# Properties

 Label 980.2.a.f Level $980$ Weight $2$ Character orbit 980.a Self dual yes Analytic conductor $7.825$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} - 2 q^{9}+O(q^{10})$$ q + q^3 - q^5 - 2 * q^9 $$q + q^{3} - q^{5} - 2 q^{9} - q^{11} - 5 q^{13} - q^{15} + q^{17} - 6 q^{19} - 4 q^{23} + q^{25} - 5 q^{27} + 3 q^{29} + 2 q^{31} - q^{33} + 8 q^{37} - 5 q^{39} - 10 q^{41} - 2 q^{43} + 2 q^{45} - 7 q^{47} + q^{51} - 2 q^{53} + q^{55} - 6 q^{57} + 14 q^{59} - 8 q^{61} + 5 q^{65} + 14 q^{67} - 4 q^{69} - 10 q^{73} + q^{75} - 11 q^{79} + q^{81} - 4 q^{83} - q^{85} + 3 q^{87} + 4 q^{89} + 2 q^{93} + 6 q^{95} - 3 q^{97} + 2 q^{99}+O(q^{100})$$ q + q^3 - q^5 - 2 * q^9 - q^11 - 5 * q^13 - q^15 + q^17 - 6 * q^19 - 4 * q^23 + q^25 - 5 * q^27 + 3 * q^29 + 2 * q^31 - q^33 + 8 * q^37 - 5 * q^39 - 10 * q^41 - 2 * q^43 + 2 * q^45 - 7 * q^47 + q^51 - 2 * q^53 + q^55 - 6 * q^57 + 14 * q^59 - 8 * q^61 + 5 * q^65 + 14 * q^67 - 4 * q^69 - 10 * q^73 + q^75 - 11 * q^79 + q^81 - 4 * q^83 - q^85 + 3 * q^87 + 4 * q^89 + 2 * q^93 + 6 * q^95 - 3 * q^97 + 2 * 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
 0
0 1.00000 0 −1.00000 0 0 0 −2.00000 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.f yes 1
3.b odd 2 1 8820.2.a.v 1
4.b odd 2 1 3920.2.a.n 1
5.b even 2 1 4900.2.a.h 1
5.c odd 4 2 4900.2.e.k 2
7.b odd 2 1 980.2.a.d 1
7.c even 3 2 980.2.i.e 2
7.d odd 6 2 980.2.i.g 2
21.c even 2 1 8820.2.a.i 1
28.d even 2 1 3920.2.a.z 1
35.c odd 2 1 4900.2.a.o 1
35.f even 4 2 4900.2.e.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.2.a.d 1 7.b odd 2 1
980.2.a.f yes 1 1.a even 1 1 trivial
980.2.i.e 2 7.c even 3 2
980.2.i.g 2 7.d odd 6 2
3920.2.a.n 1 4.b odd 2 1
3920.2.a.z 1 28.d even 2 1
4900.2.a.h 1 5.b even 2 1
4900.2.a.o 1 35.c odd 2 1
4900.2.e.j 2 35.f even 4 2
4900.2.e.k 2 5.c odd 4 2
8820.2.a.i 1 21.c even 2 1
8820.2.a.v 1 3.b odd 2 1

## Hecke kernels

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

 $$T_{3} - 1$$ T3 - 1 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T + 1$$
$13$ $$T + 5$$
$17$ $$T - 1$$
$19$ $$T + 6$$
$23$ $$T + 4$$
$29$ $$T - 3$$
$31$ $$T - 2$$
$37$ $$T - 8$$
$41$ $$T + 10$$
$43$ $$T + 2$$
$47$ $$T + 7$$
$53$ $$T + 2$$
$59$ $$T - 14$$
$61$ $$T + 8$$
$67$ $$T - 14$$
$71$ $$T$$
$73$ $$T + 10$$
$79$ $$T + 11$$
$83$ $$T + 4$$
$89$ $$T - 4$$
$97$ $$T + 3$$