LMFDB - Newform orbit 490.4.a.h

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,-2,10,4,-5,-20,0,-8,73,10,53] 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} + 10 q^{3} + 4 q^{4} - 5 q^{5} - 20 q^{6} - 8 q^{8} + 73 q^{9} + 10 q^{10} + 53 q^{11} + 40 q^{12} + 25 q^{13} - 50 q^{15} + 16 q^{16} + 14 q^{17} - 146 q^{18} - 95 q^{19} - 20 q^{20} - 106 q^{22}+ \cdots + 3869 q^{99}+O(q^{100})

q - 2 * q^2 + 10 * q^3 + 4 * q^4 - 5 * q^5 - 20 * q^6 - 8 * q^8 + 73 * q^9 + 10 * q^10 + 53 * q^11 + 40 * q^12 + 25 * q^13 - 50 * q^15 + 16 * q^16 + 14 * q^17 - 146 * q^18 - 95 * q^19 - 20 * q^20 - 106 * q^22 + q^23 - 80 * q^24 + 25 * q^25 - 50 * q^26 + 460 * q^27 - 206 * q^29 + 100 * q^30 + 108 * q^31 - 32 * q^32 + 530 * q^33 - 28 * q^34 + 292 * q^36 - 57 * q^37 + 190 * q^38 + 250 * q^39 + 40 * q^40 + 243 * q^41 + 434 * q^43 + 212 * q^44 - 365 * q^45 - 2 * q^46 - 231 * q^47 + 160 * q^48 - 50 * q^50 + 140 * q^51 + 100 * q^52 + 263 * q^53 - 920 * q^54 - 265 * q^55 - 950 * q^57 + 412 * q^58 + 24 * q^59 - 200 * q^60 + 116 * q^61 - 216 * q^62 + 64 * q^64 - 125 * q^65 - 1060 * q^66 - 204 * q^67 + 56 * q^68 + 10 * q^69 + 484 * q^71 - 584 * q^72 - 692 * q^73 + 114 * q^74 + 250 * q^75 - 380 * q^76 - 500 * q^78 + 466 * q^79 - 80 * q^80 + 2629 * q^81 - 486 * q^82 + 228 * q^83 - 70 * q^85 - 868 * q^86 - 2060 * q^87 - 424 * q^88 - 362 * q^89 + 730 * q^90 + 4 * q^92 + 1080 * q^93 + 462 * q^94 + 475 * q^95 - 320 * q^96 + 854 * q^97 + 3869 * 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

10.0000

4.00000

−5.00000

−20.0000

−8.00000

73.0000

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.h		1
5.b	even	2	1		2450.4.a.v		1
7.b	odd	2	1		490.4.a.a		1
7.c	even	3	2		70.4.e.b	✓	2
7.d	odd	6	2		490.4.e.s		2
21.h	odd	6	2		630.4.k.c		2
28.g	odd	6	2		560.4.q.g		2
35.c	odd	2	1		2450.4.a.bq		1
35.j	even	6	2		350.4.e.d		2
35.l	odd	12	4		350.4.j.a		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.e.b	✓	2	7.c	even	3	2
350.4.e.d		2	35.j	even	6	2
350.4.j.a		4	35.l	odd	12	4
490.4.a.a		1	7.b	odd	2	1
490.4.a.h		1	1.a	even	1	1	trivial
490.4.e.s		2	7.d	odd	6	2
560.4.q.g		2	28.g	odd	6	2
630.4.k.c		2	21.h	odd	6	2
2450.4.a.v		1	5.b	even	2	1
2450.4.a.bq		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} - 10

T3 - 10

T_{11} - 53

T11 - 53

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T + 2$ T + 2
$3$	$T - 10$ T - 10
$5$	$T + 5$ T + 5
$7$	$T$ T
$11$	$T - 53$ T - 53
$13$	$T - 25$ T - 25
$17$	$T - 14$ T - 14
$19$	$T + 95$ T + 95
$23$	$T - 1$ T - 1
$29$	$T + 206$ T + 206
$31$	$T - 108$ T - 108
$37$	$T + 57$ T + 57
$41$	$T - 243$ T - 243
$43$	$T - 434$ T - 434
$47$	$T + 231$ T + 231
$53$	$T - 263$ T - 263
$59$	$T - 24$ T - 24
$61$	$T - 116$ T - 116
$67$	$T + 204$ T + 204
$71$	$T - 484$ T - 484
$73$	$T + 692$ T + 692
$79$	$T - 466$ T - 466
$83$	$T - 228$ T - 228
$89$	$T + 362$ T + 362
$97$	$T - 854$ T - 854
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