LMFDB - Newform orbit 560.4.a.n

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,4,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, 4); 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, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,0,5,0,-5,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.0410696032$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 140)
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 + 5 q^{3} - 5 q^{5} - 7 q^{7} - 2 q^{9} - 15 q^{11} + 17 q^{13} - 25 q^{15} + 123 q^{17} - 86 q^{19} - 35 q^{21} - 54 q^{23} + 25 q^{25} - 145 q^{27} - 177 q^{29} - 212 q^{31} - 75 q^{33} + 35 q^{35}+ \cdots + 30 q^{99}+O(q^{100}) $

q + 5 * q^3 - 5 * q^5 - 7 * q^7 - 2 * q^9 - 15 * q^11 + 17 * q^13 - 25 * q^15 + 123 * q^17 - 86 * q^19 - 35 * q^21 - 54 * q^23 + 25 * q^25 - 145 * q^27 - 177 * q^29 - 212 * q^31 - 75 * q^33 + 35 * q^35 + 74 * q^37 + 85 * q^39 - 444 * q^41 + 46 * q^43 + 10 * q^45 - 471 * q^47 + 49 * q^49 + 615 * q^51 - 180 * q^53 + 75 * q^55 - 430 * q^57 - 144 * q^59 - 376 * q^61 + 14 * q^63 - 85 * q^65 - 356 * q^67 - 270 * q^69 + 48 * q^71 + 818 * q^73 + 125 * q^75 + 105 * q^77 - 89 * q^79 - 671 * q^81 + 780 * q^83 - 615 * q^85 - 885 * q^87 + 1140 * q^89 - 119 * q^91 - 1060 * q^93 + 430 * q^95 - 169 * q^97 + 30 * 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

5.00000

−5.00000

−7.00000

−2.00000

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	560.4.a.n		1
4.b	odd	2	1		140.4.a.a	✓	1
8.b	even	2	1		2240.4.a.i		1
8.d	odd	2	1		2240.4.a.bf		1
12.b	even	2	1		1260.4.a.k		1
20.d	odd	2	1		700.4.a.k		1
20.e	even	4	2		700.4.e.e		2
28.d	even	2	1		980.4.a.m		1
28.f	even	6	2		980.4.i.c		2
28.g	odd	6	2		980.4.i.p		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.4.a.a	✓	1	4.b	odd	2	1
560.4.a.n		1	1.a	even	1	1	trivial
700.4.a.k		1	20.d	odd	2	1
700.4.e.e		2	20.e	even	4	2
980.4.a.m		1	28.d	even	2	1
980.4.i.c		2	28.f	even	6	2
980.4.i.p		2	28.g	odd	6	2
1260.4.a.k		1	12.b	even	2	1
2240.4.a.i		1	8.b	even	2	1
2240.4.a.bf		1	8.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(560))$:

$ T_{3} - 5 $

T3 - 5

$ T_{11} + 15 $

T11 + 15

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 5 $ T - 5
$5$	$ T + 5 $ T + 5
$7$	$ T + 7 $ T + 7
$11$	$ T + 15 $ T + 15
$13$	$ T - 17 $ T - 17
$17$	$ T - 123 $ T - 123
$19$	$ T + 86 $ T + 86
$23$	$ T + 54 $ T + 54
$29$	$ T + 177 $ T + 177
$31$	$ T + 212 $ T + 212
$37$	$ T - 74 $ T - 74
$41$	$ T + 444 $ T + 444
$43$	$ T - 46 $ T - 46
$47$	$ T + 471 $ T + 471
$53$	$ T + 180 $ T + 180
$59$	$ T + 144 $ T + 144
$61$	$ T + 376 $ T + 376
$67$	$ T + 356 $ T + 356
$71$	$ T - 48 $ T - 48
$73$	$ T - 818 $ T - 818
$79$	$ T + 89 $ T + 89
$83$	$ T - 780 $ T - 780
$89$	$ T - 1140 $ T - 1140
$97$	$ T + 169 $ T + 169
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Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

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