LMFDB - Newform orbit 200.8.a.f

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 = [1,0,34,0,0,0,106] 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:	no (minimal twist has level 40)
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 + 34 q^{3} + 106 q^{7} - 1031 q^{9} - 1324 q^{11} + 8828 q^{13} - 24000 q^{17} - 4876 q^{19} + 3604 q^{21} + 46646 q^{23} - 109412 q^{27} - 110902 q^{29} - 247680 q^{31} - 45016 q^{33} + 360092 q^{37}+ \cdots + 1365044 q^{99}+O(q^{100})

q + 34 * q^3 + 106 * q^7 - 1031 * q^9 - 1324 * q^11 + 8828 * q^13 - 24000 * q^17 - 4876 * q^19 + 3604 * q^21 + 46646 * q^23 - 109412 * q^27 - 110902 * q^29 - 247680 * q^31 - 45016 * q^33 + 360092 * q^37 + 300152 * q^39 + 104402 * q^41 - 713622 * q^43 + 156882 * q^47 - 812307 * q^49 - 816000 * q^51 - 1066268 * q^53 - 165784 * q^57 + 832572 * q^59 + 529070 * q^61 - 109286 * q^63 - 4174418 * q^67 + 1585964 * q^69 + 5176568 * q^71 + 237976 * q^73 - 140344 * q^77 - 3742736 * q^79 - 1465211 * q^81 - 7861886 * q^83 - 3770668 * q^87 + 4300854 * q^89 + 935768 * q^91 - 8421120 * q^93 - 1147792 * q^97 + 1365044 * 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

34.0000

106.000

−1031.00

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.f		1
4.b	odd	2	1		400.8.a.h		1
5.b	even	2	1		200.8.a.c		1
5.c	odd	4	2		40.8.c.a	✓	2
15.e	even	4	2		360.8.f.a		2
20.d	odd	2	1		400.8.a.n		1
20.e	even	4	2		80.8.c.b		2
40.i	odd	4	2		320.8.c.b		2
40.k	even	4	2		320.8.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
40.8.c.a	✓	2	5.c	odd	4	2
80.8.c.b		2	20.e	even	4	2
200.8.a.c		1	5.b	even	2	1
200.8.a.f		1	1.a	even	1	1	trivial
320.8.c.a		2	40.k	even	4	2
320.8.c.b		2	40.i	odd	4	2
360.8.f.a		2	15.e	even	4	2
400.8.a.h		1	4.b	odd	2	1
400.8.a.n		1	20.d	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3} - 34$ T3 - 34 acting on $S_{8}^{\mathrm{new}}(\Gamma_0(200))$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T$ T
$3$	$T - 34$ T - 34
$5$	$T$ T
$7$	$T - 106$ T - 106
$11$	$T + 1324$ T + 1324
$13$	$T - 8828$ T - 8828
$17$	$T + 24000$ T + 24000
$19$	$T + 4876$ T + 4876
$23$	$T - 46646$ T - 46646
$29$	$T + 110902$ T + 110902
$31$	$T + 247680$ T + 247680
$37$	$T - 360092$ T - 360092
$41$	$T - 104402$ T - 104402
$43$	$T + 713622$ T + 713622
$47$	$T - 156882$ T - 156882
$53$	$T + 1066268$ T + 1066268
$59$	$T - 832572$ T - 832572
$61$	$T - 529070$ T - 529070
$67$	$T + 4174418$ T + 4174418
$71$	$T - 5176568$ T - 5176568
$73$	$T - 237976$ T - 237976
$79$	$T + 3742736$ T + 3742736
$83$	$T + 7861886$ T + 7861886
$89$	$T - 4300854$ T - 4300854
$97$	$T + 1147792$ T + 1147792
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