LMFDB - Newform orbit 7200.2.a.ct

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7200,2,Mod(1,7200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7200, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7200.a (trivial)

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:	yes
Analytic conductor:	$57.4922894553$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 1440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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^{7}+O(q^{10}) $ q + b1 * q^7	$ q + \beta_1 q^{7} + \beta_{2} q^{11} + \beta_{2} q^{13} - \beta_{3} q^{17} - 2 \beta_{3} q^{19} + 4 q^{23} + 3 \beta_1 q^{29} + 2 \beta_{3} q^{31} + \beta_{2} q^{37} + 2 \beta_1 q^{41} - 4 \beta_1 q^{43} + 4 q^{47} + q^{49} - \beta_{3} q^{53} + \beta_{2} q^{59} - 6 q^{61} + 2 \beta_1 q^{67} - 2 \beta_{2} q^{71} + 4 \beta_{3} q^{77} - 2 \beta_{3} q^{79} + 16 q^{83} + 4 \beta_{3} q^{91} - 2 \beta_{2} q^{97}+O(q^{100}) $ q + b1 * q^7 + b2 * q^11 + b2 * q^13 - b3 * q^17 - 2b3 q^19 + 4 * q^23 + 3b1 q^29 + 2b3 q^31 + b2 * q^37 + 2b1 q^41 - 4b1 q^43 + 4 * q^47 + q^49 - b3 * q^53 + b2 * q^59 - 6 * q^61 + 2b1 q^67 - 2b2 q^71 + 4b3 q^77 - 2b3 q^79 + 16 * q^83 + 4b3 q^91 - 2b2 q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 16 q^{23} + 16 q^{47} + 4 q^{49} - 24 q^{61} + 64 q^{83}+O(q^{100}) $ 4 * q + 16 * q^23 + 16 * q^47 + 4 * q^49 - 24 * q^61 + 64 * q^83

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of $\nu = \zeta_{24} + \zeta_{24}^{-1}$:

$\beta_{1}$	$=$	$ 2\nu^{3} - 6\nu $ 2v^3 - 6v
$\beta_{2}$	$=$	$ -2\nu^{3} + 10\nu $ -2v^3 + 10v
$\beta_{3}$	$=$	$ 2\nu^{2} - 4 $ 2*v^2 - 4

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

$\nu$	$=$	$ ( \beta_{2} + \beta_1 ) / 4 $ (b2 + b1) / 4
$\nu^{2}$	$=$	$ ( \beta_{3} + 4 ) / 2 $ (b3 + 4) / 2
$\nu^{3}$	$=$	$ ( 3\beta_{2} + 5\beta_1 ) / 4 $ (3b2 + 5b1) / 4

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

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} $

1.1

−1.93185
0.517638
−0.517638
1.93185

−2.82843

1.2

−2.82843

1.3

2.82843

1.4

2.82843

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$3$	$1$
$5$	$-1$

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
12.b	even	2	1	inner
15.d	odd	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	7200.2.a.ct		4
3.b	odd	2	1		7200.2.a.cs		4
4.b	odd	2	1		7200.2.a.cs		4
5.b	even	2	1		7200.2.a.cs		4
5.c	odd	4	2		1440.2.f.j	✓	8
12.b	even	2	1	inner	7200.2.a.ct		4
15.d	odd	2	1	inner	7200.2.a.ct		4
15.e	even	4	2		1440.2.f.j	✓	8
20.d	odd	2	1	inner	7200.2.a.ct		4
20.e	even	4	2		1440.2.f.j	✓	8
40.i	odd	4	2		2880.2.f.x		8
40.k	even	4	2		2880.2.f.x		8
60.h	even	2	1		7200.2.a.cs		4
60.l	odd	4	2		1440.2.f.j	✓	8
120.q	odd	4	2		2880.2.f.x		8
120.w	even	4	2		2880.2.f.x		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1440.2.f.j	✓	8	5.c	odd	4	2
1440.2.f.j	✓	8	15.e	even	4	2
1440.2.f.j	✓	8	20.e	even	4	2
1440.2.f.j	✓	8	60.l	odd	4	2
2880.2.f.x		8	40.i	odd	4	2
2880.2.f.x		8	40.k	even	4	2
2880.2.f.x		8	120.q	odd	4	2
2880.2.f.x		8	120.w	even	4	2
7200.2.a.cs		4	3.b	odd	2	1
7200.2.a.cs		4	4.b	odd	2	1
7200.2.a.cs		4	5.b	even	2	1
7200.2.a.cs		4	60.h	even	2	1
7200.2.a.ct		4	1.a	even	1	1	trivial
7200.2.a.ct		4	12.b	even	2	1	inner
7200.2.a.ct		4	15.d	odd	2	1	inner
7200.2.a.ct		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 $S_{2}^{\mathrm{new}}(\Gamma_0(7200))$:

