# Properties

 Label 3120.1.be.e Level $3120$ Weight $1$ Character orbit 3120.be Analytic conductor $1.557$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -39 Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,1,Mod(1169,3120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 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("3120.1169");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3120.be (of order $$2$$, degree $$1$$, 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: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 195) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.12675.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{3} + \zeta_{8}^{3} q^{5} - q^{9}+O(q^{10})$$ q + z^2 * q^3 + z^3 * q^5 - q^9 $$q + \zeta_{8}^{2} q^{3} + \zeta_{8}^{3} q^{5} - q^{9} + (\zeta_{8}^{3} - \zeta_{8}) q^{11} + \zeta_{8}^{2} q^{13} - \zeta_{8} q^{15} - \zeta_{8}^{2} q^{25} - \zeta_{8}^{2} q^{27} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{33} - q^{39} + (\zeta_{8}^{3} - \zeta_{8}) q^{41} - \zeta_{8}^{3} q^{45} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{47} - q^{49} + ( - \zeta_{8}^{2} + 1) q^{55} + (\zeta_{8}^{3} - \zeta_{8}) q^{59} - \zeta_{8} q^{65} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{71} + q^{75} + q^{81} + (\zeta_{8}^{3} + \zeta_{8}) q^{83} + (\zeta_{8}^{3} - \zeta_{8}) q^{89} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{99} +O(q^{100})$$ q + z^2 * q^3 + z^3 * q^5 - q^9 + (z^3 - z) * q^11 + z^2 * q^13 - z * q^15 - z^2 * q^25 - z^2 * q^27 + (-z^3 - z) * q^33 - q^39 + (z^3 - z) * q^41 - z^3 * q^45 + (-z^3 - z) * q^47 - q^49 + (-z^2 + 1) * q^55 + (z^3 - z) * q^59 - z * q^65 + (-z^3 + z) * q^71 + q^75 + q^81 + (z^3 + z) * q^83 + (z^3 - z) * q^89 + (-z^3 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^9 $$4 q - 4 q^{9} - 4 q^{39} - 4 q^{49} + 4 q^{55} + 4 q^{75} + 4 q^{81}+O(q^{100})$$ 4 * q - 4 * q^9 - 4 * q^39 - 4 * q^49 + 4 * q^55 + 4 * q^75 + 4 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$$.

 $$n$$ $$1951$$ $$2081$$ $$2341$$ $$2497$$ $$2641$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1169.1
 0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.707107 − 0.707107i
0 1.00000i 0 −0.707107 0.707107i 0 0 0 −1.00000 0
1169.2 0 1.00000i 0 0.707107 + 0.707107i 0 0 0 −1.00000 0
1169.3 0 1.00000i 0 −0.707107 + 0.707107i 0 0 0 −1.00000 0
1169.4 0 1.00000i 0 0.707107 0.707107i 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
39.d odd 2 1 CM by $$\Q(\sqrt{-39})$$
3.b odd 2 1 inner
5.b even 2 1 inner
13.b even 2 1 inner
15.d odd 2 1 inner
65.d even 2 1 inner
195.e odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.1.be.e 4
3.b odd 2 1 inner 3120.1.be.e 4
4.b odd 2 1 195.1.e.a 4
5.b even 2 1 inner 3120.1.be.e 4
12.b even 2 1 195.1.e.a 4
13.b even 2 1 inner 3120.1.be.e 4
15.d odd 2 1 inner 3120.1.be.e 4
20.d odd 2 1 195.1.e.a 4
20.e even 4 1 975.1.g.b 2
20.e even 4 1 975.1.g.c 2
39.d odd 2 1 CM 3120.1.be.e 4
52.b odd 2 1 195.1.e.a 4
52.f even 4 2 2535.1.f.e 4
52.i odd 6 2 2535.1.y.a 8
52.j odd 6 2 2535.1.y.a 8
52.l even 12 4 2535.1.x.e 8
60.h even 2 1 195.1.e.a 4
60.l odd 4 1 975.1.g.b 2
60.l odd 4 1 975.1.g.c 2
65.d even 2 1 inner 3120.1.be.e 4
156.h even 2 1 195.1.e.a 4
156.l odd 4 2 2535.1.f.e 4
156.p even 6 2 2535.1.y.a 8
156.r even 6 2 2535.1.y.a 8
156.v odd 12 4 2535.1.x.e 8
195.e odd 2 1 inner 3120.1.be.e 4
260.g odd 2 1 195.1.e.a 4
260.p even 4 1 975.1.g.b 2
260.p even 4 1 975.1.g.c 2
260.u even 4 2 2535.1.f.e 4
260.v odd 6 2 2535.1.y.a 8
260.w odd 6 2 2535.1.y.a 8
260.bc even 12 4 2535.1.x.e 8
780.d even 2 1 195.1.e.a 4
780.w odd 4 1 975.1.g.b 2
780.w odd 4 1 975.1.g.c 2
780.bb odd 4 2 2535.1.f.e 4
780.br even 6 2 2535.1.y.a 8
780.cb even 6 2 2535.1.y.a 8
780.cr odd 12 4 2535.1.x.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.1.e.a 4 4.b odd 2 1
195.1.e.a 4 12.b even 2 1
195.1.e.a 4 20.d odd 2 1
195.1.e.a 4 52.b odd 2 1
195.1.e.a 4 60.h even 2 1
195.1.e.a 4 156.h even 2 1
195.1.e.a 4 260.g odd 2 1
195.1.e.a 4 780.d even 2 1
975.1.g.b 2 20.e even 4 1
975.1.g.b 2 60.l odd 4 1
975.1.g.b 2 260.p even 4 1
975.1.g.b 2 780.w odd 4 1
975.1.g.c 2 20.e even 4 1
975.1.g.c 2 60.l odd 4 1
975.1.g.c 2 260.p even 4 1
975.1.g.c 2 780.w odd 4 1
2535.1.f.e 4 52.f even 4 2
2535.1.f.e 4 156.l odd 4 2
2535.1.f.e 4 260.u even 4 2
2535.1.f.e 4 780.bb odd 4 2
2535.1.x.e 8 52.l even 12 4
2535.1.x.e 8 156.v odd 12 4
2535.1.x.e 8 260.bc even 12 4
2535.1.x.e 8 780.cr odd 12 4
2535.1.y.a 8 52.i odd 6 2
2535.1.y.a 8 52.j odd 6 2
2535.1.y.a 8 156.p even 6 2
2535.1.y.a 8 156.r even 6 2
2535.1.y.a 8 260.v odd 6 2
2535.1.y.a 8 260.w odd 6 2
2535.1.y.a 8 780.br even 6 2
2535.1.y.a 8 780.cb even 6 2
3120.1.be.e 4 1.a even 1 1 trivial
3120.1.be.e 4 3.b odd 2 1 inner
3120.1.be.e 4 5.b even 2 1 inner
3120.1.be.e 4 13.b even 2 1 inner
3120.1.be.e 4 15.d odd 2 1 inner
3120.1.be.e 4 39.d odd 2 1 CM
3120.1.be.e 4 65.d even 2 1 inner
3120.1.be.e 4 195.e odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3120, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} - 2$$ T11^2 - 2

## Hecke characteristic polynomials

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