# Properties

 Label 20.3.d.c Level $20$ Weight $3$ Character orbit 20.d Analytic conductor $0.545$ 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] = [20,3,Mod(19,20)]

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([1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("20.19");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$20 = 2^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 20.d (of order $$2$$, degree $$1$$, 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: $$0.544960528721$$ 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

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

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 4 q^{4} + ( - 2 \beta + 3) q^{5} - 4 \beta q^{8} - 9 q^{9} +O(q^{10})$$ q + b * q^2 - 4 * q^4 + (-2*b + 3) * q^5 - 4*b * q^8 - 9 * q^9 $$q + \beta q^{2} - 4 q^{4} + ( - 2 \beta + 3) q^{5} - 4 \beta q^{8} - 9 q^{9} + (3 \beta + 8) q^{10} + 12 \beta q^{13} + 16 q^{16} - 8 \beta q^{17} - 9 \beta q^{18} + (8 \beta - 12) q^{20} + ( - 12 \beta - 7) q^{25} - 48 q^{26} + 42 q^{29} + 16 \beta q^{32} + 32 q^{34} + 36 q^{36} + 12 \beta q^{37} + ( - 12 \beta - 32) q^{40} - 18 q^{41} + (18 \beta - 27) q^{45} - 49 q^{49} + ( - 7 \beta + 48) q^{50} - 48 \beta q^{52} - 28 \beta q^{53} + 42 \beta q^{58} + 22 q^{61} - 64 q^{64} + (36 \beta + 96) q^{65} + 32 \beta q^{68} + 36 \beta q^{72} - 48 \beta q^{73} - 48 q^{74} + ( - 32 \beta + 48) q^{80} + 81 q^{81} - 18 \beta q^{82} + ( - 24 \beta - 64) q^{85} - 78 q^{89} + ( - 27 \beta - 72) q^{90} + 72 \beta q^{97} - 49 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 4 * q^4 + (-2*b + 3) * q^5 - 4*b * q^8 - 9 * q^9 + (3*b + 8) * q^10 + 12*b * q^13 + 16 * q^16 - 8*b * q^17 - 9*b * q^18 + (8*b - 12) * q^20 + (-12*b - 7) * q^25 - 48 * q^26 + 42 * q^29 + 16*b * q^32 + 32 * q^34 + 36 * q^36 + 12*b * q^37 + (-12*b - 32) * q^40 - 18 * q^41 + (18*b - 27) * q^45 - 49 * q^49 + (-7*b + 48) * q^50 - 48*b * q^52 - 28*b * q^53 + 42*b * q^58 + 22 * q^61 - 64 * q^64 + (36*b + 96) * q^65 + 32*b * q^68 + 36*b * q^72 - 48*b * q^73 - 48 * q^74 + (-32*b + 48) * q^80 + 81 * q^81 - 18*b * q^82 + (-24*b - 64) * q^85 - 78 * q^89 + (-27*b - 72) * q^90 + 72*b * q^97 - 49*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} + 6 q^{5} - 18 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 + 6 * q^5 - 18 * q^9 $$2 q - 8 q^{4} + 6 q^{5} - 18 q^{9} + 16 q^{10} + 32 q^{16} - 24 q^{20} - 14 q^{25} - 96 q^{26} + 84 q^{29} + 64 q^{34} + 72 q^{36} - 64 q^{40} - 36 q^{41} - 54 q^{45} - 98 q^{49} + 96 q^{50} + 44 q^{61} - 128 q^{64} + 192 q^{65} - 96 q^{74} + 96 q^{80} + 162 q^{81} - 128 q^{85} - 156 q^{89} - 144 q^{90}+O(q^{100})$$ 2 * q - 8 * q^4 + 6 * q^5 - 18 * q^9 + 16 * q^10 + 32 * q^16 - 24 * q^20 - 14 * q^25 - 96 * q^26 + 84 * q^29 + 64 * q^34 + 72 * q^36 - 64 * q^40 - 36 * q^41 - 54 * q^45 - 98 * q^49 + 96 * q^50 + 44 * q^61 - 128 * q^64 + 192 * q^65 - 96 * q^74 + 96 * q^80 + 162 * q^81 - 128 * q^85 - 156 * q^89 - 144 * q^90

