LMFDB - Newform orbit 490.4.e.p

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [490,4,Mod(361,490)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(490, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("490.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$490 = 2 \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	490.e (of order $3$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.9109359028$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 70)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$ -expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$ -expansion are expressed in terms of a primitive root of unity $\zeta_{6}$ . We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + 2 \zeta_{6} q^{2} + ( - 3 \zeta_{6} + 3) q^{3} + (4 \zeta_{6} - 4) q^{4} - 5 \zeta_{6} q^{5} + 6 q^{6} - 8 q^{8} + 18 \zeta_{6} q^{9} + ( - 10 \zeta_{6} + 10) q^{10} + ( - 17 \zeta_{6} + 17) q^{11} + \cdots + 306 q^{99} +O(q^{100})$ q + 2z q^2 + (-3z + 3) q^3 + (4z - 4) q^4 - 5z q^5 + 6 * q^6 - 8 * q^8 + 18z q^9 + (-10z + 10) q^10 + (-17z + 17) q^11 + 12z q^12 - 81 * q^13 - 15 * q^15 - 16z q^16 + (-91z + 91) q^17 + (36z - 36) q^18 - 102z q^19 + 20 * q^20 + 34 * q^22 + 90z q^23 + (24z - 24) q^24 + (25z - 25) q^25 - 162z q^26 + 135 * q^27 - 129 * q^29 - 30z q^30 + (116z - 116) q^31 + (-32z + 32) q^32 - 51z q^33 + 182 * q^34 - 72 * q^36 - 314z q^37 + (-204z + 204) q^38 + (243z - 243) q^39 + 40z q^40 - 124 * q^41 - 434 * q^43 + 68z q^44 + (-90z + 90) q^45 + (180z - 180) q^46 - 497z q^47 - 48 * q^48 - 50 * q^50 - 273z q^51 + (-324z + 324) q^52 + (-584z + 584) q^53 + 270z q^54 - 85 * q^55 - 306 * q^57 - 258z q^58 + (-332z + 332) q^59 + (-60z + 60) q^60 - 220z q^61 - 232 * q^62 + 64 * q^64 + 405z q^65 + (-102z + 102) q^66 + (384z - 384) q^67 + 364z q^68 + 270 * q^69 - 664 * q^71 - 144z q^72 + (230z - 230) q^73 + (-628z + 628) q^74 + 75z q^75 + 408 * q^76 - 486 * q^78 - 361z q^79 + (80z - 80) q^80 + (81z - 81) q^81 - 248z q^82 + 1172 * q^83 - 455 * q^85 - 868z q^86 + (387z - 387) q^87 + (136z - 136) q^88 - 40z q^89 + 180 * q^90 - 360 * q^92 + 348z q^93 + (-994z + 994) q^94 + (510z - 510) q^95 - 96z q^96 - 175 * q^97 + 306 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 3 q^{3} - 4 q^{4} - 5 q^{5} + 12 q^{6} - 16 q^{8} + 18 q^{9} + 10 q^{10} + 17 q^{11} + 12 q^{12} - 162 q^{13} - 30 q^{15} - 16 q^{16} + 91 q^{17} - 36 q^{18} - 102 q^{19} + 40 q^{20} + 68 q^{22}+ \cdots + 612 q^{99}+O(q^{100})$ 2 * q + 2 * q^2 + 3 * q^3 - 4 * q^4 - 5 * q^5 + 12 * q^6 - 16 * q^8 + 18 * q^9 + 10 * q^10 + 17 * q^11 + 12 * q^12 - 162 * q^13 - 30 * q^15 - 16 * q^16 + 91 * q^17 - 36 * q^18 - 102 * q^19 + 40 * q^20 + 68 * q^22 + 90 * q^23 - 24 * q^24 - 25 * q^25 - 162 * q^26 + 270 * q^27 - 258 * q^29 - 30 * q^30 - 116 * q^31 + 32 * q^32 - 51 * q^33 + 364 * q^34 - 144 * q^36 - 314 * q^37 + 204 * q^38 - 243 * q^39 + 40 * q^40 - 248 * q^41 - 868 * q^43 + 68 * q^44 + 90 * q^45 - 180 * q^46 - 497 * q^47 - 96 * q^48 - 100 * q^50 - 273 * q^51 + 324 * q^52 + 584 * q^53 + 270 * q^54 - 170 * q^55 - 612 * q^57 - 258 * q^58 + 332 * q^59 + 60 * q^60 - 220 * q^61 - 464 * q^62 + 128 * q^64 + 405 * q^65 + 102 * q^66 - 384 * q^67 + 364 * q^68 + 540 * q^69 - 1328 * q^71 - 144 * q^72 - 230 * q^73 + 628 * q^74 + 75 * q^75 + 816 * q^76 - 972 * q^78 - 361 * q^79 - 80 * q^80 - 81 * q^81 - 248 * q^82 + 2344 * q^83 - 910 * q^85 - 868 * q^86 - 387 * q^87 - 136 * q^88 - 40 * q^89 + 360 * q^90 - 720 * q^92 + 348 * q^93 + 994 * q^94 - 510 * q^95 - 96 * q^96 - 350 * q^97 + 612 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/490\mathbb{Z}\right)^\times$ .

