# Properties

 Label 3920.1.br.b Level $3920$ Weight $1$ Character orbit 3920.br Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -35 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,1,Mod(129,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

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

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

chi := DirichletCharacter("3920.129");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3920.br (of order $$6$$, degree $$2$$, not 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: $$1.95633484952$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 140) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.140.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} - \zeta_{6}^{2} q^{5} +O(q^{10})$$ q + z * q^3 - z^2 * q^5 $$q + \zeta_{6} q^{3} - \zeta_{6}^{2} q^{5} - \zeta_{6} q^{11} + q^{13} + q^{15} - \zeta_{6} q^{17} - \zeta_{6} q^{25} + q^{27} - q^{29} - \zeta_{6}^{2} q^{33} + \zeta_{6} q^{39} - \zeta_{6}^{2} q^{47} - \zeta_{6}^{2} q^{51} - q^{55} - \zeta_{6}^{2} q^{65} - q^{71} + \zeta_{6} q^{73} - \zeta_{6}^{2} q^{75} + \zeta_{6}^{2} q^{79} + \zeta_{6} q^{81} + q^{83} - q^{85} - \zeta_{6} q^{87} + q^{97} +O(q^{100})$$ q + z * q^3 - z^2 * q^5 - z * q^11 + q^13 + q^15 - z * q^17 - z * q^25 + q^27 - q^29 - z^2 * q^33 + z * q^39 - z^2 * q^47 - z^2 * q^51 - q^55 - z^2 * q^65 - q^71 + z * q^73 - z^2 * q^75 + z^2 * q^79 + z * q^81 + q^83 - q^85 - z * q^87 + q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{5}+O(q^{10})$$ 2 * q + q^3 + q^5 $$2 q + q^{3} + q^{5} - q^{11} + 2 q^{13} + 2 q^{15} - q^{17} - q^{25} + 2 q^{27} - 2 q^{29} + q^{33} + q^{39} + q^{47} + q^{51} - 2 q^{55} + q^{65} - 4 q^{71} + 2 q^{73} + q^{75} - q^{79} + q^{81} + 4 q^{83} - 2 q^{85} - q^{87} + 2 q^{97}+O(q^{100})$$ 2 * q + q^3 + q^5 - q^11 + 2 * q^13 + 2 * q^15 - q^17 - q^25 + 2 * q^27 - 2 * q^29 + q^33 + q^39 + q^47 + q^51 - 2 * q^55 + q^65 - 4 * q^71 + 2 * q^73 + q^75 - q^79 + q^81 + 4 * q^83 - 2 * q^85 - q^87 + 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$$.

 $$n$$ $$981$$ $$1471$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0 0 0 0
1489.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0 0 0 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$
7.c even 3 1 inner
35.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3920.1.br.b 2
4.b odd 2 1 980.1.n.a 2
5.b even 2 1 3920.1.br.a 2
7.b odd 2 1 3920.1.br.a 2
7.c even 3 1 560.1.p.a 1
7.c even 3 1 inner 3920.1.br.b 2
7.d odd 6 1 560.1.p.b 1
7.d odd 6 1 3920.1.br.a 2
20.d odd 2 1 980.1.n.b 2
28.d even 2 1 980.1.n.b 2
28.f even 6 1 140.1.h.a 1
28.f even 6 1 980.1.n.b 2
28.g odd 6 1 140.1.h.b yes 1
28.g odd 6 1 980.1.n.a 2
35.c odd 2 1 CM 3920.1.br.b 2
35.i odd 6 1 560.1.p.a 1
35.i odd 6 1 inner 3920.1.br.b 2
35.j even 6 1 560.1.p.b 1
35.j even 6 1 3920.1.br.a 2
35.k even 12 2 2800.1.f.c 2
35.l odd 12 2 2800.1.f.c 2
56.j odd 6 1 2240.1.p.a 1
56.k odd 6 1 2240.1.p.b 1
56.m even 6 1 2240.1.p.c 1
56.p even 6 1 2240.1.p.d 1
84.j odd 6 1 1260.1.p.a 1
84.n even 6 1 1260.1.p.b 1
140.c even 2 1 980.1.n.a 2
140.p odd 6 1 140.1.h.a 1
140.p odd 6 1 980.1.n.b 2
140.s even 6 1 140.1.h.b yes 1
140.s even 6 1 980.1.n.a 2
140.w even 12 2 700.1.d.a 2
140.x odd 12 2 700.1.d.a 2
280.ba even 6 1 2240.1.p.b 1
280.bf even 6 1 2240.1.p.a 1
280.bi odd 6 1 2240.1.p.c 1
280.bk odd 6 1 2240.1.p.d 1
420.ba even 6 1 1260.1.p.a 1
420.be odd 6 1 1260.1.p.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.1.h.a 1 28.f even 6 1
140.1.h.a 1 140.p odd 6 1
140.1.h.b yes 1 28.g odd 6 1
140.1.h.b yes 1 140.s even 6 1
560.1.p.a 1 7.c even 3 1
560.1.p.a 1 35.i odd 6 1
560.1.p.b 1 7.d odd 6 1
560.1.p.b 1 35.j even 6 1
700.1.d.a 2 140.w even 12 2
700.1.d.a 2 140.x odd 12 2
980.1.n.a 2 4.b odd 2 1
980.1.n.a 2 28.g odd 6 1
980.1.n.a 2 140.c even 2 1
980.1.n.a 2 140.s even 6 1
980.1.n.b 2 20.d odd 2 1
980.1.n.b 2 28.d even 2 1
980.1.n.b 2 28.f even 6 1
980.1.n.b 2 140.p odd 6 1
1260.1.p.a 1 84.j odd 6 1
1260.1.p.a 1 420.ba even 6 1
1260.1.p.b 1 84.n even 6 1
1260.1.p.b 1 420.be odd 6 1
2240.1.p.a 1 56.j odd 6 1
2240.1.p.a 1 280.bf even 6 1
2240.1.p.b 1 56.k odd 6 1
2240.1.p.b 1 280.ba even 6 1
2240.1.p.c 1 56.m even 6 1
2240.1.p.c 1 280.bi odd 6 1
2240.1.p.d 1 56.p even 6 1
2240.1.p.d 1 280.bk odd 6 1
2800.1.f.c 2 35.k even 12 2
2800.1.f.c 2 35.l odd 12 2
3920.1.br.a 2 5.b even 2 1
3920.1.br.a 2 7.b odd 2 1
3920.1.br.a 2 7.d odd 6 1
3920.1.br.a 2 35.j even 6 1
3920.1.br.b 2 1.a even 1 1 trivial
3920.1.br.b 2 7.c even 3 1 inner
3920.1.br.b 2 35.c odd 2 1 CM
3920.1.br.b 2 35.i odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3920, [\chi])$$.

## Hecke characteristic polynomials

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