# Properties

 Label 980.1.f.d Level $980$ Weight $1$ Character orbit 980.f Self dual yes Analytic conductor $0.489$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -20 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,1,Mod(99,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("980.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.f (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.980.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.6722800.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8}+O(q^{10})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} - q^{10} + q^{12} - q^{15} + q^{16} - q^{20} - q^{23} + q^{24} + q^{25} - q^{27} - q^{29} - q^{30} + q^{32} - q^{40} + q^{41} - q^{43} - q^{46} - 2 q^{47} + q^{48} + q^{50} - q^{54} - q^{58} - q^{60} + q^{61} + q^{64} - q^{67} - q^{69} + q^{75} - q^{80} - q^{81} + q^{82} + q^{83} - q^{86} - q^{87} + q^{89} - q^{92} - 2 q^{94} + q^{96}+O(q^{100})$$ q + q^2 + q^3 + q^4 - q^5 + q^6 + q^8 - q^10 + q^12 - q^15 + q^16 - q^20 - q^23 + q^24 + q^25 - q^27 - q^29 - q^30 + q^32 - q^40 + q^41 - q^43 - q^46 - 2 * q^47 + q^48 + q^50 - q^54 - q^58 - q^60 + q^61 + q^64 - q^67 - q^69 + q^75 - q^80 - q^81 + q^82 + q^83 - q^86 - q^87 + q^89 - q^92 - 2 * q^94 + q^96

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$0$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
99.1
 0
1.00000 1.00000 1.00000 −1.00000 1.00000 0 1.00000 0 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.f.d 1
4.b odd 2 1 980.1.f.a 1
5.b even 2 1 980.1.f.a 1
7.b odd 2 1 980.1.f.c 1
7.c even 3 2 980.1.p.a 2
7.d odd 6 2 140.1.p.a 2
20.d odd 2 1 CM 980.1.f.d 1
21.g even 6 2 1260.1.ci.b 2
28.d even 2 1 980.1.f.b 1
28.f even 6 2 140.1.p.b yes 2
28.g odd 6 2 980.1.p.b 2
35.c odd 2 1 980.1.f.b 1
35.i odd 6 2 140.1.p.b yes 2
35.j even 6 2 980.1.p.b 2
35.k even 12 4 700.1.u.a 4
56.j odd 6 2 2240.1.bt.a 2
56.m even 6 2 2240.1.bt.b 2
84.j odd 6 2 1260.1.ci.a 2
105.p even 6 2 1260.1.ci.a 2
140.c even 2 1 980.1.f.c 1
140.p odd 6 2 980.1.p.a 2
140.s even 6 2 140.1.p.a 2
140.x odd 12 4 700.1.u.a 4
280.ba even 6 2 2240.1.bt.a 2
280.bk odd 6 2 2240.1.bt.b 2
420.be odd 6 2 1260.1.ci.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.p.a 2 7.d odd 6 2
140.1.p.a 2 140.s even 6 2
140.1.p.b yes 2 28.f even 6 2
140.1.p.b yes 2 35.i odd 6 2
700.1.u.a 4 35.k even 12 4
700.1.u.a 4 140.x odd 12 4
980.1.f.a 1 4.b odd 2 1
980.1.f.a 1 5.b even 2 1
980.1.f.b 1 28.d even 2 1
980.1.f.b 1 35.c odd 2 1
980.1.f.c 1 7.b odd 2 1
980.1.f.c 1 140.c even 2 1
980.1.f.d 1 1.a even 1 1 trivial
980.1.f.d 1 20.d odd 2 1 CM
980.1.p.a 2 7.c even 3 2
980.1.p.a 2 140.p odd 6 2
980.1.p.b 2 28.g odd 6 2
980.1.p.b 2 35.j even 6 2
1260.1.ci.a 2 84.j odd 6 2
1260.1.ci.a 2 105.p even 6 2
1260.1.ci.b 2 21.g even 6 2
1260.1.ci.b 2 420.be odd 6 2
2240.1.bt.a 2 56.j odd 6 2
2240.1.bt.a 2 280.ba even 6 2
2240.1.bt.b 2 56.m even 6 2
2240.1.bt.b 2 280.bk odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(980, [\chi])$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{23} + 1$$ T23 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 1$$
$29$ $$T + 1$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T - 1$$
$43$ $$T + 1$$
$47$ $$T + 2$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 1$$
$67$ $$T + 1$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T - 1$$
$89$ $$T - 1$$
$97$ $$T$$