LMFDB - Newform orbit 490.4.a.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [490,4,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, 4, 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, 4); N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	490.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-2,1,4,5,-2,0,-8,-26,-10,-2] 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:	$28.9109359028$
Analytic rank:	$0$
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 - 2 q^{2} + q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} - 8 q^{8} - 26 q^{9} - 10 q^{10} - 2 q^{11} + 4 q^{12} + 8 q^{13} + 5 q^{15} + 16 q^{16} + 52 q^{17} + 52 q^{18} - 26 q^{19} + 20 q^{20} + 4 q^{22} + 67 q^{23}+ \cdots + 52 q^{99}+O(q^{100})

q - 2 * q^2 + q^3 + 4 * q^4 + 5 * q^5 - 2 * q^6 - 8 * q^8 - 26 * q^9 - 10 * q^10 - 2 * q^11 + 4 * q^12 + 8 * q^13 + 5 * q^15 + 16 * q^16 + 52 * q^17 + 52 * q^18 - 26 * q^19 + 20 * q^20 + 4 * q^22 + 67 * q^23 - 8 * q^24 + 25 * q^25 - 16 * q^26 - 53 * q^27 + 69 * q^29 - 10 * q^30 + 332 * q^31 - 32 * q^32 - 2 * q^33 - 104 * q^34 - 104 * q^36 + 196 * q^37 + 52 * q^38 + 8 * q^39 - 40 * q^40 - 353 * q^41 - 369 * q^43 - 8 * q^44 - 130 * q^45 - 134 * q^46 - 88 * q^47 + 16 * q^48 - 50 * q^50 + 52 * q^51 + 32 * q^52 + 582 * q^53 + 106 * q^54 - 10 * q^55 - 26 * q^57 - 138 * q^58 + 350 * q^59 + 20 * q^60 + 467 * q^61 - 664 * q^62 + 64 * q^64 + 40 * q^65 + 4 * q^66 + 291 * q^67 + 208 * q^68 + 67 * q^69 + 770 * q^71 + 208 * q^72 - 628 * q^73 - 392 * q^74 + 25 * q^75 - 104 * q^76 - 16 * q^78 + 1170 * q^79 + 80 * q^80 + 649 * q^81 + 706 * q^82 - 525 * q^83 + 260 * q^85 + 738 * q^86 + 69 * q^87 + 16 * q^88 - 89 * q^89 + 260 * q^90 + 268 * q^92 + 332 * q^93 + 176 * q^94 - 130 * q^95 - 32 * q^96 + 290 * q^97 + 52 * 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

−2.00000

1.00000

4.00000

5.00000

−2.00000

−8.00000

−26.0000

−10.0000

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	490.4.a.e		1
5.b	even	2	1		2450.4.a.be		1
7.b	odd	2	1		490.4.a.c		1
7.c	even	3	2		490.4.e.m		2
7.d	odd	6	2		70.4.e.c	✓	2
21.g	even	6	2		630.4.k.b		2
28.f	even	6	2		560.4.q.d		2
35.c	odd	2	1		2450.4.a.bg		1
35.i	odd	6	2		350.4.e.a		2
35.k	even	12	4		350.4.j.e		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.e.c	✓	2	7.d	odd	6	2
350.4.e.a		2	35.i	odd	6	2
350.4.j.e		4	35.k	even	12	4
490.4.a.c		1	7.b	odd	2	1
490.4.a.e		1	1.a	even	1	1	trivial
490.4.e.m		2	7.c	even	3	2
560.4.q.d		2	28.f	even	6	2
630.4.k.b		2	21.g	even	6	2
2450.4.a.be		1	5.b	even	2	1
2450.4.a.bg		1	35.c	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(\Gamma_0(490))$ :

T_{3} - 1

T3 - 1

T_{11} + 2

T11 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$ T + 2
$3$	$T - 1$ T - 1
$5$	$T - 5$ T - 5
$7$	$T$ T
$11$	$T + 2$ T + 2
$13$	$T - 8$ T - 8
$17$	$T - 52$ T - 52
$19$	$T + 26$ T + 26
$23$	$T - 67$ T - 67
$29$	$T - 69$ T - 69
$31$	$T - 332$ T - 332
$37$	$T - 196$ T - 196
$41$	$T + 353$ T + 353
$43$	$T + 369$ T + 369
$47$	$T + 88$ T + 88
$53$	$T - 582$ T - 582
$59$	$T - 350$ T - 350
$61$	$T - 467$ T - 467
$67$	$T - 291$ T - 291
$71$	$T - 770$ T - 770
$73$	$T + 628$ T + 628
$79$	$T - 1170$ T - 1170
$83$	$T + 525$ T + 525
$89$	$T + 89$ T + 89
$97$	$T - 290$ T - 290
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