# Properties

 Label 1520.1.m.b Level $1520$ Weight $1$ Character orbit 1520.m Self dual yes Analytic conductor $0.759$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -95 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $D_8$ Artin field: Galois closure of 8.2.1097440000.5

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} - q^{5} + q^{9} +O(q^{10})$$ q - b * q^3 - q^5 + q^9 $$q - \beta q^{3} - q^{5} + q^{9} - \beta q^{13} + \beta q^{15} + q^{19} + q^{25} + \beta q^{37} + 2 q^{39} - q^{45} + q^{49} + \beta q^{53} - \beta q^{57} + \beta q^{65} + \beta q^{67} - \beta q^{75} - q^{81} - q^{95} + \beta q^{97} +O(q^{100})$$ q - b * q^3 - q^5 + q^9 - b * q^13 + b * q^15 + q^19 + q^25 + b * q^37 + 2 * q^39 - q^45 + q^49 + b * q^53 - b * q^57 + b * q^65 + b * q^67 - b * q^75 - q^81 - q^95 + b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^5 + 2 * q^9 $$2 q - 2 q^{5} + 2 q^{9} + 2 q^{19} + 2 q^{25} + 4 q^{39} - 2 q^{45} + 2 q^{49} - 2 q^{81} - 2 q^{95}+O(q^{100})$$ 2 * q - 2 * q^5 + 2 * q^9 + 2 * q^19 + 2 * q^25 + 4 * q^39 - 2 * q^45 + 2 * q^49 - 2 * q^81 - 2 * q^95

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1329.1
 1.41421 −1.41421
0 −1.41421 0 −1.00000 0 0 0 1.00000 0
1329.2 0 1.41421 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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 4.b odd 2 1
95.1.d.b 2 20.d odd 2 1
95.1.d.b 2 76.d even 2 1
95.1.d.b 2 380.d even 2 1
475.1.c.b 2 20.e even 4 2
475.1.c.b 2 380.j odd 4 2
855.1.g.c 2 12.b even 2 1
855.1.g.c 2 60.h even 2 1
855.1.g.c 2 228.b odd 2 1
855.1.g.c 2 1140.p odd 2 1
1520.1.m.b 2 1.a even 1 1 trivial
1520.1.m.b 2 5.b even 2 1 inner
1520.1.m.b 2 19.b odd 2 1 inner
1520.1.m.b 2 95.d odd 2 1 CM
1805.1.h.b 4 76.f even 6 2
1805.1.h.b 4 76.g odd 6 2
1805.1.h.b 4 380.p odd 6 2
1805.1.h.b 4 380.s even 6 2
1805.1.o.b 12 76.k even 18 6
1805.1.o.b 12 76.l odd 18 6
1805.1.o.b 12 380.ba odd 18 6
1805.1.o.b 12 380.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 $$S_{1}^{\mathrm{new}}(1520, [\chi])$$:

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

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