LMFDB - Newform orbit 980.2.i.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(361,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("980.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$980 = 2^{2} \cdot 5 \cdot 7^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	980.i (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:	$7.82533939809$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 140)
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.

$f(q)$	$=$	$q + (3 \zeta_{6} - 3) q^{3} - \zeta_{6} q^{5} - 6 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 2) q^{11} + 6 q^{13} + 3 q^{15} + ( - 2 \zeta_{6} + 2) q^{17} + 9 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + 9 q^{27} + \cdots - 12 q^{99} +O(q^{100})$ q + (3z - 3) q^3 - z * q^5 - 6z q^9 + (-2z + 2) q^11 + 6 * q^13 + 3 * q^15 + (-2z + 2) q^17 + 9z q^23 + (z - 1) * q^25 + 9 * q^27 + 3 * q^29 + (-2z + 2) q^31 + 6z q^33 - 8z q^37 + (18z - 18) q^39 - 5 * q^41 + q^43 + (6z - 6) q^45 + 8z q^47 + 6z q^51 + (4z - 4) q^53 - 2 * q^55 + (8z - 8) q^59 + 7z q^61 - 6z q^65 + (-3z + 3) q^67 - 27 * q^69 + 8 * q^71 + (-14z + 14) q^73 - 3z q^75 - 4z q^79 + (9z - 9) q^81 + q^83 - 2 * q^85 + (9z - 9) q^87 + 13z q^89 + 6z q^93 + 10 * q^97 - 12 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 3 q^{3} - q^{5} - 6 q^{9} + 2 q^{11} + 12 q^{13} + 6 q^{15} + 2 q^{17} + 9 q^{23} - q^{25} + 18 q^{27} + 6 q^{29} + 2 q^{31} + 6 q^{33} - 8 q^{37} - 18 q^{39} - 10 q^{41} + 2 q^{43} - 6 q^{45} + 8 q^{47}+ \cdots - 24 q^{99}+O(q^{100})$ 2 * q - 3 * q^3 - q^5 - 6 * q^9 + 2 * q^11 + 12 * q^13 + 6 * q^15 + 2 * q^17 + 9 * q^23 - q^25 + 18 * q^27 + 6 * q^29 + 2 * q^31 + 6 * q^33 - 8 * q^37 - 18 * q^39 - 10 * q^41 + 2 * q^43 - 6 * q^45 + 8 * q^47 + 6 * q^51 - 4 * q^53 - 4 * q^55 - 8 * q^59 + 7 * q^61 - 6 * q^65 + 3 * q^67 - 54 * q^69 + 16 * q^71 + 14 * q^73 - 3 * q^75 - 4 * q^79 - 9 * q^81 + 2 * q^83 - 4 * q^85 - 9 * q^87 + 13 * q^89 + 6 * q^93 + 20 * q^97 - 24 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$ .

$n$	$101$	$197$	$491$
$\chi(n)$	$-\zeta_{6}$	$1$	$1$

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

−1.50000

2.59808i

−0.500000

−

0.866025i

−3.00000

−

5.19615i

961.1

−1.50000

−

2.59808i

−0.500000

0.866025i

−3.00000

5.19615i

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	980.2.i.a		2
7.b	odd	2	1		140.2.i.b	✓	2
7.c	even	3	1		980.2.a.i		1
7.c	even	3	1	inner	980.2.i.a		2
7.d	odd	6	1		140.2.i.b	✓	2
7.d	odd	6	1		980.2.a.a		1
21.c	even	2	1		1260.2.s.b		2
21.g	even	6	1		1260.2.s.b		2
21.g	even	6	1		8820.2.a.w		1
21.h	odd	6	1		8820.2.a.k		1
28.d	even	2	1		560.2.q.a		2
28.f	even	6	1		560.2.q.a		2
28.f	even	6	1		3920.2.a.bi		1
28.g	odd	6	1		3920.2.a.d		1
35.c	odd	2	1		700.2.i.a		2
35.f	even	4	2		700.2.r.c		4
35.i	odd	6	1		700.2.i.a		2
35.i	odd	6	1		4900.2.a.v		1
35.j	even	6	1		4900.2.a.a		1
35.k	even	12	2		700.2.r.c		4
35.k	even	12	2		4900.2.e.c		2
35.l	odd	12	2		4900.2.e.b		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
140.2.i.b	✓	2	7.b	odd	2	1
140.2.i.b	✓	2	7.d	odd	6	1
560.2.q.a		2	28.d	even	2	1
560.2.q.a		2	28.f	even	6	1
700.2.i.a		2	35.c	odd	2	1
700.2.i.a		2	35.i	odd	6	1
700.2.r.c		4	35.f	even	4	2
700.2.r.c		4	35.k	even	12	2
980.2.a.a		1	7.d	odd	6	1
980.2.a.i		1	7.c	even	3	1
980.2.i.a		2	1.a	even	1	1	trivial
980.2.i.a		2	7.c	even	3	1	inner
1260.2.s.b		2	21.c	even	2	1
1260.2.s.b		2	21.g	even	6	1
3920.2.a.d		1	28.g	odd	6	1
3920.2.a.bi		1	28.f	even	6	1
4900.2.a.a		1	35.j	even	6	1
4900.2.a.v		1	35.i	odd	6	1
4900.2.e.b		2	35.l	odd	12	2
4900.2.e.c		2	35.k	even	12	2
8820.2.a.k		1	21.h	odd	6	1
8820.2.a.w		1	21.g	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(980, [\chi])$ :

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

T_{11}^{2} - 2T_{11} + 4

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{2}$ T^2
$3$	$T^{2} + 3T + 9$ T^2 + 3*T + 9
$5$	$T^{2} + T + 1$ T^2 + T + 1
$7$	$T^{2}$ T^2
$11$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$13$	$(T - 6)^{2}$ (T - 6)^2
$17$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$19$	$T^{2}$ T^2
$23$	$T^{2} - 9T + 81$ T^2 - 9*T + 81
$29$	$(T - 3)^{2}$ (T - 3)^2
$31$	$T^{2} - 2T + 4$ T^2 - 2*T + 4
$37$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$41$	$(T + 5)^{2}$ (T + 5)^2
$43$	$(T - 1)^{2}$ (T - 1)^2
$47$	$T^{2} - 8T + 64$ T^2 - 8*T + 64
$53$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$59$	$T^{2} + 8T + 64$ T^2 + 8*T + 64
$61$	$T^{2} - 7T + 49$ T^2 - 7*T + 49
$67$	$T^{2} - 3T + 9$ T^2 - 3*T + 9
$71$	$(T - 8)^{2}$ (T - 8)^2
$73$	$T^{2} - 14T + 196$ T^2 - 14*T + 196
$79$	$T^{2} + 4T + 16$ T^2 + 4*T + 16
$83$	$(T - 1)^{2}$ (T - 1)^2
$89$	$T^{2} - 13T + 169$ T^2 - 13*T + 169
$97$	$(T - 10)^{2}$ (T - 10)^2
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