# Properties

 Label 1520.1.m.a Level $1520$ Weight $1$ Character orbit 1520.m Self dual yes Analytic conductor $0.759$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -19, -95, 5 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,1,Mod(1329,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1520.1329");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1520.m (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: $$0.758578819202$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-19})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.7600.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{9}+O(q^{10})$$ q + q^5 - q^9 $$q + q^{5} - q^{9} + 2 q^{11} - q^{19} + q^{25} - q^{45} + q^{49} + 2 q^{55} - 2 q^{61} + q^{81} - q^{95} - 2 q^{99}+O(q^{100})$$ q + q^5 - q^9 + 2 * q^11 - q^19 + q^25 - q^45 + q^49 + 2 * q^55 - 2 * q^61 + q^81 - q^95 - 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$401$$ $$1141$$ $$1217$$ $$\chi(n)$$ $$0$$ $$1$$ $$0$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1329.1
 0
0 0 0 1.00000 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
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1520.1.m.a 1
4.b odd 2 1 95.1.d.a 1
5.b even 2 1 RM 1520.1.m.a 1
12.b even 2 1 855.1.g.a 1
19.b odd 2 1 CM 1520.1.m.a 1
20.d odd 2 1 95.1.d.a 1
20.e even 4 2 475.1.c.a 1
60.h even 2 1 855.1.g.a 1
76.d even 2 1 95.1.d.a 1
76.f even 6 2 1805.1.h.a 2
76.g odd 6 2 1805.1.h.a 2
76.k even 18 6 1805.1.o.a 6
76.l odd 18 6 1805.1.o.a 6
95.d odd 2 1 CM 1520.1.m.a 1
228.b odd 2 1 855.1.g.a 1
380.d even 2 1 95.1.d.a 1
380.j odd 4 2 475.1.c.a 1
380.p odd 6 2 1805.1.h.a 2
380.s even 6 2 1805.1.h.a 2
380.ba odd 18 6 1805.1.o.a 6
380.bb even 18 6 1805.1.o.a 6
1140.p odd 2 1 855.1.g.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.a 1 4.b odd 2 1
95.1.d.a 1 20.d odd 2 1
95.1.d.a 1 76.d even 2 1
95.1.d.a 1 380.d even 2 1
475.1.c.a 1 20.e even 4 2
475.1.c.a 1 380.j odd 4 2
855.1.g.a 1 12.b even 2 1
855.1.g.a 1 60.h even 2 1
855.1.g.a 1 228.b odd 2 1
855.1.g.a 1 1140.p odd 2 1
1520.1.m.a 1 1.a even 1 1 trivial
1520.1.m.a 1 5.b even 2 1 RM
1520.1.m.a 1 19.b odd 2 1 CM
1520.1.m.a 1 95.d odd 2 1 CM
1805.1.h.a 2 76.f even 6 2
1805.1.h.a 2 76.g odd 6 2
1805.1.h.a 2 380.p odd 6 2
1805.1.h.a 2 380.s even 6 2
1805.1.o.a 6 76.k even 18 6
1805.1.o.a 6 76.l odd 18 6
1805.1.o.a 6 380.ba odd 18 6
1805.1.o.a 6 380.bb even 18 6

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}$$ T3 $$T_{7}$$ T7

## Hecke characteristic polynomials

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