# Properties

 Label 189.2.e.a Level $189$ Weight $2$ Character orbit 189.e 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(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, 2, names="a")

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ 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: $$1.50917259820$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} - 3 q^{8} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^4 + 4*z * q^5 + (2*z + 1) * q^7 - 3 * q^8 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + 4 \zeta_{6} q^{5} + (2 \zeta_{6} + 1) q^{7} - 3 q^{8} + ( - 4 \zeta_{6} + 4) q^{10} + (2 \zeta_{6} - 2) q^{11} + q^{13} + ( - 3 \zeta_{6} + 2) q^{14} + \zeta_{6} q^{16} + ( - 6 \zeta_{6} + 6) q^{17} - 4 \zeta_{6} q^{19} + 4 q^{20} + 2 q^{22} - 6 \zeta_{6} q^{23} + (11 \zeta_{6} - 11) q^{25} - \zeta_{6} q^{26} + ( - \zeta_{6} + 3) q^{28} + 2 q^{29} + (3 \zeta_{6} - 3) q^{31} + (5 \zeta_{6} - 5) q^{32} - 6 q^{34} + (12 \zeta_{6} - 8) q^{35} - 3 \zeta_{6} q^{37} + (4 \zeta_{6} - 4) q^{38} - 12 \zeta_{6} q^{40} - 2 q^{41} - q^{43} + 2 \zeta_{6} q^{44} + (6 \zeta_{6} - 6) q^{46} - 6 \zeta_{6} q^{47} + (8 \zeta_{6} - 3) q^{49} + 11 q^{50} + ( - \zeta_{6} + 1) q^{52} + ( - 6 \zeta_{6} + 6) q^{53} - 8 q^{55} + ( - 6 \zeta_{6} - 3) q^{56} - 2 \zeta_{6} q^{58} + (6 \zeta_{6} - 6) q^{59} + 5 \zeta_{6} q^{61} + 3 q^{62} + 7 q^{64} + 4 \zeta_{6} q^{65} + (7 \zeta_{6} - 7) q^{67} - 6 \zeta_{6} q^{68} + ( - 4 \zeta_{6} + 12) q^{70} + ( - 6 \zeta_{6} + 6) q^{73} + (3 \zeta_{6} - 3) q^{74} - 4 q^{76} + (2 \zeta_{6} - 6) q^{77} - 11 \zeta_{6} q^{79} + (4 \zeta_{6} - 4) q^{80} + 2 \zeta_{6} q^{82} + 6 q^{83} + 24 q^{85} + \zeta_{6} q^{86} + ( - 6 \zeta_{6} + 6) q^{88} + 4 \zeta_{6} q^{89} + (2 \zeta_{6} + 1) q^{91} - 6 q^{92} + (6 \zeta_{6} - 6) q^{94} + ( - 16 \zeta_{6} + 16) q^{95} + 9 q^{97} + ( - 5 \zeta_{6} + 8) q^{98} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^4 + 4*z * q^5 + (2*z + 1) * q^7 - 3 * q^8 + (-4*z + 4) * q^10 + (2*z - 2) * q^11 + q^13 + (-3*z + 2) * q^14 + z * q^16 + (-6*z + 6) * q^17 - 4*z * q^19 + 4 * q^20 + 2 * q^22 - 6*z * q^23 + (11*z - 11) * q^25 - z * q^26 + (-z + 3) * q^28 + 2 * q^29 + (3*z - 3) * q^31 + (5*z - 5) * q^32 - 6 * q^34 + (12*z - 8) * q^35 - 3*z * q^37 + (4*z - 4) * q^38 - 12*z * q^40 - 2 * q^41 - q^43 + 2*z * q^44 + (6*z - 6) * q^46 - 6*z * q^47 + (8*z - 3) * q^49 + 11 * q^50 + (-z + 1) * q^52 + (-6*z + 6) * q^53 - 8 * q^55 + (-6*z - 3) * q^56 - 2*z * q^58 + (6*z - 6) * q^59 + 5*z * q^61 + 3 * q^62 + 7 * q^64 + 4*z * q^65 + (7*z - 7) * q^67 - 6*z * q^68 + (-4*z + 12) * q^70 + (-6*z + 6) * q^73 + (3*z - 3) * q^74 - 4 * q^76 + (2*z - 6) * q^77 - 11*z * q^79 + (4*z - 4) * q^80 + 2*z * q^82 + 6 * q^83 + 24 * q^85 + z * q^86 + (-6*z + 6) * q^88 + 4*z * q^89 + (2*z + 1) * q^91 - 6 * q^92 + (6*z - 6) * q^94 + (-16*z + 16) * q^95 + 9 * q^97 + (-5*z + 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} + 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 + 4 * q^5 + 4 * q^7 - 6 * q^8 $$2 q - q^{2} + q^{4} + 4 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{10} - 2 q^{11} + 2 q^{13} + q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + 8 q^{20} + 4 q^{22} - 6 q^{23} - 11 q^{25} - q^{26} + 5 q^{28} + 4 q^{29} - 3 q^{31} - 5 q^{32} - 12 q^{34} - 4 q^{35} - 3 q^{37} - 4 q^{38} - 12 q^{40} - 4 q^{41} - 2 q^{43} + 2 q^{44} - 6 q^{46} - 6 q^{47} + 2 q^{49} + 22 q^{50} + q^{52} + 6 q^{53} - 16 q^{55} - 12 q^{56} - 2 q^{58} - 6 q^{59} + 5 q^{61} + 6 q^{62} + 14 q^{64} + 4 q^{65} - 7 q^{67} - 6 q^{68} + 20 q^{70} + 6 q^{73} - 3 q^{74} - 8 q^{76} - 10 q^{77} - 11 q^{79} - 4 q^{80} + 2 q^{82} + 12 q^{83} + 48 q^{85} + q^{86} + 6 q^{88} + 4 q^{89} + 4 q^{91} - 12 q^{92} - 6 q^{94} + 16 q^{95} + 18 q^{97} + 11 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^4 + 4 * q^5 + 4 * q^7 - 6 * q^8 + 4 * q^10 - 2 * q^11 + 2 * q^13 + q^14 + q^16 + 6 * q^17 - 4 * q^19 + 8 * q^20 + 4 * q^22 - 6 * q^23 - 11 * q^25 - q^26 + 5 * q^28 + 4 * q^29 - 3 * q^31 - 5 * q^32 - 12 * q^34 - 4 * q^35 - 3 * q^37 - 4 * q^38 - 12 * q^40 - 4 * q^41 - 2 * q^43 + 2 * q^44 - 6 * q^46 - 6 * q^47 + 2 * q^49 + 22 * q^50 + q^52 + 6 * q^53 - 16 * q^55 - 12 * q^56 - 2 * q^58 - 6 * q^59 + 5 * q^61 + 6 * q^62 + 14 * q^64 + 4 * q^65 - 7 * q^67 - 6 * q^68 + 20 * q^70 + 6 * q^73 - 3 * q^74 - 8 * q^76 - 10 * q^77 - 11 * q^79 - 4 * q^80 + 2 * q^82 + 12 * q^83 + 48 * q^85 + q^86 + 6 * q^88 + 4 * q^89 + 4 * q^91 - 12 * q^92 - 6 * q^94 + 16 * q^95 + 18 * q^97 + 11 * q^98

