# Properties

 Label 189.2.p.a Level $189$ Weight $2$ Character orbit 189.p 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(26,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("189.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.p (of order $$6$$, degree $$2$$, 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{4} + (3 \zeta_{6} - 1) q^{7}+O(q^{10})$$ q + (2*z - 2) * q^4 + (3*z - 1) * q^7 $$q + (2 \zeta_{6} - 2) q^{4} + (3 \zeta_{6} - 1) q^{7} + (8 \zeta_{6} - 4) q^{13} - 4 \zeta_{6} q^{16} + ( - 3 \zeta_{6} + 6) q^{19} + ( - 5 \zeta_{6} + 5) q^{25} + ( - 2 \zeta_{6} - 4) q^{28} + ( - \zeta_{6} - 1) q^{31} - 10 \zeta_{6} q^{37} + 13 q^{43} + (3 \zeta_{6} - 8) q^{49} + ( - 8 \zeta_{6} - 8) q^{52} + (5 \zeta_{6} - 10) q^{61} + 8 q^{64} + ( - 16 \zeta_{6} + 16) q^{67} + (9 \zeta_{6} + 9) q^{73} + (12 \zeta_{6} - 6) q^{76} + 4 \zeta_{6} q^{79} + (4 \zeta_{6} - 20) q^{91} + (22 \zeta_{6} - 11) q^{97} +O(q^{100})$$ q + (2*z - 2) * q^4 + (3*z - 1) * q^7 + (8*z - 4) * q^13 - 4*z * q^16 + (-3*z + 6) * q^19 + (-5*z + 5) * q^25 + (-2*z - 4) * q^28 + (-z - 1) * q^31 - 10*z * q^37 + 13 * q^43 + (3*z - 8) * q^49 + (-8*z - 8) * q^52 + (5*z - 10) * q^61 + 8 * q^64 + (-16*z + 16) * q^67 + (9*z + 9) * q^73 + (12*z - 6) * q^76 + 4*z * q^79 + (4*z - 20) * q^91 + (22*z - 11) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + q^{7}+O(q^{10})$$ 2 * q - 2 * q^4 + q^7 $$2 q - 2 q^{4} + q^{7} - 4 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 3 q^{31} - 10 q^{37} + 26 q^{43} - 13 q^{49} - 24 q^{52} - 15 q^{61} + 16 q^{64} + 16 q^{67} + 27 q^{73} + 4 q^{79} - 36 q^{91}+O(q^{100})$$ 2 * q - 2 * q^4 + q^7 - 4 * q^16 + 9 * q^19 + 5 * q^25 - 10 * q^28 - 3 * q^31 - 10 * q^37 + 26 * q^43 - 13 * q^49 - 24 * q^52 - 15 * q^61 + 16 * q^64 + 16 * q^67 + 27 * q^73 + 4 * q^79 - 36 * q^91

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1$$ $$\zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 −1.00000 1.73205i 0 0 0.500000 2.59808i 0 0 0
80.1 0 0 −1.00000 + 1.73205i 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.p.a 2
3.b odd 2 1 CM 189.2.p.a 2
7.c even 3 1 1323.2.c.a 2
7.d odd 6 1 inner 189.2.p.a 2
7.d odd 6 1 1323.2.c.a 2
9.c even 3 1 567.2.i.a 2
9.c even 3 1 567.2.s.b 2
9.d odd 6 1 567.2.i.a 2
9.d odd 6 1 567.2.s.b 2
21.g even 6 1 inner 189.2.p.a 2
21.g even 6 1 1323.2.c.a 2
21.h odd 6 1 1323.2.c.a 2
63.i even 6 1 567.2.s.b 2
63.k odd 6 1 567.2.i.a 2
63.s even 6 1 567.2.i.a 2
63.t odd 6 1 567.2.s.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.a 2 1.a even 1 1 trivial
189.2.p.a 2 3.b odd 2 1 CM
189.2.p.a 2 7.d odd 6 1 inner
189.2.p.a 2 21.g even 6 1 inner
567.2.i.a 2 9.c even 3 1
567.2.i.a 2 9.d odd 6 1
567.2.i.a 2 63.k odd 6 1
567.2.i.a 2 63.s even 6 1
567.2.s.b 2 9.c even 3 1
567.2.s.b 2 9.d odd 6 1
567.2.s.b 2 63.i even 6 1
567.2.s.b 2 63.t odd 6 1
1323.2.c.a 2 7.c even 3 1
1323.2.c.a 2 7.d odd 6 1
1323.2.c.a 2 21.g even 6 1
1323.2.c.a 2 21.h odd 6 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 48$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 9T + 27$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} + 3T + 3$$
$37$ $$T^{2} + 10T + 100$$
$41$ $$T^{2}$$
$43$ $$(T - 13)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 15T + 75$$
$67$ $$T^{2} - 16T + 256$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 27T + 243$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 363$$