Properties

 Label 6552.2.a.bi Level $6552$ Weight $2$ Character orbit 6552.a Self dual yes Analytic conductor $52.318$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{5} - q^{7}+O(q^{10})$$ q + (b + 1) * q^5 - q^7 $$q + (\beta + 1) q^{5} - q^{7} + (\beta + 1) q^{11} + q^{13} + 2 q^{17} + 4 q^{19} + 2 q^{23} + (2 \beta + 1) q^{25} + ( - 2 \beta + 2) q^{29} + (2 \beta + 2) q^{31} + ( - \beta - 1) q^{35} + 2 \beta q^{37} + ( - \beta - 5) q^{41} + 4 \beta q^{43} + ( - 3 \beta + 3) q^{47} + q^{49} - 4 q^{53} + (2 \beta + 6) q^{55} + ( - \beta + 1) q^{59} - 6 q^{61} + (\beta + 1) q^{65} + ( - 2 \beta - 2) q^{67} + (5 \beta - 3) q^{71} + ( - 2 \beta - 12) q^{73} + ( - \beta - 1) q^{77} - 4 \beta q^{79} + (\beta + 11) q^{83} + (2 \beta + 2) q^{85} + (\beta + 9) q^{89} - q^{91} + (4 \beta + 4) q^{95} + 2 q^{97} +O(q^{100})$$ q + (b + 1) * q^5 - q^7 + (b + 1) * q^11 + q^13 + 2 * q^17 + 4 * q^19 + 2 * q^23 + (2*b + 1) * q^25 + (-2*b + 2) * q^29 + (2*b + 2) * q^31 + (-b - 1) * q^35 + 2*b * q^37 + (-b - 5) * q^41 + 4*b * q^43 + (-3*b + 3) * q^47 + q^49 - 4 * q^53 + (2*b + 6) * q^55 + (-b + 1) * q^59 - 6 * q^61 + (b + 1) * q^65 + (-2*b - 2) * q^67 + (5*b - 3) * q^71 + (-2*b - 12) * q^73 + (-b - 1) * q^77 - 4*b * q^79 + (b + 11) * q^83 + (2*b + 2) * q^85 + (b + 9) * q^89 - q^91 + (4*b + 4) * q^95 + 2 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} - 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 - 2 * q^7 $$2 q + 2 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} + 4 q^{17} + 8 q^{19} + 4 q^{23} + 2 q^{25} + 4 q^{29} + 4 q^{31} - 2 q^{35} - 10 q^{41} + 6 q^{47} + 2 q^{49} - 8 q^{53} + 12 q^{55} + 2 q^{59} - 12 q^{61} + 2 q^{65} - 4 q^{67} - 6 q^{71} - 24 q^{73} - 2 q^{77} + 22 q^{83} + 4 q^{85} + 18 q^{89} - 2 q^{91} + 8 q^{95} + 4 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 - 2 * q^7 + 2 * q^11 + 2 * q^13 + 4 * q^17 + 8 * q^19 + 4 * q^23 + 2 * q^25 + 4 * q^29 + 4 * q^31 - 2 * q^35 - 10 * q^41 + 6 * q^47 + 2 * q^49 - 8 * q^53 + 12 * q^55 + 2 * q^59 - 12 * q^61 + 2 * q^65 - 4 * q^67 - 6 * q^71 - 24 * q^73 - 2 * q^77 + 22 * q^83 + 4 * q^85 + 18 * q^89 - 2 * q^91 + 8 * q^95 + 4 * 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
 −0.618034 1.61803
0 0 0 −1.23607 0 −1.00000 0 0 0
1.2 0 0 0 3.23607 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.bi yes 2
3.b odd 2 1 6552.2.a.bc 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6552.2.a.bc 2 3.b odd 2 1
6552.2.a.bi yes 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6552))$$:

 $$T_{5}^{2} - 2T_{5} - 4$$ T5^2 - 2*T5 - 4 $$T_{11}^{2} - 2T_{11} - 4$$ T11^2 - 2*T11 - 4 $$T_{17} - 2$$ T17 - 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T - 4$$
$7$ $$(T + 1)^{2}$$
$11$ $$T^{2} - 2T - 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} - 4T - 16$$
$31$ $$T^{2} - 4T - 16$$
$37$ $$T^{2} - 20$$
$41$ $$T^{2} + 10T + 20$$
$43$ $$T^{2} - 80$$
$47$ $$T^{2} - 6T - 36$$
$53$ $$(T + 4)^{2}$$
$59$ $$T^{2} - 2T - 4$$
$61$ $$(T + 6)^{2}$$
$67$ $$T^{2} + 4T - 16$$
$71$ $$T^{2} + 6T - 116$$
$73$ $$T^{2} + 24T + 124$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} - 22T + 116$$
$89$ $$T^{2} - 18T + 76$$
$97$ $$(T - 2)^{2}$$