# Properties

 Label 980.2.q.g Level $980$ Weight $2$ Character orbit 980.q 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(569,980)]

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, 3, 2]))

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

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

chi := DirichletCharacter("980.569");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.q (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: $$7.82533939809$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 2) q^{3} + \beta_{3} q^{5}+O(q^{10})$$ q + (-b2 + 2) * q^3 + b3 * q^5 $$q + ( - \beta_{2} + 2) q^{3} + \beta_{3} q^{5} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{11} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 2) q^{13} + (2 \beta_{3} - \beta_1) q^{15} + ( - \beta_{3} + \beta_{2} - 3) q^{17} + (2 \beta_{3} - \beta_1 + 1) q^{19} + ( - \beta_{2} + 2 \beta_1 - 3) q^{23} + ( - 5 \beta_{2} - \beta_1 + 5) q^{25} + (6 \beta_{2} - 3) q^{27} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{29} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{31} + ( - 3 \beta_1 + 3) q^{33} + (4 \beta_{2} - \beta_1 + 5) q^{37} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{39} + (\beta_{3} + \beta_{2} + \beta_1 + 7) q^{41} + ( - \beta_{3} + 5 \beta_{2} + \beta_1 - 3) q^{43} + (2 \beta_{2} - \beta_1 + 3) q^{47} + ( - 2 \beta_{3} + 4 \beta_{2} + \beta_1 - 5) q^{51} + ( - 3 \beta_{3} - \beta_{2} - 1) q^{53} + ( - \beta_{3} - 5 \beta_{2} + 2 \beta_1 - 5) q^{55} + (3 \beta_{3} - \beta_{2} - 3 \beta_1 + 2) q^{57} + (\beta_{3} - \beta_{2} - 2 \beta_1 + 1) q^{59} + (4 \beta_{3} + 7 \beta_{2} - 2 \beta_1 - 5) q^{61} + ( - 10 \beta_{2} - 2 \beta_1) q^{65} + ( - 4 \beta_{3} + \beta_{2} - 6) q^{67} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 7) q^{69} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{71} + ( - \beta_{3} - 3 \beta_{2} + 5) q^{73} + ( - \beta_{3} - 10 \beta_{2} - \beta_1 + 5) q^{75} + ( - 2 \beta_{3} - 4 \beta_{2} + \beta_1 + 3) q^{79} + 9 \beta_{2} q^{81} + ( - 3 \beta_{3} - \beta_{2} + 3 \beta_1 - 1) q^{83} + ( - 3 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 5) q^{85} + (3 \beta_{3} + 2 \beta_{2} - 1) q^{87} + ( - 7 \beta_{2} + 7) q^{89} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{93} + ( - 10 \beta_{2} - \beta_1 + 5) q^{95} + (8 \beta_{2} - 4) q^{97}+O(q^{100})$$ q + (-b2 + 2) * q^3 + b3 * q^5 + (b3 + b2 - 2*b1 + 1) * q^11 + (2*b3 - 2*b2 - 2*b1 + 2) * q^13 + (2*b3 - b1) * q^15 + (-b3 + b2 - 3) * q^17 + (2*b3 - b1 + 1) * q^19 + (-b2 + 2*b1 - 3) * q^23 + (-5*b2 - b1 + 5) * q^25 + (6*b2 - 3) * q^27 + (b3 + b2 + b1 - 1) * q^29 + (b3 - b2 - 2*b1 + 1) * q^31 + (-3*b1 + 3) * q^33 + (4*b2 - b1 + 5) * q^37 + (2*b3 - 4*b2 - 4*b1 + 2) * q^39 + (b3 + b2 + b1 + 7) * q^41 + (-b3 + 5*b2 + b1 - 3) * q^43 + (2*b2 - b1 + 3) * q^47 + (-2*b3 + 4*b2 + b1 - 5) * q^51 + (-3*b3 - b2 - 1) * q^53 + (-b3 - 5*b2 + 2*b1 - 5) * q^55 + (3*b3 - b2 - 3*b1 + 2) * q^57 + (b3 - b2 - 2*b1 + 1) * q^59 + (4*b3 + 7*b2 - 2*b1 - 5) * q^61 + (-10*b2 - 2*b1) * q^65 + (-4*b3 + b2 - 6) * q^67 + (2*b3 + 2*b2 + 2*b1 - 7) * q^69 + (2*b3 + 2*b2 + 2*b1 + 2) * q^71 + (-b3 - 3*b2 + 5) * q^73 + (-b3 - 10*b2 - b1 + 5) * q^75 + (-2*b3 - 4*b2 + b1 + 3) * q^79 + 9*b2 * q^81 + (-3*b3 - b2 + 3*b1 - 1) * q^83 + (-3*b3 + 5*b2 + 2*b1 - 5) * q^85 + (3*b3 + 2*b2 - 1) * q^87 + (-7*b2 + 7) * q^89 + (-2*b2 - 3*b1 + 1) * q^93 + (-10*b2 - b1 + 5) * q^95 + (8*b2 - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} - q^{5}+O(q^{10})$$ 4 * q + 6 * q^3 - q^5 $$4 q + 6 q^{3} - q^{5} + 3 q^{11} - 3 q^{15} - 9 q^{17} + q^{19} - 12 q^{23} + 9 q^{25} - 2 q^{29} - q^{31} + 9 q^{33} + 27 q^{37} - 6 q^{39} + 30 q^{41} + 15 q^{47} - 9 q^{51} - 3 q^{53} - 27 q^{55} - q^{59} - 12 q^{61} - 22 q^{65} - 18 q^{67} - 24 q^{69} + 12 q^{71} + 15 q^{73} + 7 q^{79} + 18 q^{81} - 5 q^{85} - 3 q^{87} + 14 q^{89} - 3 q^{93} - q^{95}+O(q^{100})$$ 4 * q + 6 * q^3 - q^5 + 3 * q^11 - 3 * q^15 - 9 * q^17 + q^19 - 12 * q^23 + 9 * q^25 - 2 * q^29 - q^31 + 9 * q^33 + 27 * q^37 - 6 * q^39 + 30 * q^41 + 15 * q^47 - 9 * q^51 - 3 * q^53 - 27 * q^55 - q^59 - 12 * q^61 - 22 * q^65 - 18 * q^67 - 24 * q^69 + 12 * q^71 + 15 * q^73 + 7 * q^79 + 18 * q^81 - 5 * q^85 - 3 * q^87 + 14 * q^89 - 3 * q^93 - q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20$$ (v^3 + 4*v^2 - 4*v - 5) / 20 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu + 5 ) / 4$$ (-v^3 + 4*v + 5) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + 5\beta_{2}$$ b3 + 5*b2 $$\nu^{3}$$ $$=$$ $$-4\beta_{3} + 4\beta _1 + 5$$ -4*b3 + 4*b1 + 5

