LMFDB - Newform orbit 400.6.a.k

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
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 + 14 q^{3} + 158 q^{7} - 47 q^{9}+O(q^{10}) $

$ q + 14 q^{3} + 158 q^{7} - 47 q^{9} + 148 q^{11} - 684 q^{13} - 2048 q^{17} - 2220 q^{19} + 2212 q^{21} - 1246 q^{23} - 4060 q^{27} - 270 q^{29} + 2048 q^{31} + 2072 q^{33} + 4372 q^{37} - 9576 q^{39} - 2398 q^{41} + 2294 q^{43} - 10682 q^{47} + 8157 q^{49} - 28672 q^{51} - 2964 q^{53} - 31080 q^{57} + 39740 q^{59} - 42298 q^{61} - 7426 q^{63} + 32098 q^{67} - 17444 q^{69} + 4248 q^{71} - 30104 q^{73} + 23384 q^{77} - 35280 q^{79} - 45419 q^{81} - 27826 q^{83} - 3780 q^{87} - 85210 q^{89} - 108072 q^{91} + 28672 q^{93} + 97232 q^{97} - 6956 q^{99}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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

14.0000

158.000

−47.0000

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	400.6.a.k		1
4.b	odd	2	1		50.6.a.e		1
5.b	even	2	1		400.6.a.c		1
5.c	odd	4	2		80.6.c.c		2
12.b	even	2	1		450.6.a.c		1
15.e	even	4	2		720.6.f.a		2
20.d	odd	2	1		50.6.a.c		1
20.e	even	4	2		10.6.b.a	✓	2
40.i	odd	4	2		320.6.c.a		2
40.k	even	4	2		320.6.c.b		2
60.h	even	2	1		450.6.a.w		1
60.l	odd	4	2		90.6.c.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.6.b.a	✓	2	20.e	even	4	2
50.6.a.c		1	20.d	odd	2	1
50.6.a.e		1	4.b	odd	2	1
80.6.c.c		2	5.c	odd	4	2
90.6.c.a		2	60.l	odd	4	2
320.6.c.a		2	40.i	odd	4	2
320.6.c.b		2	40.k	even	4	2
400.6.a.c		1	5.b	even	2	1
400.6.a.k		1	1.a	even	1	1	trivial
450.6.a.c		1	12.b	even	2	1
450.6.a.w		1	60.h	even	2	1
720.6.f.a		2	15.e	even	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} - 14 $ T3 - 14 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(400))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 14 $ T - 14
$5$	$ T $ T
$7$	$ T - 158 $ T - 158
$11$	$ T - 148 $ T - 148
$13$	$ T + 684 $ T + 684
$17$	$ T + 2048 $ T + 2048
$19$	$ T + 2220 $ T + 2220
$23$	$ T + 1246 $ T + 1246
$29$	$ T + 270 $ T + 270
$31$	$ T - 2048 $ T - 2048
$37$	$ T - 4372 $ T - 4372
$41$	$ T + 2398 $ T + 2398
$43$	$ T - 2294 $ T - 2294
$47$	$ T + 10682 $ T + 10682
$53$	$ T + 2964 $ T + 2964
$59$	$ T - 39740 $ T - 39740
$61$	$ T + 42298 $ T + 42298
$67$	$ T - 32098 $ T - 32098
$71$	$ T - 4248 $ T - 4248
$73$	$ T + 30104 $ T + 30104
$79$	$ T + 35280 $ T + 35280
$83$	$ T + 27826 $ T + 27826
$89$	$ T + 85210 $ T + 85210
$97$	$ T - 97232 $ T - 97232
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Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

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