LMFDB - Newform orbit 560.6.a.c

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,-1,0,25,0,-49,0,-242] 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 35)
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 - q^{3} + 25 q^{5} - 49 q^{7} - 242 q^{9} + 453 q^{11} - 969 q^{13} - 25 q^{15} + 1637 q^{17} + 1550 q^{19} + 49 q^{21} + 1654 q^{23} + 625 q^{25} + 485 q^{27} - 4985 q^{29} - 1192 q^{31} - 453 q^{33}+ \cdots - 109626 q^{99}+O(q^{100}) $

q - q^3 + 25 * q^5 - 49 * q^7 - 242 * q^9 + 453 * q^11 - 969 * q^13 - 25 * q^15 + 1637 * q^17 + 1550 * q^19 + 49 * q^21 + 1654 * q^23 + 625 * q^25 + 485 * q^27 - 4985 * q^29 - 1192 * q^31 - 453 * q^33 - 1225 * q^35 - 11018 * q^37 + 969 * q^39 - 1728 * q^41 + 10814 * q^43 - 6050 * q^45 - 26237 * q^47 + 2401 * q^49 - 1637 * q^51 + 25936 * q^53 + 11325 * q^55 - 1550 * q^57 + 4580 * q^59 - 12488 * q^61 + 11858 * q^63 - 24225 * q^65 + 15848 * q^67 - 1654 * q^69 - 51792 * q^71 + 4846 * q^73 - 625 * q^75 - 22197 * q^77 - 62765 * q^79 + 58321 * q^81 + 23644 * q^83 + 40925 * q^85 + 4985 * q^87 - 147300 * q^89 + 47481 * q^91 + 1192 * q^93 + 38750 * q^95 - 8343 * q^97 - 109626 * 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

−1.00000

25.0000

−49.0000

−242.000

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.c		1
4.b	odd	2	1		35.6.a.a	✓	1
12.b	even	2	1		315.6.a.a		1
20.d	odd	2	1		175.6.a.a		1
20.e	even	4	2		175.6.b.b		2
28.d	even	2	1		245.6.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
35.6.a.a	✓	1	4.b	odd	2	1
175.6.a.a		1	20.d	odd	2	1
175.6.b.b		2	20.e	even	4	2
245.6.a.a		1	28.d	even	2	1
315.6.a.a		1	12.b	even	2	1
560.6.a.c		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 1 $ T3 + 1 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(560))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 1 $ T + 1
$5$	$ T - 25 $ T - 25
$7$	$ T + 49 $ T + 49
$11$	$ T - 453 $ T - 453
$13$	$ T + 969 $ T + 969
$17$	$ T - 1637 $ T - 1637
$19$	$ T - 1550 $ T - 1550
$23$	$ T - 1654 $ T - 1654
$29$	$ T + 4985 $ T + 4985
$31$	$ T + 1192 $ T + 1192
$37$	$ T + 11018 $ T + 11018
$41$	$ T + 1728 $ T + 1728
$43$	$ T - 10814 $ T - 10814
$47$	$ T + 26237 $ T + 26237
$53$	$ T - 25936 $ T - 25936
$59$	$ T - 4580 $ T - 4580
$61$	$ T + 12488 $ T + 12488
$67$	$ T - 15848 $ T - 15848
$71$	$ T + 51792 $ T + 51792
$73$	$ T - 4846 $ T - 4846
$79$	$ T + 62765 $ T + 62765
$83$	$ T - 23644 $ T - 23644
$89$	$ T + 147300 $ T + 147300
$97$	$ T + 8343 $ T + 8343
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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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