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
2.00000i 0 −4.00000 3.00000 + 4.00000i 0 0 8.00000i −9.00000 8.00000 6.00000i
19.2 2.00000i 0 −4.00000 3.00000 4.00000i 0 0 8.00000i −9.00000 8.00000 + 6.00000i
 $$n$$: 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.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.3.d.c 2
3.b odd 2 1 180.3.f.c 2
4.b odd 2 1 CM 20.3.d.c 2
5.b even 2 1 inner 20.3.d.c 2
5.c odd 4 1 100.3.b.a 1
5.c odd 4 1 100.3.b.b 1
8.b even 2 1 320.3.h.d 2
8.d odd 2 1 320.3.h.d 2
12.b even 2 1 180.3.f.c 2
15.d odd 2 1 180.3.f.c 2
15.e even 4 1 900.3.c.a 1
15.e even 4 1 900.3.c.d 1
16.e even 4 1 1280.3.e.a 2
16.e even 4 1 1280.3.e.d 2
16.f odd 4 1 1280.3.e.a 2
16.f odd 4 1 1280.3.e.d 2
20.d odd 2 1 inner 20.3.d.c 2
20.e even 4 1 100.3.b.a 1
20.e even 4 1 100.3.b.b 1
40.e odd 2 1 320.3.h.d 2
40.f even 2 1 320.3.h.d 2
40.i odd 4 1 1600.3.b.a 1
40.i odd 4 1 1600.3.b.c 1
40.k even 4 1 1600.3.b.a 1
40.k even 4 1 1600.3.b.c 1
60.h even 2 1 180.3.f.c 2
60.l odd 4 1 900.3.c.a 1
60.l odd 4 1 900.3.c.d 1
80.k odd 4 1 1280.3.e.a 2
80.k odd 4 1 1280.3.e.d 2
80.q even 4 1 1280.3.e.a 2
80.q even 4 1 1280.3.e.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.d.c 2 1.a even 1 1 trivial
20.3.d.c 2 4.b odd 2 1 CM
20.3.d.c 2 5.b even 2 1 inner
20.3.d.c 2 20.d odd 2 1 inner
100.3.b.a 1 5.c odd 4 1
100.3.b.a 1 20.e even 4 1
100.3.b.b 1 5.c odd 4 1
100.3.b.b 1 20.e even 4 1
180.3.f.c 2 3.b odd 2 1
180.3.f.c 2 12.b even 2 1
180.3.f.c 2 15.d odd 2 1
180.3.f.c 2 60.h even 2 1
320.3.h.d 2 8.b even 2 1
320.3.h.d 2 8.d odd 2 1
320.3.h.d 2 40.e odd 2 1
320.3.h.d 2 40.f even 2 1
900.3.c.a 1 15.e even 4 1
900.3.c.a 1 60.l odd 4 1
900.3.c.d 1 15.e even 4 1
900.3.c.d 1 60.l odd 4 1
1280.3.e.a 2 16.e even 4 1
1280.3.e.a 2 16.f odd 4 1
1280.3.e.a 2 80.k odd 4 1
1280.3.e.a 2 80.q even 4 1
1280.3.e.d 2 16.e even 4 1
1280.3.e.d 2 16.f odd 4 1
1280.3.e.d 2 80.k odd 4 1
1280.3.e.d 2 80.q even 4 1
1600.3.b.a 1 40.i odd 4 1
1600.3.b.a 1 40.k even 4 1
1600.3.b.c 1 40.i odd 4 1
1600.3.b.c 1 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{3}^{\mathrm{new}}(20, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 25$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 576$$
$17$ $$T^{2} + 256$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T - 42)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 576$$
$41$ $$(T + 18)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 3136$$
$59$ $$T^{2}$$
$61$ $$(T - 22)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 9216$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 78)^{2}$$
$97$ $$T^{2} + 20736$$