# Properties

 Label 3120.1.be.a Level $3120$ Weight $1$ Character orbit 3120.be Self dual yes Analytic conductor $1.557$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -195 Inner twists $2$

# 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: yes Analytic conductor: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 780) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.780.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.126547200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 - q^5 - q^7 + q^9 $$q - q^{3} - q^{5} - q^{7} + q^{9} - q^{11} - q^{13} + q^{15} - q^{17} + q^{21} + q^{23} + q^{25} - q^{27} + q^{33} + q^{35} + q^{37} + q^{39} + q^{41} - q^{45} + q^{51} - q^{53} + q^{55} + 2 q^{59} - q^{61} - q^{63} + q^{65} + 2 q^{67} - q^{69} - q^{71} - 2 q^{73} - q^{75} + q^{77} + q^{79} + q^{81} + q^{85} + q^{89} + q^{91} + q^{97} - q^{99}+O(q^{100})$$ q - q^3 - q^5 - q^7 + q^9 - q^11 - q^13 + q^15 - q^17 + q^21 + q^23 + q^25 - q^27 + q^33 + q^35 + q^37 + q^39 + q^41 - q^45 + q^51 - q^53 + q^55 + 2 * q^59 - q^61 - q^63 + q^65 + 2 * q^67 - q^69 - q^71 - 2 * q^73 - q^75 + q^77 + q^79 + q^81 + q^85 + q^89 + q^91 + q^97 - q^99

## 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
0 −1.00000 0 −1.00000 0 −1.00000 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
195.e odd 2 1 CM by $$\Q(\sqrt{-195})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.1.be.a 1
3.b odd 2 1 3120.1.be.d 1
4.b odd 2 1 780.1.o.c yes 1
5.b even 2 1 3120.1.be.c 1
12.b even 2 1 780.1.o.b yes 1
13.b even 2 1 3120.1.be.b 1
15.d odd 2 1 3120.1.be.b 1
20.d odd 2 1 780.1.o.a 1
20.e even 4 2 3900.1.f.d 2
39.d odd 2 1 3120.1.be.c 1
52.b odd 2 1 780.1.o.d yes 1
60.h even 2 1 780.1.o.d yes 1
60.l odd 4 2 3900.1.f.c 2
65.d even 2 1 3120.1.be.d 1
156.h even 2 1 780.1.o.a 1
195.e odd 2 1 CM 3120.1.be.a 1
260.g odd 2 1 780.1.o.b yes 1
260.p even 4 2 3900.1.f.c 2
780.d even 2 1 780.1.o.c yes 1
780.w odd 4 2 3900.1.f.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.1.o.a 1 20.d odd 2 1
780.1.o.a 1 156.h even 2 1
780.1.o.b yes 1 12.b even 2 1
780.1.o.b yes 1 260.g odd 2 1
780.1.o.c yes 1 4.b odd 2 1
780.1.o.c yes 1 780.d even 2 1
780.1.o.d yes 1 52.b odd 2 1
780.1.o.d yes 1 60.h even 2 1
3120.1.be.a 1 1.a even 1 1 trivial
3120.1.be.a 1 195.e odd 2 1 CM
3120.1.be.b 1 13.b even 2 1
3120.1.be.b 1 15.d odd 2 1
3120.1.be.c 1 5.b even 2 1
3120.1.be.c 1 39.d odd 2 1
3120.1.be.d 1 3.b odd 2 1
3120.1.be.d 1 65.d even 2 1
3900.1.f.c 2 60.l odd 4 2
3900.1.f.c 2 260.p even 4 2
3900.1.f.d 2 20.e even 4 2
3900.1.f.d 2 780.w odd 4 2

## Hecke kernels

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

 $$T_{7} + 1$$ T7 + 1 $$T_{11} + 1$$ T11 + 1

## Hecke characteristic polynomials

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