LMFDB - Newform orbit 490.4.a.d

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,-65] 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} - 65 q^{11} + 4 q^{12} - 13 q^{13} + 5 q^{15} + 16 q^{16} + 73 q^{17} + 52 q^{18} + 142 q^{19} + 20 q^{20} + 130 q^{22}+ \cdots + 1690 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 - 65 * q^11 + 4 * q^12 - 13 * q^13 + 5 * q^15 + 16 * q^16 + 73 * q^17 + 52 * q^18 + 142 * q^19 + 20 * q^20 + 130 * q^22 + 130 * q^23 - 8 * q^24 + 25 * q^25 + 26 * q^26 - 53 * q^27 + 111 * q^29 - 10 * q^30 - 256 * q^31 - 32 * q^32 - 65 * q^33 - 146 * q^34 - 104 * q^36 - 266 * q^37 - 284 * q^38 - 13 * q^39 - 40 * q^40 + 424 * q^41 + 534 * q^43 - 260 * q^44 - 130 * q^45 - 260 * q^46 + 269 * q^47 + 16 * q^48 - 50 * q^50 + 73 * q^51 - 52 * q^52 - 132 * q^53 + 106 * q^54 - 325 * q^55 + 142 * q^57 - 222 * q^58 + 224 * q^59 + 20 * q^60 + 572 * q^61 + 512 * q^62 + 64 * q^64 - 65 * q^65 + 130 * q^66 - 108 * q^67 + 292 * q^68 + 130 * q^69 + 560 * q^71 + 208 * q^72 - 586 * q^73 + 532 * q^74 + 25 * q^75 + 568 * q^76 + 26 * q^78 + 57 * q^79 + 80 * q^80 + 649 * q^81 - 848 * q^82 - 252 * q^83 + 365 * q^85 - 1068 * q^86 + 111 * q^87 + 520 * q^88 + 184 * q^89 + 260 * q^90 + 520 * q^92 - 256 * q^93 - 538 * q^94 + 710 * q^95 - 32 * q^96 + 605 * q^97 + 1690 * 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.d		1
5.b	even	2	1		2450.4.a.bc		1
7.b	odd	2	1		70.4.a.c	✓	1
7.c	even	3	2		490.4.e.n		2
7.d	odd	6	2		490.4.e.o		2
21.c	even	2	1		630.4.a.x		1
28.d	even	2	1		560.4.a.i		1
35.c	odd	2	1		350.4.a.r		1
35.f	even	4	2		350.4.c.h		2
56.e	even	2	1		2240.4.a.r		1
56.h	odd	2	1		2240.4.a.v		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.c	✓	1	7.b	odd	2	1
350.4.a.r		1	35.c	odd	2	1
350.4.c.h		2	35.f	even	4	2
490.4.a.d		1	1.a	even	1	1	trivial
490.4.e.n		2	7.c	even	3	2
490.4.e.o		2	7.d	odd	6	2
560.4.a.i		1	28.d	even	2	1
630.4.a.x		1	21.c	even	2	1
2240.4.a.r		1	56.e	even	2	1
2240.4.a.v		1	56.h	odd	2	1
2450.4.a.bc		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} + 65

T11 + 65

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
$11$	$T + 65$ T + 65
$13$	$T + 13$ T + 13
$17$	$T - 73$ T - 73
$19$	$T - 142$ T - 142
$23$	$T - 130$ T - 130
$29$	$T - 111$ T - 111
$31$	$T + 256$ T + 256
$37$	$T + 266$ T + 266
$41$	$T - 424$ T - 424
$43$	$T - 534$ T - 534
$47$	$T - 269$ T - 269
$53$	$T + 132$ T + 132
$59$	$T - 224$ T - 224
$61$	$T - 572$ T - 572
$67$	$T + 108$ T + 108
$71$	$T - 560$ T - 560
$73$	$T + 586$ T + 586
$79$	$T - 57$ T - 57
$83$	$T + 252$ T + 252
$89$	$T - 184$ T - 184
$97$	$T - 605$ T - 605
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