LMFDB - Newform orbit 5292.2.f.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(2645,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("5292.2645");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.f (of order $2$, degree $1$, 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 756)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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 basis $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q - \beta_1 q^{5}+O(q^{10}) $ q - b1 * q^5	$ q - \beta_1 q^{5} - 2 \beta_{2} q^{13} - \beta_1 q^{17} + \beta_{2} q^{19} - \beta_{3} q^{23} + 13 q^{25} - \beta_{3} q^{29} - \beta_{2} q^{31} - 8 q^{37} + \beta_1 q^{41} - 5 q^{43} - \beta_1 q^{47} - \beta_{3} q^{53} - 2 \beta_1 q^{59} + 3 \beta_{2} q^{61} - 2 \beta_{3} q^{65} + 2 q^{67} + 2 \beta_{3} q^{71} - 5 \beta_{2} q^{73} - 4 q^{79} + 4 \beta_1 q^{83} + 18 q^{85} - 3 \beta_1 q^{89} + \beta_{3} q^{95} - 9 \beta_{2} q^{97}+O(q^{100}) $ q - b1 * q^5 - 2b2 q^13 - b1 * q^17 + b2 * q^19 - b3 * q^23 + 13 * q^25 - b3 * q^29 - b2 * q^31 - 8 * q^37 + b1 * q^41 - 5 * q^43 - b1 * q^47 - b3 * q^53 - 2b1 q^59 + 3b2 q^61 - 2b3 q^65 + 2 * q^67 + 2b3 q^71 - 5b2 q^73 - 4 * q^79 + 4b1 q^83 + 18 * q^85 - 3b1 q^89 + b3 * q^95 - 9b2 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 52 q^{25} - 32 q^{37} - 20 q^{43} + 8 q^{67} - 16 q^{79} + 72 q^{85}+O(q^{100}) $ 4 * q + 52 * q^25 - 32 * q^37 - 20 * q^43 + 8 * q^67 - 16 * q^79 + 72 * q^85

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4 :

$\beta_{1}$	$=$	$ ( 3\nu^{3} ) / 2 $ (3*v^3) / 2
$\beta_{2}$	$=$	$ \nu^{2} + 1 $ v^2 + 1
$\beta_{3}$	$=$	$ ( 3\nu^{3} + 12\nu ) / 2 $ (3v^3 + 12v) / 2

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{3} - \beta_1 ) / 6 $ (b3 - b1) / 6
$\nu^{2}$	$=$	$ \beta_{2} - 1 $ b2 - 1
$\nu^{3}$	$=$	$ ( 2\beta_1 ) / 3 $ (2*b1) / 3

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$-1$	$-1$	$1$

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} $

2645.1

−0.707107	−	1.22474i
−0.707107	+	1.22474i
0.707107	+	1.22474i
0.707107	−	1.22474i

−4.24264

2645.2

−4.24264

2645.3

4.24264

2645.4

4.24264

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.b	odd	2	1	inner
21.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	5292.2.f.d		4
3.b	odd	2	1	inner	5292.2.f.d		4
7.b	odd	2	1	inner	5292.2.f.d		4
7.c	even	3	1		756.2.t.d	✓	4
7.d	odd	6	1		756.2.t.d	✓	4
21.c	even	2	1	inner	5292.2.f.d		4
21.g	even	6	1		756.2.t.d	✓	4
21.h	odd	6	1		756.2.t.d	✓	4
63.g	even	3	1		2268.2.bm.h		4
63.h	even	3	1		2268.2.w.g		4
63.i	even	6	1		2268.2.w.g		4
63.j	odd	6	1		2268.2.w.g		4
63.k	odd	6	1		2268.2.bm.h		4
63.n	odd	6	1		2268.2.bm.h		4
63.s	even	6	1		2268.2.bm.h		4
63.t	odd	6	1		2268.2.w.g		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
756.2.t.d	✓	4	7.c	even	3	1
756.2.t.d	✓	4	7.d	odd	6	1
756.2.t.d	✓	4	21.g	even	6	1
756.2.t.d	✓	4	21.h	odd	6	1
2268.2.w.g		4	63.h	even	3	1
2268.2.w.g		4	63.i	even	6	1
2268.2.w.g		4	63.j	odd	6	1
2268.2.w.g		4	63.t	odd	6	1
2268.2.bm.h		4	63.g	even	3	1
2268.2.bm.h		4	63.k	odd	6	1
2268.2.bm.h		4	63.n	odd	6	1
2268.2.bm.h		4	63.s	even	6	1
5292.2.f.d		4	1.a	even	1	1	trivial
5292.2.f.d		4	3.b	odd	2	1	inner
5292.2.f.d		4	7.b	odd	2	1	inner
5292.2.f.d		4	21.c	even	2	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(5292, [\chi])$:

