LMFDB - Newform orbit 1200.4.f.h

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 120)
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.

$f(q)$	$=$	$q - 3 i q^{3} + 20 i q^{7} - 9 q^{9} - 16 q^{11} + 58 i q^{13} - 38 i q^{17} + 4 q^{19} + 60 q^{21} + 80 i q^{23} + 27 i q^{27} - 82 q^{29} + 8 q^{31} + 48 i q^{33} - 426 i q^{37} + 174 q^{39} - 246 q^{41} + \cdots + 144 q^{99} +O(q^{100})$ q - 3i q^3 + 20i q^7 - 9 * q^9 - 16 * q^11 + 58i q^13 - 38i q^17 + 4 * q^19 + 60 * q^21 + 80i q^23 + 27i q^27 - 82 * q^29 + 8 * q^31 + 48i q^33 - 426i q^37 + 174 * q^39 - 246 * q^41 + 524i q^43 - 464i q^47 - 57 * q^49 - 114 * q^51 - 702i q^53 - 12i q^57 - 592 * q^59 + 574 * q^61 - 180i q^63 - 172i q^67 + 240 * q^69 - 768 * q^71 - 558i q^73 - 320i q^77 + 408 * q^79 + 81 * q^81 - 164i q^83 + 246i q^87 + 510 * q^89 - 1160 * q^91 - 24i q^93 - 514i q^97 + 144 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 18 q^{9} - 32 q^{11} + 8 q^{19} + 120 q^{21} - 164 q^{29} + 16 q^{31} + 348 q^{39} - 492 q^{41} - 114 q^{49} - 228 q^{51} - 1184 q^{59} + 1148 q^{61} + 480 q^{69} - 1536 q^{71} + 816 q^{79} + 162 q^{81}+ \cdots + 288 q^{99}+O(q^{100})$ 2 * q - 18 * q^9 - 32 * q^11 + 8 * q^19 + 120 * q^21 - 164 * q^29 + 16 * q^31 + 348 * q^39 - 492 * q^41 - 114 * q^49 - 228 * q^51 - 1184 * q^59 + 1148 * q^61 + 480 * q^69 - 1536 * q^71 + 816 * q^79 + 162 * q^81 + 1020 * q^89 - 2320 * q^91 + 288 * 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

20.0000i

−9.00000

49.2

3.00000i

−

20.0000i

−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.h		2
4.b	odd	2	1		600.4.f.f		2
5.b	even	2	1	inner	1200.4.f.h		2
5.c	odd	4	1		240.4.a.a		1
5.c	odd	4	1		1200.4.a.bj		1
12.b	even	2	1		1800.4.f.k		2
15.e	even	4	1		720.4.a.s		1
20.d	odd	2	1		600.4.f.f		2
20.e	even	4	1		120.4.a.e	✓	1
20.e	even	4	1		600.4.a.a		1
40.i	odd	4	1		960.4.a.bd		1
40.k	even	4	1		960.4.a.q		1
60.h	even	2	1		1800.4.f.k		2
60.l	odd	4	1		360.4.a.m		1
60.l	odd	4	1		1800.4.a.e		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
120.4.a.e	✓	1	20.e	even	4	1
240.4.a.a		1	5.c	odd	4	1
360.4.a.m		1	60.l	odd	4	1
600.4.a.a		1	20.e	even	4	1
600.4.f.f		2	4.b	odd	2	1
600.4.f.f		2	20.d	odd	2	1
720.4.a.s		1	15.e	even	4	1
960.4.a.q		1	40.k	even	4	1
960.4.a.bd		1	40.i	odd	4	1
1200.4.a.bj		1	5.c	odd	4	1
1200.4.f.h		2	1.a	even	1	1	trivial
1200.4.f.h		2	5.b	even	2	1	inner
1800.4.a.e		1	60.l	odd	4	1
1800.4.f.k		2	12.b	even	2	1
1800.4.f.k		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} + 400

T_{11} + 16

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 9$ T^2 + 9
$5$	$T^{2}$ T^2
$7$	$T^{2} + 400$ T^2 + 400
$11$	$(T + 16)^{2}$ (T + 16)^2
$13$	$T^{2} + 3364$ T^2 + 3364
$17$	$T^{2} + 1444$ T^2 + 1444
$19$	$(T - 4)^{2}$ (T - 4)^2
$23$	$T^{2} + 6400$ T^2 + 6400
$29$	$(T + 82)^{2}$ (T + 82)^2
$31$	$(T - 8)^{2}$ (T - 8)^2
$37$	$T^{2} + 181476$ T^2 + 181476
$41$	$(T + 246)^{2}$ (T + 246)^2
$43$	$T^{2} + 274576$ T^2 + 274576
$47$	$T^{2} + 215296$ T^2 + 215296
$53$	$T^{2} + 492804$ T^2 + 492804
$59$	$(T + 592)^{2}$ (T + 592)^2
$61$	$(T - 574)^{2}$ (T - 574)^2
$67$	$T^{2} + 29584$ T^2 + 29584
$71$	$(T + 768)^{2}$ (T + 768)^2
$73$	$T^{2} + 311364$ T^2 + 311364
$79$	$(T - 408)^{2}$ (T - 408)^2
$83$	$T^{2} + 26896$ T^2 + 26896
$89$	$(T - 510)^{2}$ (T - 510)^2
$97$	$T^{2} + 264196$ T^2 + 264196
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