# Properties

 Label 3120.2.a.w 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 390) 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 $$q + q^{3} + q^{5} + q^{9} - 4 q^{11} + q^{13} + q^{15} - 6 q^{17} - 4 q^{19} - 8 q^{23} + q^{25} + q^{27} + 6 q^{29} + 8 q^{31} - 4 q^{33} - 10 q^{37} + q^{39} - 6 q^{41} - 4 q^{43} + q^{45} - 7 q^{49} - 6 q^{51} - 10 q^{53} - 4 q^{55} - 4 q^{57} - 4 q^{59} - 2 q^{61} + q^{65} + 12 q^{67} - 8 q^{69} - 16 q^{71} + 2 q^{73} + q^{75} + 16 q^{79} + q^{81} + 12 q^{83} - 6 q^{85} + 6 q^{87} + 10 q^{89} + 8 q^{93} - 4 q^{95} - 6 q^{97} - 4 q^{99}+O(q^{100})$$ q + q^3 + q^5 + q^9 - 4 * q^11 + q^13 + q^15 - 6 * q^17 - 4 * q^19 - 8 * q^23 + q^25 + q^27 + 6 * q^29 + 8 * q^31 - 4 * q^33 - 10 * q^37 + q^39 - 6 * q^41 - 4 * q^43 + q^45 - 7 * q^49 - 6 * q^51 - 10 * q^53 - 4 * q^55 - 4 * q^57 - 4 * q^59 - 2 * q^61 + q^65 + 12 * q^67 - 8 * q^69 - 16 * q^71 + 2 * q^73 + q^75 + 16 * q^79 + q^81 + 12 * q^83 - 6 * q^85 + 6 * q^87 + 10 * q^89 + 8 * q^93 - 4 * q^95 - 6 * 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 3120.2.a.w 1
3.b odd 2 1 9360.2.a.p 1
4.b odd 2 1 390.2.a.f 1
12.b even 2 1 1170.2.a.a 1
20.d odd 2 1 1950.2.a.k 1
20.e even 4 2 1950.2.e.g 2
52.b odd 2 1 5070.2.a.a 1
52.f even 4 2 5070.2.b.d 2
60.h even 2 1 5850.2.a.bo 1
60.l odd 4 2 5850.2.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
390.2.a.f 1 4.b odd 2 1
1170.2.a.a 1 12.b even 2 1
1950.2.a.k 1 20.d odd 2 1
1950.2.e.g 2 20.e even 4 2
3120.2.a.w 1 1.a even 1 1 trivial
5070.2.a.a 1 52.b odd 2 1
5070.2.b.d 2 52.f even 4 2
5850.2.a.bo 1 60.h even 2 1
5850.2.e.e 2 60.l odd 4 2
9360.2.a.p 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 $$T_{11} + 4$$ T11 + 4 $$T_{17} + 6$$ T17 + 6 $$T_{19} + 4$$ T19 + 4 $$T_{31} - 8$$ T31 - 8

## Hecke characteristic polynomials

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