$ T_{5}^{2} - 18 $

$ T_{13}^{2} + 12 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ (T^{2} - 18)^{2} $ (T^2 - 18)^2
$7$	$ T^{4} $ T^4
$11$	$ T^{4} $ T^4
$13$	$ (T^{2} + 12)^{2} $ (T^2 + 12)^2
$17$	$ (T^{2} - 18)^{2} $ (T^2 - 18)^2
$19$	$ (T^{2} + 3)^{2} $ (T^2 + 3)^2
$23$	$ (T^{2} + 54)^{2} $ (T^2 + 54)^2
$29$	$ (T^{2} + 54)^{2} $ (T^2 + 54)^2
$31$	$ (T^{2} + 3)^{2} $ (T^2 + 3)^2
$37$	$ (T + 8)^{4} $ (T + 8)^4
$41$	$ (T^{2} - 18)^{2} $ (T^2 - 18)^2
$43$	$ (T + 5)^{4} $ (T + 5)^4
$47$	$ (T^{2} - 18)^{2} $ (T^2 - 18)^2
$53$	$ (T^{2} + 54)^{2} $ (T^2 + 54)^2
$59$	$ (T^{2} - 72)^{2} $ (T^2 - 72)^2
$61$	$ (T^{2} + 27)^{2} $ (T^2 + 27)^2
$67$	$ (T - 2)^{4} $ (T - 2)^4
$71$	$ (T^{2} + 216)^{2} $ (T^2 + 216)^2
$73$	$ (T^{2} + 75)^{2} $ (T^2 + 75)^2
$79$	$ (T + 4)^{4} $ (T + 4)^4
$83$	$ (T^{2} - 288)^{2} $ (T^2 - 288)^2
$89$	$ (T^{2} - 162)^{2} $ (T^2 - 162)^2
$97$	$ (T^{2} + 243)^{2} $ (T^2 + 243)^2
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Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q - \beta_1 q^{5}+O(q^{10}) \) q - b1 * q^5	\( q - \beta_1 q^{5} - 2 \beta_{2} q^{13} - \beta_1 q^{17} + \beta_{2} q^{19} - \beta_{3} q^{23} + 13 q^{25} - \beta_{3} q^{29} - \beta_{2} q^{31} - 8 q^{37} + \beta_1 q^{41} - 5 q^{43} - \beta_1 q^{47} - \beta_{3} q^{53} - 2 \beta_1 q^{59} + 3 \beta_{2} q^{61} - 2 \beta_{3} q^{65} + 2 q^{67} + 2 \beta_{3} q^{71} - 5 \beta_{2} q^{73} - 4 q^{79} + 4 \beta_1 q^{83} + 18 q^{85} - 3 \beta_1 q^{89} + \beta_{3} q^{95} - 9 \beta_{2} q^{97}+O(q^{100}) \) q - b1 * q^5 - 2b2 q^13 - b1 * q^17 + b2 * q^19 - b3 * q^23 + 13 * q^25 - b3 * q^29 - b2 * q^31 - 8 * q^37 + b1 * q^41 - 5 * q^43 - b1 * q^47 - b3 * q^53 - 2b1 q^59 + 3b2 q^61 - 2b3 q^65 + 2 * q^67 + 2b3 q^71 - 5b2 q^73 - 4 * q^79 + 4b1 q^83 + 18 * q^85 - 3b1 q^89 + b3 * q^95 - 9b2 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 52 q^{25} - 32 q^{37} - 20 q^{43} + 8 q^{67} - 16 q^{79} + 72 q^{85}+O(q^{100}) \) 4 * q + 52 * q^25 - 32 * q^37 - 20 * q^43 + 8 * q^67 - 16 * q^79 + 72 * q^85

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