# Properties

 Label 3920.1.bt.b Level $3920$ Weight $1$ Character orbit 3920.bt Analytic conductor $1.956$ Analytic rank $0$ Dimension $2$ Projective image $D_{2}$ CM/RM discs -4, -20, 5 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,1,Mod(79,3920)]

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([3, 0, 3, 2]))

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

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

chi := DirichletCharacter("3920.79");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3920 = 2^{4} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3920.bt (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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{5})$$ Artin image: $C_3\times D_4$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{5} + \zeta_{6} q^{9}+O(q^{10})$$ q + z * q^5 + z * q^9 $$q + \zeta_{6} q^{5} + \zeta_{6} q^{9} + \zeta_{6}^{2} q^{25} + q^{29} - q^{41} + \zeta_{6}^{2} q^{45} + \zeta_{6} q^{61} + \zeta_{6}^{2} q^{81} - \zeta_{6} q^{89} +O(q^{100})$$ q + z * q^5 + z * q^9 + z^2 * q^25 + q^29 - q^41 + z^2 * q^45 + z * q^61 + z^2 * q^81 - z * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} + q^{9}+O(q^{10})$$ 2 * q + q^5 + q^9 $$2 q + q^{5} + q^{9} - q^{25} + 4 q^{29} - 4 q^{41} - q^{45} + 2 q^{61} - q^{81} - 2 q^{89}+O(q^{100})$$ 2 * q + q^5 + q^9 - q^25 + 4 * q^29 - 4 * q^41 - q^45 + 2 * q^61 - q^81 - 2 * q^89

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0 0.500000 0.866025i 0 0 0 0.500000 0.866025i 0
1439.1 0 0 0 0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
7.c even 3 1 inner
28.g odd 6 1 inner
35.j even 6 1 inner
140.p 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.bt.b 2
4.b odd 2 1 CM 3920.1.bt.b 2
5.b even 2 1 RM 3920.1.bt.b 2
7.b odd 2 1 3920.1.bt.a 2
7.c even 3 1 80.1.h.a 1
7.c even 3 1 inner 3920.1.bt.b 2
7.d odd 6 1 3920.1.j.a 1
7.d odd 6 1 3920.1.bt.a 2
20.d odd 2 1 CM 3920.1.bt.b 2
21.h odd 6 1 720.1.j.a 1
28.d even 2 1 3920.1.bt.a 2
28.f even 6 1 3920.1.j.a 1
28.f even 6 1 3920.1.bt.a 2
28.g odd 6 1 80.1.h.a 1
28.g odd 6 1 inner 3920.1.bt.b 2
35.c odd 2 1 3920.1.bt.a 2
35.i odd 6 1 3920.1.j.a 1
35.i odd 6 1 3920.1.bt.a 2
35.j even 6 1 80.1.h.a 1
35.j even 6 1 inner 3920.1.bt.b 2
35.l odd 12 2 400.1.b.a 1
56.k odd 6 1 320.1.h.a 1
56.p even 6 1 320.1.h.a 1
84.n even 6 1 720.1.j.a 1
105.o odd 6 1 720.1.j.a 1
105.x even 12 2 3600.1.e.a 1
112.u odd 12 2 1280.1.e.a 2
112.w even 12 2 1280.1.e.a 2
140.c even 2 1 3920.1.bt.a 2
140.p odd 6 1 80.1.h.a 1
140.p odd 6 1 inner 3920.1.bt.b 2
140.s even 6 1 3920.1.j.a 1
140.s even 6 1 3920.1.bt.a 2
140.w even 12 2 400.1.b.a 1
168.s odd 6 1 2880.1.j.a 1
168.v even 6 1 2880.1.j.a 1
280.bf even 6 1 320.1.h.a 1
280.bi odd 6 1 320.1.h.a 1
280.br even 12 2 1600.1.b.a 1
280.bt odd 12 2 1600.1.b.a 1
420.ba even 6 1 720.1.j.a 1
420.bp odd 12 2 3600.1.e.a 1
560.cr even 12 2 1280.1.e.a 2
560.cs odd 12 2 1280.1.e.a 2
840.cg odd 6 1 2880.1.j.a 1
840.cv even 6 1 2880.1.j.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 7.c even 3 1
80.1.h.a 1 28.g odd 6 1
80.1.h.a 1 35.j even 6 1
80.1.h.a 1 140.p odd 6 1
320.1.h.a 1 56.k odd 6 1
320.1.h.a 1 56.p even 6 1
320.1.h.a 1 280.bf even 6 1
320.1.h.a 1 280.bi odd 6 1
400.1.b.a 1 35.l odd 12 2
400.1.b.a 1 140.w even 12 2
720.1.j.a 1 21.h odd 6 1
720.1.j.a 1 84.n even 6 1
720.1.j.a 1 105.o odd 6 1
720.1.j.a 1 420.ba even 6 1
1280.1.e.a 2 112.u odd 12 2
1280.1.e.a 2 112.w even 12 2
1280.1.e.a 2 560.cr even 12 2
1280.1.e.a 2 560.cs odd 12 2
1600.1.b.a 1 280.br even 12 2
1600.1.b.a 1 280.bt odd 12 2
2880.1.j.a 1 168.s odd 6 1
2880.1.j.a 1 168.v even 6 1
2880.1.j.a 1 840.cg odd 6 1
2880.1.j.a 1 840.cv even 6 1
3600.1.e.a 1 105.x even 12 2
3600.1.e.a 1 420.bp odd 12 2
3920.1.j.a 1 7.d odd 6 1
3920.1.j.a 1 28.f even 6 1
3920.1.j.a 1 35.i odd 6 1
3920.1.j.a 1 140.s even 6 1
3920.1.bt.a 2 7.b odd 2 1
3920.1.bt.a 2 7.d odd 6 1
3920.1.bt.a 2 28.d even 2 1
3920.1.bt.a 2 28.f even 6 1
3920.1.bt.a 2 35.c odd 2 1
3920.1.bt.a 2 35.i odd 6 1
3920.1.bt.a 2 140.c even 2 1
3920.1.bt.a 2 140.s even 6 1
3920.1.bt.b 2 1.a even 1 1 trivial
3920.1.bt.b 2 4.b odd 2 1 CM
3920.1.bt.b 2 5.b even 2 1 RM
3920.1.bt.b 2 7.c even 3 1 inner
3920.1.bt.b 2 20.d odd 2 1 CM
3920.1.bt.b 2 28.g odd 6 1 inner
3920.1.bt.b 2 35.j even 6 1 inner
3920.1.bt.b 2 140.p odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3920, [\chi])$$:

 $$T_{3}$$ T3 $$T_{11}$$ T11 $$T_{13}$$ T13 $$T_{41} + 2$$ T41 + 2

## Hecke characteristic polynomials

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