# Properties

 Label 980.2.i.k Level $980$ Weight $2$ Character orbit 980.i Analytic conductor $7.825$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + \beta_1 - 1) q^{3} - \beta_{2} q^{5} + ( - 2 \beta_{3} - 2 \beta_1) q^{9}+O(q^{10})$$ q + (-b2 + b1 - 1) * q^3 - b2 * q^5 + (-2*b3 - 2*b1) * q^9 $$q + ( - \beta_{2} + \beta_1 - 1) q^{3} - \beta_{2} q^{5} + ( - 2 \beta_{3} - 2 \beta_1) q^{9} + (\beta_{2} + 2 \beta_1 + 1) q^{11} + (\beta_{3} + 5) q^{13} + ( - \beta_{3} - 1) q^{15} + ( - 5 \beta_{2} - \beta_1 - 5) q^{17} + (4 \beta_{3} - 2 \beta_{2} + 4 \beta_1) q^{19} + ( - \beta_{3} + 2 \beta_{2} - \beta_1) q^{23} + ( - \beta_{2} - 1) q^{25} + ( - \beta_{3} + 1) q^{27} + ( - 4 \beta_{3} + 1) q^{29} + ( - 6 \beta_{2} + \beta_1 - 6) q^{31} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{33} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{37} + ( - 7 \beta_{2} + 6 \beta_1 - 7) q^{39} + (\beta_{3} + 2) q^{41} + ( - 4 \beta_{3} + 6) q^{43} - 2 \beta_1 q^{45} + ( - 7 \beta_{3} + \beta_{2} - 7 \beta_1) q^{47} + ( - 4 \beta_{3} + 3 \beta_{2} - 4 \beta_1) q^{51} + (8 \beta_{2} - 3 \beta_1 + 8) q^{53} + ( - 2 \beta_{3} + 1) q^{55} + ( - 6 \beta_{3} - 10) q^{57} + ( - 2 \beta_{2} - \beta_1 - 2) q^{59} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{61} + (\beta_{3} - 5 \beta_{2} + \beta_1) q^{65} + (4 \beta_{2} + 5 \beta_1 + 4) q^{67} + (3 \beta_{3} + 4) q^{69} + ( - 6 \beta_{3} - 2) q^{71} + ( - 8 \beta_{2} + 2 \beta_1 - 8) q^{73} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{75} + (10 \beta_{3} - \beta_{2} + 10 \beta_1) q^{79} + (\beta_{2} - 6 \beta_1 + 1) q^{81} - 8 q^{83} + (\beta_{3} - 5) q^{85} + (7 \beta_{2} - 3 \beta_1 + 7) q^{87} + (12 \beta_{3} + 12 \beta_1) q^{89} + ( - 7 \beta_{3} + 8 \beta_{2} - 7 \beta_1) q^{93} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{95} + ( - 9 \beta_{3} + 3) q^{97} + ( - 2 \beta_{3} + 8) q^{99}+O(q^{100})$$ q + (-b2 + b1 - 1) * q^3 - b2 * q^5 + (-2*b3 - 2*b1) * q^9 + (b2 + 2*b1 + 1) * q^11 + (b3 + 5) * q^13 + (-b3 - 1) * q^15 + (-5*b2 - b1 - 5) * q^17 + (4*b3 - 2*b2 + 4*b1) * q^19 + (-b3 + 2*b2 - b1) * q^23 + (-b2 - 1) * q^25 + (-b3 + 1) * q^27 + (-4*b3 + 1) * q^29 + (-6*b2 + b1 - 6) * q^31 + (-b3 + 3*b2 - b1) * q^33 + (-b3 - 2*b2 - b1) * q^37 + (-7*b2 + 6*b1 - 7) * q^39 + (b3 + 2) * q^41 + (-4*b3 + 6) * q^43 - 2*b1 * q^45 + (-7*b3 + b2 - 7*b1) * q^47 + (-4*b3 + 3*b2 - 4*b1) * q^51 + (8*b2 - 3*b1 + 8) * q^53 + (-2*b3 + 1) * q^55 + (-6*b3 - 10) * q^57 + (-2*b2 - b1 - 2) * q^59 + (-2*b3 + 8*b2 - 2*b1) * q^61 + (b3 - 5*b2 + b1) * q^65 + (4*b2 + 5*b1 + 4) * q^67 + (3*b3 + 4) * q^69 + (-6*b3 - 2) * q^71 + (-8*b2 + 2*b1 - 8) * q^73 + (-b3 + b2 - b1) * q^75 + (10*b3 - b2 + 10*b1) * q^79 + (b2 - 6*b1 + 1) * q^81 - 8 * q^83 + (b3 - 5) * q^85 + (7*b2 - 3*b1 + 7) * q^87 + (12*b3 + 12*b1) * q^89 + (-7*b3 + 8*b2 - 7*b1) * q^93 + (-2*b2 + 4*b1 - 2) * q^95 + (-9*b3 + 3) * q^97 + (-2*b3 + 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 2 q^{5}+O(q^{10})$$ 4 * q - 2 * q^3 + 2 * q^5 $$4 q - 2 q^{3} + 2 q^{5} + 2 q^{11} + 20 q^{13} - 4 q^{15} - 10 q^{17} + 4 q^{19} - 4 q^{23} - 2 q^{25} + 4 q^{27} + 4 q^{29} - 12 q^{31} - 6 q^{33} + 4 q^{37} - 14 q^{39} + 8 q^{41} + 24 q^{43} - 2 q^{47} - 6 q^{51} + 16 q^{53} + 4 q^{55} - 40 q^{57} - 4 q^{59} - 16 q^{61} + 10 q^{65} + 8 q^{67} + 16 q^{69} - 8 q^{71} - 16 q^{73} - 2 q^{75} + 2 q^{79} + 2 q^{81} - 32 q^{83} - 20 q^{85} + 14 q^{87} - 16 q^{93} - 4 q^{95} + 12 q^{97} + 32 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 + 2 * q^5 + 2 * q^11 + 20 * q^13 - 4 * q^15 - 10 * q^17 + 4 * q^19 - 4 * q^23 - 2 * q^25 + 4 * q^27 + 4 * q^29 - 12 * q^31 - 6 * q^33 + 4 * q^37 - 14 * q^39 + 8 * q^41 + 24 * q^43 - 2 * q^47 - 6 * q^51 + 16 * q^53 + 4 * q^55 - 40 * q^57 - 4 * q^59 - 16 * q^61 + 10 * q^65 + 8 * q^67 + 16 * q^69 - 8 * q^71 - 16 * q^73 - 2 * q^75 + 2 * q^79 + 2 * q^81 - 32 * q^83 - 20 * q^85 + 14 * q^87 - 16 * q^93 - 4 * q^95 + 12 * q^97 + 32 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 −1.20711 + 2.09077i 0 0.500000 + 0.866025i 0 0 0 −1.41421 2.44949i 0
361.2 0 0.207107 0.358719i 0 0.500000 + 0.866025i 0 0 0 1.41421 + 2.44949i 0
961.1 0 −1.20711 2.09077i 0 0.500000 0.866025i 0 0 0 −1.41421 + 2.44949i 0
961.2 0 0.207107 + 0.358719i 0 0.500000 0.866025i 0 0 0 1.41421 2.44949i 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.k 4
7.b odd 2 1 980.2.i.l 4
7.c even 3 1 980.2.a.k yes 2
7.c even 3 1 inner 980.2.i.k 4
7.d odd 6 1 980.2.a.j 2
7.d odd 6 1 980.2.i.l 4
21.g even 6 1 8820.2.a.bg 2
21.h odd 6 1 8820.2.a.bl 2
28.f even 6 1 3920.2.a.bx 2
28.g odd 6 1 3920.2.a.bo 2
35.i odd 6 1 4900.2.a.z 2
35.j even 6 1 4900.2.a.x 2
35.k even 12 2 4900.2.e.q 4
35.l odd 12 2 4900.2.e.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 7.d odd 6 1
980.2.a.k yes 2 7.c even 3 1
980.2.i.k 4 1.a even 1 1 trivial
980.2.i.k 4 7.c even 3 1 inner
980.2.i.l 4 7.b odd 2 1
980.2.i.l 4 7.d odd 6 1
3920.2.a.bo 2 28.g odd 6 1
3920.2.a.bx 2 28.f even 6 1
4900.2.a.x 2 35.j even 6 1
4900.2.a.z 2 35.i odd 6 1
4900.2.e.q 4 35.k even 12 2
4900.2.e.r 4 35.l odd 12 2
8820.2.a.bg 2 21.g even 6 1
8820.2.a.bl 2 21.h odd 6 1

