LMFDB - Newform orbit 400.10.a.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,10,Mod(1,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.1"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-156,0,0,0,-952,0,4653] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$206.014334466$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
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 - 156 q^{3} - 952 q^{7} + 4653 q^{9} + 56148 q^{11} - 178094 q^{13} + 247662 q^{17} - 315380 q^{19} + 148512 q^{21} + 204504 q^{23} + 2344680 q^{27} - 3840450 q^{29} + 1309408 q^{31} - 8759088 q^{33}+ \cdots + 261256644 q^{99}+O(q^{100}) $

q - 156 * q^3 - 952 * q^7 + 4653 * q^9 + 56148 * q^11 - 178094 * q^13 + 247662 * q^17 - 315380 * q^19 + 148512 * q^21 + 204504 * q^23 + 2344680 * q^27 - 3840450 * q^29 + 1309408 * q^31 - 8759088 * q^33 - 4307078 * q^37 + 27782664 * q^39 + 1512042 * q^41 + 33670604 * q^43 - 10581072 * q^47 - 39447303 * q^49 - 38635272 * q^51 - 16616214 * q^53 + 49199280 * q^57 - 112235100 * q^59 - 33197218 * q^61 - 4429656 * q^63 - 121372252 * q^67 - 31902624 * q^69 + 387172728 * q^71 - 255240074 * q^73 - 53452896 * q^77 - 492101840 * q^79 - 457355079 * q^81 - 457420236 * q^83 + 599110200 * q^87 - 31809510 * q^89 + 169545488 * q^91 - 204267648 * q^93 + 673532062 * q^97 + 261256644 * 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

−156.000

−952.000

4653.00

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	400.10.a.b		1
4.b	odd	2	1		50.10.a.c		1
5.b	even	2	1		16.10.a.d		1
5.c	odd	4	2		400.10.c.d		2
15.d	odd	2	1		144.10.a.d		1
20.d	odd	2	1		2.10.a.a	✓	1
20.e	even	4	2		50.10.b.a		2
40.e	odd	2	1		64.10.a.h		1
40.f	even	2	1		64.10.a.b		1
60.h	even	2	1		18.10.a.a		1
80.k	odd	4	2		256.10.b.g		2
80.q	even	4	2		256.10.b.e		2
140.c	even	2	1		98.10.a.c		1
140.p	odd	6	2		98.10.c.c		2
140.s	even	6	2		98.10.c.b		2
180.n	even	6	2		162.10.c.i		2
180.p	odd	6	2		162.10.c.b		2
220.g	even	2	1		242.10.a.a		1
260.g	odd	2	1		338.10.a.a		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
2.10.a.a	✓	1	20.d	odd	2	1
16.10.a.d		1	5.b	even	2	1
18.10.a.a		1	60.h	even	2	1
50.10.a.c		1	4.b	odd	2	1
50.10.b.a		2	20.e	even	4	2
64.10.a.b		1	40.f	even	2	1
64.10.a.h		1	40.e	odd	2	1
98.10.a.c		1	140.c	even	2	1
98.10.c.b		2	140.s	even	6	2
98.10.c.c		2	140.p	odd	6	2
144.10.a.d		1	15.d	odd	2	1
162.10.c.b		2	180.p	odd	6	2
162.10.c.i		2	180.n	even	6	2
242.10.a.a		1	220.g	even	2	1
256.10.b.e		2	80.q	even	4	2
256.10.b.g		2	80.k	odd	4	2
338.10.a.a		1	260.g	odd	2	1
400.10.a.b		1	1.a	even	1	1	trivial
400.10.c.d		2	5.c	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3} + 156 $ T3 + 156 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(400))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 156 $ T + 156
$5$	$ T $ T
$7$	$ T + 952 $ T + 952
$11$	$ T - 56148 $ T - 56148
$13$	$ T + 178094 $ T + 178094
$17$	$ T - 247662 $ T - 247662
$19$	$ T + 315380 $ T + 315380
$23$	$ T - 204504 $ T - 204504
$29$	$ T + 3840450 $ T + 3840450
$31$	$ T - 1309408 $ T - 1309408
$37$	$ T + 4307078 $ T + 4307078
$41$	$ T - 1512042 $ T - 1512042
$43$	$ T - 33670604 $ T - 33670604
$47$	$ T + 10581072 $ T + 10581072
$53$	$ T + 16616214 $ T + 16616214
$59$	$ T + 112235100 $ T + 112235100
$61$	$ T + 33197218 $ T + 33197218
$67$	$ T + 121372252 $ T + 121372252
$71$	$ T - 387172728 $ T - 387172728
$73$	$ T + 255240074 $ T + 255240074
$79$	$ T + 492101840 $ T + 492101840
$83$	$ T + 457420236 $ T + 457420236
$89$	$ T + 31809510 $ T + 31809510
$97$	$ T - 673532062 $ T - 673532062
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Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

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