# Properties

 Label 9072.2.a.ca Level $9072$ Weight $2$ Character orbit 9072.a Self dual yes Analytic conductor $72.440$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

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: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{5} - q^{7}+O(q^{10})$$ q + (-b1 + 1) * q^5 - q^7 $$q + ( - \beta_1 + 1) q^{5} - q^{7} + ( - \beta_1 - 2) q^{11} + (2 \beta_{2} - 4 \beta_1 - 1) q^{13} + (\beta_{2} - \beta_1 + 2) q^{17} + ( - \beta_{2} - \beta_1 + 1) q^{19} + (3 \beta_{2} - 2 \beta_1 - 4) q^{23} + (\beta_{2} - 2 \beta_1 - 2) q^{25} + ( - \beta_{2} - 4 \beta_1 + 3) q^{29} + (6 \beta_{2} - 3 \beta_1 + 1) q^{31} + (\beta_1 - 1) q^{35} + ( - 6 \beta_{2} + 3 \beta_1 - 1) q^{37} + ( - 2 \beta_{2} + \beta_1) q^{41} + ( - 2 \beta_{2} + \beta_1 + 1) q^{43} + (5 \beta_{2} - 3 \beta_1 - 1) q^{47} + q^{49} + (\beta_{2} + 2 \beta_1 + 2) q^{53} + (\beta_{2} + \beta_1) q^{55} + ( - 5 \beta_{2} + 1) q^{59} + ( - 3 \beta_{2} + 2) q^{61} + (6 \beta_{2} - 5 \beta_1 + 5) q^{65} + (3 \beta_{2} - 3 \beta_1 + 4) q^{67} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{71} + ( - 5 \beta_{2} + \beta_1 - 7) q^{73} + (\beta_1 + 2) q^{77} + (3 \beta_{2} - 3 \beta_1 + 7) q^{79} + ( - 5 \beta_{2} + \beta_1 + 6) q^{83} + (2 \beta_{2} - 4 \beta_1 + 3) q^{85} + ( - 4 \beta_{2} + 7 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} + 4 \beta_1 + 1) q^{91} + ( - \beta_1 + 4) q^{95} + ( - \beta_{2} + 8 \beta_1 - 1) q^{97}+O(q^{100})$$ q + (-b1 + 1) * q^5 - q^7 + (-b1 - 2) * q^11 + (2*b2 - 4*b1 - 1) * q^13 + (b2 - b1 + 2) * q^17 + (-b2 - b1 + 1) * q^19 + (3*b2 - 2*b1 - 4) * q^23 + (b2 - 2*b1 - 2) * q^25 + (-b2 - 4*b1 + 3) * q^29 + (6*b2 - 3*b1 + 1) * q^31 + (b1 - 1) * q^35 + (-6*b2 + 3*b1 - 1) * q^37 + (-2*b2 + b1) * q^41 + (-2*b2 + b1 + 1) * q^43 + (5*b2 - 3*b1 - 1) * q^47 + q^49 + (b2 + 2*b1 + 2) * q^53 + (b2 + b1) * q^55 + (-5*b2 + 1) * q^59 + (-3*b2 + 2) * q^61 + (6*b2 - 5*b1 + 5) * q^65 + (3*b2 - 3*b1 + 4) * q^67 + (-3*b2 + 6*b1 - 3) * q^71 + (-5*b2 + b1 - 7) * q^73 + (b1 + 2) * q^77 + (3*b2 - 3*b1 + 7) * q^79 + (-5*b2 + b1 + 6) * q^83 + (2*b2 - 4*b1 + 3) * q^85 + (-4*b2 + 7*b1 + 4) * q^89 + (-2*b2 + 4*b1 + 1) * q^91 + (-b1 + 4) * q^95 + (-b2 + 8*b1 - 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{5} - 3 q^{7}+O(q^{10})$$ 3 * q + 3 * q^5 - 3 * q^7 $$3 q + 3 q^{5} - 3 q^{7} - 6 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 12 q^{23} - 6 q^{25} + 9 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 3 q^{43} - 3 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} + 6 q^{61} + 15 q^{65} + 12 q^{67} - 9 q^{71} - 21 q^{73} + 6 q^{77} + 21 q^{79} + 18 q^{83} + 9 q^{85} + 12 q^{89} + 3 q^{91} + 12 q^{95} - 3 q^{97}+O(q^{100})$$ 3 * q + 3 * q^5 - 3 * q^7 - 6 * q^11 - 3 * q^13 + 6 * q^17 + 3 * q^19 - 12 * q^23 - 6 * q^25 + 9 * q^29 + 3 * q^31 - 3 * q^35 - 3 * q^37 + 3 * q^43 - 3 * q^47 + 3 * q^49 + 6 * q^53 + 3 * q^59 + 6 * q^61 + 15 * q^65 + 12 * q^67 - 9 * q^71 - 21 * q^73 + 6 * q^77 + 21 * q^79 + 18 * q^83 + 9 * q^85 + 12 * q^89 + 3 * q^91 + 12 * q^95 - 3 * q^97

