LMFDB - Newform orbit 3600.3.j.h

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} $
Twist minimal:	no (minimal twist has level 1200)
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_{3} q^{7}+O(q^{10}) $ q - b3 * q^7	$ q - \beta_{3} q^{7} + 2 \beta_{2} q^{11} + 11 \beta_1 q^{13} - 6 \beta_1 q^{17} - 11 \beta_{2} q^{19} - 10 \beta_{3} q^{23} + 30 q^{29} - 3 \beta_{2} q^{31} + 14 \beta_1 q^{37} - 36 q^{41} - 3 \beta_{3} q^{43} + 26 \beta_{3} q^{47} - 46 q^{49} - 72 \beta_1 q^{53} + 22 \beta_{2} q^{59} + 35 q^{61} - 17 \beta_{3} q^{67} - 52 \beta_{2} q^{71} + 62 \beta_1 q^{73} - 6 \beta_1 q^{77} - 44 \beta_{2} q^{79} + 42 \beta_{3} q^{83} + 144 q^{89} - 11 \beta_{2} q^{91} + 181 \beta_1 q^{97}+O(q^{100}) $ q - b3 * q^7 + 2b2 q^11 + 11b1 q^13 - 6b1 q^17 - 11b2 q^19 - 10b3 q^23 + 30 * q^29 - 3b2 q^31 + 14b1 q^37 - 36 * q^41 - 3b3 q^43 + 26b3 q^47 - 46 * q^49 - 72b1 q^53 + 22b2 q^59 + 35 * q^61 - 17b3 q^67 - 52b2 q^71 + 62b1 q^73 - 6b1 q^77 - 44b2 q^79 + 42b3 q^83 + 144 * q^89 - 11b2 q^91 + 181b1 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 120 q^{29} - 144 q^{41} - 184 q^{49} + 140 q^{61} + 576 q^{89}+O(q^{100}) $ 4 * q + 120 * q^29 - 144 * q^41 - 184 * q^49 + 140 * q^61 + 576 * q^89

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

$\zeta_{12}$	$=$	$ ( \beta_{3} + \beta_1 ) / 2 $ (b3 + b1) / 2
$\zeta_{12}^{2}$	$=$	$ ( \beta_{2} + 1 ) / 2 $ (b2 + 1) / 2
$\zeta_{12}^{3}$	$=$	$ \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

−1.73205

1999.2

−1.73205

1999.3

1.73205

1999.4

1.73205

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.b	even	2	1	inner
20.d	odd	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.h		4
3.b	odd	2	1		1200.3.j.c		4
4.b	odd	2	1	inner	3600.3.j.h		4
5.b	even	2	1	inner	3600.3.j.h		4
5.c	odd	4	1		3600.3.e.l		2
5.c	odd	4	1		3600.3.e.r		2
12.b	even	2	1		1200.3.j.c		4
15.d	odd	2	1		1200.3.j.c		4
15.e	even	4	1		1200.3.e.c	✓	2
15.e	even	4	1		1200.3.e.f	yes	2
20.d	odd	2	1	inner	3600.3.j.h		4
20.e	even	4	1		3600.3.e.l		2
20.e	even	4	1		3600.3.e.r		2
60.h	even	2	1		1200.3.j.c		4
60.l	odd	4	1		1200.3.e.c	✓	2
60.l	odd	4	1		1200.3.e.f	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1200.3.e.c	✓	2	15.e	even	4	1
1200.3.e.c	✓	2	60.l	odd	4	1
1200.3.e.f	yes	2	15.e	even	4	1
1200.3.e.f	yes	2	60.l	odd	4	1
1200.3.j.c		4	3.b	odd	2	1
1200.3.j.c		4	12.b	even	2	1
1200.3.j.c		4	15.d	odd	2	1
1200.3.j.c		4	60.h	even	2	1
3600.3.e.l		2	5.c	odd	4	1
3600.3.e.l		2	20.e	even	4	1
3600.3.e.r		2	5.c	odd	4	1
3600.3.e.r		2	20.e	even	4	1
3600.3.j.h		4	1.a	even	1	1	trivial
3600.3.j.h		4	4.b	odd	2	1	inner
3600.3.j.h		4	5.b	even	2	1	inner
3600.3.j.h		4	20.d	odd	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} - 3 $

