LMFDB - Newform orbit 1200.4.f.f

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(49,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1200.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1200 = 2^{4} \cdot 3 \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1200.f (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:	$70.8022920069$
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 600)
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.

$f(q)$	$=$	$q + 3 i q^{3} + 19 i q^{7} - 9 q^{9} - 22 q^{11} - i q^{13} - 58 i q^{17} - 53 q^{19} - 57 q^{21} + 58 i q^{23} - 27 i q^{27} - 22 q^{29} + 35 q^{31} - 66 i q^{33} - 270 i q^{37} + 3 q^{39} - 468 q^{41} + \cdots + 198 q^{99} +O(q^{100})$ q + 3i q^3 + 19i q^7 - 9 * q^9 - 22 * q^11 - i * q^13 - 58i q^17 - 53 * q^19 - 57 * q^21 + 58i q^23 - 27i q^27 - 22 * q^29 + 35 * q^31 - 66i q^33 - 270i q^37 + 3 * q^39 - 468 * q^41 - 431i q^43 + 230i q^47 - 18 * q^49 + 174 * q^51 - 159i q^57 + 446 * q^59 + 127 * q^61 - 171i q^63 + 811i q^67 - 174 * q^69 - 36 * q^71 - 522i q^73 - 418i q^77 + 1368 * q^79 + 81 * q^81 - 1138i q^83 - 66i q^87 - 144 * q^89 + 19 * q^91 + 105i q^93 - 1079i q^97 + 198 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 18 q^{9} - 44 q^{11} - 106 q^{19} - 114 q^{21} - 44 q^{29} + 70 q^{31} + 6 q^{39} - 936 q^{41} - 36 q^{49} + 348 q^{51} + 892 q^{59} + 254 q^{61} - 348 q^{69} - 72 q^{71} + 2736 q^{79} + 162 q^{81}+ \cdots + 396 q^{99}+O(q^{100})$ 2 * q - 18 * q^9 - 44 * q^11 - 106 * q^19 - 114 * q^21 - 44 * q^29 + 70 * q^31 + 6 * q^39 - 936 * q^41 - 36 * q^49 + 348 * q^51 + 892 * q^59 + 254 * q^61 - 348 * q^69 - 72 * q^71 + 2736 * q^79 + 162 * q^81 - 288 * q^89 + 38 * q^91 + 396 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$ .

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$-1$	$1$	$1$

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

−

3.00000i

−

19.0000i

−9.00000

49.2

3.00000i

19.0000i

−9.00000

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	1200.4.f.f		2
4.b	odd	2	1		600.4.f.h		2
5.b	even	2	1	inner	1200.4.f.f		2
5.c	odd	4	1		1200.4.a.n		1
5.c	odd	4	1		1200.4.a.x		1
12.b	even	2	1		1800.4.f.g		2
20.d	odd	2	1		600.4.f.h		2
20.e	even	4	1		600.4.a.g	✓	1
20.e	even	4	1		600.4.a.j	yes	1
60.h	even	2	1		1800.4.f.g		2
60.l	odd	4	1		1800.4.a.g		1
60.l	odd	4	1		1800.4.a.bf		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.4.a.g	✓	1	20.e	even	4	1
600.4.a.j	yes	1	20.e	even	4	1
600.4.f.h		2	4.b	odd	2	1
600.4.f.h		2	20.d	odd	2	1
1200.4.a.n		1	5.c	odd	4	1
1200.4.a.x		1	5.c	odd	4	1
1200.4.f.f		2	1.a	even	1	1	trivial
1200.4.f.f		2	5.b	even	2	1	inner
1800.4.a.g		1	60.l	odd	4	1
1800.4.a.bf		1	60.l	odd	4	1
1800.4.f.g		2	12.b	even	2	1
1800.4.f.g		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 $S_{4}^{\mathrm{new}}(1200, [\chi])$ :

T_{7}^{2} + 361

T_{11} + 22

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 361$ T^2 + 361
$11$	$(T + 22)^{2}$ (T + 22)^2
$13$	$T^{2} + 1$ T^2 + 1
$17$	$T^{2} + 3364$ T^2 + 3364
$19$	$(T + 53)^{2}$ (T + 53)^2
$23$	$T^{2} + 3364$ T^2 + 3364
$29$	$(T + 22)^{2}$ (T + 22)^2
$31$	$(T - 35)^{2}$ (T - 35)^2
$37$	$T^{2} + 72900$ T^2 + 72900
$41$	$(T + 468)^{2}$ (T + 468)^2
$43$	$T^{2} + 185761$ T^2 + 185761
$47$	$T^{2} + 52900$ T^2 + 52900
$53$	$T^{2}$ T^2
$59$	$(T - 446)^{2}$ (T - 446)^2
$61$	$(T - 127)^{2}$ (T - 127)^2
$67$	$T^{2} + 657721$ T^2 + 657721
$71$	$(T + 36)^{2}$ (T + 36)^2
$73$	$T^{2} + 272484$ T^2 + 272484
$79$	$(T - 1368)^{2}$ (T - 1368)^2
$83$	$T^{2} + 1295044$ T^2 + 1295044
$89$	$(T + 144)^{2}$ (T + 144)^2
$97$	$T^{2} + 1164241$ T^2 + 1164241
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