## Hecke kernels

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

 $$T_{3}^{4} + 2T_{3}^{3} + 5T_{3}^{2} - 2T_{3} + 1$$ T3^4 + 2*T3^3 + 5*T3^2 - 2*T3 + 1 $$T_{11}^{4} - 2T_{11}^{3} + 11T_{11}^{2} + 14T_{11} + 49$$ T11^4 - 2*T11^3 + 11*T11^2 + 14*T11 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} + \cdots + 1$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 2 T^{3} + \cdots + 49$$
$13$ $$(T^{2} - 10 T + 23)^{2}$$
$17$ $$T^{4} + 10 T^{3} + \cdots + 529$$
$19$ $$T^{4} - 4 T^{3} + \cdots + 784$$
$23$ $$T^{4} + 4 T^{3} + \cdots + 4$$
$29$ $$(T^{2} - 2 T - 31)^{2}$$
$31$ $$T^{4} + 12 T^{3} + \cdots + 1156$$
$37$ $$T^{4} - 4 T^{3} + \cdots + 4$$
$41$ $$(T^{2} - 4 T + 2)^{2}$$
$43$ $$(T^{2} - 12 T + 4)^{2}$$
$47$ $$T^{4} + 2 T^{3} + \cdots + 9409$$
$53$ $$T^{4} - 16 T^{3} + \cdots + 2116$$
$59$ $$T^{4} + 4 T^{3} + \cdots + 4$$
$61$ $$T^{4} + 16 T^{3} + \cdots + 3136$$
$67$ $$T^{4} - 8 T^{3} + \cdots + 1156$$
$71$ $$(T^{2} + 4 T - 68)^{2}$$
$73$ $$T^{4} + 16 T^{3} + \cdots + 3136$$
$79$ $$T^{4} - 2 T^{3} + \cdots + 39601$$
$83$ $$(T + 8)^{4}$$
$89$ $$T^{4} + 288 T^{2} + 82944$$
$97$ $$(T^{2} - 6 T - 153)^{2}$$