LMFDB - Newform orbit 416.7.h.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,7,Mod(207,416)] 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("416.207"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	416.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [1,0,50,0,-218] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$95.7024987859$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$.

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$-1$

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} $

207.1

50.0000

−218.000

614.000

1771.00

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by $\Q(\sqrt{-26}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.7.h.a		1
4.b	odd	2	1		104.7.h.b	yes	1
8.b	even	2	1		104.7.h.a	✓	1
8.d	odd	2	1		416.7.h.b		1
13.b	even	2	1		416.7.h.b		1
52.b	odd	2	1		104.7.h.a	✓	1
104.e	even	2	1		104.7.h.b	yes	1
104.h	odd	2	1	CM	416.7.h.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.7.h.a	✓	1	8.b	even	2	1
104.7.h.a	✓	1	52.b	odd	2	1
104.7.h.b	yes	1	4.b	odd	2	1
104.7.h.b	yes	1	104.e	even	2	1
416.7.h.a		1	1.a	even	1	1	trivial
416.7.h.a		1	104.h	odd	2	1	CM
416.7.h.b		1	8.d	odd	2	1
416.7.h.b		1	13.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{7}^{\mathrm{new}}(416, [\chi])$:

$ T_{3} - 50 $

T3 - 50

$ T_{5} + 218 $

T5 + 218

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 50 $ T - 50
$5$	$ T + 218 $ T + 218
$7$	$ T - 614 $ T - 614
$11$	$ T $ T
$13$	$ T + 2197 $ T + 2197
$17$	$ T - 3170 $ T - 3170
$19$	$ T $ T
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T - 27830 $ T - 27830
$37$	$ T - 13894 $ T - 13894
$41$	$ T $ T
$43$	$ T - 111490 $ T - 111490
$47$	$ T + 128554 $ T + 128554
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T $ T
$67$	$ T $ T
$71$	$ T - 317990 $ T - 317990
$73$	$ T $ T
$79$	$ T $ T
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T $ T
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Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	416.h (of order \(2\), degree \(1\), not minimal)

\(n\)	\(261\)	\(287\)	\(353\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)

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