$ T_{7}^{2} - 8 $

$ T_{11}^{2} - 24 $

$ T_{13}^{2} - 24 $

$ T_{17}^{2} - 12 $

$ T_{19}^{2} - 48 $

$ T_{23} - 4 $

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} - 8)^{2} $ (T^2 - 8)^2
$11$	$ (T^{2} - 24)^{2} $ (T^2 - 24)^2
$13$	$ (T^{2} - 24)^{2} $ (T^2 - 24)^2
$17$	$ (T^{2} - 12)^{2} $ (T^2 - 12)^2
$19$	$ (T^{2} - 48)^{2} $ (T^2 - 48)^2
$23$	$ (T - 4)^{4} $ (T - 4)^4
$29$	$ (T^{2} - 72)^{2} $ (T^2 - 72)^2
$31$	$ (T^{2} - 48)^{2} $ (T^2 - 48)^2
$37$	$ (T^{2} - 24)^{2} $ (T^2 - 24)^2
$41$	$ (T^{2} - 32)^{2} $ (T^2 - 32)^2
$43$	$ (T^{2} - 128)^{2} $ (T^2 - 128)^2
$47$	$ (T - 4)^{4} $ (T - 4)^4
$53$	$ (T^{2} - 12)^{2} $ (T^2 - 12)^2
$59$	$ (T^{2} - 24)^{2} $ (T^2 - 24)^2
$61$	$ (T + 6)^{4} $ (T + 6)^4
$67$	$ (T^{2} - 32)^{2} $ (T^2 - 32)^2
$71$	$ (T^{2} - 96)^{2} $ (T^2 - 96)^2
$73$	$ T^{4} $ T^4
$79$	$ (T^{2} - 48)^{2} $ (T^2 - 48)^2
$83$	$ (T - 16)^{4} $ (T - 16)^4
$89$	$ T^{4} $ T^4
$97$	$ (T^{2} - 96)^{2} $ (T^2 - 96)^2
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Level:	\( N \)	\(=\)	\( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7200.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{7}+O(q^{10}) \) q + b1 * q^7	\( q + \beta_1 q^{7} + \beta_{2} q^{11} + \beta_{2} q^{13} - \beta_{3} q^{17} - 2 \beta_{3} q^{19} + 4 q^{23} + 3 \beta_1 q^{29} + 2 \beta_{3} q^{31} + \beta_{2} q^{37} + 2 \beta_1 q^{41} - 4 \beta_1 q^{43} + 4 q^{47} + q^{49} - \beta_{3} q^{53} + \beta_{2} q^{59} - 6 q^{61} + 2 \beta_1 q^{67} - 2 \beta_{2} q^{71} + 4 \beta_{3} q^{77} - 2 \beta_{3} q^{79} + 16 q^{83} + 4 \beta_{3} q^{91} - 2 \beta_{2} q^{97}+O(q^{100}) \) q + b1 * q^7 + b2 * q^11 + b2 * q^13 - b3 * q^17 - 2b3 q^19 + 4 * q^23 + 3b1 q^29 + 2b3 q^31 + b2 * q^37 + 2b1 q^41 - 4b1 q^43 + 4 * q^47 + q^49 - b3 * q^53 + b2 * q^59 - 6 * q^61 + 2b1 q^67 - 2b2 q^71 + 4b3 q^77 - 2b3 q^79 + 16 * q^83 + 4b3 q^91 - 2b2 q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 16 q^{23} + 16 q^{47} + 4 q^{49} - 24 q^{61} + 64 q^{83}+O(q^{100}) \) 4 * q + 16 * q^23 + 16 * q^47 + 4 * q^49 - 24 * q^61 + 64 * q^83

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