# Properties

 Label 39.1.d.a Level $39$ Weight $1$ Character orbit 39.d Self dual yes Analytic conductor $0.019$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -3, -39, 13 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

This is the first weight $1$ newform with projective image $D_2$.

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,1,Mod(38,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("39.38");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 39.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N # Warning: the index may be different

gp: f = lf \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$0.0194635354927$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.117.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{4} + q^{9}+O(q^{10})$$ q - q^3 - q^4 + q^9 $$q - q^{3} - q^{4} + q^{9} + q^{12} - q^{13} + q^{16} - q^{25} - q^{27} - q^{36} + q^{39} + 2 q^{43} - q^{48} + q^{49} + q^{52} - 2 q^{61} - q^{64} + q^{75} - 2 q^{79} + q^{81}+O(q^{100})$$ q - q^3 - q^4 + q^9 + q^12 - q^13 + q^16 - q^25 - q^27 - q^36 + q^39 + 2 * q^43 - q^48 + q^49 + q^52 - 2 * q^61 - q^64 + q^75 - 2 * q^79 + q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1
 0
0 −1.00000 −1.00000 0 0 0 0 1.00000 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.b even 2 1 RM by $$\Q(\sqrt{13})$$
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.1.d.a 1
3.b odd 2 1 CM 39.1.d.a 1
4.b odd 2 1 624.1.l.a 1
5.b even 2 1 975.1.g.a 1
5.c odd 4 2 975.1.e.a 2
7.b odd 2 1 1911.1.h.a 1
7.c even 3 2 1911.1.w.b 2
7.d odd 6 2 1911.1.w.a 2
8.b even 2 1 2496.1.l.b 1
8.d odd 2 1 2496.1.l.a 1
9.c even 3 2 1053.1.n.b 2
9.d odd 6 2 1053.1.n.b 2
12.b even 2 1 624.1.l.a 1
13.b even 2 1 RM 39.1.d.a 1
13.c even 3 2 507.1.h.a 2
13.d odd 4 2 507.1.c.a 1
13.e even 6 2 507.1.h.a 2
13.f odd 12 4 507.1.i.a 2
15.d odd 2 1 975.1.g.a 1
15.e even 4 2 975.1.e.a 2
21.c even 2 1 1911.1.h.a 1
21.g even 6 2 1911.1.w.a 2
21.h odd 6 2 1911.1.w.b 2
24.f even 2 1 2496.1.l.a 1
24.h odd 2 1 2496.1.l.b 1
39.d odd 2 1 CM 39.1.d.a 1
39.f even 4 2 507.1.c.a 1
39.h odd 6 2 507.1.h.a 2
39.i odd 6 2 507.1.h.a 2
39.k even 12 4 507.1.i.a 2
52.b odd 2 1 624.1.l.a 1
65.d even 2 1 975.1.g.a 1
65.h odd 4 2 975.1.e.a 2
91.b odd 2 1 1911.1.h.a 1
91.r even 6 2 1911.1.w.b 2
91.s odd 6 2 1911.1.w.a 2
104.e even 2 1 2496.1.l.b 1
104.h odd 2 1 2496.1.l.a 1
117.n odd 6 2 1053.1.n.b 2
117.t even 6 2 1053.1.n.b 2
156.h even 2 1 624.1.l.a 1
195.e odd 2 1 975.1.g.a 1
195.s even 4 2 975.1.e.a 2
273.g even 2 1 1911.1.h.a 1
273.w odd 6 2 1911.1.w.b 2
273.ba even 6 2 1911.1.w.a 2
312.b odd 2 1 2496.1.l.b 1
312.h even 2 1 2496.1.l.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.1.d.a 1 1.a even 1 1 trivial
39.1.d.a 1 3.b odd 2 1 CM
39.1.d.a 1 13.b even 2 1 RM
39.1.d.a 1 39.d odd 2 1 CM
507.1.c.a 1 13.d odd 4 2
507.1.c.a 1 39.f even 4 2
507.1.h.a 2 13.c even 3 2
507.1.h.a 2 13.e even 6 2
507.1.h.a 2 39.h odd 6 2
507.1.h.a 2 39.i odd 6 2
507.1.i.a 2 13.f odd 12 4
507.1.i.a 2 39.k even 12 4
624.1.l.a 1 4.b odd 2 1
624.1.l.a 1 12.b even 2 1
624.1.l.a 1 52.b odd 2 1
624.1.l.a 1 156.h even 2 1
975.1.e.a 2 5.c odd 4 2
975.1.e.a 2 15.e even 4 2
975.1.e.a 2 65.h odd 4 2
975.1.e.a 2 195.s even 4 2
975.1.g.a 1 5.b even 2 1
975.1.g.a 1 15.d odd 2 1
975.1.g.a 1 65.d even 2 1
975.1.g.a 1 195.e odd 2 1
1053.1.n.b 2 9.c even 3 2
1053.1.n.b 2 9.d odd 6 2
1053.1.n.b 2 117.n odd 6 2
1053.1.n.b 2 117.t even 6 2
1911.1.h.a 1 7.b odd 2 1
1911.1.h.a 1 21.c even 2 1
1911.1.h.a 1 91.b odd 2 1
1911.1.h.a 1 273.g even 2 1
1911.1.w.a 2 7.d odd 6 2
1911.1.w.a 2 21.g even 6 2
1911.1.w.a 2 91.s odd 6 2
1911.1.w.a 2 273.ba even 6 2
1911.1.w.b 2 7.c even 3 2
1911.1.w.b 2 21.h odd 6 2
1911.1.w.b 2 91.r even 6 2
1911.1.w.b 2 273.w odd 6 2
2496.1.l.a 1 8.d odd 2 1
2496.1.l.a 1 24.f even 2 1
2496.1.l.a 1 104.h odd 2 1
2496.1.l.a 1 312.h even 2 1
2496.1.l.b 1 8.b even 2 1
2496.1.l.b 1 24.h odd 2 1
2496.1.l.b 1 104.e even 2 1
2496.1.l.b 1 312.b odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

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