# Properties

 Label 3920.1.bt.e Level $3920$ Weight $1$ Character orbit 3920.bt Analytic conductor $1.956$ Analytic rank $0$ Dimension $4$ Projective image $D_{6}$ CM discriminant -35 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{6}$$ Projective field: Galois closure of 6.0.313600.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{9}+O(q^{10})$$ q + (-z^3 - z) * q^3 + z * q^5 + (z^4 + z^2 - 1) * q^9 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + \zeta_{12} q^{5} + (\zeta_{12}^{4} + \zeta_{12}^{2} - 1) q^{9} + ( - \zeta_{12}^{4} + 1) q^{11} - \zeta_{12}^{3} q^{13} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{15} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{2} q^{25} + ( - \zeta_{12}^{5} - \zeta_{12}) q^{27} + q^{29} + (\zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{33} + (\zeta_{12}^{4} - 1) q^{39} + (\zeta_{12}^{5} + \zeta_{12}^{3} - \zeta_{12}) q^{45} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{47} + (\zeta_{12}^{2} + 1) q^{51} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{55} - \zeta_{12}^{4} q^{65} - \zeta_{12}^{5} q^{73} + ( - \zeta_{12}^{5} - \zeta_{12}^{3}) q^{75} + ( - \zeta_{12}^{2} - 1) q^{79} + (\zeta_{12}^{2} + 1) q^{81} - q^{85} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{87} - \zeta_{12}^{3} q^{97} + (2 \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{99} +O(q^{100})$$ q + (-z^3 - z) * q^3 + z * q^5 + (z^4 + z^2 - 1) * q^9 + (-z^4 + 1) * q^11 - z^3 * q^13 + (-z^4 - z^2) * q^15 + z^5 * q^17 + z^2 * q^25 + (-z^5 - z) * q^27 + q^29 + (z^5 - z^3 - z) * q^33 + (z^4 - 1) * q^39 + (z^5 + z^3 - z) * q^45 + (-z^5 - z^3) * q^47 + (z^2 + 1) * q^51 + (-z^5 + z) * q^55 - z^4 * q^65 - z^5 * q^73 + (-z^5 - z^3) * q^75 + (-z^2 - 1) * q^79 + (z^2 + 1) * q^81 - q^85 + (-z^3 - z) * q^87 - z^3 * q^97 + (2*z^4 + z^2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} + 6 q^{11} + 2 q^{25} + 4 q^{29} - 6 q^{39} + 6 q^{51} + 2 q^{65} - 6 q^{79} - 2 q^{81} - 4 q^{85}+O(q^{100})$$ 4 * q - 4 * q^9 + 6 * q^11 + 2 * q^25 + 4 * q^29 - 6 * q^39 + 6 * q^51 + 2 * q^65 - 6 * q^79 - 2 * q^81 - 4 * q^85

## Character values

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

 $$n$$ $$981$$ $$1471$$ $$3041$$ $$3137$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 0 0.866025 + 0.500000i 0 0 0 −1.00000 + 1.73205i 0
79.2 0 0.866025 + 1.50000i 0 −0.866025 0.500000i 0 0 0 −1.00000 + 1.73205i 0
1439.1 0 −0.866025 + 1.50000i 0 0.866025 0.500000i 0 0 0 −1.00000 1.73205i 0
1439.2 0 0.866025 1.50000i 0 −0.866025 + 0.500000i 0 0 0 −1.00000 1.73205i 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 $$\Q(\sqrt{-35})$$
5.b even 2 1 inner
7.b odd 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner
140.p odd 6 1 inner
140.s even 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.e 4
4.b odd 2 1 3920.1.bt.d 4
5.b even 2 1 inner 3920.1.bt.e 4
7.b odd 2 1 inner 3920.1.bt.e 4
7.c even 3 1 3920.1.j.e 4
7.c even 3 1 3920.1.bt.d 4
7.d odd 6 1 3920.1.j.e 4
7.d odd 6 1 3920.1.bt.d 4
20.d odd 2 1 3920.1.bt.d 4
28.d even 2 1 3920.1.bt.d 4
28.f even 6 1 3920.1.j.e 4
28.f even 6 1 inner 3920.1.bt.e 4
28.g odd 6 1 3920.1.j.e 4
28.g odd 6 1 inner 3920.1.bt.e 4
35.c odd 2 1 CM 3920.1.bt.e 4
35.i odd 6 1 3920.1.j.e 4
35.i odd 6 1 3920.1.bt.d 4
35.j even 6 1 3920.1.j.e 4
35.j even 6 1 3920.1.bt.d 4
140.c even 2 1 3920.1.bt.d 4
140.p odd 6 1 3920.1.j.e 4
140.p odd 6 1 inner 3920.1.bt.e 4
140.s even 6 1 3920.1.j.e 4
140.s even 6 1 inner 3920.1.bt.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3920.1.j.e 4 7.c even 3 1
3920.1.j.e 4 7.d odd 6 1
3920.1.j.e 4 28.f even 6 1
3920.1.j.e 4 28.g odd 6 1
3920.1.j.e 4 35.i odd 6 1
3920.1.j.e 4 35.j even 6 1
3920.1.j.e 4 140.p odd 6 1
3920.1.j.e 4 140.s even 6 1
3920.1.bt.d 4 4.b odd 2 1
3920.1.bt.d 4 7.c even 3 1
3920.1.bt.d 4 7.d odd 6 1
3920.1.bt.d 4 20.d odd 2 1
3920.1.bt.d 4 28.d even 2 1
3920.1.bt.d 4 35.i odd 6 1
3920.1.bt.d 4 35.j even 6 1
3920.1.bt.d 4 140.c even 2 1
3920.1.bt.e 4 1.a even 1 1 trivial
3920.1.bt.e 4 5.b even 2 1 inner
3920.1.bt.e 4 7.b odd 2 1 inner
3920.1.bt.e 4 28.f even 6 1 inner
3920.1.bt.e 4 28.g odd 6 1 inner
3920.1.bt.e 4 35.c odd 2 1 CM
3920.1.bt.e 4 140.p odd 6 1 inner
3920.1.bt.e 4 140.s even 6 1 inner

## Hecke kernels

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

 $$T_{3}^{4} + 3T_{3}^{2} + 9$$ T3^4 + 3*T3^2 + 9 $$T_{11}^{2} - 3T_{11} + 3$$ T11^2 - 3*T11 + 3 $$T_{13}^{2} + 1$$ T13^2 + 1 $$T_{41}$$ T41

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4} - T^{2} + 1$$
$7$ $$T^{4}$$
$11$ $$(T^{2} - 3 T + 3)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} - T^{2} + 1$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T - 1)^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 3T^{2} + 9$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 4T^{2} + 16$$
$79$ $$(T^{2} + 3 T + 3)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 1)^{2}$$