# Properties

 Label 1089.2.a.k Level $1089$ Weight $2$ Character orbit 1089.a Self dual yes Analytic conductor $8.696$ 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] = [1089,2,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1089 = 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1089.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: $$8.69570878012$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 363) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7}+O(q^{10})$$ q + 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 $$q + 2 q^{2} + 2 q^{4} - 4 q^{5} + q^{7} - 8 q^{10} - 2 q^{13} + 2 q^{14} - 4 q^{16} - 4 q^{17} - 3 q^{19} - 8 q^{20} - 2 q^{23} + 11 q^{25} - 4 q^{26} + 2 q^{28} - 6 q^{29} - 5 q^{31} - 8 q^{32} - 8 q^{34} - 4 q^{35} + 3 q^{37} - 6 q^{38} + 2 q^{41} + 12 q^{43} - 4 q^{46} - 2 q^{47} - 6 q^{49} + 22 q^{50} - 4 q^{52} - 6 q^{53} - 12 q^{58} + 10 q^{59} + 3 q^{61} - 10 q^{62} - 8 q^{64} + 8 q^{65} - q^{67} - 8 q^{68} - 8 q^{70} - 11 q^{73} + 6 q^{74} - 6 q^{76} + 11 q^{79} + 16 q^{80} + 4 q^{82} - 6 q^{83} + 16 q^{85} + 24 q^{86} - 12 q^{89} - 2 q^{91} - 4 q^{92} - 4 q^{94} + 12 q^{95} + 5 q^{97} - 12 q^{98}+O(q^{100})$$ q + 2 * q^2 + 2 * q^4 - 4 * q^5 + q^7 - 8 * q^10 - 2 * q^13 + 2 * q^14 - 4 * q^16 - 4 * q^17 - 3 * q^19 - 8 * q^20 - 2 * q^23 + 11 * q^25 - 4 * q^26 + 2 * q^28 - 6 * q^29 - 5 * q^31 - 8 * q^32 - 8 * q^34 - 4 * q^35 + 3 * q^37 - 6 * q^38 + 2 * q^41 + 12 * q^43 - 4 * q^46 - 2 * q^47 - 6 * q^49 + 22 * q^50 - 4 * q^52 - 6 * q^53 - 12 * q^58 + 10 * q^59 + 3 * q^61 - 10 * q^62 - 8 * q^64 + 8 * q^65 - q^67 - 8 * q^68 - 8 * q^70 - 11 * q^73 + 6 * q^74 - 6 * q^76 + 11 * q^79 + 16 * q^80 + 4 * q^82 - 6 * q^83 + 16 * q^85 + 24 * q^86 - 12 * q^89 - 2 * q^91 - 4 * q^92 - 4 * q^94 + 12 * q^95 + 5 * q^97 - 12 * q^98

## 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
2.00000 0 2.00000 −4.00000 0 1.00000 0 0 −8.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 1089.2.a.k 1
3.b odd 2 1 363.2.a.a 1
11.b odd 2 1 1089.2.a.a 1
12.b even 2 1 5808.2.a.bh 1
15.d odd 2 1 9075.2.a.t 1
33.d even 2 1 363.2.a.c yes 1
33.f even 10 4 363.2.e.d 4
33.h odd 10 4 363.2.e.i 4
132.d odd 2 1 5808.2.a.bi 1
165.d even 2 1 9075.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.a 1 3.b odd 2 1
363.2.a.c yes 1 33.d even 2 1
363.2.e.d 4 33.f even 10 4
363.2.e.i 4 33.h odd 10 4
1089.2.a.a 1 11.b odd 2 1
1089.2.a.k 1 1.a even 1 1 trivial
5808.2.a.bh 1 12.b even 2 1
5808.2.a.bi 1 132.d odd 2 1
9075.2.a.b 1 165.d even 2 1
9075.2.a.t 1 15.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1089))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{5} + 4$$ T5 + 4 $$T_{7} - 1$$ T7 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 4$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T + 2$$
$17$ $$T + 4$$
$19$ $$T + 3$$
$23$ $$T + 2$$
$29$ $$T + 6$$
$31$ $$T + 5$$
$37$ $$T - 3$$
$41$ $$T - 2$$
$43$ $$T - 12$$
$47$ $$T + 2$$
$53$ $$T + 6$$
$59$ $$T - 10$$
$61$ $$T - 3$$
$67$ $$T + 1$$
$71$ $$T$$
$73$ $$T + 11$$
$79$ $$T - 11$$
$83$ $$T + 6$$
$89$ $$T + 12$$
$97$ $$T - 5$$
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