# Properties

 Label 189.2.a.e Level $189$ Weight $2$ Character orbit 189.a Self dual yes Analytic conductor $1.509$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(1,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, 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("189.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.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: $$1.50917259820$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + q^{4} + \beta q^{5} + q^{7} - \beta q^{8} +O(q^{10})$$ q + b * q^2 + q^4 + b * q^5 + q^7 - b * q^8 $$q + \beta q^{2} + q^{4} + \beta q^{5} + q^{7} - \beta q^{8} + 3 q^{10} - \beta q^{11} + 2 q^{13} + \beta q^{14} - 5 q^{16} - 4 \beta q^{17} + 5 q^{19} + \beta q^{20} - 3 q^{22} + \beta q^{23} - 2 q^{25} + 2 \beta q^{26} + q^{28} - 6 \beta q^{29} + 5 q^{31} - 3 \beta q^{32} - 12 q^{34} + \beta q^{35} - 7 q^{37} + 5 \beta q^{38} - 3 q^{40} + 3 \beta q^{41} - 4 q^{43} - \beta q^{44} + 3 q^{46} + 4 \beta q^{47} + q^{49} - 2 \beta q^{50} + 2 q^{52} + 8 \beta q^{53} - 3 q^{55} - \beta q^{56} - 18 q^{58} - 4 \beta q^{59} + 8 q^{61} + 5 \beta q^{62} + q^{64} + 2 \beta q^{65} + 14 q^{67} - 4 \beta q^{68} + 3 q^{70} + 3 \beta q^{71} - 4 q^{73} - 7 \beta q^{74} + 5 q^{76} - \beta q^{77} + 8 q^{79} - 5 \beta q^{80} + 9 q^{82} - 6 \beta q^{83} - 12 q^{85} - 4 \beta q^{86} + 3 q^{88} - 5 \beta q^{89} + 2 q^{91} + \beta q^{92} + 12 q^{94} + 5 \beta q^{95} - 4 q^{97} + \beta q^{98} +O(q^{100})$$ q + b * q^2 + q^4 + b * q^5 + q^7 - b * q^8 + 3 * q^10 - b * q^11 + 2 * q^13 + b * q^14 - 5 * q^16 - 4*b * q^17 + 5 * q^19 + b * q^20 - 3 * q^22 + b * q^23 - 2 * q^25 + 2*b * q^26 + q^28 - 6*b * q^29 + 5 * q^31 - 3*b * q^32 - 12 * q^34 + b * q^35 - 7 * q^37 + 5*b * q^38 - 3 * q^40 + 3*b * q^41 - 4 * q^43 - b * q^44 + 3 * q^46 + 4*b * q^47 + q^49 - 2*b * q^50 + 2 * q^52 + 8*b * q^53 - 3 * q^55 - b * q^56 - 18 * q^58 - 4*b * q^59 + 8 * q^61 + 5*b * q^62 + q^64 + 2*b * q^65 + 14 * q^67 - 4*b * q^68 + 3 * q^70 + 3*b * q^71 - 4 * q^73 - 7*b * q^74 + 5 * q^76 - b * q^77 + 8 * q^79 - 5*b * q^80 + 9 * q^82 - 6*b * q^83 - 12 * q^85 - 4*b * q^86 + 3 * q^88 - 5*b * q^89 + 2 * q^91 + b * q^92 + 12 * q^94 + 5*b * q^95 - 4 * q^97 + b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{7}+O(q^{10})$$ 2 * q + 2 * q^4 + 2 * q^7 $$2 q + 2 q^{4} + 2 q^{7} + 6 q^{10} + 4 q^{13} - 10 q^{16} + 10 q^{19} - 6 q^{22} - 4 q^{25} + 2 q^{28} + 10 q^{31} - 24 q^{34} - 14 q^{37} - 6 q^{40} - 8 q^{43} + 6 q^{46} + 2 q^{49} + 4 q^{52} - 6 q^{55} - 36 q^{58} + 16 q^{61} + 2 q^{64} + 28 q^{67} + 6 q^{70} - 8 q^{73} + 10 q^{76} + 16 q^{79} + 18 q^{82} - 24 q^{85} + 6 q^{88} + 4 q^{91} + 24 q^{94} - 8 q^{97}+O(q^{100})$$ 2 * q + 2 * q^4 + 2 * q^7 + 6 * q^10 + 4 * q^13 - 10 * q^16 + 10 * q^19 - 6 * q^22 - 4 * q^25 + 2 * q^28 + 10 * q^31 - 24 * q^34 - 14 * q^37 - 6 * q^40 - 8 * q^43 + 6 * q^46 + 2 * q^49 + 4 * q^52 - 6 * q^55 - 36 * q^58 + 16 * q^61 + 2 * q^64 + 28 * q^67 + 6 * q^70 - 8 * q^73 + 10 * q^76 + 16 * q^79 + 18 * q^82 - 24 * q^85 + 6 * q^88 + 4 * q^91 + 24 * q^94 - 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
−1.73205 0 1.00000 −1.73205 0 1.00000 1.73205 0 3.00000
1.2 1.73205 0 1.00000 1.73205 0 1.00000 −1.73205 0 3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.a.e 2
3.b odd 2 1 inner 189.2.a.e 2
4.b odd 2 1 3024.2.a.bg 2
5.b even 2 1 4725.2.a.ba 2
7.b odd 2 1 1323.2.a.t 2
9.c even 3 2 567.2.f.k 4
9.d odd 6 2 567.2.f.k 4
12.b even 2 1 3024.2.a.bg 2
15.d odd 2 1 4725.2.a.ba 2
21.c even 2 1 1323.2.a.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.e 2 1.a even 1 1 trivial
189.2.a.e 2 3.b odd 2 1 inner
567.2.f.k 4 9.c even 3 2
567.2.f.k 4 9.d odd 6 2
1323.2.a.t 2 7.b odd 2 1
1323.2.a.t 2 21.c even 2 1
3024.2.a.bg 2 4.b odd 2 1
3024.2.a.bg 2 12.b even 2 1
4725.2.a.ba 2 5.b even 2 1
4725.2.a.ba 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(189))$$:

 $$T_{2}^{2} - 3$$ T2^2 - 3 $$T_{5}^{2} - 3$$ T5^2 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 3$$
$7$ $$(T - 1)^{2}$$
$11$ $$T^{2} - 3$$
$13$ $$(T - 2)^{2}$$
$17$ $$T^{2} - 48$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} - 3$$
$29$ $$T^{2} - 108$$
$31$ $$(T - 5)^{2}$$
$37$ $$(T + 7)^{2}$$
$41$ $$T^{2} - 27$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} - 48$$
$53$ $$T^{2} - 192$$
$59$ $$T^{2} - 48$$
$61$ $$(T - 8)^{2}$$
$67$ $$(T - 14)^{2}$$
$71$ $$T^{2} - 27$$
$73$ $$(T + 4)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} - 108$$
$89$ $$T^{2} - 75$$
$97$ $$(T + 4)^{2}$$