LMFDB - Newform orbit 768.6.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,6,Mod(1,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.1"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	768.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,9,0,-80,0,-36,0,81,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$123.174773616$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 384)
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 + 9 q^{3} - 80 q^{5} - 36 q^{7} + 81 q^{9} - 36 q^{11} - 404 q^{13} - 720 q^{15} - 2 q^{17} + 2916 q^{19} - 324 q^{21} + 2808 q^{23} + 3275 q^{25} + 729 q^{27} + 4408 q^{29} - 5796 q^{31} - 324 q^{33}+ \cdots - 2916 q^{99}+O(q^{100}) $

q + 9 * q^3 - 80 * q^5 - 36 * q^7 + 81 * q^9 - 36 * q^11 - 404 * q^13 - 720 * q^15 - 2 * q^17 + 2916 * q^19 - 324 * q^21 + 2808 * q^23 + 3275 * q^25 + 729 * q^27 + 4408 * q^29 - 5796 * q^31 - 324 * q^33 + 2880 * q^35 - 2316 * q^37 - 3636 * q^39 - 6874 * q^41 + 17244 * q^43 - 6480 * q^45 + 5688 * q^47 - 15511 * q^49 - 18 * q^51 - 32072 * q^53 + 2880 * q^55 + 26244 * q^57 + 43308 * q^59 - 18012 * q^61 - 2916 * q^63 + 32320 * q^65 - 11628 * q^67 + 25272 * q^69 + 8712 * q^71 - 15846 * q^73 + 29475 * q^75 + 1296 * q^77 - 40356 * q^79 + 6561 * q^81 - 66204 * q^83 + 160 * q^85 + 39672 * q^87 - 66086 * q^89 + 14544 * q^91 - 52164 * q^93 - 233280 * q^95 - 119682 * q^97 - 2916 * 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

9.00000

−80.0000

−36.0000

81.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -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	768.6.a.h		1
4.b	odd	2	1		768.6.a.b		1
8.b	even	2	1		768.6.a.e		1
8.d	odd	2	1		768.6.a.k		1
16.e	even	4	2		384.6.d.d	yes	2
16.f	odd	4	2		384.6.d.c	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
384.6.d.c	✓	2	16.f	odd	4	2
384.6.d.d	yes	2	16.e	even	4	2
768.6.a.b		1	4.b	odd	2	1
768.6.a.e		1	8.b	even	2	1
768.6.a.h		1	1.a	even	1	1	trivial
768.6.a.k		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 $S_{6}^{\mathrm{new}}(\Gamma_0(768))$:

$ T_{5} + 80 $

T5 + 80

$ T_{7} + 36 $

T7 + 36

$ T_{11} + 36 $

T11 + 36

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 9 $ T - 9
$5$	$ T + 80 $ T + 80
$7$	$ T + 36 $ T + 36
$11$	$ T + 36 $ T + 36
$13$	$ T + 404 $ T + 404
$17$	$ T + 2 $ T + 2
$19$	$ T - 2916 $ T - 2916
$23$	$ T - 2808 $ T - 2808
$29$	$ T - 4408 $ T - 4408
$31$	$ T + 5796 $ T + 5796
$37$	$ T + 2316 $ T + 2316
$41$	$ T + 6874 $ T + 6874
$43$	$ T - 17244 $ T - 17244
$47$	$ T - 5688 $ T - 5688
$53$	$ T + 32072 $ T + 32072
$59$	$ T - 43308 $ T - 43308
$61$	$ T + 18012 $ T + 18012
$67$	$ T + 11628 $ T + 11628
$71$	$ T - 8712 $ T - 8712
$73$	$ T + 15846 $ T + 15846
$79$	$ T + 40356 $ T + 40356
$83$	$ T + 66204 $ T + 66204
$89$	$ T + 66086 $ T + 66086
$97$	$ T + 119682 $ T + 119682
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Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	768.a (trivial)

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