# Properties

 Label 189.4.e.c Level $189$ Weight $4$ Character orbit 189.e Analytic conductor $11.151$ 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] = [189,4,Mod(109,189)]

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([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 189.e (of order $$3$$, 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: $$11.1513609911$$ 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: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 8 \zeta_{6} + 8) q^{4} + (18 \zeta_{6} + 1) q^{7}+O(q^{10})$$ q + (-8*z + 8) * q^4 + (18*z + 1) * q^7 $$q + ( - 8 \zeta_{6} + 8) q^{4} + (18 \zeta_{6} + 1) q^{7} + 89 q^{13} - 64 \zeta_{6} q^{16} - 56 \zeta_{6} q^{19} + ( - 125 \zeta_{6} + 125) q^{25} + ( - 8 \zeta_{6} + 152) q^{28} + ( - 289 \zeta_{6} + 289) q^{31} + 433 \zeta_{6} q^{37} + 71 q^{43} + (360 \zeta_{6} - 323) q^{49} + ( - 712 \zeta_{6} + 712) q^{52} - 719 \zeta_{6} q^{61} - 512 q^{64} + (1007 \zeta_{6} - 1007) q^{67} + (1190 \zeta_{6} - 1190) q^{73} - 448 q^{76} - 503 \zeta_{6} q^{79} + (1602 \zeta_{6} + 89) q^{91} - 523 q^{97} +O(q^{100})$$ q + (-8*z + 8) * q^4 + (18*z + 1) * q^7 + 89 * q^13 - 64*z * q^16 - 56*z * q^19 + (-125*z + 125) * q^25 + (-8*z + 152) * q^28 + (-289*z + 289) * q^31 + 433*z * q^37 + 71 * q^43 + (360*z - 323) * q^49 + (-712*z + 712) * q^52 - 719*z * q^61 - 512 * q^64 + (1007*z - 1007) * q^67 + (1190*z - 1190) * q^73 - 448 * q^76 - 503*z * q^79 + (1602*z + 89) * q^91 - 523 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 8 q^{4} + 20 q^{7}+O(q^{10})$$ 2 * q + 8 * q^4 + 20 * q^7 $$2 q + 8 q^{4} + 20 q^{7} + 178 q^{13} - 64 q^{16} - 56 q^{19} + 125 q^{25} + 296 q^{28} + 289 q^{31} + 433 q^{37} + 142 q^{43} - 286 q^{49} + 712 q^{52} - 719 q^{61} - 1024 q^{64} - 1007 q^{67} - 1190 q^{73} - 896 q^{76} - 503 q^{79} + 1780 q^{91} - 1046 q^{97}+O(q^{100})$$ 2 * q + 8 * q^4 + 20 * q^7 + 178 * q^13 - 64 * q^16 - 56 * q^19 + 125 * q^25 + 296 * q^28 + 289 * q^31 + 433 * q^37 + 142 * q^43 - 286 * q^49 + 712 * q^52 - 719 * q^61 - 1024 * q^64 - 1007 * q^67 - 1190 * q^73 - 896 * q^76 - 503 * q^79 + 1780 * q^91 - 1046 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 4.00000 6.92820i 0 0 10.0000 + 15.5885i 0 0 0
163.1 0 0 4.00000 + 6.92820i 0 0 10.0000 15.5885i 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.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.e.c 2
3.b odd 2 1 CM 189.4.e.c 2
7.c even 3 1 inner 189.4.e.c 2
7.c even 3 1 1323.4.a.i 1
7.d odd 6 1 1323.4.a.f 1
21.g even 6 1 1323.4.a.f 1
21.h odd 6 1 inner 189.4.e.c 2
21.h odd 6 1 1323.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.c 2 1.a even 1 1 trivial
189.4.e.c 2 3.b odd 2 1 CM
189.4.e.c 2 7.c even 3 1 inner
189.4.e.c 2 21.h odd 6 1 inner
1323.4.a.f 1 7.d odd 6 1
1323.4.a.f 1 21.g even 6 1
1323.4.a.i 1 7.c even 3 1
1323.4.a.i 1 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(189, [\chi])$$:

 $$T_{2}$$ T2 $$T_{13} - 89$$ T13 - 89

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 20T + 343$$
$11$ $$T^{2}$$
$13$ $$(T - 89)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 56T + 3136$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 289T + 83521$$
$37$ $$T^{2} - 433T + 187489$$
$41$ $$T^{2}$$
$43$ $$(T - 71)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 719T + 516961$$
$67$ $$T^{2} + 1007 T + 1014049$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 1190 T + 1416100$$
$79$ $$T^{2} + 503T + 253009$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 523)^{2}$$