# Properties

 Label 20.2.e.a Level $20$ Weight $2$ Character orbit 20.e Analytic conductor $0.160$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [20,2,Mod(3,20)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(20, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("20.3");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 20.e (of order $$4$$, degree $$2$$, 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: $$0.159700804043$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{2} + 2 i q^{4} + (i - 2) q^{5} + ( - 2 i + 2) q^{8} - 3 i q^{9} +O(q^{10})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (i - 2) * q^5 + (-2*i + 2) * q^8 - 3*i * q^9 $$q + ( - i - 1) q^{2} + 2 i q^{4} + (i - 2) q^{5} + ( - 2 i + 2) q^{8} - 3 i q^{9} + (i + 3) q^{10} + (i - 1) q^{13} - 4 q^{16} + (3 i + 3) q^{17} + (3 i - 3) q^{18} + ( - 4 i - 2) q^{20} + ( - 4 i + 3) q^{25} + 2 q^{26} + 4 i q^{29} + (4 i + 4) q^{32} - 6 i q^{34} + 6 q^{36} + ( - 7 i - 7) q^{37} + (6 i - 2) q^{40} - 8 q^{41} + (6 i + 3) q^{45} + 7 i q^{49} + (i - 7) q^{50} + ( - 2 i - 2) q^{52} + ( - 9 i + 9) q^{53} + ( - 4 i + 4) q^{58} + 12 q^{61} - 8 i q^{64} + ( - 3 i + 1) q^{65} + (6 i - 6) q^{68} + ( - 6 i - 6) q^{72} + (11 i - 11) q^{73} + 14 i q^{74} + ( - 4 i + 8) q^{80} - 9 q^{81} + (8 i + 8) q^{82} + ( - 3 i - 9) q^{85} - 16 i q^{89} + ( - 9 i + 3) q^{90} + (13 i + 13) q^{97} + ( - 7 i + 7) q^{98} +O(q^{100})$$ q + (-i - 1) * q^2 + 2*i * q^4 + (i - 2) * q^5 + (-2*i + 2) * q^8 - 3*i * q^9 + (i + 3) * q^10 + (i - 1) * q^13 - 4 * q^16 + (3*i + 3) * q^17 + (3*i - 3) * q^18 + (-4*i - 2) * q^20 + (-4*i + 3) * q^25 + 2 * q^26 + 4*i * q^29 + (4*i + 4) * q^32 - 6*i * q^34 + 6 * q^36 + (-7*i - 7) * q^37 + (6*i - 2) * q^40 - 8 * q^41 + (6*i + 3) * q^45 + 7*i * q^49 + (i - 7) * q^50 + (-2*i - 2) * q^52 + (-9*i + 9) * q^53 + (-4*i + 4) * q^58 + 12 * q^61 - 8*i * q^64 + (-3*i + 1) * q^65 + (6*i - 6) * q^68 + (-6*i - 6) * q^72 + (11*i - 11) * q^73 + 14*i * q^74 + (-4*i + 8) * q^80 - 9 * q^81 + (8*i + 8) * q^82 + (-3*i - 9) * q^85 - 16*i * q^89 + (-9*i + 3) * q^90 + (13*i + 13) * q^97 + (-7*i + 7) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{5} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 $$2 q - 2 q^{2} - 4 q^{5} + 4 q^{8} + 6 q^{10} - 2 q^{13} - 8 q^{16} + 6 q^{17} - 6 q^{18} - 4 q^{20} + 6 q^{25} + 4 q^{26} + 8 q^{32} + 12 q^{36} - 14 q^{37} - 4 q^{40} - 16 q^{41} + 6 q^{45} - 14 q^{50} - 4 q^{52} + 18 q^{53} + 8 q^{58} + 24 q^{61} + 2 q^{65} - 12 q^{68} - 12 q^{72} - 22 q^{73} + 16 q^{80} - 18 q^{81} + 16 q^{82} - 18 q^{85} + 6 q^{90} + 26 q^{97} + 14 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^5 + 4 * q^8 + 6 * q^10 - 2 * q^13 - 8 * q^16 + 6 * q^17 - 6 * q^18 - 4 * q^20 + 6 * q^25 + 4 * q^26 + 8 * q^32 + 12 * q^36 - 14 * q^37 - 4 * q^40 - 16 * q^41 + 6 * q^45 - 14 * q^50 - 4 * q^52 + 18 * q^53 + 8 * q^58 + 24 * q^61 + 2 * q^65 - 12 * q^68 - 12 * q^72 - 22 * q^73 + 16 * q^80 - 18 * q^81 + 16 * q^82 - 18 * q^85 + 6 * q^90 + 26 * q^97 + 14 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/20\mathbb{Z}\right)^\times$$.

