# Properties

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

# Learn more

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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 126) 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)

 $$f(q)$$ $$=$$ $$q + 3 q^{5} - q^{7}+O(q^{10})$$ q + 3 * q^5 - q^7 $$q + 3 q^{5} - q^{7} - 6 q^{11} + 2 q^{13} - 6 q^{17} + 7 q^{19} + 3 q^{23} + 4 q^{25} - 6 q^{29} - 2 q^{31} - 3 q^{35} + 2 q^{37} - 2 q^{43} + q^{49} - 6 q^{53} - 18 q^{55} + 5 q^{61} + 6 q^{65} - 8 q^{67} + 3 q^{71} + 2 q^{73} + 6 q^{77} - 5 q^{79} + 12 q^{83} - 18 q^{85} - 2 q^{91} + 21 q^{95} + 2 q^{97}+O(q^{100})$$ q + 3 * q^5 - q^7 - 6 * q^11 + 2 * q^13 - 6 * q^17 + 7 * q^19 + 3 * q^23 + 4 * q^25 - 6 * q^29 - 2 * q^31 - 3 * q^35 + 2 * q^37 - 2 * q^43 + q^49 - 6 * q^53 - 18 * q^55 + 5 * q^61 + 6 * q^65 - 8 * q^67 + 3 * q^71 + 2 * q^73 + 6 * q^77 - 5 * q^79 + 12 * q^83 - 18 * q^85 - 2 * q^91 + 21 * q^95 + 2 * 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
0 0 0 3.00000 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.w 1
3.b odd 2 1 9072.2.a.c 1
4.b odd 2 1 1134.2.a.h 1
9.c even 3 2 3024.2.r.a 2
9.d odd 6 2 1008.2.r.d 2
12.b even 2 1 1134.2.a.a 1
28.d even 2 1 7938.2.a.u 1
36.f odd 6 2 378.2.f.a 2
36.h even 6 2 126.2.f.a 2
84.h odd 2 1 7938.2.a.l 1
252.n even 6 2 2646.2.h.a 2
252.o even 6 2 882.2.h.j 2
252.r odd 6 2 882.2.e.d 2
252.s odd 6 2 882.2.f.h 2
252.u odd 6 2 2646.2.e.f 2
252.bb even 6 2 882.2.e.b 2
252.bi even 6 2 2646.2.f.c 2
252.bj even 6 2 2646.2.e.j 2
252.bl odd 6 2 2646.2.h.e 2
252.bn odd 6 2 882.2.h.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.a 2 36.h even 6 2
378.2.f.a 2 36.f odd 6 2
882.2.e.b 2 252.bb even 6 2
882.2.e.d 2 252.r odd 6 2
882.2.f.h 2 252.s odd 6 2
882.2.h.f 2 252.bn odd 6 2
882.2.h.j 2 252.o even 6 2
1008.2.r.d 2 9.d odd 6 2
1134.2.a.a 1 12.b even 2 1
1134.2.a.h 1 4.b odd 2 1
2646.2.e.f 2 252.u odd 6 2
2646.2.e.j 2 252.bj even 6 2
2646.2.f.c 2 252.bi even 6 2
2646.2.h.a 2 252.n even 6 2
2646.2.h.e 2 252.bl odd 6 2
3024.2.r.a 2 9.c even 3 2
7938.2.a.l 1 84.h odd 2 1
7938.2.a.u 1 28.d even 2 1
9072.2.a.c 1 3.b odd 2 1
9072.2.a.w 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(9072))$$:

 $$T_{5} - 3$$ T5 - 3 $$T_{11} + 6$$ T11 + 6 $$T_{13} - 2$$ T13 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 3$$
$7$ $$T + 1$$
$11$ $$T + 6$$
$13$ $$T - 2$$
$17$ $$T + 6$$
$19$ $$T - 7$$
$23$ $$T - 3$$
$29$ $$T + 6$$
$31$ $$T + 2$$
$37$ $$T - 2$$
$41$ $$T$$
$43$ $$T + 2$$
$47$ $$T$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T - 5$$
$67$ $$T + 8$$
$71$ $$T - 3$$
$73$ $$T - 2$$
$79$ $$T + 5$$
$83$ $$T - 12$$
$89$ $$T$$
$97$ $$T - 2$$
show more
show less