# Properties

 Label 1520.2.a.f Level $1520$ Weight $2$ Character orbit 1520.a Self dual yes Analytic conductor $12.137$ Analytic rank $1$ 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] = [1520,2,Mod(1,1520)]

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, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 760) 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 3 q^{9}+O(q^{10})$$ q + q^5 - 3 * q^9 $$q + q^{5} - 3 q^{9} + 4 q^{11} - 6 q^{13} - 6 q^{17} + q^{19} - 8 q^{23} + q^{25} - 2 q^{29} + 2 q^{37} + 2 q^{41} - 4 q^{43} - 3 q^{45} + 8 q^{47} - 7 q^{49} - 6 q^{53} + 4 q^{55} + 4 q^{59} - 2 q^{61} - 6 q^{65} - 8 q^{67} - 8 q^{71} + 2 q^{73} + 8 q^{79} + 9 q^{81} - 4 q^{83} - 6 q^{85} - 14 q^{89} + q^{95} + 14 q^{97} - 12 q^{99}+O(q^{100})$$ q + q^5 - 3 * q^9 + 4 * q^11 - 6 * q^13 - 6 * q^17 + q^19 - 8 * q^23 + q^25 - 2 * q^29 + 2 * q^37 + 2 * q^41 - 4 * q^43 - 3 * q^45 + 8 * q^47 - 7 * q^49 - 6 * q^53 + 4 * q^55 + 4 * q^59 - 2 * q^61 - 6 * q^65 - 8 * q^67 - 8 * q^71 + 2 * q^73 + 8 * q^79 + 9 * q^81 - 4 * q^83 - 6 * q^85 - 14 * q^89 + q^95 + 14 * q^97 - 12 * 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 0 0 1.00000 0 0 0 −3.00000 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
$$2$$ $$+1$$
$$5$$ $$-1$$
$$19$$ $$-1$$

## 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 1520.2.a.f 1
4.b odd 2 1 760.2.a.c 1
5.b even 2 1 7600.2.a.k 1
8.b even 2 1 6080.2.a.j 1
8.d odd 2 1 6080.2.a.k 1
12.b even 2 1 6840.2.a.h 1
20.d odd 2 1 3800.2.a.e 1
20.e even 4 2 3800.2.d.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.a.c 1 4.b odd 2 1
1520.2.a.f 1 1.a even 1 1 trivial
3800.2.a.e 1 20.d odd 2 1
3800.2.d.g 2 20.e even 4 2
6080.2.a.j 1 8.b even 2 1
6080.2.a.k 1 8.d odd 2 1
6840.2.a.h 1 12.b even 2 1
7600.2.a.k 1 5.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1520))$$:

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

## Hecke characteristic polynomials

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