LMFDB - Newform orbit 448.8.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,8,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-46,0,160,0,-343] 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:	$139.948491417$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
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 - 46 q^{3} + 160 q^{5} - 343 q^{7} - 71 q^{9} + 6840 q^{11} + 2900 q^{13} - 7360 q^{15} + 16566 q^{17} + 6718 q^{19} + 15778 q^{21} - 976 q^{23} - 52525 q^{25} + 103868 q^{27} + 61662 q^{29} - 69236 q^{31}+ \cdots - 485640 q^{99}+O(q^{100}) $

q - 46 * q^3 + 160 * q^5 - 343 * q^7 - 71 * q^9 + 6840 * q^11 + 2900 * q^13 - 7360 * q^15 + 16566 * q^17 + 6718 * q^19 + 15778 * q^21 - 976 * q^23 - 52525 * q^25 + 103868 * q^27 + 61662 * q^29 - 69236 * q^31 - 314640 * q^33 - 54880 * q^35 + 533062 * q^37 - 133400 * q^39 + 183158 * q^41 - 966864 * q^43 - 11360 * q^45 - 190268 * q^47 + 117649 * q^49 - 762036 * q^51 + 785010 * q^53 + 1094400 * q^55 - 309028 * q^57 - 2893594 * q^59 + 95896 * q^61 + 24353 * q^63 + 464000 * q^65 + 991644 * q^67 + 44896 * q^69 + 1068160 * q^71 + 2523458 * q^73 + 2416150 * q^75 - 2346120 * q^77 + 285848 * q^79 - 4622651 * q^81 - 7094938 * q^83 + 2650560 * q^85 - 2836452 * q^87 - 252390 * q^89 - 994700 * q^91 + 3184856 * q^93 + 1074880 * q^95 - 1824794 * q^97 - 485640 * 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

−46.0000

160.000

−343.000

−71.0000

Atkin-Lehner signs

$ p $	Sign
$2$	$ +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	448.8.a.c		1
4.b	odd	2	1		448.8.a.h		1
8.b	even	2	1		56.8.a.b	✓	1
8.d	odd	2	1		112.8.a.a		1
24.h	odd	2	1		504.8.a.b		1
56.h	odd	2	1		392.8.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
56.8.a.b	✓	1	8.b	even	2	1
112.8.a.a		1	8.d	odd	2	1
392.8.a.a		1	56.h	odd	2	1
448.8.a.c		1	1.a	even	1	1	trivial
448.8.a.h		1	4.b	odd	2	1
504.8.a.b		1	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 46 $ T3 + 46 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(448))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 46 $ T + 46
$5$	$ T - 160 $ T - 160
$7$	$ T + 343 $ T + 343
$11$	$ T - 6840 $ T - 6840
$13$	$ T - 2900 $ T - 2900
$17$	$ T - 16566 $ T - 16566
$19$	$ T - 6718 $ T - 6718
$23$	$ T + 976 $ T + 976
$29$	$ T - 61662 $ T - 61662
$31$	$ T + 69236 $ T + 69236
$37$	$ T - 533062 $ T - 533062
$41$	$ T - 183158 $ T - 183158
$43$	$ T + 966864 $ T + 966864
$47$	$ T + 190268 $ T + 190268
$53$	$ T - 785010 $ T - 785010
$59$	$ T + 2893594 $ T + 2893594
$61$	$ T - 95896 $ T - 95896
$67$	$ T - 991644 $ T - 991644
$71$	$ T - 1068160 $ T - 1068160
$73$	$ T - 2523458 $ T - 2523458
$79$	$ T - 285848 $ T - 285848
$83$	$ T + 7094938 $ T + 7094938
$89$	$ T + 252390 $ T + 252390
$97$	$ T + 1824794 $ T + 1824794
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Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

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