LMFDB - Newform orbit 490.4.a.k

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)

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);

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")

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,-30] 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:	$1$
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 + 2 q^{2} - q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} + 8 q^{8} - 26 q^{9} + 10 q^{10} - 30 q^{11} - 4 q^{12} - 44 q^{13} - 5 q^{15} + 16 q^{16} + 24 q^{17} - 52 q^{18} - 2 q^{19} + 20 q^{20} - 60 q^{22}+ \cdots + 780 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 - 30 * q^11 - 4 * q^12 - 44 * q^13 - 5 * q^15 + 16 * q^16 + 24 * q^17 - 52 * q^18 - 2 * q^19 + 20 * q^20 - 60 * q^22 - 183 * q^23 - 8 * q^24 + 25 * q^25 - 88 * q^26 + 53 * q^27 - 279 * q^29 - 10 * q^30 + 40 * q^31 + 32 * q^32 + 30 * q^33 + 48 * q^34 - 104 * q^36 - 76 * q^37 - 4 * q^38 + 44 * q^39 + 40 * q^40 + 423 * q^41 + 305 * q^43 - 120 * q^44 - 130 * q^45 - 366 * q^46 - 456 * q^47 - 16 * q^48 + 50 * q^50 - 24 * q^51 - 176 * q^52 - 198 * q^53 + 106 * q^54 - 150 * q^55 + 2 * q^57 - 558 * q^58 + 462 * q^59 - 20 * q^60 - 281 * q^61 + 80 * q^62 + 64 * q^64 - 220 * q^65 + 60 * q^66 - 499 * q^67 + 96 * q^68 + 183 * q^69 - 534 * q^71 - 208 * q^72 - 800 * q^73 - 152 * q^74 - 25 * q^75 - 8 * q^76 + 88 * q^78 - 790 * q^79 + 80 * q^80 + 649 * q^81 + 846 * q^82 + 597 * q^83 + 120 * q^85 + 610 * q^86 + 279 * q^87 - 240 * q^88 - 1017 * q^89 - 260 * q^90 - 732 * q^92 - 40 * q^93 - 912 * q^94 - 10 * q^95 - 32 * q^96 + 1330 * q^97 + 780 * 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.k		1
5.b	even	2	1		2450.4.a.m		1
7.b	odd	2	1		490.4.a.m		1
7.c	even	3	2		490.4.e.f		2
7.d	odd	6	2		70.4.e.a	✓	2
21.g	even	6	2		630.4.k.i		2
28.f	even	6	2		560.4.q.e		2
35.c	odd	2	1		2450.4.a.j		1
35.i	odd	6	2		350.4.e.g		2
35.k	even	12	4		350.4.j.f		4

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

Hecke kernels

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

T_{3} + 1

T3 + 1

T_{11} + 30

T11 + 30

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
$11$	$T + 30$ T + 30
$13$	$T + 44$ T + 44
$17$	$T - 24$ T - 24
$19$	$T + 2$ T + 2
$23$	$T + 183$ T + 183
$29$	$T + 279$ T + 279
$31$	$T - 40$ T - 40
$37$	$T + 76$ T + 76
$41$	$T - 423$ T - 423
$43$	$T - 305$ T - 305
$47$	$T + 456$ T + 456
$53$	$T + 198$ T + 198
$59$	$T - 462$ T - 462
$61$	$T + 281$ T + 281
$67$	$T + 499$ T + 499
$71$	$T + 534$ T + 534
$73$	$T + 800$ T + 800
$79$	$T + 790$ T + 790
$83$	$T - 597$ T - 597
$89$	$T + 1017$ T + 1017
$97$	$T - 1330$ T - 1330
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