 $$n$$ $$11$$ $$17$$ $$\chi(n)$$ $$-1$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 1.00000i 1.00000i
−1.00000 + 1.00000i 0 2.00000i −2.00000 1.00000i 0 0 2.00000 + 2.00000i 3.00000i 3.00000 1.00000i
7.1 −1.00000 1.00000i 0 2.00000i −2.00000 + 1.00000i 0 0 2.00000 2.00000i 3.00000i 3.00000 + 1.00000i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.2.e.a 2
3.b odd 2 1 180.2.k.c 2
4.b odd 2 1 CM 20.2.e.a 2
5.b even 2 1 100.2.e.b 2
5.c odd 4 1 inner 20.2.e.a 2
5.c odd 4 1 100.2.e.b 2
7.b odd 2 1 980.2.k.a 2
7.c even 3 2 980.2.x.d 4
7.d odd 6 2 980.2.x.c 4
8.b even 2 1 320.2.n.e 2
8.d odd 2 1 320.2.n.e 2
12.b even 2 1 180.2.k.c 2
15.d odd 2 1 900.2.k.c 2
15.e even 4 1 180.2.k.c 2
15.e even 4 1 900.2.k.c 2
16.e even 4 1 1280.2.o.g 2
16.e even 4 1 1280.2.o.j 2
16.f odd 4 1 1280.2.o.g 2
16.f odd 4 1 1280.2.o.j 2
20.d odd 2 1 100.2.e.b 2
20.e even 4 1 inner 20.2.e.a 2
20.e even 4 1 100.2.e.b 2
28.d even 2 1 980.2.k.a 2
28.f even 6 2 980.2.x.c 4
28.g odd 6 2 980.2.x.d 4
35.f even 4 1 980.2.k.a 2
35.k even 12 2 980.2.x.c 4
35.l odd 12 2 980.2.x.d 4
40.e odd 2 1 1600.2.n.h 2
40.f even 2 1 1600.2.n.h 2
40.i odd 4 1 320.2.n.e 2
40.i odd 4 1 1600.2.n.h 2
40.k even 4 1 320.2.n.e 2
40.k even 4 1 1600.2.n.h 2
60.h even 2 1 900.2.k.c 2
60.l odd 4 1 180.2.k.c 2
60.l odd 4 1 900.2.k.c 2
80.i odd 4 1 1280.2.o.g 2
80.j even 4 1 1280.2.o.j 2
80.s even 4 1 1280.2.o.g 2
80.t odd 4 1 1280.2.o.j 2
140.j odd 4 1 980.2.k.a 2
140.w even 12 2 980.2.x.d 4
140.x odd 12 2 980.2.x.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.2.e.a 2 1.a even 1 1 trivial
20.2.e.a 2 4.b odd 2 1 CM
20.2.e.a 2 5.c odd 4 1 inner
20.2.e.a 2 20.e even 4 1 inner
100.2.e.b 2 5.b even 2 1
100.2.e.b 2 5.c odd 4 1
100.2.e.b 2 20.d odd 2 1
100.2.e.b 2 20.e even 4 1
180.2.k.c 2 3.b odd 2 1
180.2.k.c 2 12.b even 2 1
180.2.k.c 2 15.e even 4 1
180.2.k.c 2 60.l odd 4 1
320.2.n.e 2 8.b even 2 1
320.2.n.e 2 8.d odd 2 1
320.2.n.e 2 40.i odd 4 1
320.2.n.e 2 40.k even 4 1
900.2.k.c 2 15.d odd 2 1
900.2.k.c 2 15.e even 4 1
900.2.k.c 2 60.h even 2 1
900.2.k.c 2 60.l odd 4 1
980.2.k.a 2 7.b odd 2 1
980.2.k.a 2 28.d even 2 1
980.2.k.a 2 35.f even 4 1
980.2.k.a 2 140.j odd 4 1
980.2.x.c 4 7.d odd 6 2
980.2.x.c 4 28.f even 6 2
980.2.x.c 4 35.k even 12 2
980.2.x.c 4 140.x odd 12 2
980.2.x.d 4 7.c even 3 2
980.2.x.d 4 28.g odd 6 2
980.2.x.d 4 35.l odd 12 2
980.2.x.d 4 140.w even 12 2
1280.2.o.g 2 16.e even 4 1
1280.2.o.g 2 16.f odd 4 1
1280.2.o.g 2 80.i odd 4 1
1280.2.o.g 2 80.s even 4 1
1280.2.o.j 2 16.e even 4 1
1280.2.o.j 2 16.f odd 4 1
1280.2.o.j 2 80.j even 4 1
1280.2.o.j 2 80.t odd 4 1
1600.2.n.h 2 40.e odd 2 1
1600.2.n.h 2 40.f even 2 1
1600.2.n.h 2 40.i odd 4 1
1600.2.n.h 2 40.k even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 2T + 2$$
$17$ $$T^{2} - 6T + 18$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 16$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 14T + 98$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 18T + 162$$
$59$ $$T^{2}$$
$61$ $$(T - 12)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 22T + 242$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 256$$
$97$ $$T^{2} - 26T + 338$$