# Properties

 Label 3960.2.a.p Level $3960$ Weight $2$ Character orbit 3960.a Self dual yes Analytic conductor $31.621$ 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] = [3960,2,Mod(1,3960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3960, 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("3960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3960.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: $$31.6207592004$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 440) 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)

 $$f(q)$$ $$=$$ $$q + q^{5} - 2 q^{7}+O(q^{10})$$ q + q^5 - 2 * q^7 $$q + q^{5} - 2 q^{7} + q^{11} - 8 q^{19} + 8 q^{23} + q^{25} - 10 q^{29} + 8 q^{31} - 2 q^{35} - 10 q^{37} + 2 q^{41} - 6 q^{43} + 8 q^{47} - 3 q^{49} - 14 q^{53} + q^{55} + 4 q^{59} + 10 q^{61} + 4 q^{67} - 8 q^{73} - 2 q^{77} - 4 q^{79} - 10 q^{83} - 6 q^{89} - 8 q^{95} - 10 q^{97}+O(q^{100})$$ q + q^5 - 2 * q^7 + q^11 - 8 * q^19 + 8 * q^23 + q^25 - 10 * q^29 + 8 * q^31 - 2 * q^35 - 10 * q^37 + 2 * q^41 - 6 * q^43 + 8 * q^47 - 3 * q^49 - 14 * q^53 + q^55 + 4 * q^59 + 10 * q^61 + 4 * q^67 - 8 * q^73 - 2 * q^77 - 4 * q^79 - 10 * q^83 - 6 * q^89 - 8 * q^95 - 10 * 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 0 0 1.00000 0 −2.00000 0 0 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$11$$ $$-1$$

## 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 3960.2.a.p 1
3.b odd 2 1 440.2.a.a 1
4.b odd 2 1 7920.2.a.bg 1
12.b even 2 1 880.2.a.f 1
15.d odd 2 1 2200.2.a.f 1
15.e even 4 2 2200.2.b.d 2
24.f even 2 1 3520.2.a.u 1
24.h odd 2 1 3520.2.a.t 1
33.d even 2 1 4840.2.a.c 1
60.h even 2 1 4400.2.a.o 1
60.l odd 4 2 4400.2.b.m 2
132.d odd 2 1 9680.2.a.n 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.a 1 3.b odd 2 1
880.2.a.f 1 12.b even 2 1
2200.2.a.f 1 15.d odd 2 1
2200.2.b.d 2 15.e even 4 2
3520.2.a.t 1 24.h odd 2 1
3520.2.a.u 1 24.f even 2 1
3960.2.a.p 1 1.a even 1 1 trivial
4400.2.a.o 1 60.h even 2 1
4400.2.b.m 2 60.l odd 4 2
4840.2.a.c 1 33.d even 2 1
7920.2.a.bg 1 4.b odd 2 1
9680.2.a.n 1 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3960))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{13}$$ T13 $$T_{17}$$ T17 $$T_{19} + 8$$ T19 + 8 $$T_{23} - 8$$ T23 - 8 $$T_{29} + 10$$ T29 + 10

## Hecke characteristic polynomials

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