# Properties

 Label 6552.2.a.br Level $6552$ Weight $2$ Character orbit 6552.a Self dual yes Analytic conductor $52.318$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.64268.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 12x^{2} + 6x + 32$$ x^4 - x^3 - 12*x^2 + 6*x + 32 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} - 1) q^{5} - q^{7}+O(q^{10})$$ q + (-b2 - 1) * q^5 - q^7 $$q + ( - \beta_{2} - 1) q^{5} - q^{7} + (2 \beta_{3} + \beta_1) q^{11} - q^{13} + (\beta_{3} - 4) q^{17} + ( - 2 \beta_{3} - \beta_{2} - 3) q^{19} + (\beta_{2} - \beta_1 - 1) q^{23} + (\beta_{3} + \beta_{2} + 4) q^{25} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{29} + (\beta_{3} - \beta_{2} + \beta_1 - 1) q^{31} + (\beta_{2} + 1) q^{35} + ( - 2 \beta_{3} - 3 \beta_1 + 2) q^{37} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{41} + (\beta_{3} - \beta_{2} + 3) q^{43} + (\beta_{3} - 3 \beta_{2} + 3 \beta_1 + 1) q^{47} + q^{49} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{53} + ( - \beta_{3} - 6 \beta_1 + 2) q^{55} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{59} + (3 \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{61} + (\beta_{2} + 1) q^{65} + (\beta_{3} - 3 \beta_1 + 2) q^{67} + (\beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{71} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{73} + ( - 2 \beta_{3} - \beta_1) q^{77} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 3) q^{79} + ( - 3 \beta_{2} + 2 \beta_1 - 5) q^{83} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 + 6) q^{85} + ( - 4 \beta_{3} - \beta_{2} - 2 \beta_1 - 1) q^{89} + q^{91} + ( - \beta_{3} + 3 \beta_{2} + 4 \beta_1 + 7) q^{95} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 5) q^{97}+O(q^{100})$$ q + (-b2 - 1) * q^5 - q^7 + (2*b3 + b1) * q^11 - q^13 + (b3 - 4) * q^17 + (-2*b3 - b2 - 3) * q^19 + (b2 - b1 - 1) * q^23 + (b3 + b2 + 4) * q^25 + (b3 + b2 + 2*b1 - 1) * q^29 + (b3 - b2 + b1 - 1) * q^31 + (b2 + 1) * q^35 + (-2*b3 - 3*b1 + 2) * q^37 + (-b3 - 2*b2 + b1 - 6) * q^41 + (b3 - b2 + 3) * q^43 + (b3 - 3*b2 + 3*b1 + 1) * q^47 + q^49 + (b3 - b2 + 2*b1 + 1) * q^53 + (-b3 - 6*b1 + 2) * q^55 + (-2*b3 - 2*b2 - 2*b1 + 6) * q^59 + (3*b3 - 2*b2 + 3*b1 - 2) * q^61 + (b2 + 1) * q^65 + (b3 - 3*b1 + 2) * q^67 + (b3 - 4*b2 + 2*b1 + 2) * q^71 + (b3 - b2 + b1 + 1) * q^73 + (-2*b3 - b1) * q^77 + (-2*b3 + b2 - b1 + 3) * q^79 + (-3*b2 + 2*b1 - 5) * q^83 + (b3 + 4*b2 - 2*b1 + 6) * q^85 + (-4*b3 - b2 - 2*b1 - 1) * q^89 + q^91 + (-b3 + 3*b2 + 4*b1 + 7) * q^95 + (-b3 - b2 + 3*b1 + 5) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{5} - 4 q^{7}+O(q^{10})$$ 4 * q - 2 * q^5 - 4 * q^7 $$4 q - 2 q^{5} - 4 q^{7} + q^{11} - 4 q^{13} - 16 q^{17} - 10 q^{19} - 7 q^{23} + 14 q^{25} - 4 q^{29} - q^{31} + 2 q^{35} + 5 q^{37} - 19 q^{41} + 14 q^{43} + 13 q^{47} + 4 q^{49} + 8 q^{53} + 2 q^{55} + 26 q^{59} - q^{61} + 2 q^{65} + 5 q^{67} + 18 q^{71} + 7 q^{73} - q^{77} + 9 q^{79} - 12 q^{83} + 14 q^{85} - 4 q^{89} + 4 q^{91} + 26 q^{95} + 25 q^{97}+O(q^{100})$$ 4 * q - 2 * q^5 - 4 * q^7 + q^11 - 4 * q^13 - 16 * q^17 - 10 * q^19 - 7 * q^23 + 14 * q^25 - 4 * q^29 - q^31 + 2 * q^35 + 5 * q^37 - 19 * q^41 + 14 * q^43 + 13 * q^47 + 4 * q^49 + 8 * q^53 + 2 * q^55 + 26 * q^59 - q^61 + 2 * q^65 + 5 * q^67 + 18 * q^71 + 7 * q^73 - q^77 + 9 * q^79 - 12 * q^83 + 14 * q^85 - 4 * q^89 + 4 * q^91 + 26 * q^95 + 25 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 12x^{2} + 6x + 32$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} - \nu^{2} - 6\nu + 2 ) / 2$$ (v^3 - v^2 - 6*v + 2) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 6$$ v^2 - v - 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta _1 + 6$$ b3 + b1 + 6 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2\beta_{2} + 7\beta _1 + 4$$ b3 + 2*b2 + 7*b1 + 4

