LMFDB - Newform orbit 200.8.a.d

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [200,8,Mod(1,200)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("200.1"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-9,0,0,0,694] 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:	$62.4770050968$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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} + 694 q^{7} - 2106 q^{9} + 4901 q^{11} - 7228 q^{13} - 15925 q^{17} + 8749 q^{19} - 6246 q^{21} + 53554 q^{23} + 38637 q^{27} - 98752 q^{29} - 65030 q^{31} - 44109 q^{33} + 323958 q^{37} + 65052 q^{39}+ \cdots - 10321506 q^{99}+O(q^{100}) $

q - 9 * q^3 + 694 * q^7 - 2106 * q^9 + 4901 * q^11 - 7228 * q^13 - 15925 * q^17 + 8749 * q^19 - 6246 * q^21 + 53554 * q^23 + 38637 * q^27 - 98752 * q^29 - 65030 * q^31 - 44109 * q^33 + 323958 * q^37 + 65052 * q^39 + 170277 * q^41 - 226228 * q^43 - 399932 * q^47 - 341907 * q^49 + 143325 * q^51 + 217218 * q^53 - 78741 * q^57 - 2733128 * q^59 - 1410230 * q^61 - 1461564 * q^63 + 3163693 * q^67 - 481986 * q^69 - 3458532 * q^71 - 4113051 * q^73 + 3401294 * q^77 - 34386 * q^79 + 4258089 * q^81 - 1798039 * q^83 + 888768 * q^87 - 12630321 * q^89 - 5016232 * q^91 + 585270 * q^93 + 7118942 * q^97 - 10321506 * 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

694.000

−2106.00

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$5$	$ +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	200.8.a.d	✓	1
4.b	odd	2	1		400.8.a.k		1
5.b	even	2	1		200.8.a.e	yes	1
5.c	odd	4	2		200.8.c.e		2
20.d	odd	2	1		400.8.a.i		1
20.e	even	4	2		400.8.c.k		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.8.a.d	✓	1	1.a	even	1	1	trivial
200.8.a.e	yes	1	5.b	even	2	1
200.8.c.e		2	5.c	odd	4	2
400.8.a.i		1	20.d	odd	2	1
400.8.a.k		1	4.b	odd	2	1
400.8.c.k		2	20.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 9 $ T3 + 9 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(200))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 9 $ T + 9
$5$	$ T $ T
$7$	$ T - 694 $ T - 694
$11$	$ T - 4901 $ T - 4901
$13$	$ T + 7228 $ T + 7228
$17$	$ T + 15925 $ T + 15925
$19$	$ T - 8749 $ T - 8749
$23$	$ T - 53554 $ T - 53554
$29$	$ T + 98752 $ T + 98752
$31$	$ T + 65030 $ T + 65030
$37$	$ T - 323958 $ T - 323958
$41$	$ T - 170277 $ T - 170277
$43$	$ T + 226228 $ T + 226228
$47$	$ T + 399932 $ T + 399932
$53$	$ T - 217218 $ T - 217218
$59$	$ T + 2733128 $ T + 2733128
$61$	$ T + 1410230 $ T + 1410230
$67$	$ T - 3163693 $ T - 3163693
$71$	$ T + 3458532 $ T + 3458532
$73$	$ T + 4113051 $ T + 4113051
$79$	$ T + 34386 $ T + 34386
$83$	$ T + 1798039 $ T + 1798039
$89$	$ T + 12630321 $ T + 12630321
$97$	$ T - 7118942 $ T - 7118942
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Level:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

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