LMFDB - Newform orbit 400.4.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$400 = 2^{4} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	400.c (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of $i = \sqrt{-1}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 9 i q^{3} - 26 i q^{7} - 54 q^{9} + 59 q^{11} - 28 i q^{13} + 5 i q^{17} + 109 q^{19} + 234 q^{21} - 194 i q^{23} - 243 i q^{27} + 32 q^{29} - 10 q^{31} + 531 i q^{33} - 198 i q^{37} + 252 q^{39} + 117 q^{41} + \cdots - 3186 q^{99} +O(q^{100})$ q + 9i q^3 - 26i q^7 - 54 * q^9 + 59 * q^11 - 28i q^13 + 5i q^17 + 109 * q^19 + 234 * q^21 - 194i q^23 - 243i q^27 + 32 * q^29 - 10 * q^31 + 531i q^33 - 198i q^37 + 252 * q^39 + 117 * q^41 + 388i q^43 + 68i q^47 - 333 * q^49 - 45 * q^51 + 18i q^53 + 981i q^57 + 392 * q^59 - 710 * q^61 + 1404i q^63 + 253i q^67 + 1746 * q^69 + 612 * q^71 + 549i q^73 - 1534i q^77 + 414 * q^79 + 729 * q^81 - 121i q^83 + 288i q^87 + 81 * q^89 - 728 * q^91 - 90i q^93 - 1502i q^97 - 3186 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 108 q^{9} + 118 q^{11} + 218 q^{19} + 468 q^{21} + 64 q^{29} - 20 q^{31} + 504 q^{39} + 234 q^{41} - 666 q^{49} - 90 q^{51} + 784 q^{59} - 1420 q^{61} + 3492 q^{69} + 1224 q^{71} + 828 q^{79} + 1458 q^{81}+ \cdots - 6372 q^{99}+O(q^{100})$ 2 * q - 108 * q^9 + 118 * q^11 + 218 * q^19 + 468 * q^21 + 64 * q^29 - 20 * q^31 + 504 * q^39 + 234 * q^41 - 666 * q^49 - 90 * q^51 + 784 * q^59 - 1420 * q^61 + 3492 * q^69 + 1224 * q^71 + 828 * q^79 + 1458 * q^81 + 162 * q^89 - 1456 * q^91 - 6372 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$ .

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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}

49.1

	−	1.00000i
		1.00000i

−

9.00000i

26.0000i

−54.0000

49.2

9.00000i

−

26.0000i

−54.0000

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	400.4.c.b		2
4.b	odd	2	1		200.4.c.b		2
5.b	even	2	1	inner	400.4.c.b		2
5.c	odd	4	1		400.4.a.a		1
5.c	odd	4	1		400.4.a.t		1
12.b	even	2	1		1800.4.f.w		2
20.d	odd	2	1		200.4.c.b		2
20.e	even	4	1		200.4.a.b	✓	1
20.e	even	4	1		200.4.a.j	yes	1
40.i	odd	4	1		1600.4.a.b		1
40.i	odd	4	1		1600.4.a.by		1
40.k	even	4	1		1600.4.a.c		1
40.k	even	4	1		1600.4.a.bz		1
60.h	even	2	1		1800.4.f.w		2
60.l	odd	4	1		1800.4.a.c		1
60.l	odd	4	1		1800.4.a.bh		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
200.4.a.b	✓	1	20.e	even	4	1
200.4.a.j	yes	1	20.e	even	4	1
200.4.c.b		2	4.b	odd	2	1
200.4.c.b		2	20.d	odd	2	1
400.4.a.a		1	5.c	odd	4	1
400.4.a.t		1	5.c	odd	4	1
400.4.c.b		2	1.a	even	1	1	trivial
400.4.c.b		2	5.b	even	2	1	inner
1600.4.a.b		1	40.i	odd	4	1
1600.4.a.c		1	40.k	even	4	1
1600.4.a.by		1	40.i	odd	4	1
1600.4.a.bz		1	40.k	even	4	1
1800.4.a.c		1	60.l	odd	4	1
1800.4.a.bh		1	60.l	odd	4	1
1800.4.f.w		2	12.b	even	2	1
1800.4.f.w		2	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(400, [\chi])$ :

T_{3}^{2} + 81

T_{7}^{2} + 676

T_{11} - 59

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 81$ T^2 + 81
$5$	$T^{2}$ T^2
$7$	$T^{2} + 676$ T^2 + 676
$11$	$(T - 59)^{2}$ (T - 59)^2
$13$	$T^{2} + 784$ T^2 + 784
$17$	$T^{2} + 25$ T^2 + 25
$19$	$(T - 109)^{2}$ (T - 109)^2
$23$	$T^{2} + 37636$ T^2 + 37636
$29$	$(T - 32)^{2}$ (T - 32)^2
$31$	$(T + 10)^{2}$ (T + 10)^2
$37$	$T^{2} + 39204$ T^2 + 39204
$41$	$(T - 117)^{2}$ (T - 117)^2
$43$	$T^{2} + 150544$ T^2 + 150544
$47$	$T^{2} + 4624$ T^2 + 4624
$53$	$T^{2} + 324$ T^2 + 324
$59$	$(T - 392)^{2}$ (T - 392)^2
$61$	$(T + 710)^{2}$ (T + 710)^2
$67$	$T^{2} + 64009$ T^2 + 64009
$71$	$(T - 612)^{2}$ (T - 612)^2
$73$	$T^{2} + 301401$ T^2 + 301401
$79$	$(T - 414)^{2}$ (T - 414)^2
$83$	$T^{2} + 14641$ T^2 + 14641
$89$	$(T - 81)^{2}$ (T - 81)^2
$97$	$T^{2} + 2256004$ T^2 + 2256004
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