# Properties

 Label 20.2.a.a Level $20$ Weight $2$ Character orbit 20.a Self dual yes Analytic conductor $0.160$ Analytic rank $0$ 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] = [20,2,Mod(1,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("20.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 20.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: $$0.159700804043$$ Analytic rank: $$0$$ 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)

 $$f(q)$$ $$=$$ $$q - 2 q^{3} - q^{5} + 2 q^{7} + q^{9}+O(q^{10})$$ q - 2 * q^3 - q^5 + 2 * q^7 + q^9 $$q - 2 q^{3} - q^{5} + 2 q^{7} + q^{9} + 2 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 4 q^{21} + 6 q^{23} + q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} - 2 q^{35} + 2 q^{37} - 4 q^{39} + 6 q^{41} - 10 q^{43} - q^{45} - 6 q^{47} - 3 q^{49} + 12 q^{51} - 6 q^{53} + 8 q^{57} + 12 q^{59} + 2 q^{61} + 2 q^{63} - 2 q^{65} + 2 q^{67} - 12 q^{69} - 12 q^{71} + 2 q^{73} - 2 q^{75} + 8 q^{79} - 11 q^{81} + 6 q^{83} + 6 q^{85} - 12 q^{87} - 6 q^{89} + 4 q^{91} + 8 q^{93} + 4 q^{95} + 2 q^{97}+O(q^{100})$$ q - 2 * q^3 - q^5 + 2 * q^7 + q^9 + 2 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^19 - 4 * q^21 + 6 * q^23 + q^25 + 4 * q^27 + 6 * q^29 - 4 * q^31 - 2 * q^35 + 2 * q^37 - 4 * q^39 + 6 * q^41 - 10 * q^43 - q^45 - 6 * q^47 - 3 * q^49 + 12 * q^51 - 6 * q^53 + 8 * q^57 + 12 * q^59 + 2 * q^61 + 2 * q^63 - 2 * q^65 + 2 * q^67 - 12 * q^69 - 12 * q^71 + 2 * q^73 - 2 * q^75 + 8 * q^79 - 11 * q^81 + 6 * q^83 + 6 * q^85 - 12 * q^87 - 6 * q^89 + 4 * q^91 + 8 * q^93 + 4 * q^95 + 2 * q^97

## Expression as an eta quotient

$$f(z) = \eta(2z)^{2}\eta(10z)^{2}=q\prod_{n=1}^\infty(1 - q^{2n})^{2}(1 - q^{10n})^{2}$$

## 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 −2.00000 0 −1.00000 0 2.00000 0 1.00000 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
$$2$$ $$-1$$
$$5$$ $$1$$

## 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 20.2.a.a 1
3.b odd 2 1 180.2.a.a 1
4.b odd 2 1 80.2.a.b 1
5.b even 2 1 100.2.a.a 1
5.c odd 4 2 100.2.c.a 2
7.b odd 2 1 980.2.a.h 1
7.c even 3 2 980.2.i.i 2
7.d odd 6 2 980.2.i.c 2
8.b even 2 1 320.2.a.f 1
8.d odd 2 1 320.2.a.a 1
9.c even 3 2 1620.2.i.h 2
9.d odd 6 2 1620.2.i.b 2
11.b odd 2 1 2420.2.a.a 1
12.b even 2 1 720.2.a.h 1
13.b even 2 1 3380.2.a.c 1
13.d odd 4 2 3380.2.f.b 2
15.d odd 2 1 900.2.a.b 1
15.e even 4 2 900.2.d.c 2
16.e even 4 2 1280.2.d.c 2
16.f odd 4 2 1280.2.d.g 2
17.b even 2 1 5780.2.a.f 1
17.c even 4 2 5780.2.c.a 2
19.b odd 2 1 7220.2.a.f 1
20.d odd 2 1 400.2.a.c 1
20.e even 4 2 400.2.c.b 2
21.c even 2 1 8820.2.a.g 1
24.f even 2 1 2880.2.a.f 1
24.h odd 2 1 2880.2.a.m 1
28.d even 2 1 3920.2.a.h 1
35.c odd 2 1 4900.2.a.e 1
35.f even 4 2 4900.2.e.f 2
40.e odd 2 1 1600.2.a.w 1
40.f even 2 1 1600.2.a.c 1
40.i odd 4 2 1600.2.c.d 2
40.k even 4 2 1600.2.c.e 2
44.c even 2 1 9680.2.a.ba 1
60.h even 2 1 3600.2.a.be 1
60.l odd 4 2 3600.2.f.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 1.a even 1 1 trivial
80.2.a.b 1 4.b odd 2 1
100.2.a.a 1 5.b even 2 1
100.2.c.a 2 5.c odd 4 2
180.2.a.a 1 3.b odd 2 1
320.2.a.a 1 8.d odd 2 1
320.2.a.f 1 8.b even 2 1
400.2.a.c 1 20.d odd 2 1
400.2.c.b 2 20.e even 4 2
720.2.a.h 1 12.b even 2 1
900.2.a.b 1 15.d odd 2 1
900.2.d.c 2 15.e even 4 2
980.2.a.h 1 7.b odd 2 1
980.2.i.c 2 7.d odd 6 2
980.2.i.i 2 7.c even 3 2
1280.2.d.c 2 16.e even 4 2
1280.2.d.g 2 16.f odd 4 2
1600.2.a.c 1 40.f even 2 1
1600.2.a.w 1 40.e odd 2 1
1600.2.c.d 2 40.i odd 4 2
1600.2.c.e 2 40.k even 4 2
1620.2.i.b 2 9.d odd 6 2
1620.2.i.h 2 9.c even 3 2
2420.2.a.a 1 11.b odd 2 1
2880.2.a.f 1 24.f even 2 1
2880.2.a.m 1 24.h odd 2 1
3380.2.a.c 1 13.b even 2 1
3380.2.f.b 2 13.d odd 4 2
3600.2.a.be 1 60.h even 2 1
3600.2.f.j 2 60.l odd 4 2
3920.2.a.h 1 28.d even 2 1
4900.2.a.e 1 35.c odd 2 1
4900.2.e.f 2 35.f even 4 2
5780.2.a.f 1 17.b even 2 1
5780.2.c.a 2 17.c even 4 2
7220.2.a.f 1 19.b odd 2 1
8820.2.a.g 1 21.c even 2 1
9680.2.a.ba 1 44.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(20))$$.

## Hecke characteristic polynomials

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