# Properties

 Label 4056.2.a.l Level $4056$ Weight $2$ Character orbit 4056.a Self dual yes Analytic conductor $32.387$ 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] = [4056,2,Mod(1,4056)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4056, 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("4056.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4056 = 2^{3} \cdot 3 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4056.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: $$32.3873230598$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 312) 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)

 $$f(q)$$ $$=$$ $$q + q^{3} - 3 q^{5} + q^{9}+O(q^{10})$$ q + q^3 - 3 * q^5 + q^9 $$q + q^{3} - 3 q^{5} + q^{9} - 3 q^{15} - q^{17} + 4 q^{23} + 4 q^{25} + q^{27} + 3 q^{29} - 8 q^{31} + 5 q^{37} - 3 q^{41} + 4 q^{43} - 3 q^{45} + 8 q^{47} - 7 q^{49} - q^{51} - 13 q^{53} - 12 q^{59} + 15 q^{61} - 12 q^{67} + 4 q^{69} - 8 q^{71} - 3 q^{73} + 4 q^{75} - 4 q^{79} + q^{81} - 12 q^{83} + 3 q^{85} + 3 q^{87} - 10 q^{89} - 8 q^{93} - 2 q^{97}+O(q^{100})$$ q + q^3 - 3 * q^5 + q^9 - 3 * q^15 - q^17 + 4 * q^23 + 4 * q^25 + q^27 + 3 * q^29 - 8 * q^31 + 5 * q^37 - 3 * q^41 + 4 * q^43 - 3 * q^45 + 8 * q^47 - 7 * q^49 - q^51 - 13 * q^53 - 12 * q^59 + 15 * q^61 - 12 * q^67 + 4 * q^69 - 8 * q^71 - 3 * q^73 + 4 * q^75 - 4 * q^79 + q^81 - 12 * q^83 + 3 * q^85 + 3 * q^87 - 10 * q^89 - 8 * q^93 - 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 −3.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$13$$ $$1$$

## 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 4056.2.a.l 1
4.b odd 2 1 8112.2.a.b 1
13.b even 2 1 4056.2.a.q 1
13.d odd 4 2 4056.2.c.i 2
13.e even 6 2 312.2.q.a 2
39.h odd 6 2 936.2.t.a 2
52.b odd 2 1 8112.2.a.n 1
52.i odd 6 2 624.2.q.f 2
156.r even 6 2 1872.2.t.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.2.q.a 2 13.e even 6 2
624.2.q.f 2 52.i odd 6 2
936.2.t.a 2 39.h odd 6 2
1872.2.t.b 2 156.r even 6 2
4056.2.a.l 1 1.a even 1 1 trivial
4056.2.a.q 1 13.b even 2 1
4056.2.c.i 2 13.d odd 4 2
8112.2.a.b 1 4.b odd 2 1
8112.2.a.n 1 52.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4056))$$:

 $$T_{5} + 3$$ T5 + 3 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T + 3$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T + 1$$
$19$ $$T$$
$23$ $$T - 4$$
$29$ $$T - 3$$
$31$ $$T + 8$$
$37$ $$T - 5$$
$41$ $$T + 3$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T + 13$$
$59$ $$T + 12$$
$61$ $$T - 15$$
$67$ $$T + 12$$
$71$ $$T + 8$$
$73$ $$T + 3$$
$79$ $$T + 4$$
$83$ $$T + 12$$
$89$ $$T + 10$$
$97$ $$T + 2$$