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
569.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 1.50000 0.866025i 0 −2.13746 + 0.656712i 0 0 0 0 0
569.2 0 1.50000 0.866025i 0 1.63746 1.52274i 0 0 0 0 0
949.1 0 1.50000 + 0.866025i 0 −2.13746 0.656712i 0 0 0 0 0
949.2 0 1.50000 + 0.866025i 0 1.63746 + 1.52274i 0 0 0 0 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
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.2.q.g 4
5.b even 2 1 980.2.q.b 4
7.b odd 2 1 140.2.q.a 4
7.c even 3 1 980.2.e.c 4
7.c even 3 1 980.2.q.b 4
7.d odd 6 1 140.2.q.b yes 4
7.d odd 6 1 980.2.e.f 4
21.c even 2 1 1260.2.bm.a 4
21.g even 6 1 1260.2.bm.b 4
28.d even 2 1 560.2.bw.e 4
28.f even 6 1 560.2.bw.a 4
35.c odd 2 1 140.2.q.b yes 4
35.f even 4 2 700.2.i.f 8
35.i odd 6 1 140.2.q.a 4
35.i odd 6 1 980.2.e.f 4
35.j even 6 1 980.2.e.c 4
35.j even 6 1 inner 980.2.q.g 4
35.k even 12 2 700.2.i.f 8
35.k even 12 2 4900.2.a.be 4
35.l odd 12 2 4900.2.a.bf 4
105.g even 2 1 1260.2.bm.b 4
105.p even 6 1 1260.2.bm.a 4
140.c even 2 1 560.2.bw.a 4
140.s even 6 1 560.2.bw.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.q.a 4 7.b odd 2 1
140.2.q.a 4 35.i odd 6 1
140.2.q.b yes 4 7.d odd 6 1
140.2.q.b yes 4 35.c odd 2 1
560.2.bw.a 4 28.f even 6 1
560.2.bw.a 4 140.c even 2 1
560.2.bw.e 4 28.d even 2 1
560.2.bw.e 4 140.s even 6 1
700.2.i.f 8 35.f even 4 2
700.2.i.f 8 35.k even 12 2
980.2.e.c 4 7.c even 3 1
980.2.e.c 4 35.j even 6 1
980.2.e.f 4 7.d odd 6 1
980.2.e.f 4 35.i odd 6 1
980.2.q.b 4 5.b even 2 1
980.2.q.b 4 7.c even 3 1
980.2.q.g 4 1.a even 1 1 trivial
980.2.q.g 4 35.j even 6 1 inner
1260.2.bm.a 4 21.c even 2 1
1260.2.bm.a 4 105.p even 6 1
1260.2.bm.b 4 21.g even 6 1
1260.2.bm.b 4 105.g even 2 1
4900.2.a.be 4 35.k even 12 2
4900.2.a.bf 4 35.l odd 12 2

