LMFDB - Newform orbit 3600.3.j.d

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(1999,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3600.1999");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.j (of order $2$, degree $1$, 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:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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_{3} q^{7}+O(q^{10}) $ q + b3 * q^7	$ q + \beta_{3} q^{7} - \beta_1 q^{13} - \beta_{2} q^{19} + 7 \beta_{2} q^{31} + 26 \beta_1 q^{37} + 7 \beta_{3} q^{43} + 26 q^{49} + 47 q^{61} + 9 \beta_{3} q^{67} - 46 \beta_1 q^{73} - 8 \beta_{2} q^{79} - \beta_{2} q^{91} + 169 \beta_1 q^{97}+O(q^{100}) $ q + b3 * q^7 - b1 * q^13 - b2 * q^19 + 7b2 q^31 + 26b1 q^37 + 7b3 q^43 + 26 * q^49 + 47 * q^61 + 9b3 q^67 - 46b1 q^73 - 8b2 q^79 - b2 * q^91 + 169b1 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 104 q^{49} + 188 q^{61}+O(q^{100}) $ 4 * q + 104 * q^49 + 188 * q^61

Display 100 coefficients 10 coefficients

Basis of coefficient ring

$\beta_{1}$	$=$	$ \zeta_{12}^{3} $ v^3
$\beta_{2}$	$=$	$ 10\zeta_{12}^{2} - 5 $ 10*v^2 - 5
$\beta_{3}$	$=$	$ -5\zeta_{12}^{3} + 10\zeta_{12} $ -5v^3 + 10v

Display $\zeta_{12}^j$ in terms of $\beta_i$

$\zeta_{12}$	$=$	$ ( \beta_{3} + 5\beta_1 ) / 10 $ (b3 + 5*b1) / 10
$\zeta_{12}^{2}$	$=$	$ ( \beta_{2} + 5 ) / 10 $ (b2 + 5) / 10
$\zeta_{12}^{3}$	$=$	$ \beta_1 $ b1

Display $\beta_i$ in terms of $\zeta_{12}^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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} $

1999.1

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

−8.66025

1999.2

−8.66025

1999.3

8.66025

1999.4

8.66025

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $
4.b	odd	2	1	inner
5.b	even	2	1	inner
12.b	even	2	1	inner
15.d	odd	2	1	inner
20.d	odd	2	1	inner
60.h	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	3600.3.j.d		4
3.b	odd	2	1	CM	3600.3.j.d		4
4.b	odd	2	1	inner	3600.3.j.d		4
5.b	even	2	1	inner	3600.3.j.d		4
5.c	odd	4	1		3600.3.e.n	✓	2
5.c	odd	4	1		3600.3.e.p	yes	2
12.b	even	2	1	inner	3600.3.j.d		4
15.d	odd	2	1	inner	3600.3.j.d		4
15.e	even	4	1		3600.3.e.n	✓	2
15.e	even	4	1		3600.3.e.p	yes	2
20.d	odd	2	1	inner	3600.3.j.d		4
20.e	even	4	1		3600.3.e.n	✓	2
20.e	even	4	1		3600.3.e.p	yes	2
60.h	even	2	1	inner	3600.3.j.d		4
60.l	odd	4	1		3600.3.e.n	✓	2
60.l	odd	4	1		3600.3.e.p	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
3600.3.e.n	✓	2	5.c	odd	4	1
3600.3.e.n	✓	2	15.e	even	4	1
3600.3.e.n	✓	2	20.e	even	4	1
3600.3.e.n	✓	2	60.l	odd	4	1
3600.3.e.p	yes	2	5.c	odd	4	1
3600.3.e.p	yes	2	15.e	even	4	1
3600.3.e.p	yes	2	20.e	even	4	1
3600.3.e.p	yes	2	60.l	odd	4	1
3600.3.j.d		4	1.a	even	1	1	trivial
3600.3.j.d		4	3.b	odd	2	1	CM
3600.3.j.d		4	4.b	odd	2	1	inner
3600.3.j.d		4	5.b	even	2	1	inner
3600.3.j.d		4	12.b	even	2	1	inner
3600.3.j.d		4	15.d	odd	2	1	inner
3600.3.j.d		4	20.d	odd	2	1	inner
3600.3.j.d		4	60.h	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 $S_{3}^{\mathrm{new}}(3600, [\chi])$:

$ T_{7}^{2} - 75 $

$ T_{11} $

$ T_{13}^{2} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} $ T^4
$7$	$ (T^{2} - 75)^{2} $ (T^2 - 75)^2
$11$	$ T^{4} $ T^4
$13$	$ (T^{2} + 1)^{2} $ (T^2 + 1)^2
$17$	$ T^{4} $ T^4
$19$	$ (T^{2} + 75)^{2} $ (T^2 + 75)^2
$23$	$ T^{4} $ T^4
$29$	$ T^{4} $ T^4
$31$	$ (T^{2} + 3675)^{2} $ (T^2 + 3675)^2
$37$	$ (T^{2} + 676)^{2} $ (T^2 + 676)^2
$41$	$ T^{4} $ T^4
$43$	$ (T^{2} - 3675)^{2} $ (T^2 - 3675)^2
$47$	$ T^{4} $ T^4
$53$	$ T^{4} $ T^4
$59$	$ T^{4} $ T^4
$61$	$ (T - 47)^{4} $ (T - 47)^4
$67$	$ (T^{2} - 6075)^{2} $ (T^2 - 6075)^2
$71$	$ T^{4} $ T^4
$73$	$ (T^{2} + 2116)^{2} $ (T^2 + 2116)^2
$79$	$ (T^{2} + 4800)^{2} $ (T^2 + 4800)^2
$83$	$ T^{4} $ T^4
$89$	$ T^{4} $ T^4
$97$	$ (T^{2} + 28561)^{2} $ (T^2 + 28561)^2
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Level:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + \beta_{3} q^{7}+O(q^{10}) \) q + b3 * q^7	\( q + \beta_{3} q^{7} - \beta_1 q^{13} - \beta_{2} q^{19} + 7 \beta_{2} q^{31} + 26 \beta_1 q^{37} + 7 \beta_{3} q^{43} + 26 q^{49} + 47 q^{61} + 9 \beta_{3} q^{67} - 46 \beta_1 q^{73} - 8 \beta_{2} q^{79} - \beta_{2} q^{91} + 169 \beta_1 q^{97}+O(q^{100}) \) q + b3 * q^7 - b1 * q^13 - b2 * q^19 + 7b2 q^31 + 26b1 q^37 + 7b3 q^43 + 26 * q^49 + 47 * q^61 + 9b3 q^67 - 46b1 q^73 - 8b2 q^79 - b2 * q^91 + 169b1 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 104 q^{49} + 188 q^{61}+O(q^{100}) \) 4 * q + 104 * q^49 + 188 * q^61

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