LMFDB - Newform orbit 1323.2.a.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.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:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
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)

$f(q)$

$=$

$ q + q^{2} - q^{4} + 4 q^{5} - 3 q^{8}+O(q^{10}) $

$ q + q^{2} - q^{4} + 4 q^{5} - 3 q^{8} + 4 q^{10} + 2 q^{11} - q^{13} - q^{16} + 6 q^{17} - 4 q^{19} - 4 q^{20} + 2 q^{22} + 6 q^{23} + 11 q^{25} - q^{26} + 2 q^{29} - 3 q^{31} + 5 q^{32} + 6 q^{34} + 3 q^{37} - 4 q^{38} - 12 q^{40} + 2 q^{41} - q^{43} - 2 q^{44} + 6 q^{46} - 6 q^{47} + 11 q^{50} + q^{52} - 6 q^{53} + 8 q^{55} + 2 q^{58} - 6 q^{59} + 5 q^{61} - 3 q^{62} + 7 q^{64} - 4 q^{65} + 7 q^{67} - 6 q^{68} + 6 q^{73} + 3 q^{74} + 4 q^{76} + 11 q^{79} - 4 q^{80} + 2 q^{82} - 6 q^{83} + 24 q^{85} - q^{86} - 6 q^{88} + 4 q^{89} - 6 q^{92} - 6 q^{94} - 16 q^{95} - 9 q^{97}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

1.00000

−1.00000

4.00000

−3.00000

4.00000

Atkin-Lehner signs

$ p $	Sign
$3$	$ +1 $
$7$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	1323.2.a.p		1
3.b	odd	2	1		1323.2.a.d		1
7.b	odd	2	1		1323.2.a.m		1
7.d	odd	6	2		189.2.e.a	✓	2
21.c	even	2	1		1323.2.a.g		1
21.g	even	6	2		189.2.e.c	yes	2
63.i	even	6	2		567.2.h.b		2
63.k	odd	6	2		567.2.g.b		2
63.s	even	6	2		567.2.g.e		2
63.t	odd	6	2		567.2.h.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
189.2.e.a	✓	2	7.d	odd	6	2
189.2.e.c	yes	2	21.g	even	6	2
567.2.g.b		2	63.k	odd	6	2
567.2.g.e		2	63.s	even	6	2
567.2.h.b		2	63.i	even	6	2
567.2.h.e		2	63.t	odd	6	2
1323.2.a.d		1	3.b	odd	2	1
1323.2.a.g		1	21.c	even	2	1
1323.2.a.m		1	7.b	odd	2	1
1323.2.a.p		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(1323))$:

$ T_{2} - 1 $

$ T_{5} - 4 $

$ T_{13} + 1 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 1 $ T - 1
$3$	$ T $ T
$5$	$ T - 4 $ T - 4
$7$	$ T $ T
$11$	$ T - 2 $ T - 2
$13$	$ T + 1 $ T + 1
$17$	$ T - 6 $ T - 6
$19$	$ T + 4 $ T + 4
$23$	$ T - 6 $ T - 6
$29$	$ T - 2 $ T - 2
$31$	$ T + 3 $ T + 3
$37$	$ T - 3 $ T - 3
$41$	$ T - 2 $ T - 2
$43$	$ T + 1 $ T + 1
$47$	$ T + 6 $ T + 6
$53$	$ T + 6 $ T + 6
$59$	$ T + 6 $ T + 6
$61$	$ T - 5 $ T - 5
$67$	$ T - 7 $ T - 7
$71$	$ T $ T
$73$	$ T - 6 $ T - 6
$79$	$ T - 11 $ T - 11
$83$	$ T + 6 $ T + 6
$89$	$ T - 4 $ T - 4
$97$	$ T + 9 $ T + 9
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Classical	Maass
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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