LMFDB - Newform orbit 528.8.a.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [528,8,Mod(1,528)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(528, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("528.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	528.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:	$164.939293456$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
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 - 27 q^{3} - 410 q^{5} + 1028 q^{7} + 729 q^{9}+O(q^{10}) $

$ q - 27 q^{3} - 410 q^{5} + 1028 q^{7} + 729 q^{9} + 1331 q^{11} + 12958 q^{13} + 11070 q^{15} + 17062 q^{17} + 54168 q^{19} - 27756 q^{21} + 11488 q^{23} + 89975 q^{25} - 19683 q^{27} - 186654 q^{29} + 188672 q^{31} - 35937 q^{33} - 421480 q^{35} + 395886 q^{37} - 349866 q^{39} - 47546 q^{41} - 602088 q^{43} - 298890 q^{45} + 647200 q^{47} + 233241 q^{49} - 460674 q^{51} - 1312722 q^{53} - 545710 q^{55} - 1462536 q^{57} + 2681140 q^{59} + 551190 q^{61} + 749412 q^{63} - 5312780 q^{65} - 459260 q^{67} - 310176 q^{69} + 18072 q^{71} - 426062 q^{73} - 2429325 q^{75} + 1368268 q^{77} - 297764 q^{79} + 531441 q^{81} - 5684028 q^{83} - 6995420 q^{85} + 5039658 q^{87} - 6342966 q^{89} + 13320824 q^{91} - 5094144 q^{93} - 22208880 q^{95} + 16651586 q^{97} + 970299 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

−27.0000

−410.000

1028.00

729.000

Atkin-Lehner signs

$ p $	Sign
$2$	$-1$
$3$	$1$
$11$	$-1$

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	528.8.a.a		1
4.b	odd	2	1		33.8.a.a	✓	1
12.b	even	2	1		99.8.a.a		1
44.c	even	2	1		363.8.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
33.8.a.a	✓	1	4.b	odd	2	1
99.8.a.a		1	12.b	even	2	1
363.8.a.a		1	44.c	even	2	1
528.8.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_{5} + 410 $ T5 + 410 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(528))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 27 $ T + 27
$5$	$ T + 410 $ T + 410
$7$	$ T - 1028 $ T - 1028
$11$	$ T - 1331 $ T - 1331
$13$	$ T - 12958 $ T - 12958
$17$	$ T - 17062 $ T - 17062
$19$	$ T - 54168 $ T - 54168
$23$	$ T - 11488 $ T - 11488
$29$	$ T + 186654 $ T + 186654
$31$	$ T - 188672 $ T - 188672
$37$	$ T - 395886 $ T - 395886
$41$	$ T + 47546 $ T + 47546
$43$	$ T + 602088 $ T + 602088
$47$	$ T - 647200 $ T - 647200
$53$	$ T + 1312722 $ T + 1312722
$59$	$ T - 2681140 $ T - 2681140
$61$	$ T - 551190 $ T - 551190
$67$	$ T + 459260 $ T + 459260
$71$	$ T - 18072 $ T - 18072
$73$	$ T + 426062 $ T + 426062
$79$	$ T + 297764 $ T + 297764
$83$	$ T + 5684028 $ T + 5684028
$89$	$ T + 6342966 $ T + 6342966
$97$	$ T - 16651586 $ T - 16651586
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	528.a (trivial)

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