# Properties

 Label 9072.2.a.y Level $9072$ Weight $2$ Character orbit 9072.a Self dual yes Analytic conductor $72.440$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9072 = 2^{4} \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9072.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: $$72.4402847137$$ Analytic rank: $$0$$ 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 1134) 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.

 $$f(q)$$ $$=$$ $$q + (\beta - 2) q^{5} + q^{7}+O(q^{10})$$ q + (b - 2) * q^5 + q^7 $$q + (\beta - 2) q^{5} + q^{7} + (3 \beta + 1) q^{11} + ( - 2 \beta - 3) q^{13} - 7 q^{17} + ( - \beta + 1) q^{19} + ( - 3 \beta + 1) q^{23} + ( - 4 \beta + 2) q^{25} + (2 \beta - 5) q^{29} + ( - 3 \beta - 3) q^{31} + (\beta - 2) q^{35} + (5 \beta + 2) q^{37} + (2 \beta - 6) q^{41} + (2 \beta - 2) q^{43} + (\beta + 3) q^{47} + q^{49} + (2 \beta + 6) q^{53} + ( - 5 \beta + 7) q^{55} + (3 \beta - 1) q^{59} + (4 \beta - 3) q^{61} + \beta q^{65} + (\beta + 5) q^{67} + ( - 2 \beta + 10) q^{71} + ( - \beta + 10) q^{73} + (3 \beta + 1) q^{77} + (7 \beta - 3) q^{79} + (9 \beta + 1) q^{83} + ( - 7 \beta + 14) q^{85} + ( - 4 \beta - 3) q^{89} + ( - 2 \beta - 3) q^{91} + (3 \beta - 5) q^{95} + (4 \beta + 4) q^{97}+O(q^{100})$$ q + (b - 2) * q^5 + q^7 + (3*b + 1) * q^11 + (-2*b - 3) * q^13 - 7 * q^17 + (-b + 1) * q^19 + (-3*b + 1) * q^23 + (-4*b + 2) * q^25 + (2*b - 5) * q^29 + (-3*b - 3) * q^31 + (b - 2) * q^35 + (5*b + 2) * q^37 + (2*b - 6) * q^41 + (2*b - 2) * q^43 + (b + 3) * q^47 + q^49 + (2*b + 6) * q^53 + (-5*b + 7) * q^55 + (3*b - 1) * q^59 + (4*b - 3) * q^61 + b * q^65 + (b + 5) * q^67 + (-2*b + 10) * q^71 + (-b + 10) * q^73 + (3*b + 1) * q^77 + (7*b - 3) * q^79 + (9*b + 1) * q^83 + (-7*b + 14) * q^85 + (-4*b - 3) * q^89 + (-2*b - 3) * q^91 + (3*b - 5) * q^95 + (4*b + 4) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{5} + 2 q^{7}+O(q^{10})$$ 2 * q - 4 * q^5 + 2 * q^7 $$2 q - 4 q^{5} + 2 q^{7} + 2 q^{11} - 6 q^{13} - 14 q^{17} + 2 q^{19} + 2 q^{23} + 4 q^{25} - 10 q^{29} - 6 q^{31} - 4 q^{35} + 4 q^{37} - 12 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} + 12 q^{53} + 14 q^{55} - 2 q^{59} - 6 q^{61} + 10 q^{67} + 20 q^{71} + 20 q^{73} + 2 q^{77} - 6 q^{79} + 2 q^{83} + 28 q^{85} - 6 q^{89} - 6 q^{91} - 10 q^{95} + 8 q^{97}+O(q^{100})$$ 2 * q - 4 * q^5 + 2 * q^7 + 2 * q^11 - 6 * q^13 - 14 * q^17 + 2 * q^19 + 2 * q^23 + 4 * q^25 - 10 * q^29 - 6 * q^31 - 4 * q^35 + 4 * q^37 - 12 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 + 12 * q^53 + 14 * q^55 - 2 * q^59 - 6 * q^61 + 10 * q^67 + 20 * q^71 + 20 * q^73 + 2 * q^77 - 6 * q^79 + 2 * q^83 + 28 * q^85 - 6 * q^89 - 6 * q^91 - 10 * 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 −3.73205 0 1.00000 0 0 0
1.2 0 0 0 −0.267949 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

## 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 9072.2.a.y 2
3.b odd 2 1 9072.2.a.bp 2
4.b odd 2 1 1134.2.a.m yes 2
12.b even 2 1 1134.2.a.l 2
28.d even 2 1 7938.2.a.bt 2
36.f odd 6 2 1134.2.f.r 4
36.h even 6 2 1134.2.f.s 4
84.h odd 2 1 7938.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.a.l 2 12.b even 2 1
1134.2.a.m yes 2 4.b odd 2 1
1134.2.f.r 4 36.f odd 6 2
1134.2.f.s 4 36.h even 6 2
7938.2.a.bg 2 84.h odd 2 1
7938.2.a.bt 2 28.d even 2 1
9072.2.a.y 2 1.a even 1 1 trivial
9072.2.a.bp 2 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5}^{2} + 4T_{5} + 1$$ T5^2 + 4*T5 + 1 $$T_{11}^{2} - 2T_{11} - 26$$ T11^2 - 2*T11 - 26 $$T_{13}^{2} + 6T_{13} - 3$$ T13^2 + 6*T13 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4T + 1$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 2T - 26$$
$13$ $$T^{2} + 6T - 3$$
$17$ $$(T + 7)^{2}$$
$19$ $$T^{2} - 2T - 2$$
$23$ $$T^{2} - 2T - 26$$
$29$ $$T^{2} + 10T + 13$$
$31$ $$T^{2} + 6T - 18$$
$37$ $$T^{2} - 4T - 71$$
$41$ $$T^{2} + 12T + 24$$
$43$ $$T^{2} + 4T - 8$$
$47$ $$T^{2} - 6T + 6$$
$53$ $$T^{2} - 12T + 24$$
$59$ $$T^{2} + 2T - 26$$
$61$ $$T^{2} + 6T - 39$$
$67$ $$T^{2} - 10T + 22$$
$71$ $$T^{2} - 20T + 88$$
$73$ $$T^{2} - 20T + 97$$
$79$ $$T^{2} + 6T - 138$$
$83$ $$T^{2} - 2T - 242$$
$89$ $$T^{2} + 6T - 39$$
$97$ $$T^{2} - 8T - 32$$
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