## 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
 3.18587 −1.72110 2.19455 −2.65932
0 0 0 −3.53544 0 −1.00000 0 0 0
1.2 0 0 0 −3.13311 0 −1.00000 0 0 0
1.3 0 0 0 1.70715 0 −1.00000 0 0 0
1.4 0 0 0 2.96141 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.br 4
3.b odd 2 1 728.2.a.i 4
12.b even 2 1 1456.2.a.v 4
21.c even 2 1 5096.2.a.s 4
24.f even 2 1 5824.2.a.ce 4
24.h odd 2 1 5824.2.a.cb 4
39.d odd 2 1 9464.2.a.z 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.a.i 4 3.b odd 2 1
1456.2.a.v 4 12.b even 2 1
5096.2.a.s 4 21.c even 2 1
5824.2.a.cb 4 24.h odd 2 1
5824.2.a.ce 4 24.f even 2 1
6552.2.a.br 4 1.a even 1 1 trivial
9464.2.a.z 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6552))$$:

 $$T_{5}^{4} + 2T_{5}^{3} - 15T_{5}^{2} - 18T_{5} + 56$$ T5^4 + 2*T5^3 - 15*T5^2 - 18*T5 + 56 $$T_{11}^{4} - T_{11}^{3} - 44T_{11}^{2} + 22T_{11} + 488$$ T11^4 - T11^3 - 44*T11^2 + 22*T11 + 488 $$T_{17}^{4} + 16T_{17}^{3} + 82T_{17}^{2} + 140T_{17} + 32$$ T17^4 + 16*T17^3 + 82*T17^2 + 140*T17 + 32

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 2 T^{3} - 15 T^{2} - 18 T + 56$$
$7$ $$(T + 1)^{4}$$
$11$ $$T^{4} - T^{3} - 44 T^{2} + 22 T + 488$$
$13$ $$(T + 1)^{4}$$
$17$ $$T^{4} + 16 T^{3} + 82 T^{2} + 140 T + 32$$
$19$ $$T^{4} + 10 T^{3} - 23 T^{2} + \cdots - 784$$
$23$ $$T^{4} + 7 T^{3} - T^{2} - 55 T - 64$$
$29$ $$T^{4} + 4 T^{3} - 63 T^{2} - 390 T - 568$$
$31$ $$T^{4} + T^{3} - 27 T^{2} + 29 T - 8$$
$37$ $$T^{4} - 5 T^{3} - 86 T^{2} + 434 T - 508$$
$41$ $$T^{4} + 19 T^{3} + 62 T^{2} + \cdots - 2104$$
$43$ $$T^{4} - 14 T^{3} + 37 T^{2} - 16 T - 16$$
$47$ $$T^{4} - 13 T^{3} - 107 T^{2} + \cdots - 4952$$
$53$ $$T^{4} - 8 T^{3} - 19 T^{2} + 278 T - 544$$
$59$ $$T^{4} - 26 T^{3} + 116 T^{2} + \cdots - 9344$$
$61$ $$T^{4} + T^{3} - 174 T^{2} + 732 T + 104$$
$67$ $$T^{4} - 5 T^{3} - 152 T^{2} + \cdots + 4208$$
$71$ $$T^{4} - 18 T^{3} - 130 T^{2} + \cdots + 2048$$
$73$ $$T^{4} - 7 T^{3} - 9 T^{2} + 117 T - 166$$
$79$ $$T^{4} - 9 T^{3} - 33 T^{2} + 281 T - 112$$
$83$ $$T^{4} + 12 T^{3} - 87 T^{2} + \cdots + 1088$$
$89$ $$T^{4} + 4 T^{3} - 183 T^{2} + \cdots + 5524$$
$97$ $$T^{4} - 25 T^{3} + 91 T^{2} + \cdots + 526$$