## Hecke kernels

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

 $$T_{3}^{2} - 3T_{3} + 3$$ T3^2 - 3*T3 + 3 $$T_{11}^{4} - 3T_{11}^{3} + 21T_{11}^{2} + 36T_{11} + 144$$ T11^4 - 3*T11^3 + 21*T11^2 + 36*T11 + 144 $$T_{19}^{4} - T_{19}^{3} + 15T_{19}^{2} + 14T_{19} + 196$$ T19^4 - T19^3 + 15*T19^2 + 14*T19 + 196

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3 T + 3)^{2}$$
$5$ $$T^{4} + T^{3} - 4 T^{2} + 5 T + 25$$
$7$ $$T^{4}$$
$11$ $$T^{4} - 3 T^{3} + 21 T^{2} + 36 T + 144$$
$13$ $$T^{4} + 44T^{2} + 256$$
$17$ $$T^{4} + 9 T^{3} + 29 T^{2} + 18 T + 4$$
$19$ $$T^{4} - T^{3} + 15 T^{2} + 14 T + 196$$
$23$ $$T^{4} + 12 T^{3} + 41 T^{2} - 84 T + 49$$
$29$ $$(T^{2} + T - 14)^{2}$$
$31$ $$T^{4} + T^{3} + 15 T^{2} - 14 T + 196$$
$37$ $$T^{4} - 27 T^{3} + 299 T^{2} + \cdots + 3136$$
$41$ $$(T^{2} - 15 T + 42)^{2}$$
$43$ $$T^{4} + 47T^{2} + 196$$
$47$ $$T^{4} - 15 T^{3} + 89 T^{2} + \cdots + 196$$
$53$ $$T^{4} + 3 T^{3} - 39 T^{2} + \cdots + 1764$$
$59$ $$T^{4} + T^{3} + 15 T^{2} - 14 T + 196$$
$61$ $$T^{4} + 12 T^{3} + 165 T^{2} + \cdots + 441$$
$67$ $$T^{4} + 18 T^{3} + 59 T^{2} + \cdots + 2401$$
$71$ $$(T^{2} - 6 T - 48)^{2}$$
$73$ $$T^{4} - 15 T^{3} + 89 T^{2} + \cdots + 196$$
$79$ $$T^{4} - 7 T^{3} + 51 T^{2} + 14 T + 4$$
$83$ $$T^{4} + 87T^{2} + 1764$$
$89$ $$(T^{2} - 7 T + 49)^{2}$$
$97$ $$(T^{2} + 48)^{2}$$