# Properties

 Label 980.1.p.b Level $980$ Weight $1$ Character orbit 980.p Analytic conductor $0.489$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -20 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,1,Mod(79,980)]

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([3, 3, 2]))

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

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

chi := DirichletCharacter("980.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.p (of order $$6$$, 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: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, a_2]$$ 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: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

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

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$\zeta_{6}^{2}$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 0.866025i 1.00000 0 −1.00000 0 −0.500000 0.866025i
459.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 0 −1.00000 0 −0.500000 + 0.866025i
 $$n$$: 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 $$\Q(\sqrt{-5})$$
7.c even 3 1 inner
140.p odd 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} + T + 1$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 1)^{2}$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} + 2T + 4$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} + T + 1$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$(T + 1)^{2}$$
$89$ $$T^{2} + T + 1$$
$97$ $$T^{2}$$