# Properties

 Label 1560.2.a.b Level $1560$ Weight $2$ Character orbit 1560.a Self dual yes Analytic conductor $12.457$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,2,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1560.a (trivial)

## 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: $$12.4566627153$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} + q^{9}+O(q^{10})$$ q - q^3 - q^5 + q^9 $$q - q^{3} - q^{5} + q^{9} + 4 q^{11} + q^{13} + q^{15} - 2 q^{17} - 4 q^{19} + 4 q^{23} + q^{25} - q^{27} - 6 q^{29} - 4 q^{31} - 4 q^{33} + 6 q^{37} - q^{39} - 2 q^{41} + 12 q^{43} - q^{45} + 8 q^{47} - 7 q^{49} + 2 q^{51} + 14 q^{53} - 4 q^{55} + 4 q^{57} + 12 q^{59} - 2 q^{61} - q^{65} - 4 q^{69} + 8 q^{71} - 2 q^{73} - q^{75} + 8 q^{79} + q^{81} + 12 q^{83} + 2 q^{85} + 6 q^{87} + 6 q^{89} + 4 q^{93} + 4 q^{95} - 10 q^{97} + 4 q^{99}+O(q^{100})$$ q - q^3 - q^5 + q^9 + 4 * q^11 + q^13 + q^15 - 2 * q^17 - 4 * q^19 + 4 * q^23 + q^25 - q^27 - 6 * q^29 - 4 * q^31 - 4 * q^33 + 6 * q^37 - q^39 - 2 * q^41 + 12 * q^43 - q^45 + 8 * q^47 - 7 * q^49 + 2 * q^51 + 14 * q^53 - 4 * q^55 + 4 * q^57 + 12 * q^59 - 2 * q^61 - q^65 - 4 * q^69 + 8 * q^71 - 2 * q^73 - q^75 + 8 * q^79 + q^81 + 12 * q^83 + 2 * q^85 + 6 * q^87 + 6 * q^89 + 4 * q^93 + 4 * q^95 - 10 * q^97 + 4 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1560.2.a.b 1
3.b odd 2 1 4680.2.a.p 1
4.b odd 2 1 3120.2.a.p 1
5.b even 2 1 7800.2.a.u 1
12.b even 2 1 9360.2.a.bq 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.b 1 1.a even 1 1 trivial
3120.2.a.p 1 4.b odd 2 1
4680.2.a.p 1 3.b odd 2 1
7800.2.a.u 1 5.b even 2 1
9360.2.a.bq 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1560))$$:

 $$T_{7}$$ T7 $$T_{11} - 4$$ T11 - 4 $$T_{17} + 2$$ T17 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T + 2$$
$19$ $$T + 4$$
$23$ $$T - 4$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T - 6$$
$41$ $$T + 2$$
$43$ $$T - 12$$
$47$ $$T - 8$$
$53$ $$T - 14$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T$$
$71$ $$T - 8$$
$73$ $$T + 2$$
$79$ $$T - 8$$
$83$ $$T - 12$$
$89$ $$T - 6$$
$97$ $$T + 10$$