$ T_{11}^{2} + 12 $

$ T_{13}^{2} + 121 $

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} - 3)^{2} $ (T^2 - 3)^2
$11$	$ (T^{2} + 12)^{2} $ (T^2 + 12)^2
$13$	$ (T^{2} + 121)^{2} $ (T^2 + 121)^2
$17$	$ (T^{2} + 36)^{2} $ (T^2 + 36)^2
$19$	$ (T^{2} + 363)^{2} $ (T^2 + 363)^2
$23$	$ (T^{2} - 300)^{2} $ (T^2 - 300)^2
$29$	$ (T - 30)^{4} $ (T - 30)^4
$31$	$ (T^{2} + 27)^{2} $ (T^2 + 27)^2
$37$	$ (T^{2} + 196)^{2} $ (T^2 + 196)^2
$41$	$ (T + 36)^{4} $ (T + 36)^4
$43$	$ (T^{2} - 27)^{2} $ (T^2 - 27)^2
$47$	$ (T^{2} - 2028)^{2} $ (T^2 - 2028)^2
$53$	$ (T^{2} + 5184)^{2} $ (T^2 + 5184)^2
$59$	$ (T^{2} + 1452)^{2} $ (T^2 + 1452)^2
$61$	$ (T - 35)^{4} $ (T - 35)^4
$67$	$ (T^{2} - 867)^{2} $ (T^2 - 867)^2
$71$	$ (T^{2} + 8112)^{2} $ (T^2 + 8112)^2
$73$	$ (T^{2} + 3844)^{2} $ (T^2 + 3844)^2
$79$	$ (T^{2} + 5808)^{2} $ (T^2 + 5808)^2
$83$	$ (T^{2} - 5292)^{2} $ (T^2 - 5292)^2
$89$	$ (T - 144)^{4} $ (T - 144)^4
$97$	$ (T^{2} + 32761)^{2} $ (T^2 + 32761)^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	\( q - \beta_{3} q^{7} + 2 \beta_{2} q^{11} + 11 \beta_1 q^{13} - 6 \beta_1 q^{17} - 11 \beta_{2} q^{19} - 10 \beta_{3} q^{23} + 30 q^{29} - 3 \beta_{2} q^{31} + 14 \beta_1 q^{37} - 36 q^{41} - 3 \beta_{3} q^{43} + 26 \beta_{3} q^{47} - 46 q^{49} - 72 \beta_1 q^{53} + 22 \beta_{2} q^{59} + 35 q^{61} - 17 \beta_{3} q^{67} - 52 \beta_{2} q^{71} + 62 \beta_1 q^{73} - 6 \beta_1 q^{77} - 44 \beta_{2} q^{79} + 42 \beta_{3} q^{83} + 144 q^{89} - 11 \beta_{2} q^{91} + 181 \beta_1 q^{97}+O(q^{100}) \) q - b3 * q^7 + 2b2 q^11 + 11b1 q^13 - 6b1 q^17 - 11b2 q^19 - 10b3 q^23 + 30 * q^29 - 3b2 q^31 + 14b1 q^37 - 36 * q^41 - 3b3 q^43 + 26b3 q^47 - 46 * q^49 - 72b1 q^53 + 22b2 q^59 + 35 * q^61 - 17b3 q^67 - 52b2 q^71 + 62b1 q^73 - 6b1 q^77 - 44b2 q^79 + 42b3 q^83 + 144 * q^89 - 11b2 q^91 + 181b1 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 120 q^{29} - 144 q^{41} - 184 q^{49} + 140 q^{61} + 576 q^{89}+O(q^{100}) \) 4 * q + 120 * q^29 - 144 * q^41 - 184 * q^49 + 140 * q^61 + 576 * q^89

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