	1
648.2.n.l		4	12.b	even	2	1
648.2.n.l		4	72.l	even	6	1
648.2.n.m		4	8.d	odd	2	1
648.2.n.m		4	36.f	odd	6	1
2592.2.d.g		4	9.d	odd	6	1
2592.2.d.g		4	72.j	odd	6	1
2592.2.d.h		4	9.c	even	3	1
2592.2.d.h		4	72.n	even	6	1
2592.2.r.a		4	9.d	odd	6	1
2592.2.r.a		4	24.h	odd	2	1
2592.2.r.b		4	1.a	even	1	1	trivial
2592.2.r.b		4	72.n	even	6	1	inner
2592.2.r.j		4	8.b	even	2	1
2592.2.r.j		4	9.c	even	3	1
2592.2.r.k		4	3.b	odd	2	1
2592.2.r.k		4	72.j	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2592, [\chi])$:

$ T_{5}^{4} + 6T_{5}^{3} + 11T_{5}^{2} - 6T_{5} + 1 $

$ T_{13}^{4} - 12T_{13}^{3} + 59T_{13}^{2} - 132T_{13} + 121 $

$ T_{17}^{2} - 4T_{17} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 6 T^{3} + \cdots + 1 $ T^4 + 6T^3 + 11T^2 - 6*T + 1
$7$	$ T^{4} - 2 T^{3} + \cdots + 4 $ T^4 - 2T^3 + 6T^2 + 4*T + 4
$11$	$ T^{4} + 6 T^{3} + \cdots + 36 $ T^4 + 6T^3 + 6T^2 - 36*T + 36
$13$	$ T^{4} - 12 T^{3} + \cdots + 121 $ T^4 - 12T^3 + 59T^2 - 132*T + 121
$17$	$ (T^{2} - 4 T + 1)^{2} $ (T^2 - 4*T + 1)^2
$19$	$ T^{4} + 56T^{2} + 484 $ T^4 + 56*T^2 + 484
$23$	$ T^{4} + 14 T^{3} + \cdots + 2116 $ T^4 + 14T^3 + 150T^2 + 644*T + 2116
$29$	$ (T^{2} + 9 T + 27)^{2} $ (T^2 + 9*T + 27)^2
$31$	$ (T^{2} - 2 T + 4)^{2} $ (T^2 - 2*T + 4)^2
$37$	$ (T^{2} + 25)^{2} $ (T^2 + 25)^2
$41$	$ T^{4} - 16 T^{3} + \cdots + 2704 $ T^4 - 16T^3 + 204T^2 - 832*T + 2704
$43$	$ T^{4} + 30 T^{3} + \cdots + 5476 $ T^4 + 30T^3 + 374T^2 + 2220*T + 5476
$47$	$ T^{4} + 12T^{2} + 144 $ T^4 + 12*T^2 + 144
$53$	$ T^{4} + 168T^{2} + 144 $ T^4 + 168*T^2 + 144
$59$	$ T^{4} - 36 T^{3} + \cdots + 10816 $ T^4 - 36T^3 + 536T^2 - 3744*T + 10816
$61$	$ T^{4} - 81T^{2} + 6561 $ T^4 - 81*T^2 + 6561
$67$	$ T^{4} - 6 T^{3} + \cdots + 36 $ T^4 - 6T^3 + 6T^2 + 36*T + 36
$71$	$ (T^{2} + 6 T - 66)^{2} $ (T^2 + 6*T - 66)^2
$73$	$ (T - 9)^{4} $ (T - 9)^4
$79$	$ T^{4} - 14 T^{3} + \cdots + 2116 $ T^4 - 14T^3 + 150T^2 - 644*T + 2116
$83$	$ T^{4} - 196 T^{2} + 38416 $ T^4 - 196*T^2 + 38416
$89$	$ (T^{2} - 20 T + 97)^{2} $ (T^2 - 20*T + 97)^2
$97$	$ (T^{2} - 8 T + 64)^{2} $ (T^2 - 8*T + 64)^2
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Level:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{7}+O(q^{10}) \) q + (-z^2 + 2z - 1) q^5 + (z^3 + z^2 + z) * q^7	\( q + ( - \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{5} + (\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{7} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + \cdots - 2) q^{11}+ \cdots + 8 \zeta_{12}^{2} q^{97}+O(q^{100}) \) q + (-z^2 + 2z - 1) q^5 + (z^3 + z^2 + z) * q^7 + (-3z^3 + z^2 + 3z - 2) * q^11 + (2z^2 + z + 2) q^13 + (z^3 - 2z + 2) q^17 + (5z^3 + 2z^2 - 1) * q^19 + (-2z^3 + 7z^2 + z - 7) * q^23 + (-4z^3 + 2z^2 - 4z) q^25 + (3z^2 - 6) q^29 + (-2z^2 + 2) q^31 + (-z^3 + 2z^2 - 1) q^35 + 5z^3 q^37 + (-4z^3 - 8z^2 + 2z + 8) q^41 + (-z^3 + 5z^2 + z - 10) q^43 + (2z^3 + 2z) * q^47 + (4z^3 - 3z^2 - 2z + 3) q^49 + (-6z^3 - 8z^2 + 4) * q^53 + (5z^3 - 10z + 9) * q^55 + (6z^2 + 2z + 6) * q^59 + (9z^3 - 9z) * q^61 + (3z^3 - 4z^2 + 3z) q^65 + (z^2 - 3z + 1) q^67 + (5z^3 - 10z - 3) * q^71 + 9 * q^73 + (2z^2 + 2) q^77 + (z^3 + 7z^2 + z) q^79 + (14z^3 - 14z) * q^83 + (-4z^2 + 7z - 4) * q^85 + (z^3 - 2z + 10) q^89 + (7z^3 + 6z^2 - 3) * q^91 + (-6z^3 + 7z^2 + 3z - 7) q^95 + 8z^2 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q - 6 * q^5 + 2 * q^7	\( 4 q - 6 q^{5} + 2 q^{7} - 6 q^{11} + 12 q^{13} + 8 q^{17} - 14 q^{23} + 4 q^{25} - 18 q^{29} + 4 q^{31} + 16 q^{41} - 30 q^{43} + 6 q^{49} + 36 q^{55} + 36 q^{59} - 8 q^{65} + 6 q^{67} - 12 q^{71} + 36 q^{73} + 12 q^{77} + 14 q^{79} - 24 q^{85} + 40 q^{89} - 14 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 + 2 * q^7 - 6 * q^11 + 12 * q^13 + 8 * q^17 - 14 * q^23 + 4 * q^25 - 18 * q^29 + 4 * q^31 + 16 * q^41 - 30 * q^43 + 6 * q^49 + 36 * q^55 + 36 * q^59 - 8 * q^65 + 6 * q^67 - 12 * q^71 + 36 * q^73 + 12 * q^77 + 14 * q^79 - 24 * q^85 + 40 * q^89 - 14 * q^95 + 16 * q^97

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