LMFDB - Newform orbit 882.6.a.i

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
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} + 16 q^{4} + 24 q^{5} - 64 q^{8}+O(q^{10}) $

$ q - 4 q^{2} + 16 q^{4} + 24 q^{5} - 64 q^{8} - 96 q^{10} - 66 q^{11} - 98 q^{13} + 256 q^{16} - 216 q^{17} + 340 q^{19} + 384 q^{20} + 264 q^{22} + 1038 q^{23} - 2549 q^{25} + 392 q^{26} + 2490 q^{29} + 7048 q^{31} - 1024 q^{32} + 864 q^{34} - 12238 q^{37} - 1360 q^{38} - 1536 q^{40} + 6468 q^{41} - 15412 q^{43} - 1056 q^{44} - 4152 q^{46} + 20604 q^{47} + 10196 q^{50} - 1568 q^{52} - 32490 q^{53} - 1584 q^{55} - 9960 q^{58} + 34224 q^{59} - 35654 q^{61} - 28192 q^{62} + 4096 q^{64} - 2352 q^{65} + 12680 q^{67} - 3456 q^{68} + 42642 q^{71} - 33734 q^{73} + 48952 q^{74} + 5440 q^{76} - 85108 q^{79} + 6144 q^{80} - 25872 q^{82} - 106764 q^{83} - 5184 q^{85} + 61648 q^{86} + 4224 q^{88} + 34884 q^{89} + 16608 q^{92} - 82416 q^{94} + 8160 q^{95} - 18662 q^{97}+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

16.0000

24.0000

−64.0000

−96.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$1$
$3$	$-1$
$7$	$-1$

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	882.6.a.i		1
3.b	odd	2	1		294.6.a.i		1
7.b	odd	2	1		126.6.a.b		1
21.c	even	2	1		42.6.a.f	✓	1
21.g	even	6	2		294.6.e.b		2
21.h	odd	6	2		294.6.e.f		2
28.d	even	2	1		1008.6.a.k		1
84.h	odd	2	1		336.6.a.g		1
105.g	even	2	1		1050.6.a.a		1
105.k	odd	4	2		1050.6.g.m		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.6.a.f	✓	1	21.c	even	2	1
126.6.a.b		1	7.b	odd	2	1
294.6.a.i		1	3.b	odd	2	1
294.6.e.b		2	21.g	even	6	2
294.6.e.f		2	21.h	odd	6	2
336.6.a.g		1	84.h	odd	2	1
882.6.a.i		1	1.a	even	1	1	trivial
1008.6.a.k		1	28.d	even	2	1
1050.6.a.a		1	105.g	even	2	1
1050.6.g.m		2	105.k	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{6}^{\mathrm{new}}(\Gamma_0(882))$:

$ T_{5} - 24 $

$ T_{11} + 66 $

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 4 $ T + 4
$3$	$ T $ T
$5$	$ T - 24 $ T - 24
$7$	$ T $ T
$11$	$ T + 66 $ T + 66
$13$	$ T + 98 $ T + 98
$17$	$ T + 216 $ T + 216
$19$	$ T - 340 $ T - 340
$23$	$ T - 1038 $ T - 1038
$29$	$ T - 2490 $ T - 2490
$31$	$ T - 7048 $ T - 7048
$37$	$ T + 12238 $ T + 12238
$41$	$ T - 6468 $ T - 6468
$43$	$ T + 15412 $ T + 15412
$47$	$ T - 20604 $ T - 20604
$53$	$ T + 32490 $ T + 32490
$59$	$ T - 34224 $ T - 34224
$61$	$ T + 35654 $ T + 35654
$67$	$ T - 12680 $ T - 12680
$71$	$ T - 42642 $ T - 42642
$73$	$ T + 33734 $ T + 33734
$79$	$ T + 85108 $ T + 85108
$83$	$ T + 106764 $ T + 106764
$89$	$ T - 34884 $ T - 34884
$97$	$ T + 18662 $ T + 18662
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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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

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