LMFDB - Newform orbit 490.4.a.o

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 = [1,2,8,4,-5,16,0,8,37,-10,12] 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 10)
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} + 8 q^{3} + 4 q^{4} - 5 q^{5} + 16 q^{6} + 8 q^{8} + 37 q^{9} - 10 q^{10} + 12 q^{11} + 32 q^{12} + 58 q^{13} - 40 q^{15} + 16 q^{16} - 66 q^{17} + 74 q^{18} + 100 q^{19} - 20 q^{20} + 24 q^{22}+ \cdots + 444 q^{99}+O(q^{100})

q + 2 * q^2 + 8 * q^3 + 4 * q^4 - 5 * q^5 + 16 * q^6 + 8 * q^8 + 37 * q^9 - 10 * q^10 + 12 * q^11 + 32 * q^12 + 58 * q^13 - 40 * q^15 + 16 * q^16 - 66 * q^17 + 74 * q^18 + 100 * q^19 - 20 * q^20 + 24 * q^22 + 132 * q^23 + 64 * q^24 + 25 * q^25 + 116 * q^26 + 80 * q^27 - 90 * q^29 - 80 * q^30 - 152 * q^31 + 32 * q^32 + 96 * q^33 - 132 * q^34 + 148 * q^36 - 34 * q^37 + 200 * q^38 + 464 * q^39 - 40 * q^40 + 438 * q^41 + 32 * q^43 + 48 * q^44 - 185 * q^45 + 264 * q^46 + 204 * q^47 + 128 * q^48 + 50 * q^50 - 528 * q^51 + 232 * q^52 + 222 * q^53 + 160 * q^54 - 60 * q^55 + 800 * q^57 - 180 * q^58 - 420 * q^59 - 160 * q^60 - 902 * q^61 - 304 * q^62 + 64 * q^64 - 290 * q^65 + 192 * q^66 - 1024 * q^67 - 264 * q^68 + 1056 * q^69 + 432 * q^71 + 296 * q^72 - 362 * q^73 - 68 * q^74 + 200 * q^75 + 400 * q^76 + 928 * q^78 - 160 * q^79 - 80 * q^80 - 359 * q^81 + 876 * q^82 - 72 * q^83 + 330 * q^85 + 64 * q^86 - 720 * q^87 + 96 * q^88 - 810 * q^89 - 370 * q^90 + 528 * q^92 - 1216 * q^93 + 408 * q^94 - 500 * q^95 + 256 * q^96 - 1106 * q^97 + 444 * 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

8.00000

4.00000

−5.00000

16.0000

8.00000

37.0000

−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.o		1
5.b	even	2	1		2450.4.a.b		1
7.b	odd	2	1		10.4.a.a	✓	1
7.c	even	3	2		490.4.e.a		2
7.d	odd	6	2		490.4.e.i		2
21.c	even	2	1		90.4.a.a		1
28.d	even	2	1		80.4.a.f		1
35.c	odd	2	1		50.4.a.c		1
35.f	even	4	2		50.4.b.a		2
56.e	even	2	1		320.4.a.b		1
56.h	odd	2	1		320.4.a.m		1
63.l	odd	6	2		810.4.e.c		2
63.o	even	6	2		810.4.e.w		2
77.b	even	2	1		1210.4.a.b		1
84.h	odd	2	1		720.4.a.j		1
91.b	odd	2	1		1690.4.a.a		1
105.g	even	2	1		450.4.a.q		1
105.k	odd	4	2		450.4.c.d		2
112.j	even	4	2		1280.4.d.g		2
112.l	odd	4	2		1280.4.d.j		2
140.c	even	2	1		400.4.a.b		1
140.j	odd	4	2		400.4.c.c		2
280.c	odd	2	1		1600.4.a.d		1
280.n	even	2	1		1600.4.a.bx		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
10.4.a.a	✓	1	7.b	odd	2	1
50.4.a.c		1	35.c	odd	2	1
50.4.b.a		2	35.f	even	4	2
80.4.a.f		1	28.d	even	2	1
90.4.a.a		1	21.c	even	2	1
320.4.a.b		1	56.e	even	2	1
320.4.a.m		1	56.h	odd	2	1
400.4.a.b		1	140.c	even	2	1
400.4.c.c		2	140.j	odd	4	2
450.4.a.q		1	105.g	even	2	1
450.4.c.d		2	105.k	odd	4	2
490.4.a.o		1	1.a	even	1	1	trivial
490.4.e.a		2	7.c	even	3	2
490.4.e.i		2	7.d	odd	6	2
720.4.a.j		1	84.h	odd	2	1
810.4.e.c		2	63.l	odd	6	2
810.4.e.w		2	63.o	even	6	2
1210.4.a.b		1	77.b	even	2	1
1280.4.d.g		2	112.j	even	4	2
1280.4.d.j		2	112.l	odd	4	2
1600.4.a.d		1	280.c	odd	2	1
1600.4.a.bx		1	280.n	even	2	1
1690.4.a.a		1	91.b	odd	2	1
2450.4.a.b		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} - 8

T3 - 8

T_{11} - 12

T11 - 12

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T - 2$ T - 2
$3$	$T - 8$ T - 8
$5$	$T + 5$ T + 5
$7$	$T$ T
$11$	$T - 12$ T - 12
$13$	$T - 58$ T - 58
$17$	$T + 66$ T + 66
$19$	$T - 100$ T - 100
$23$	$T - 132$ T - 132
$29$	$T + 90$ T + 90
$31$	$T + 152$ T + 152
$37$	$T + 34$ T + 34
$41$	$T - 438$ T - 438
$43$	$T - 32$ T - 32
$47$	$T - 204$ T - 204
$53$	$T - 222$ T - 222
$59$	$T + 420$ T + 420
$61$	$T + 902$ T + 902
$67$	$T + 1024$ T + 1024
$71$	$T - 432$ T - 432
$73$	$T + 362$ T + 362
$79$	$T + 160$ T + 160
$83$	$T + 72$ T + 72
$89$	$T + 810$ T + 810
$97$	$T + 1106$ T + 1106
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