# Properties

 Label 6552.2.a.w Level $6552$ Weight $2$ Character orbit 6552.a Self dual yes Analytic conductor $52.318$ Analytic rank $0$ 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] = [6552,2,Mod(1,6552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6552, 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("6552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6552.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: $$52.3179834043$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2184) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{5} + q^{7}+O(q^{10})$$ q + 3 * q^5 + q^7 $$q + 3 q^{5} + q^{7} - 6 q^{11} - q^{13} + 8 q^{17} - q^{19} - q^{23} + 4 q^{25} + 5 q^{29} + 3 q^{31} + 3 q^{35} + 12 q^{37} - 10 q^{41} - 11 q^{43} + 3 q^{47} + q^{49} + q^{53} - 18 q^{55} + 8 q^{59} + 2 q^{61} - 3 q^{65} + 2 q^{67} - 8 q^{71} + 11 q^{73} - 6 q^{77} + 13 q^{79} + 7 q^{83} + 24 q^{85} - 7 q^{89} - q^{91} - 3 q^{95} - q^{97}+O(q^{100})$$ q + 3 * q^5 + q^7 - 6 * q^11 - q^13 + 8 * q^17 - q^19 - q^23 + 4 * q^25 + 5 * q^29 + 3 * q^31 + 3 * q^35 + 12 * q^37 - 10 * q^41 - 11 * q^43 + 3 * q^47 + q^49 + q^53 - 18 * q^55 + 8 * q^59 + 2 * q^61 - 3 * q^65 + 2 * q^67 - 8 * q^71 + 11 * q^73 - 6 * q^77 + 13 * q^79 + 7 * q^83 + 24 * q^85 - 7 * q^89 - q^91 - 3 * q^95 - 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 3.00000 0 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$1$$

## 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 6552.2.a.w 1
3.b odd 2 1 2184.2.a.b 1
12.b even 2 1 4368.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2184.2.a.b 1 3.b odd 2 1
4368.2.a.m 1 12.b even 2 1
6552.2.a.w 1 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6552))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11} + 6$$ T11 + 6 $$T_{17} - 8$$ T17 - 8

## Hecke characteristic polynomials

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