LMFDB - Newform orbit 560.6.a.g

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 = [1,0,12,0,25,0,49,0,-99] 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:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
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 + 12 q^{3} + 25 q^{5} + 49 q^{7} - 99 q^{9} - 556 q^{11} - 354 q^{13} + 300 q^{15} + 770 q^{17} + 2684 q^{19} + 588 q^{21} + 1528 q^{23} + 625 q^{25} - 4104 q^{27} - 2418 q^{29} - 7840 q^{31} - 6672 q^{33}+ \cdots + 55044 q^{99}+O(q^{100}) $

q + 12 * q^3 + 25 * q^5 + 49 * q^7 - 99 * q^9 - 556 * q^11 - 354 * q^13 + 300 * q^15 + 770 * q^17 + 2684 * q^19 + 588 * q^21 + 1528 * q^23 + 625 * q^25 - 4104 * q^27 - 2418 * q^29 - 7840 * q^31 - 6672 * q^33 + 1225 * q^35 - 314 * q^37 - 4248 * q^39 - 17878 * q^41 - 16476 * q^43 - 2475 * q^45 - 5376 * q^47 + 2401 * q^49 + 9240 * q^51 + 1654 * q^53 - 13900 * q^55 + 32208 * q^57 + 29492 * q^59 + 27630 * q^61 - 4851 * q^63 - 8850 * q^65 - 57716 * q^67 + 18336 * q^69 - 70648 * q^71 + 74202 * q^73 + 7500 * q^75 - 27244 * q^77 - 74336 * q^79 - 25191 * q^81 - 44068 * q^83 + 19250 * q^85 - 29016 * q^87 + 129306 * q^89 - 17346 * q^91 - 94080 * q^93 + 67100 * q^95 - 137646 * q^97 + 55044 * 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

12.0000

25.0000

49.0000

−99.0000

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.g		1
4.b	odd	2	1		280.6.a.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
280.6.a.a	✓	1	4.b	odd	2	1
560.6.a.g		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 12 $ T3 - 12 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(560))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 12 $ T - 12
$5$	$ T - 25 $ T - 25
$7$	$ T - 49 $ T - 49
$11$	$ T + 556 $ T + 556
$13$	$ T + 354 $ T + 354
$17$	$ T - 770 $ T - 770
$19$	$ T - 2684 $ T - 2684
$23$	$ T - 1528 $ T - 1528
$29$	$ T + 2418 $ T + 2418
$31$	$ T + 7840 $ T + 7840
$37$	$ T + 314 $ T + 314
$41$	$ T + 17878 $ T + 17878
$43$	$ T + 16476 $ T + 16476
$47$	$ T + 5376 $ T + 5376
$53$	$ T - 1654 $ T - 1654
$59$	$ T - 29492 $ T - 29492
$61$	$ T - 27630 $ T - 27630
$67$	$ T + 57716 $ T + 57716
$71$	$ T + 70648 $ T + 70648
$73$	$ T - 74202 $ T - 74202
$79$	$ T + 74336 $ T + 74336
$83$	$ T + 44068 $ T + 44068
$89$	$ T - 129306 $ T - 129306
$97$	$ T + 137646 $ T + 137646
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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 80-90
Significant digits:
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