LMFDB - Newform orbit 490.2.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [490,2,Mod(1,490)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("490.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	490.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1,1,1,-1,-1,0,-1,-2,1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.91266969904$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
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 - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} - 2 q^{9} + q^{10} - 6 q^{11} + q^{12} - 4 q^{13} - q^{15} + q^{16} + 2 q^{18} + 2 q^{19} - q^{20} + 6 q^{22} - 3 q^{23} - q^{24} + q^{25}+ \cdots + 12 q^{99}+O(q^{100})

q - q^2 + q^3 + q^4 - q^5 - q^6 - q^8 - 2 * q^9 + q^10 - 6 * q^11 + q^12 - 4 * q^13 - q^15 + q^16 + 2 * q^18 + 2 * q^19 - q^20 + 6 * q^22 - 3 * q^23 - q^24 + q^25 + 4 * q^26 - 5 * q^27 - 3 * q^29 + q^30 + 8 * q^31 - q^32 - 6 * q^33 - 2 * q^36 - 4 * q^37 - 2 * q^38 - 4 * q^39 + q^40 + 9 * q^41 - 7 * q^43 - 6 * q^44 + 2 * q^45 + 3 * q^46 + q^48 - q^50 - 4 * q^52 - 6 * q^53 + 5 * q^54 + 6 * q^55 + 2 * q^57 + 3 * q^58 - 6 * q^59 - q^60 + 5 * q^61 - 8 * q^62 + q^64 + 4 * q^65 + 6 * q^66 + 5 * q^67 - 3 * q^69 - 6 * q^71 + 2 * q^72 - 16 * q^73 + 4 * q^74 + q^75 + 2 * q^76 + 4 * q^78 + 2 * q^79 - q^80 + q^81 - 9 * q^82 + 3 * q^83 + 7 * q^86 - 3 * q^87 + 6 * q^88 - 15 * q^89 - 2 * q^90 - 3 * q^92 + 8 * q^93 - 2 * q^95 - q^96 + 14 * q^97 + 12 * 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

−1.00000

1.00000

−1.00000

−2.00000

1.00000

Atkin-Lehner signs

$p$	Sign
$2$	$+1$
$5$	$+1$
$7$	$+1$

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	490.2.a.c		1
3.b	odd	2	1		4410.2.a.bm		1
4.b	odd	2	1		3920.2.a.p		1
5.b	even	2	1		2450.2.a.w		1
5.c	odd	4	2		2450.2.c.g		2
7.b	odd	2	1		490.2.a.b		1
7.c	even	3	2		70.2.e.c	✓	2
7.d	odd	6	2		490.2.e.h		2
21.c	even	2	1		4410.2.a.bd		1
21.h	odd	6	2		630.2.k.b		2
28.d	even	2	1		3920.2.a.bc		1
28.g	odd	6	2		560.2.q.g		2
35.c	odd	2	1		2450.2.a.bc		1
35.f	even	4	2		2450.2.c.l		2
35.j	even	6	2		350.2.e.e		2
35.l	odd	12	4		350.2.j.b		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.2.e.c	✓	2	7.c	even	3	2
350.2.e.e		2	35.j	even	6	2
350.2.j.b		4	35.l	odd	12	4
490.2.a.b		1	7.b	odd	2	1
490.2.a.c		1	1.a	even	1	1	trivial
490.2.e.h		2	7.d	odd	6	2
560.2.q.g		2	28.g	odd	6	2
630.2.k.b		2	21.h	odd	6	2
2450.2.a.w		1	5.b	even	2	1
2450.2.a.bc		1	35.c	odd	2	1
2450.2.c.g		2	5.c	odd	4	2
2450.2.c.l		2	35.f	even	4	2
3920.2.a.p		1	4.b	odd	2	1
3920.2.a.bc		1	28.d	even	2	1
4410.2.a.bd		1	21.c	even	2	1
4410.2.a.bm		1	3.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(490))$ :

T_{3} - 1

T3 - 1

T_{11} + 6

T11 + 6

T_{13} + 4

T13 + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 1$ T + 1
$3$	$T - 1$ T - 1
$5$	$T + 1$ T + 1
$7$	$T$ T
$11$	$T + 6$ T + 6
$13$	$T + 4$ T + 4
$17$	$T$ T
$19$	$T - 2$ T - 2
$23$	$T + 3$ T + 3
$29$	$T + 3$ T + 3
$31$	$T - 8$ T - 8
$37$	$T + 4$ T + 4
$41$	$T - 9$ T - 9
$43$	$T + 7$ T + 7
$47$	$T$ T
$53$	$T + 6$ T + 6
$59$	$T + 6$ T + 6
$61$	$T - 5$ T - 5
$67$	$T - 5$ T - 5
$71$	$T + 6$ T + 6
$73$	$T + 16$ T + 16
$79$	$T - 2$ T - 2
$83$	$T - 3$ T - 3
$89$	$T + 15$ T + 15
$97$	$T - 14$ T - 14
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