# Properties

 Label 80.2.a.b Level $80$ Weight $2$ Character orbit 80.a Self dual yes Analytic conductor $0.639$ 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] = [80,2,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, 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("80.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 80.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: $$0.638803216170$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 20) 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

$$f(z) = \dfrac{\eta(4z)^{6}\eta(20z)^{6}}{\eta(2z)^{2}\eta(8z)^{2}\eta(10z)^{2}\eta(40z)^{2}}=q\prod_{n=1}^\infty(1 - q^{2n})^{-2}(1 - q^{4n})^{6}(1 - q^{8n})^{-2}(1 - q^{10n})^{-2}(1 - q^{20n})^{6}(1 - q^{40n})^{-2}$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.a.a 1 4.b odd 2 1
80.2.a.b 1 1.a even 1 1 trivial
100.2.a.a 1 20.d odd 2 1
100.2.c.a 2 20.e even 4 2
180.2.a.a 1 12.b even 2 1
320.2.a.a 1 8.b even 2 1
320.2.a.f 1 8.d odd 2 1
400.2.a.c 1 5.b even 2 1
400.2.c.b 2 5.c odd 4 2
720.2.a.h 1 3.b odd 2 1
900.2.a.b 1 60.h even 2 1
900.2.d.c 2 60.l odd 4 2
980.2.a.h 1 28.d even 2 1
980.2.i.c 2 28.f even 6 2
980.2.i.i 2 28.g odd 6 2
1280.2.d.c 2 16.f odd 4 2
1280.2.d.g 2 16.e even 4 2
1600.2.a.c 1 40.e odd 2 1
1600.2.a.w 1 40.f even 2 1
1600.2.c.d 2 40.k even 4 2
1600.2.c.e 2 40.i odd 4 2
1620.2.i.b 2 36.h even 6 2
1620.2.i.h 2 36.f odd 6 2
2420.2.a.a 1 44.c even 2 1
2880.2.a.f 1 24.h odd 2 1
2880.2.a.m 1 24.f even 2 1
3380.2.a.c 1 52.b odd 2 1
3380.2.f.b 2 52.f even 4 2
3600.2.a.be 1 15.d odd 2 1
3600.2.f.j 2 15.e even 4 2
3920.2.a.h 1 7.b odd 2 1
4900.2.a.e 1 140.c even 2 1
4900.2.e.f 2 140.j odd 4 2
5780.2.a.f 1 68.d odd 2 1
5780.2.c.a 2 68.f odd 4 2
7220.2.a.f 1 76.d even 2 1
8820.2.a.g 1 84.h odd 2 1
9680.2.a.ba 1 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(80))$$.

## Hecke characteristic polynomials

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