# Properties

 Label 980.2.x.c Level $980$ Weight $2$ Character orbit 980.x Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(67,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 3, 8]))

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

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

chi := DirichletCharacter("980.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.x (of order $$12$$, degree $$4$$, 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: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9}+O(q^{10})$$ q + (-z^2 + z + 1) * q^2 + (-2*z^3 + 2*z) * q^4 + (2*z^2 + z - 2) * q^5 + (-2*z^3 + 2) * q^8 + 3*z * q^9 $$q + ( - \zeta_{12}^{2} + \zeta_{12} + 1) q^{2} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{4} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + ( - 2 \zeta_{12}^{3} + 2) q^{8} + 3 \zeta_{12} q^{9} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - \zeta_{12}) q^{10} + ( - \zeta_{12}^{3} + 1) q^{13} + ( - 4 \zeta_{12}^{2} + 4) q^{16} + (3 \zeta_{12}^{3} + \cdots - 3 \zeta_{12}) q^{17}+ \cdots + ( - 13 \zeta_{12}^{3} - 13) q^{97}+O(q^{100})$$ q + (-z^2 + z + 1) * q^2 + (-2*z^3 + 2*z) * q^4 + (2*z^2 + z - 2) * q^5 + (-2*z^3 + 2) * q^8 + 3*z * q^9 + (z^3 + 3*z^2 - z) * q^10 + (-z^3 + 1) * q^13 + (-4*z^2 + 4) * q^16 + (3*z^3 + 3*z^2 - 3*z) * q^17 + (-3*z^3 + 3*z^2 + 3*z) * q^18 + (4*z^3 + 2) * q^20 + (4*z^3 - 3*z^2 - 4*z) * q^25 + (-2*z^2 + 2) * q^26 + 4*z^3 * q^29 + (-4*z^3 - 4*z^2 + 4*z) * q^32 + 6*z^3 * q^34 + 6 * q^36 + (-7*z^2 + 7*z + 7) * q^37 + (2*z^2 + 6*z - 2) * q^40 + 8 * q^41 + (6*z^3 + 3*z^2 - 6*z) * q^45 + (z^3 - 7) * q^50 + (-2*z^3 - 2*z^2 + 2*z) * q^52 + (9*z^3 - 9*z^2 - 9*z) * q^53 + (4*z^2 + 4*z - 4) * q^58 + (-12*z^2 + 12) * q^61 - 8*z^3 * q^64 + (z^2 + 3*z - 1) * q^65 + (6*z^2 + 6*z - 6) * q^68 + (-6*z^2 + 6*z + 6) * q^72 + (11*z^3 - 11*z^2 - 11*z) * q^73 + (-14*z^3 + 14*z) * q^74 + (-4*z^3 + 8*z^2 + 4*z) * q^80 + 9*z^2 * q^81 + (-8*z^2 + 8*z + 8) * q^82 + (-3*z^3 - 9) * q^85 - 16*z * q^89 + (9*z^3 - 3) * q^90 + (-13*z^3 - 13) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 4 q^{5} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 $$4 q + 2 q^{2} - 4 q^{5} + 8 q^{8} + 6 q^{10} + 4 q^{13} + 8 q^{16} + 6 q^{17} + 6 q^{18} + 8 q^{20} - 6 q^{25} + 4 q^{26} - 8 q^{32} + 24 q^{36} + 14 q^{37} - 4 q^{40} + 32 q^{41} + 6 q^{45} - 28 q^{50} - 4 q^{52} - 18 q^{53} - 8 q^{58} + 24 q^{61} - 2 q^{65} - 12 q^{68} + 12 q^{72} - 22 q^{73} + 16 q^{80} + 18 q^{81} + 16 q^{82} - 36 q^{85} - 12 q^{90} - 52 q^{97}+O(q^{100})$$ 4 * q + 2 * q^2 - 4 * q^5 + 8 * q^8 + 6 * q^10 + 4 * q^13 + 8 * q^16 + 6 * q^17 + 6 * q^18 + 8 * q^20 - 6 * q^25 + 4 * q^26 - 8 * q^32 + 24 * q^36 + 14 * q^37 - 4 * q^40 + 32 * q^41 + 6 * q^45 - 28 * q^50 - 4 * q^52 - 18 * q^53 - 8 * q^58 + 24 * q^61 - 2 * q^65 - 12 * q^68 + 12 * q^72 - 22 * q^73 + 16 * q^80 + 18 * q^81 + 16 * q^82 - 36 * q^85 - 12 * q^90 - 52 * q^97

