# Properties

 Label 6552.2.a.bf Level $6552$ Weight $2$ Character orbit 6552.a Self dual yes Analytic conductor $52.318$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 728) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} - q^{7}+O(q^{10})$$ q + b * q^5 - q^7 $$q + \beta q^{5} - q^{7} + ( - \beta + 1) q^{11} - q^{13} + (\beta - 1) q^{17} - \beta q^{19} + (2 \beta + 3) q^{23} - 2 q^{25} + ( - 2 \beta + 3) q^{29} + ( - \beta - 2) q^{31} - \beta q^{35} + ( - \beta - 5) q^{37} - 6 \beta q^{41} + (2 \beta - 9) q^{43} + ( - \beta + 4) q^{47} + q^{49} + ( - 4 \beta + 3) q^{53} + (\beta - 3) q^{55} + 2 \beta q^{59} - 2 q^{61} - \beta q^{65} + 6 \beta q^{67} + (3 \beta - 5) q^{71} + ( - \beta + 8) q^{73} + (\beta - 1) q^{77} + ( - 4 \beta - 3) q^{79} + (\beta + 8) q^{83} + ( - \beta + 3) q^{85} + ( - \beta - 14) q^{89} + q^{91} - 3 q^{95} + (7 \beta - 4) q^{97} +O(q^{100})$$ q + b * q^5 - q^7 + (-b + 1) * q^11 - q^13 + (b - 1) * q^17 - b * q^19 + (2*b + 3) * q^23 - 2 * q^25 + (-2*b + 3) * q^29 + (-b - 2) * q^31 - b * q^35 + (-b - 5) * q^37 - 6*b * q^41 + (2*b - 9) * q^43 + (-b + 4) * q^47 + q^49 + (-4*b + 3) * q^53 + (b - 3) * q^55 + 2*b * q^59 - 2 * q^61 - b * q^65 + 6*b * q^67 + (3*b - 5) * q^71 + (-b + 8) * q^73 + (b - 1) * q^77 + (-4*b - 3) * q^79 + (b + 8) * q^83 + (-b + 3) * q^85 + (-b - 14) * q^89 + q^91 - 3 * q^95 + (7*b - 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{7}+O(q^{10})$$ 2 * q - 2 * q^7 $$2 q - 2 q^{7} + 2 q^{11} - 2 q^{13} - 2 q^{17} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 4 q^{31} - 10 q^{37} - 18 q^{43} + 8 q^{47} + 2 q^{49} + 6 q^{53} - 6 q^{55} - 4 q^{61} - 10 q^{71} + 16 q^{73} - 2 q^{77} - 6 q^{79} + 16 q^{83} + 6 q^{85} - 28 q^{89} + 2 q^{91} - 6 q^{95} - 8 q^{97}+O(q^{100})$$ 2 * q - 2 * q^7 + 2 * q^11 - 2 * q^13 - 2 * q^17 + 6 * q^23 - 4 * q^25 + 6 * q^29 - 4 * q^31 - 10 * q^37 - 18 * q^43 + 8 * q^47 + 2 * q^49 + 6 * q^53 - 6 * q^55 - 4 * q^61 - 10 * q^71 + 16 * q^73 - 2 * q^77 - 6 * q^79 + 16 * q^83 + 6 * q^85 - 28 * q^89 + 2 * q^91 - 6 * q^95 - 8 * 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
 −1.73205 1.73205
0 0 0 −1.73205 0 −1.00000 0 0 0
1.2 0 0 0 1.73205 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.bf 2
3.b odd 2 1 728.2.a.f 2
12.b even 2 1 1456.2.a.r 2
21.c even 2 1 5096.2.a.o 2
24.f even 2 1 5824.2.a.bi 2
24.h odd 2 1 5824.2.a.bp 2
39.d odd 2 1 9464.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.a.f 2 3.b odd 2 1
1456.2.a.r 2 12.b even 2 1
5096.2.a.o 2 21.c even 2 1
5824.2.a.bi 2 24.f even 2 1
5824.2.a.bp 2 24.h odd 2 1
6552.2.a.bf 2 1.a even 1 1 trivial
9464.2.a.n 2 39.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6552))$$:

 $$T_{5}^{2} - 3$$ T5^2 - 3 $$T_{11}^{2} - 2T_{11} - 2$$ T11^2 - 2*T11 - 2 $$T_{17}^{2} + 2T_{17} - 2$$ T17^2 + 2*T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 2T - 2$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} + 2T - 2$$
$19$ $$T^{2} - 3$$
$23$ $$T^{2} - 6T - 3$$
$29$ $$T^{2} - 6T - 3$$
$31$ $$T^{2} + 4T + 1$$
$37$ $$T^{2} + 10T + 22$$
$41$ $$T^{2} - 108$$
$43$ $$T^{2} + 18T + 69$$
$47$ $$T^{2} - 8T + 13$$
$53$ $$T^{2} - 6T - 39$$
$59$ $$T^{2} - 12$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} - 108$$
$71$ $$T^{2} + 10T - 2$$
$73$ $$T^{2} - 16T + 61$$
$79$ $$T^{2} + 6T - 39$$
$83$ $$T^{2} - 16T + 61$$
$89$ $$T^{2} + 28T + 193$$
$97$ $$T^{2} + 8T - 131$$