# Properties

 Label 980.2.a.j Level $980$ Weight $2$ Character orbit 980.a Self dual yes Analytic conductor $7.825$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(1,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("980.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.a (trivial)

## 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: yes Analytic conductor: $$7.82533939809$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
0 −2.41421 0 1.00000 0 0 0 2.82843 0
1.2 0 0.414214 0 1.00000 0 0 0 −2.82843 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.j 2 1.a even 1 1 trivial
980.2.a.k yes 2 7.b odd 2 1
980.2.i.k 4 7.d odd 6 2
980.2.i.l 4 7.c even 3 2
3920.2.a.bo 2 28.d even 2 1
3920.2.a.bx 2 4.b odd 2 1
4900.2.a.x 2 35.c odd 2 1
4900.2.a.z 2 5.b even 2 1
4900.2.e.q 4 5.c odd 4 2
4900.2.e.r 4 35.f even 4 2
8820.2.a.bg 2 3.b odd 2 1
8820.2.a.bl 2 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(980))$$:

 $$T_{3}^{2} + 2T_{3} - 1$$ T3^2 + 2*T3 - 1 $$T_{11}^{2} + 2T_{11} - 7$$ T11^2 + 2*T11 - 7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T - 7$$
$13$ $$T^{2} + 10T + 23$$
$17$ $$T^{2} + 10T + 23$$
$19$ $$T^{2} - 4T - 28$$
$23$ $$T^{2} - 4T + 2$$
$29$ $$T^{2} - 2T - 31$$
$31$ $$T^{2} + 12T + 34$$
$37$ $$T^{2} + 4T + 2$$
$41$ $$T^{2} + 4T + 2$$
$43$ $$T^{2} - 12T + 4$$
$47$ $$T^{2} + 2T - 97$$
$53$ $$T^{2} + 16T + 46$$
$59$ $$T^{2} + 4T + 2$$
$61$ $$T^{2} + 16T + 56$$
$67$ $$T^{2} + 8T - 34$$
$71$ $$T^{2} + 4T - 68$$
$73$ $$T^{2} + 16T + 56$$
$79$ $$T^{2} + 2T - 199$$
$83$ $$(T - 8)^{2}$$
$89$ $$T^{2} - 288$$
$97$ $$T^{2} + 6T - 153$$