LMFDB - Newform orbit 1521.4.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
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 + 4 q^{2} + 8 q^{4} + 17 q^{5} - 20 q^{7}+O(q^{10}) $

$ q + 4 q^{2} + 8 q^{4} + 17 q^{5} - 20 q^{7} + 68 q^{10} - 32 q^{11} - 80 q^{14} - 64 q^{16} + 13 q^{17} - 30 q^{19} + 136 q^{20} - 128 q^{22} - 78 q^{23} + 164 q^{25} - 160 q^{28} - 197 q^{29} + 74 q^{31} - 256 q^{32} + 52 q^{34} - 340 q^{35} + 227 q^{37} - 120 q^{38} - 165 q^{41} - 156 q^{43} - 256 q^{44} - 312 q^{46} - 162 q^{47} + 57 q^{49} + 656 q^{50} - 93 q^{53} - 544 q^{55} - 788 q^{58} - 864 q^{59} + 145 q^{61} + 296 q^{62} - 512 q^{64} - 862 q^{67} + 104 q^{68} - 1360 q^{70} + 654 q^{71} - 215 q^{73} + 908 q^{74} - 240 q^{76} + 640 q^{77} - 76 q^{79} - 1088 q^{80} - 660 q^{82} + 628 q^{83} + 221 q^{85} - 624 q^{86} - 266 q^{89} - 624 q^{92} - 648 q^{94} - 510 q^{95} - 238 q^{97} + 228 q^{98}+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

4.00000

8.00000

17.0000

−20.0000

68.0000

Atkin-Lehner signs

$ p $	Sign
$3$	$-1$
$13$	$1$

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	1521.4.a.k		1
3.b	odd	2	1		169.4.a.a		1
13.b	even	2	1		1521.4.a.b		1
13.e	even	6	2		117.4.g.c		2
39.d	odd	2	1		169.4.a.d		1
39.f	even	4	2		169.4.b.c		2
39.h	odd	6	2		13.4.c.a	✓	2
39.i	odd	6	2		169.4.c.d		2
39.k	even	12	4		169.4.e.c		4
156.r	even	6	2		208.4.i.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.c.a	✓	2	39.h	odd	6	2
117.4.g.c		2	13.e	even	6	2
169.4.a.a		1	3.b	odd	2	1
169.4.a.d		1	39.d	odd	2	1
169.4.b.c		2	39.f	even	4	2
169.4.c.d		2	39.i	odd	6	2
169.4.e.c		4	39.k	even	12	4
208.4.i.b		2	156.r	even	6	2
1521.4.a.b		1	13.b	even	2	1
1521.4.a.k		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 $S_{4}^{\mathrm{new}}(\Gamma_0(1521))$:

$ T_{2} - 4 $

$ T_{5} - 17 $

$ T_{7} + 20 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 4 $ T - 4
$3$	$ T $ T
$5$	$ T - 17 $ T - 17
$7$	$ T + 20 $ T + 20
$11$	$ T + 32 $ T + 32
$13$	$ T $ T
$17$	$ T - 13 $ T - 13
$19$	$ T + 30 $ T + 30
$23$	$ T + 78 $ T + 78
$29$	$ T + 197 $ T + 197
$31$	$ T - 74 $ T - 74
$37$	$ T - 227 $ T - 227
$41$	$ T + 165 $ T + 165
$43$	$ T + 156 $ T + 156
$47$	$ T + 162 $ T + 162
$53$	$ T + 93 $ T + 93
$59$	$ T + 864 $ T + 864
$61$	$ T - 145 $ T - 145
$67$	$ T + 862 $ T + 862
$71$	$ T - 654 $ T - 654
$73$	$ T + 215 $ T + 215
$79$	$ T + 76 $ T + 76
$83$	$ T - 628 $ T - 628
$89$	$ T + 266 $ T + 266
$97$	$ T + 238 $ T + 238
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Overview	Random
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

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