# Properties

 Label 20.3.d.b Level $20$ Weight $3$ Character orbit 20.d Self dual yes Analytic conductor $0.545$ Analytic rank $0$ Dimension $1$ CM discriminant -20 Inner twists $2$

# 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: yes Analytic conductor: $$0.544960528721$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 5 q^{5} - 8 q^{6} + 4 q^{7} + 8 q^{8} + 7 q^{9}+O(q^{10})$$ q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 5 * q^5 - 8 * q^6 + 4 * q^7 + 8 * q^8 + 7 * q^9 $$q + 2 q^{2} - 4 q^{3} + 4 q^{4} - 5 q^{5} - 8 q^{6} + 4 q^{7} + 8 q^{8} + 7 q^{9} - 10 q^{10} - 16 q^{12} + 8 q^{14} + 20 q^{15} + 16 q^{16} + 14 q^{18} - 20 q^{20} - 16 q^{21} - 44 q^{23} - 32 q^{24} + 25 q^{25} + 8 q^{27} + 16 q^{28} - 22 q^{29} + 40 q^{30} + 32 q^{32} - 20 q^{35} + 28 q^{36} - 40 q^{40} + 62 q^{41} - 32 q^{42} + 76 q^{43} - 35 q^{45} - 88 q^{46} + 4 q^{47} - 64 q^{48} - 33 q^{49} + 50 q^{50} + 16 q^{54} + 32 q^{56} - 44 q^{58} + 80 q^{60} - 58 q^{61} + 28 q^{63} + 64 q^{64} - 116 q^{67} + 176 q^{69} - 40 q^{70} + 56 q^{72} - 100 q^{75} - 80 q^{80} - 95 q^{81} + 124 q^{82} + 76 q^{83} - 64 q^{84} + 152 q^{86} + 88 q^{87} - 142 q^{89} - 70 q^{90} - 176 q^{92} + 8 q^{94} - 128 q^{96} - 66 q^{98}+O(q^{100})$$ q + 2 * q^2 - 4 * q^3 + 4 * q^4 - 5 * q^5 - 8 * q^6 + 4 * q^7 + 8 * q^8 + 7 * q^9 - 10 * q^10 - 16 * q^12 + 8 * q^14 + 20 * q^15 + 16 * q^16 + 14 * q^18 - 20 * q^20 - 16 * q^21 - 44 * q^23 - 32 * q^24 + 25 * q^25 + 8 * q^27 + 16 * q^28 - 22 * q^29 + 40 * q^30 + 32 * q^32 - 20 * q^35 + 28 * q^36 - 40 * q^40 + 62 * q^41 - 32 * q^42 + 76 * q^43 - 35 * q^45 - 88 * q^46 + 4 * q^47 - 64 * q^48 - 33 * q^49 + 50 * q^50 + 16 * q^54 + 32 * q^56 - 44 * q^58 + 80 * q^60 - 58 * q^61 + 28 * q^63 + 64 * q^64 - 116 * q^67 + 176 * q^69 - 40 * q^70 + 56 * q^72 - 100 * q^75 - 80 * q^80 - 95 * q^81 + 124 * q^82 + 76 * q^83 - 64 * q^84 + 152 * q^86 + 88 * q^87 - 142 * q^89 - 70 * q^90 - 176 * q^92 + 8 * q^94 - 128 * q^96 - 66 * q^98

## Character values

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

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

## 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
 0
2.00000 −4.00000 4.00000 −5.00000 −8.00000 4.00000 8.00000 7.00000 −10.0000
 $$n$$: 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 4$$ acting on $$S_{3}^{\mathrm{new}}(20, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T + 4$$
$5$ $$T + 5$$
$7$ $$T - 4$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T + 44$$
$29$ $$T + 22$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T - 62$$
$43$ $$T - 76$$
$47$ $$T - 4$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T + 58$$
$67$ $$T + 116$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T - 76$$
$89$ $$T + 142$$
$97$ $$T$$