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## 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
 1.87939 −0.347296 −1.53209
0 0 0 −0.879385 0 −1.00000 0 0 0
1.2 0 0 0 1.34730 0 −1.00000 0 0 0
1.3 0 0 0 2.53209 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.ca 3
3.b odd 2 1 9072.2.a.bs 3
4.b odd 2 1 567.2.a.h 3
9.c even 3 2 1008.2.r.h 6
9.d odd 6 2 3024.2.r.k 6
12.b even 2 1 567.2.a.c 3
28.d even 2 1 3969.2.a.q 3
36.f odd 6 2 63.2.f.a 6
36.h even 6 2 189.2.f.b 6
84.h odd 2 1 3969.2.a.l 3
252.n even 6 2 441.2.g.b 6
252.o even 6 2 1323.2.g.d 6
252.r odd 6 2 1323.2.h.b 6
252.s odd 6 2 1323.2.f.d 6
252.u odd 6 2 441.2.h.d 6
252.bb even 6 2 1323.2.h.c 6
252.bi even 6 2 441.2.f.c 6
252.bj even 6 2 441.2.h.e 6
252.bl odd 6 2 441.2.g.c 6
252.bn odd 6 2 1323.2.g.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 36.f odd 6 2
189.2.f.b 6 36.h even 6 2
441.2.f.c 6 252.bi even 6 2
441.2.g.b 6 252.n even 6 2
441.2.g.c 6 252.bl odd 6 2
441.2.h.d 6 252.u odd 6 2
441.2.h.e 6 252.bj even 6 2
567.2.a.c 3 12.b even 2 1
567.2.a.h 3 4.b odd 2 1
1008.2.r.h 6 9.c even 3 2
1323.2.f.d 6 252.s odd 6 2
1323.2.g.d 6 252.o even 6 2
1323.2.g.e 6 252.bn odd 6 2
1323.2.h.b 6 252.r odd 6 2
1323.2.h.c 6 252.bb even 6 2
3024.2.r.k 6 9.d odd 6 2
3969.2.a.l 3 84.h odd 2 1
3969.2.a.q 3 28.d even 2 1
9072.2.a.bs 3 3.b odd 2 1
9072.2.a.ca 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5}^{3} - 3T_{5}^{2} + 3$$ T5^3 - 3*T5^2 + 3 $$T_{11}^{3} + 6T_{11}^{2} + 9T_{11} + 3$$ T11^3 + 6*T11^2 + 9*T11 + 3 $$T_{13}^{3} + 3T_{13}^{2} - 33T_{13} - 107$$ T13^3 + 3*T13^2 - 33*T13 - 107

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$T^{3}$$
$5$ $$T^{3} - 3T^{2} + 3$$
$7$ $$(T + 1)^{3}$$
$11$ $$T^{3} + 6 T^{2} + \cdots + 3$$
$13$ $$T^{3} + 3 T^{2} + \cdots - 107$$
$17$ $$T^{3} - 6 T^{2} + \cdots - 3$$
$19$ $$T^{3} - 3 T^{2} + \cdots + 17$$
$23$ $$T^{3} + 12 T^{2} + \cdots - 3$$
$29$ $$T^{3} - 9 T^{2} + \cdots + 333$$
$31$ $$T^{3} - 3 T^{2} + \cdots + 323$$
$37$ $$T^{3} + 3 T^{2} + \cdots - 323$$
$41$ $$T^{3} - 9T - 9$$
$43$ $$T^{3} - 3 T^{2} + \cdots - 1$$
$47$ $$T^{3} + 3 T^{2} + \cdots + 51$$
$53$ $$T^{3} - 6 T^{2} + \cdots - 3$$
$59$ $$T^{3} - 3 T^{2} + \cdots - 51$$
$61$ $$T^{3} - 6 T^{2} + \cdots + 19$$
$67$ $$T^{3} - 12 T^{2} + \cdots + 17$$
$71$ $$T^{3} + 9 T^{2} + \cdots + 27$$
$73$ $$T^{3} + 21 T^{2} + \cdots - 269$$
$79$ $$T^{3} - 21 T^{2} + \cdots - 181$$
$83$ $$T^{3} - 18 T^{2} + \cdots - 9$$
$89$ $$T^{3} - 12 T^{2} + \cdots + 813$$
$97$ $$T^{3} + 3 T^{2} + \cdots - 323$$