# Properties

 Label 980.2.i.c Level $980$ Weight $2$ Character orbit 980.i Analytic conductor $7.825$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

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: $$1$$ 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 20) 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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{5} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 - z * q^5 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} - \zeta_{6} q^{5} - \zeta_{6} q^{9} - 2 q^{13} + 2 q^{15} + (6 \zeta_{6} - 6) q^{17} - 4 \zeta_{6} q^{19} - 6 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} - 4 q^{27} + 6 q^{29} + (4 \zeta_{6} - 4) q^{31} - 2 \zeta_{6} q^{37} + ( - 4 \zeta_{6} + 4) q^{39} - 6 q^{41} - 10 q^{43} + (\zeta_{6} - 1) q^{45} - 6 \zeta_{6} q^{47} - 12 \zeta_{6} q^{51} + ( - 6 \zeta_{6} + 6) q^{53} + 8 q^{57} + ( - 12 \zeta_{6} + 12) q^{59} + 2 \zeta_{6} q^{61} + 2 \zeta_{6} q^{65} + (2 \zeta_{6} - 2) q^{67} + 12 q^{69} - 12 q^{71} + ( - 2 \zeta_{6} + 2) q^{73} - 2 \zeta_{6} q^{75} - 8 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} - 6 q^{83} + 6 q^{85} + (12 \zeta_{6} - 12) q^{87} - 6 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} + (4 \zeta_{6} - 4) q^{95} - 2 q^{97} +O(q^{100})$$ q + (2*z - 2) * q^3 - z * q^5 - z * q^9 - 2 * q^13 + 2 * q^15 + (6*z - 6) * q^17 - 4*z * q^19 - 6*z * q^23 + (z - 1) * q^25 - 4 * q^27 + 6 * q^29 + (4*z - 4) * q^31 - 2*z * q^37 + (-4*z + 4) * q^39 - 6 * q^41 - 10 * q^43 + (z - 1) * q^45 - 6*z * q^47 - 12*z * q^51 + (-6*z + 6) * q^53 + 8 * q^57 + (-12*z + 12) * q^59 + 2*z * q^61 + 2*z * q^65 + (2*z - 2) * q^67 + 12 * q^69 - 12 * q^71 + (-2*z + 2) * q^73 - 2*z * q^75 - 8*z * q^79 + (-11*z + 11) * q^81 - 6 * q^83 + 6 * q^85 + (12*z - 12) * q^87 - 6*z * q^89 - 8*z * q^93 + (4*z - 4) * q^95 - 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - q^{5} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - q^5 - q^9 $$2 q - 2 q^{3} - q^{5} - q^{9} - 4 q^{13} + 4 q^{15} - 6 q^{17} - 4 q^{19} - 6 q^{23} - q^{25} - 8 q^{27} + 12 q^{29} - 4 q^{31} - 2 q^{37} + 4 q^{39} - 12 q^{41} - 20 q^{43} - q^{45} - 6 q^{47} - 12 q^{51} + 6 q^{53} + 16 q^{57} + 12 q^{59} + 2 q^{61} + 2 q^{65} - 2 q^{67} + 24 q^{69} - 24 q^{71} + 2 q^{73} - 2 q^{75} - 8 q^{79} + 11 q^{81} - 12 q^{83} + 12 q^{85} - 12 q^{87} - 6 q^{89} - 8 q^{93} - 4 q^{95} - 4 q^{97}+O(q^{100})$$ 2 * q - 2 * q^3 - q^5 - q^9 - 4 * q^13 + 4 * q^15 - 6 * q^17 - 4 * q^19 - 6 * q^23 - q^25 - 8 * q^27 + 12 * q^29 - 4 * q^31 - 2 * q^37 + 4 * q^39 - 12 * q^41 - 20 * q^43 - q^45 - 6 * q^47 - 12 * q^51 + 6 * q^53 + 16 * q^57 + 12 * q^59 + 2 * q^61 + 2 * q^65 - 2 * q^67 + 24 * q^69 - 24 * q^71 + 2 * q^73 - 2 * q^75 - 8 * q^79 + 11 * q^81 - 12 * q^83 + 12 * q^85 - 12 * q^87 - 6 * q^89 - 8 * q^93 - 4 * q^95 - 4 * q^97