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$\zeta_{12}^{3}$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 − 0.500000i
−0.366025 + 1.36603i 0 −1.73205 1.00000i −1.86603 1.23205i 0 0 2.00000 2.00000i −2.59808 + 1.50000i 2.36603 2.09808i
263.1 1.36603 + 0.366025i 0 1.73205 + 1.00000i −0.133975 2.23205i 0 0 2.00000 + 2.00000i 2.59808 1.50000i 0.633975 3.09808i
667.1 1.36603 0.366025i 0 1.73205 1.00000i −0.133975 + 2.23205i 0 0 2.00000 2.00000i 2.59808 + 1.50000i 0.633975 + 3.09808i
863.1 −0.366025 1.36603i 0 −1.73205 + 1.00000i −1.86603 + 1.23205i 0 0 2.00000 + 2.00000i −2.59808 1.50000i 2.36603 + 2.09808i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
7.c even 3 1 inner
20.e even 4 1 inner
28.g odd 6 1 inner
35.l odd 12 1 inner
140.w even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.x.c 4
4.b odd 2 1 CM 980.2.x.c 4
5.c odd 4 1 inner 980.2.x.c 4
7.b odd 2 1 980.2.x.d 4
7.c even 3 1 980.2.k.a 2
7.c even 3 1 inner 980.2.x.c 4
7.d odd 6 1 20.2.e.a 2
7.d odd 6 1 980.2.x.d 4
20.e even 4 1 inner 980.2.x.c 4
21.g even 6 1 180.2.k.c 2
28.d even 2 1 980.2.x.d 4
28.f even 6 1 20.2.e.a 2
28.f even 6 1 980.2.x.d 4
28.g odd 6 1 980.2.k.a 2
28.g odd 6 1 inner 980.2.x.c 4
35.f even 4 1 980.2.x.d 4
35.i odd 6 1 100.2.e.b 2
35.k even 12 1 20.2.e.a 2
35.k even 12 1 100.2.e.b 2
35.k even 12 1 980.2.x.d 4
35.l odd 12 1 980.2.k.a 2
35.l odd 12 1 inner 980.2.x.c 4
56.j odd 6 1 320.2.n.e 2
56.m even 6 1 320.2.n.e 2
84.j odd 6 1 180.2.k.c 2
105.p even 6 1 900.2.k.c 2
105.w odd 12 1 180.2.k.c 2
105.w odd 12 1 900.2.k.c 2
112.v even 12 1 1280.2.o.g 2
112.v even 12 1 1280.2.o.j 2
112.x odd 12 1 1280.2.o.g 2
112.x odd 12 1 1280.2.o.j 2
140.j odd 4 1 980.2.x.d 4
140.s even 6 1 100.2.e.b 2
140.w even 12 1 980.2.k.a 2
140.w even 12 1 inner 980.2.x.c 4
140.x odd 12 1 20.2.e.a 2
140.x odd 12 1 100.2.e.b 2
140.x odd 12 1 980.2.x.d 4
280.ba even 6 1 1600.2.n.h 2
280.bk odd 6 1 1600.2.n.h 2
280.bp odd 12 1 320.2.n.e 2
280.bp odd 12 1 1600.2.n.h 2
280.bv even 12 1 320.2.n.e 2
280.bv even 12 1 1600.2.n.h 2
420.be odd 6 1 900.2.k.c 2
420.br even 12 1 180.2.k.c 2
420.br even 12 1 900.2.k.c 2
560.ce odd 12 1 1280.2.o.j 2
560.ch even 12 1 1280.2.o.g 2
560.cz even 12 1 1280.2.o.j 2
560.da odd 12 1 1280.2.o.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 7.d odd 6 1
20.2.e.a 2 28.f even 6 1
20.2.e.a 2 35.k even 12 1
20.2.e.a 2 140.x odd 12 1
100.2.e.b 2 35.i odd 6 1
100.2.e.b 2 35.k even 12 1
100.2.e.b 2 140.s even 6 1
100.2.e.b 2 140.x odd 12 1
180.2.k.c 2 21.g even 6 1
180.2.k.c 2 84.j odd 6 1
180.2.k.c 2 105.w odd 12 1
180.2.k.c 2 420.br even 12 1
320.2.n.e 2 56.j odd 6 1
320.2.n.e 2 56.m even 6 1
320.2.n.e 2 280.bp odd 12 1
320.2.n.e 2 280.bv even 12 1
900.2.k.c 2 105.p even 6 1
900.2.k.c 2 105.w odd 12 1
900.2.k.c 2 420.be odd 6 1
900.2.k.c 2 420.br even 12 1
980.2.k.a 2 7.c even 3 1
980.2.k.a 2 28.g odd 6 1
980.2.k.a 2 35.l odd 12 1
980.2.k.a 2 140.w even 12 1
980.2.x.c 4 1.a even 1 1 trivial
980.2.x.c 4 4.b odd 2 1 CM
980.2.x.c 4 5.c odd 4 1 inner
980.2.x.c 4 7.c even 3 1 inner
980.2.x.c 4 20.e even 4 1 inner
980.2.x.c 4 28.g odd 6 1 inner
980.2.x.c 4 35.l odd 12 1 inner
980.2.x.c 4 140.w even 12 1 inner
980.2.x.d 4 7.b odd 2 1
980.2.x.d 4 7.d odd 6 1
980.2.x.d 4 28.d even 2 1
980.2.x.d 4 28.f even 6 1
980.2.x.d 4 35.f even 4 1
980.2.x.d 4 35.k even 12 1
980.2.x.d 4 140.j odd 4 1
980.2.x.d 4 140.x odd 12 1
1280.2.o.g 2 112.v even 12 1
1280.2.o.g 2 112.x odd 12 1
1280.2.o.g 2 560.ch even 12 1
1280.2.o.g 2 560.da odd 12 1
1280.2.o.j 2 112.v even 12 1
1280.2.o.j 2 112.x odd 12 1
1280.2.o.j 2 560.ce odd 12 1
1280.2.o.j 2 560.cz even 12 1
1600.2.n.h 2 280.ba even 6 1
1600.2.n.h 2 280.bk odd 6 1
1600.2.n.h 2 280.bp odd 12 1
1600.2.n.h 2 280.bv even 12 1

## Hecke kernels

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

 $$T_{3}$$ T3 $$T_{11}$$ T11 $$T_{13}^{2} - 2T_{13} + 2$$ T13^2 - 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + \cdots + 4$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 4 T^{3} + \cdots + 25$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} - 2 T + 2)^{2}$$
$17$ $$T^{4} - 6 T^{3} + \cdots + 324$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 16)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4} - 14 T^{3} + \cdots + 9604$$
$41$ $$(T - 8)^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 18 T^{3} + \cdots + 26244$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 12 T + 144)^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 22 T^{3} + \cdots + 58564$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$T^{4} - 256 T^{2} + 65536$$
$97$ $$(T^{2} + 26 T + 338)^{2}$$