LMFDB - Newform orbit 560.6.a.f

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,6,Mod(1,560)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("560.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,11,0,-25,0,-49,0,-122] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$89.8149390953$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
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 + 11 q^{3} - 25 q^{5} - 49 q^{7} - 122 q^{9} + 267 q^{11} - 1087 q^{13} - 275 q^{15} - 513 q^{17} + 802 q^{19} - 539 q^{21} + 1290 q^{23} + 625 q^{25} - 4015 q^{27} + 1779 q^{29} + 2584 q^{31} + 2937 q^{33}+ \cdots - 32574 q^{99}+O(q^{100}) $

q + 11 * q^3 - 25 * q^5 - 49 * q^7 - 122 * q^9 + 267 * q^11 - 1087 * q^13 - 275 * q^15 - 513 * q^17 + 802 * q^19 - 539 * q^21 + 1290 * q^23 + 625 * q^25 - 4015 * q^27 + 1779 * q^29 + 2584 * q^31 + 2937 * q^33 + 1225 * q^35 + 13862 * q^37 - 11957 * q^39 - 11904 * q^41 + 598 * q^43 + 3050 * q^45 + 17019 * q^47 + 2401 * q^49 - 5643 * q^51 + 27852 * q^53 - 6675 * q^55 + 8822 * q^57 - 30912 * q^59 - 1780 * q^61 + 5978 * q^63 + 27175 * q^65 - 25052 * q^67 + 14190 * q^69 + 51984 * q^71 + 47690 * q^73 + 6875 * q^75 - 13083 * q^77 + 102121 * q^79 - 14519 * q^81 + 83676 * q^83 + 12825 * q^85 + 19569 * q^87 - 32400 * q^89 + 53263 * q^91 + 28424 * q^93 - 20050 * q^95 - 148645 * q^97 - 32574 * q^99

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

11.0000

−25.0000

−49.0000

−122.000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ +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	560.6.a.f		1
4.b	odd	2	1		70.6.a.f	✓	1
12.b	even	2	1		630.6.a.e		1
20.d	odd	2	1		350.6.a.d		1
20.e	even	4	2		350.6.c.b		2
28.d	even	2	1		490.6.a.l		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.6.a.f	✓	1	4.b	odd	2	1
350.6.a.d		1	20.d	odd	2	1
350.6.c.b		2	20.e	even	4	2
490.6.a.l		1	28.d	even	2	1
560.6.a.f		1	1.a	even	1	1	trivial
630.6.a.e		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 11 $ T3 - 11 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(560))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 11 $ T - 11
$5$	$ T + 25 $ T + 25
$7$	$ T + 49 $ T + 49
$11$	$ T - 267 $ T - 267
$13$	$ T + 1087 $ T + 1087
$17$	$ T + 513 $ T + 513
$19$	$ T - 802 $ T - 802
$23$	$ T - 1290 $ T - 1290
$29$	$ T - 1779 $ T - 1779
$31$	$ T - 2584 $ T - 2584
$37$	$ T - 13862 $ T - 13862
$41$	$ T + 11904 $ T + 11904
$43$	$ T - 598 $ T - 598
$47$	$ T - 17019 $ T - 17019
$53$	$ T - 27852 $ T - 27852
$59$	$ T + 30912 $ T + 30912
$61$	$ T + 1780 $ T + 1780
$67$	$ T + 25052 $ T + 25052
$71$	$ T - 51984 $ T - 51984
$73$	$ T - 47690 $ T - 47690
$79$	$ T - 102121 $ T - 102121
$83$	$ T - 83676 $ T - 83676
$89$	$ T + 32400 $ T + 32400
$97$	$ T + 148645 $ T + 148645
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Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

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