$n$	$101$	$197$
$\chi(n)$	$-\zeta_{6}$	$1$

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}

361.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.00000

1.73205i

1.50000

−

2.59808i

−2.00000

3.46410i

−2.50000

−

4.33013i

6.00000

−8.00000

9.00000

15.5885i

5.00000

−

8.66025i

471.1

1.00000

−

1.73205i

1.50000

2.59808i

−2.00000

−

3.46410i

−2.50000

4.33013i

6.00000

−8.00000

9.00000

−

15.5885i

5.00000

8.66025i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	490.4.e.p		2
7.b	odd	2	1		490.4.e.l		2
7.c	even	3	1		70.4.a.b	✓	1
7.c	even	3	1	inner	490.4.e.p		2
7.d	odd	6	1		490.4.a.f		1
7.d	odd	6	1		490.4.e.l		2
21.h	odd	6	1		630.4.a.m		1
28.g	odd	6	1		560.4.a.k		1
35.i	odd	6	1		2450.4.a.ba		1
35.j	even	6	1		350.4.a.t		1
35.l	odd	12	2		350.4.c.j		2
56.k	odd	6	1		2240.4.a.p		1
56.p	even	6	1		2240.4.a.w		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
70.4.a.b	✓	1	7.c	even	3	1
350.4.a.t		1	35.j	even	6	1
350.4.c.j		2	35.l	odd	12	2
490.4.a.f		1	7.d	odd	6	1
490.4.e.l		2	7.b	odd	2	1
490.4.e.l		2	7.d	odd	6	1
490.4.e.p		2	1.a	even	1	1	trivial
490.4.e.p		2	7.c	even	3	1	inner
560.4.a.k		1	28.g	odd	6	1
630.4.a.m		1	21.h	odd	6	1
2240.4.a.p		1	56.k	odd	6	1
2240.4.a.w		1	56.p	even	6	1
2450.4.a.ba		1	35.i	odd	6	1

Hecke kernels

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

T_{3}^{2} - 3T_{3} + 9

T_{11}^{2} - 17T_{11} + 289

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$3$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$5$	$T^{2} + 5T + 25$ T^2 + 5*T + 25
$7$	$T^{2}$ T^2
$11$	$T^{2} - 17T + 289$ T^2 - 17*T + 289
$13$	$(T + 81)^{2}$ (T + 81)^2
$17$	$T^{2} - 91T + 8281$ T^2 - 91*T + 8281
$19$	$T^{2} + 102T + 10404$ T^2 + 102*T + 10404
$23$	$T^{2} - 90T + 8100$ T^2 - 90*T + 8100
$29$	$(T + 129)^{2}$ (T + 129)^2
$31$	$T^{2} + 116T + 13456$ T^2 + 116*T + 13456
$37$	$T^{2} + 314T + 98596$ T^2 + 314*T + 98596
$41$	$(T + 124)^{2}$ (T + 124)^2
$43$	$(T + 434)^{2}$ (T + 434)^2
$47$	$T^{2} + 497T + 247009$ T^2 + 497*T + 247009
$53$	$T^{2} - 584T + 341056$ T^2 - 584*T + 341056
$59$	$T^{2} - 332T + 110224$ T^2 - 332*T + 110224
$61$	$T^{2} + 220T + 48400$ T^2 + 220*T + 48400
$67$	$T^{2} + 384T + 147456$ T^2 + 384*T + 147456
$71$	$(T + 664)^{2}$ (T + 664)^2
$73$	$T^{2} + 230T + 52900$ T^2 + 230*T + 52900
$79$	$T^{2} + 361T + 130321$ T^2 + 361*T + 130321
$83$	$(T - 1172)^{2}$ (T - 1172)^2
$89$	$T^{2} + 40T + 1600$ T^2 + 40*T + 1600
$97$	$(T + 175)^{2}$ (T + 175)^2
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more