LMFDB - Newform orbit 1521.4.a.d

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:	$0$
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 - 3 q^{2} + q^{4} + 9 q^{5} + 15 q^{7} + 21 q^{8}+O(q^{10}) $

$ q - 3 q^{2} + q^{4} + 9 q^{5} + 15 q^{7} + 21 q^{8} - 27 q^{10} + 48 q^{11} - 45 q^{14} - 71 q^{16} - 45 q^{17} + 6 q^{19} + 9 q^{20} - 144 q^{22} + 162 q^{23} - 44 q^{25} + 15 q^{28} + 144 q^{29} + 264 q^{31} + 45 q^{32} + 135 q^{34} + 135 q^{35} + 303 q^{37} - 18 q^{38} + 189 q^{40} + 192 q^{41} + 97 q^{43} + 48 q^{44} - 486 q^{46} - 111 q^{47} - 118 q^{49} + 132 q^{50} + 414 q^{53} + 432 q^{55} + 315 q^{56} - 432 q^{58} - 522 q^{59} + 376 q^{61} - 792 q^{62} + 433 q^{64} - 36 q^{67} - 45 q^{68} - 405 q^{70} - 357 q^{71} - 1098 q^{73} - 909 q^{74} + 6 q^{76} + 720 q^{77} - 830 q^{79} - 639 q^{80} - 576 q^{82} + 438 q^{83} - 405 q^{85} - 291 q^{86} + 1008 q^{88} + 438 q^{89} + 162 q^{92} + 333 q^{94} + 54 q^{95} - 852 q^{97} + 354 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

−3.00000

1.00000

9.00000

15.0000

21.0000

−27.0000

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.d		1
3.b	odd	2	1		169.4.a.c		1
13.b	even	2	1		1521.4.a.i		1
13.d	odd	4	2		117.4.b.a		2
39.d	odd	2	1		169.4.a.b		1
39.f	even	4	2		13.4.b.a	✓	2
39.h	odd	6	2		169.4.c.c		2
39.i	odd	6	2		169.4.c.b		2
39.k	even	12	4		169.4.e.d		4
156.l	odd	4	2		208.4.f.b		2
195.j	odd	4	2		325.4.d.b		2
195.n	even	4	2		325.4.c.b		2
195.u	odd	4	2		325.4.d.a		2
312.w	odd	4	2		832.4.f.c		2
312.y	even	4	2		832.4.f.e		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.b.a	✓	2	39.f	even	4	2
117.4.b.a		2	13.d	odd	4	2
169.4.a.b		1	39.d	odd	2	1
169.4.a.c		1	3.b	odd	2	1
169.4.c.b		2	39.i	odd	6	2
169.4.c.c		2	39.h	odd	6	2
169.4.e.d		4	39.k	even	12	4
208.4.f.b		2	156.l	odd	4	2
325.4.c.b		2	195.n	even	4	2
325.4.d.a		2	195.u	odd	4	2
325.4.d.b		2	195.j	odd	4	2
832.4.f.c		2	312.w	odd	4	2
832.4.f.e		2	312.y	even	4	2
1521.4.a.d		1	1.a	even	1	1	trivial
1521.4.a.i		1	13.b	even	2	1

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} + 3 $

$ T_{5} - 9 $

$ T_{7} - 15 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 3 $ T + 3
$3$	$ T $ T
$5$	$ T - 9 $ T - 9
$7$	$ T - 15 $ T - 15
$11$	$ T - 48 $ T - 48
$13$	$ T $ T
$17$	$ T + 45 $ T + 45
$19$	$ T - 6 $ T - 6
$23$	$ T - 162 $ T - 162
$29$	$ T - 144 $ T - 144
$31$	$ T - 264 $ T - 264
$37$	$ T - 303 $ T - 303
$41$	$ T - 192 $ T - 192
$43$	$ T - 97 $ T - 97
$47$	$ T + 111 $ T + 111
$53$	$ T - 414 $ T - 414
$59$	$ T + 522 $ T + 522
$61$	$ T - 376 $ T - 376
$67$	$ T + 36 $ T + 36
$71$	$ T + 357 $ T + 357
$73$	$ T + 1098 $ T + 1098
$79$	$ T + 830 $ T + 830
$83$	$ T - 438 $ T - 438
$89$	$ T - 438 $ T - 438
$97$	$ T + 852 $ T + 852
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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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