LMFDB - Newform orbit 270.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,4,Mod(1,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, 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("270.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	270.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:	$15.9305157015$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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 - 2 q^{2} + 4 q^{4} - 5 q^{5} - 34 q^{7} - 8 q^{8}+O(q^{10}) $

$ q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 34 q^{7} - 8 q^{8} + 10 q^{10} + 48 q^{11} - 70 q^{13} + 68 q^{14} + 16 q^{16} + 27 q^{17} + 119 q^{19} - 20 q^{20} - 96 q^{22} + 51 q^{23} + 25 q^{25} + 140 q^{26} - 136 q^{28} + 30 q^{29} - 133 q^{31} - 32 q^{32} - 54 q^{34} + 170 q^{35} + 218 q^{37} - 238 q^{38} + 40 q^{40} - 156 q^{41} - 88 q^{43} + 192 q^{44} - 102 q^{46} + 516 q^{47} + 813 q^{49} - 50 q^{50} - 280 q^{52} + 639 q^{53} - 240 q^{55} + 272 q^{56} - 60 q^{58} + 654 q^{59} + 461 q^{61} + 266 q^{62} + 64 q^{64} + 350 q^{65} + 182 q^{67} + 108 q^{68} - 340 q^{70} - 900 q^{71} + 704 q^{73} - 436 q^{74} + 476 q^{76} - 1632 q^{77} - 1375 q^{79} - 80 q^{80} + 312 q^{82} - 915 q^{83} - 135 q^{85} + 176 q^{86} - 384 q^{88} + 1116 q^{89} + 2380 q^{91} + 204 q^{92} - 1032 q^{94} - 595 q^{95} - 16 q^{97} - 1626 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

−2.00000

4.00000

−5.00000

−34.0000

−8.00000

10.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +1 $
$3$	$ +1 $
$5$	$ +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	270.4.a.a	✓	1
3.b	odd	2	1		270.4.a.k	yes	1
4.b	odd	2	1		2160.4.a.j		1
5.b	even	2	1		1350.4.a.bb		1
5.c	odd	4	2		1350.4.c.r		2
9.c	even	3	2		810.4.e.x		2
9.d	odd	6	2		810.4.e.d		2
12.b	even	2	1		2160.4.a.t		1
15.d	odd	2	1		1350.4.a.n		1
15.e	even	4	2		1350.4.c.c		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
270.4.a.a	✓	1	1.a	even	1	1	trivial
270.4.a.k	yes	1	3.b	odd	2	1
810.4.e.d		2	9.d	odd	6	2
810.4.e.x		2	9.c	even	3	2
1350.4.a.n		1	15.d	odd	2	1
1350.4.a.bb		1	5.b	even	2	1
1350.4.c.c		2	15.e	even	4	2
1350.4.c.r		2	5.c	odd	4	2
2160.4.a.j		1	4.b	odd	2	1
2160.4.a.t		1	12.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 $S_{4}^{\mathrm{new}}(\Gamma_0(270))$:

$ T_{7} + 34 $

$ T_{11} - 48 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 2 $ T + 2
$3$	$ T $ T
$5$	$ T + 5 $ T + 5
$7$	$ T + 34 $ T + 34
$11$	$ T - 48 $ T - 48
$13$	$ T + 70 $ T + 70
$17$	$ T - 27 $ T - 27
$19$	$ T - 119 $ T - 119
$23$	$ T - 51 $ T - 51
$29$	$ T - 30 $ T - 30
$31$	$ T + 133 $ T + 133
$37$	$ T - 218 $ T - 218
$41$	$ T + 156 $ T + 156
$43$	$ T + 88 $ T + 88
$47$	$ T - 516 $ T - 516
$53$	$ T - 639 $ T - 639
$59$	$ T - 654 $ T - 654
$61$	$ T - 461 $ T - 461
$67$	$ T - 182 $ T - 182
$71$	$ T + 900 $ T + 900
$73$	$ T - 704 $ T - 704
$79$	$ T + 1375 $ T + 1375
$83$	$ T + 915 $ T + 915
$89$	$ T - 1116 $ T - 1116
$97$	$ T + 16 $ T + 16
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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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

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