## Character values

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

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

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 + 1.73205i 0 −0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0
961.1 0 −1.00000 1.73205i 0 −0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0
 $$n$$: 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
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.c 2
7.b odd 2 1 980.2.i.i 2
7.c even 3 1 980.2.a.h 1
7.c even 3 1 inner 980.2.i.c 2
7.d odd 6 1 20.2.a.a 1
7.d odd 6 1 980.2.i.i 2
21.g even 6 1 180.2.a.a 1
21.h odd 6 1 8820.2.a.g 1
28.f even 6 1 80.2.a.b 1
28.g odd 6 1 3920.2.a.h 1
35.i odd 6 1 100.2.a.a 1
35.j even 6 1 4900.2.a.e 1
35.k even 12 2 100.2.c.a 2
35.l odd 12 2 4900.2.e.f 2
56.j odd 6 1 320.2.a.f 1
56.m even 6 1 320.2.a.a 1
63.i even 6 1 1620.2.i.b 2
63.k odd 6 1 1620.2.i.h 2
63.s even 6 1 1620.2.i.b 2
63.t odd 6 1 1620.2.i.h 2
77.i even 6 1 2420.2.a.a 1
84.j odd 6 1 720.2.a.h 1
91.s odd 6 1 3380.2.a.c 1
91.bb even 12 2 3380.2.f.b 2
105.p even 6 1 900.2.a.b 1
105.w odd 12 2 900.2.d.c 2
112.v even 12 2 1280.2.d.g 2
112.x odd 12 2 1280.2.d.c 2
119.h odd 6 1 5780.2.a.f 1
119.m odd 12 2 5780.2.c.a 2
133.o even 6 1 7220.2.a.f 1
140.s even 6 1 400.2.a.c 1
140.x odd 12 2 400.2.c.b 2
168.ba even 6 1 2880.2.a.m 1
168.be odd 6 1 2880.2.a.f 1
280.ba even 6 1 1600.2.a.w 1
280.bk odd 6 1 1600.2.a.c 1
280.bp odd 12 2 1600.2.c.e 2
280.bv even 12 2 1600.2.c.d 2
308.m odd 6 1 9680.2.a.ba 1
420.be odd 6 1 3600.2.a.be 1
420.br even 12 2 3600.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 7.d odd 6 1
80.2.a.b 1 28.f even 6 1
100.2.a.a 1 35.i odd 6 1
100.2.c.a 2 35.k even 12 2
180.2.a.a 1 21.g even 6 1
320.2.a.a 1 56.m even 6 1
320.2.a.f 1 56.j odd 6 1
400.2.a.c 1 140.s even 6 1
400.2.c.b 2 140.x odd 12 2
720.2.a.h 1 84.j odd 6 1
900.2.a.b 1 105.p even 6 1
900.2.d.c 2 105.w odd 12 2
980.2.a.h 1 7.c even 3 1
980.2.i.c 2 1.a even 1 1 trivial
980.2.i.c 2 7.c even 3 1 inner
980.2.i.i 2 7.b odd 2 1
980.2.i.i 2 7.d odd 6 1
1280.2.d.c 2 112.x odd 12 2
1280.2.d.g 2 112.v even 12 2
1600.2.a.c 1 280.bk odd 6 1
1600.2.a.w 1 280.ba even 6 1
1600.2.c.d 2 280.bv even 12 2
1600.2.c.e 2 280.bp odd 12 2
1620.2.i.b 2 63.i even 6 1
1620.2.i.b 2 63.s even 6 1
1620.2.i.h 2 63.k odd 6 1
1620.2.i.h 2 63.t odd 6 1
2420.2.a.a 1 77.i even 6 1
2880.2.a.f 1 168.be odd 6 1
2880.2.a.m 1 168.ba even 6 1
3380.2.a.c 1 91.s odd 6 1
3380.2.f.b 2 91.bb even 12 2
3600.2.a.be 1 420.be odd 6 1
3600.2.f.j 2 420.br even 12 2
3920.2.a.h 1 28.g odd 6 1
4900.2.a.e 1 35.j even 6 1
4900.2.e.f 2 35.l odd 12 2
5780.2.a.f 1 119.h odd 6 1
5780.2.c.a 2 119.m odd 12 2
7220.2.a.f 1 133.o even 6 1
8820.2.a.g 1 21.h odd 6 1
9680.2.a.ba 1 308.m odd 6 1

## Hecke kernels

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

 $$T_{3}^{2} + 2T_{3} + 4$$ T3^2 + 2*T3 + 4 $$T_{11}$$ T11

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} + T + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} + 6T + 36$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$T^{2} + 4T + 16$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T + 10)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 2T + 4$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 6T + 36$$
$97$ $$(T + 2)^{2}$$