# Properties

 Label 3120.2.a.r Level $3120$ Weight $2$ Character orbit 3120.a Self dual yes Analytic conductor $24.913$ 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] = [3120,2,Mod(1,3120)]

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, 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("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3120.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: $$24.9133254306$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1560) 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 $$q + q^{3} - q^{5} + q^{9} + q^{13} - q^{15} - 6 q^{17} - 4 q^{19} + q^{25} + q^{27} - 2 q^{29} - 2 q^{37} + q^{39} - 2 q^{41} + 4 q^{43} - q^{45} - 4 q^{47} - 7 q^{49} - 6 q^{51} - 10 q^{53} - 4 q^{57} - 8 q^{59} - 2 q^{61} - q^{65} + 4 q^{67} + 12 q^{71} - 6 q^{73} + q^{75} + q^{81} + 16 q^{83} + 6 q^{85} - 2 q^{87} - 10 q^{89} + 4 q^{95} + 2 q^{97}+O(q^{100})$$ q + q^3 - q^5 + q^9 + q^13 - q^15 - 6 * q^17 - 4 * q^19 + q^25 + q^27 - 2 * q^29 - 2 * q^37 + q^39 - 2 * q^41 + 4 * q^43 - q^45 - 4 * q^47 - 7 * q^49 - 6 * q^51 - 10 * q^53 - 4 * q^57 - 8 * q^59 - 2 * q^61 - q^65 + 4 * q^67 + 12 * q^71 - 6 * q^73 + q^75 + q^81 + 16 * q^83 + 6 * q^85 - 2 * q^87 - 10 * q^89 + 4 * q^95 + 2 * q^97

## 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 3120.2.a.r 1
3.b odd 2 1 9360.2.a.bp 1
4.b odd 2 1 1560.2.a.a 1
12.b even 2 1 4680.2.a.s 1
20.d odd 2 1 7800.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.a 1 4.b odd 2 1
3120.2.a.r 1 1.a even 1 1 trivial
4680.2.a.s 1 12.b even 2 1
7800.2.a.t 1 20.d odd 2 1
9360.2.a.bp 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3120))$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 4$$ T19 + 4 $$T_{31}$$ T31

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 1$$
$17$ $$T + 6$$
$19$ $$T + 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 2$$
$41$ $$T + 2$$
$43$ $$T - 4$$
$47$ $$T + 4$$
$53$ $$T + 10$$
$59$ $$T + 8$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T - 12$$
$73$ $$T + 6$$
$79$ $$T$$
$83$ $$T - 16$$
$89$ $$T + 10$$
$97$ $$T - 2$$