# Properties

 Label 189.2.c.a Level $189$ Weight $2$ Character orbit 189.c Analytic conductor $1.509$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(188,189)]

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([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("189.188");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.c (of order $$2$$, degree $$1$$, minimal)

## 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: no 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = \frac{1}{2}(1 + 3\sqrt{-3})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{4} + \beta q^{7}+O(q^{10})$$ q + 2 * q^4 + b * q^7 $$q + 2 q^{4} + \beta q^{7} + ( - 2 \beta + 1) q^{13} + 4 q^{16} + (2 \beta - 1) q^{19} - 5 q^{25} + 2 \beta q^{28} + ( - 4 \beta + 2) q^{31} - 11 q^{37} - 8 q^{43} + (\beta - 7) q^{49} + ( - 4 \beta + 2) q^{52} + (6 \beta - 3) q^{61} + 8 q^{64} + 5 q^{67} + ( - 6 \beta + 3) q^{73} + (4 \beta - 2) q^{76} + 17 q^{79} + ( - \beta + 14) q^{91} + ( - 2 \beta + 1) q^{97}+O(q^{100})$$ q + 2 * q^4 + b * q^7 + (-2*b + 1) * q^13 + 4 * q^16 + (2*b - 1) * q^19 - 5 * q^25 + 2*b * q^28 + (-4*b + 2) * q^31 - 11 * q^37 - 8 * q^43 + (b - 7) * q^49 + (-4*b + 2) * q^52 + (6*b - 3) * q^61 + 8 * q^64 + 5 * q^67 + (-6*b + 3) * q^73 + (4*b - 2) * q^76 + 17 * q^79 + (-b + 14) * q^91 + (-2*b + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + q^{7}+O(q^{10})$$ 2 * q + 4 * q^4 + q^7 $$2 q + 4 q^{4} + q^{7} + 8 q^{16} - 10 q^{25} + 2 q^{28} - 22 q^{37} - 16 q^{43} - 13 q^{49} + 16 q^{64} + 10 q^{67} + 34 q^{79} + 27 q^{91}+O(q^{100})$$ 2 * q + 4 * q^4 + q^7 + 8 * q^16 - 10 * q^25 + 2 * q^28 - 22 * q^37 - 16 * q^43 - 13 * q^49 + 16 * q^64 + 10 * q^67 + 34 * q^79 + 27 * q^91

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
188.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 2.00000 0 0 0.500000 2.59808i 0 0 0
188.2 0 0 2.00000 0 0 0.500000 + 2.59808i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.c.a 2
3.b odd 2 1 CM 189.2.c.a 2
4.b odd 2 1 3024.2.k.b 2
7.b odd 2 1 inner 189.2.c.a 2
9.c even 3 1 567.2.o.a 2
9.c even 3 1 567.2.o.b 2
9.d odd 6 1 567.2.o.a 2
9.d odd 6 1 567.2.o.b 2
12.b even 2 1 3024.2.k.b 2
21.c even 2 1 inner 189.2.c.a 2
28.d even 2 1 3024.2.k.b 2
63.l odd 6 1 567.2.o.a 2
63.l odd 6 1 567.2.o.b 2
63.o even 6 1 567.2.o.a 2
63.o even 6 1 567.2.o.b 2
84.h odd 2 1 3024.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.a 2 1.a even 1 1 trivial
189.2.c.a 2 3.b odd 2 1 CM
189.2.c.a 2 7.b odd 2 1 inner
189.2.c.a 2 21.c even 2 1 inner
567.2.o.a 2 9.c even 3 1
567.2.o.a 2 9.d odd 6 1
567.2.o.a 2 63.l odd 6 1
567.2.o.a 2 63.o even 6 1
567.2.o.b 2 9.c even 3 1
567.2.o.b 2 9.d odd 6 1
567.2.o.b 2 63.l odd 6 1
567.2.o.b 2 63.o even 6 1
3024.2.k.b 2 4.b odd 2 1
3024.2.k.b 2 12.b even 2 1
3024.2.k.b 2 28.d even 2 1
3024.2.k.b 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 27$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 27$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 108$$
$37$ $$(T + 11)^{2}$$
$41$ $$T^{2}$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 243$$
$67$ $$(T - 5)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 243$$
$79$ $$(T - 17)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 27$$