LMFDB - Newform orbit 900.2.e.b

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [900,2,Mod(251,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.251"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	900.e (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$7.18653618192$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 2$ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$ -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 $\beta = \sqrt{-2}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta q^{2} - 2 q^{4} - 2 \beta q^{8} + 4 q^{13} + 4 q^{16} + 5 \beta q^{17} + 4 \beta q^{26} + 7 \beta q^{29} + 4 \beta q^{32} - 10 q^{34} - 2 q^{37} + \beta q^{41} + 7 q^{49} - 8 q^{52} + 5 \beta q^{53} + \cdots + 7 \beta q^{98} +O(q^{100})$ q + b * q^2 - 2 * q^4 - 2b q^8 + 4 * q^13 + 4 * q^16 + 5b q^17 + 4b q^26 + 7b q^29 + 4b q^32 - 10 * q^34 - 2 * q^37 + b * q^41 + 7 * q^49 - 8 * q^52 + 5b q^53 - 14 * q^58 - 10 * q^61 - 8 * q^64 - 10b q^68 + 16 * q^73 - 2b q^74 - 2 * q^82 + 13b q^89 - 8 * q^97 + 7b q^98
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{4} + 8 q^{13} + 8 q^{16} - 20 q^{34} - 4 q^{37} + 14 q^{49} - 16 q^{52} - 28 q^{58} - 20 q^{61} - 16 q^{64} + 32 q^{73} - 4 q^{82} - 16 q^{97}+O(q^{100})$ 2 * q - 4 * q^4 + 8 * q^13 + 8 * q^16 - 20 * q^34 - 4 * q^37 + 14 * q^49 - 16 * q^52 - 28 * q^58 - 20 * q^61 - 16 * q^64 + 32 * q^73 - 4 * q^82 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$ .

$n$	$101$	$451$	$577$
$\chi(n)$	$-1$	$-1$	$1$

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}

251.1

	−	1.41421i
		1.41421i

−

1.41421i

−2.00000

2.82843i

251.2

1.41421i

−2.00000

−

2.82843i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	CM by $\Q(\sqrt{-1})$
3.b	odd	2	1	inner
12.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.e.b		2
3.b	odd	2	1	inner	900.2.e.b		2
4.b	odd	2	1	CM	900.2.e.b		2
5.b	even	2	1		36.2.b.a	✓	2
5.c	odd	4	2		900.2.h.a		4
12.b	even	2	1	inner	900.2.e.b		2
15.d	odd	2	1		36.2.b.a	✓	2
15.e	even	4	2		900.2.h.a		4
20.d	odd	2	1		36.2.b.a	✓	2
20.e	even	4	2		900.2.h.a		4
35.c	odd	2	1		1764.2.e.b		2
40.e	odd	2	1		576.2.c.b		2
40.f	even	2	1		576.2.c.b		2
45.h	odd	6	2		324.2.h.c		4
45.j	even	6	2		324.2.h.c		4
60.h	even	2	1		36.2.b.a	✓	2
60.l	odd	4	2		900.2.h.a		4
80.k	odd	4	2		2304.2.f.d		4
80.q	even	4	2		2304.2.f.d		4
105.g	even	2	1		1764.2.e.b		2
120.i	odd	2	1		576.2.c.b		2
120.m	even	2	1		576.2.c.b		2
140.c	even	2	1		1764.2.e.b		2
180.n	even	6	2		324.2.h.c		4
180.p	odd	6	2		324.2.h.c		4
240.t	even	4	2		2304.2.f.d		4
240.bm	odd	4	2		2304.2.f.d		4
420.o	odd	2	1		1764.2.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
36.2.b.a	✓	2	5.b	even	2	1
36.2.b.a	✓	2	15.d	odd	2	1
36.2.b.a	✓	2	20.d	odd	2	1
36.2.b.a	✓	2	60.h	even	2	1
324.2.h.c		4	45.h	odd	6	2
324.2.h.c		4	45.j	even	6	2
324.2.h.c		4	180.n	even	6	2
324.2.h.c		4	180.p	odd	6	2
576.2.c.b		2	40.e	odd	2	1
576.2.c.b		2	40.f	even	2	1
576.2.c.b		2	120.i	odd	2	1
576.2.c.b		2	120.m	even	2	1
900.2.e.b		2	1.a	even	1	1	trivial
900.2.e.b		2	3.b	odd	2	1	inner
900.2.e.b		2	4.b	odd	2	1	CM
900.2.e.b		2	12.b	even	2	1	inner
900.2.h.a		4	5.c	odd	4	2
900.2.h.a		4	15.e	even	4	2
900.2.h.a		4	20.e	even	4	2
900.2.h.a		4	60.l	odd	4	2
1764.2.e.b		2	35.c	odd	2	1
1764.2.e.b		2	105.g	even	2	1
1764.2.e.b		2	140.c	even	2	1
1764.2.e.b		2	420.o	odd	2	1
2304.2.f.d		4	80.k	odd	4	2
2304.2.f.d		4	80.q	even	4	2
2304.2.f.d		4	240.t	even	4	2
2304.2.f.d		4	240.bm	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(900, [\chi])$ :

T_{7}

T7

T_{13} - 4

T13 - 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} + 2$ T^2 + 2
$3$	$T^{2}$ T^2
$5$	$T^{2}$ T^2
$7$	$T^{2}$ T^2
$11$	$T^{2}$ T^2
$13$	$(T - 4)^{2}$ (T - 4)^2
$17$	$T^{2} + 50$ T^2 + 50
$19$	$T^{2}$ T^2
$23$	$T^{2}$ T^2
$29$	$T^{2} + 98$ T^2 + 98
$31$	$T^{2}$ T^2
$37$	$(T + 2)^{2}$ (T + 2)^2
$41$	$T^{2} + 2$ T^2 + 2
$43$	$T^{2}$ T^2
$47$	$T^{2}$ T^2
$53$	$T^{2} + 50$ T^2 + 50
$59$	$T^{2}$ T^2
$61$	$(T + 10)^{2}$ (T + 10)^2
$67$	$T^{2}$ T^2
$71$	$T^{2}$ T^2
$73$	$(T - 16)^{2}$ (T - 16)^2
$79$	$T^{2}$ T^2
$83$	$T^{2}$ T^2
$89$	$T^{2} + 338$ T^2 + 338
$97$	$(T + 8)^{2}$ (T + 8)^2
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