LMFDB - Newform orbit 1380.4.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,4,Mod(1,1380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1380.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.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:	$81.4226358079$
Analytic rank:	$1$
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 - 3 q^{3} - 5 q^{5} - 16 q^{7} + 9 q^{9}+O(q^{10}) $

$ q - 3 q^{3} - 5 q^{5} - 16 q^{7} + 9 q^{9} + 30 q^{11} - 28 q^{13} + 15 q^{15} + 78 q^{17} - 52 q^{19} + 48 q^{21} - 23 q^{23} + 25 q^{25} - 27 q^{27} - 252 q^{29} + 200 q^{31} - 90 q^{33} + 80 q^{35} + 146 q^{37} + 84 q^{39} + 438 q^{41} - 46 q^{43} - 45 q^{45} + 588 q^{47} - 87 q^{49} - 234 q^{51} + 438 q^{53} - 150 q^{55} + 156 q^{57} - 168 q^{59} - 586 q^{61} - 144 q^{63} + 140 q^{65} - 94 q^{67} + 69 q^{69} - 30 q^{71} - 466 q^{73} - 75 q^{75} - 480 q^{77} - 520 q^{79} + 81 q^{81} + 708 q^{83} - 390 q^{85} + 756 q^{87} - 600 q^{89} + 448 q^{91} - 600 q^{93} + 260 q^{95} - 340 q^{97} + 270 q^{99}+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

−5.00000

−16.0000

9.00000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$5$	$1$
$23$	$1$

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	1380.4.a.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1380.4.a.a	✓	1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7} + 16 $ T7 + 16 acting on $S_{4}^{\mathrm{new}}(\Gamma_0(1380))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 3 $ T + 3
$5$	$ T + 5 $ T + 5
$7$	$ T + 16 $ T + 16
$11$	$ T - 30 $ T - 30
$13$	$ T + 28 $ T + 28
$17$	$ T - 78 $ T - 78
$19$	$ T + 52 $ T + 52
$23$	$ T + 23 $ T + 23
$29$	$ T + 252 $ T + 252
$31$	$ T - 200 $ T - 200
$37$	$ T - 146 $ T - 146
$41$	$ T - 438 $ T - 438
$43$	$ T + 46 $ T + 46
$47$	$ T - 588 $ T - 588
$53$	$ T - 438 $ T - 438
$59$	$ T + 168 $ T + 168
$61$	$ T + 586 $ T + 586
$67$	$ T + 94 $ T + 94
$71$	$ T + 30 $ T + 30
$73$	$ T + 466 $ T + 466
$79$	$ T + 520 $ T + 520
$83$	$ T - 708 $ T - 708
$89$	$ T + 600 $ T + 600
$97$	$ T + 340 $ T + 340
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

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