## 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
−0.500000 0.866025i 0 0.500000 0.866025i 2.00000 + 3.46410i 0 2.00000 + 1.73205i −3.00000 0 2.00000 3.46410i
163.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 2.00000 3.46410i 0 2.00000 1.73205i −3.00000 0 2.00000 + 3.46410i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.e.a 2
3.b odd 2 1 189.2.e.c yes 2
7.c even 3 1 inner 189.2.e.a 2
7.c even 3 1 1323.2.a.m 1
7.d odd 6 1 1323.2.a.p 1
9.c even 3 1 567.2.g.b 2
9.c even 3 1 567.2.h.e 2
9.d odd 6 1 567.2.g.e 2
9.d odd 6 1 567.2.h.b 2
21.g even 6 1 1323.2.a.d 1
21.h odd 6 1 189.2.e.c yes 2
21.h odd 6 1 1323.2.a.g 1
63.g even 3 1 567.2.h.e 2
63.h even 3 1 567.2.g.b 2
63.j odd 6 1 567.2.g.e 2
63.n odd 6 1 567.2.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.a 2 1.a even 1 1 trivial
189.2.e.a 2 7.c even 3 1 inner
189.2.e.c yes 2 3.b odd 2 1
189.2.e.c yes 2 21.h odd 6 1
567.2.g.b 2 9.c even 3 1
567.2.g.b 2 63.h even 3 1
567.2.g.e 2 9.d odd 6 1
567.2.g.e 2 63.j odd 6 1
567.2.h.b 2 9.d odd 6 1
567.2.h.b 2 63.n odd 6 1
567.2.h.e 2 9.c even 3 1
567.2.h.e 2 63.g even 3 1
1323.2.a.d 1 21.g even 6 1
1323.2.a.g 1 21.h odd 6 1
1323.2.a.m 1 7.c even 3 1
1323.2.a.p 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 16$$
$7$ $$T^{2} - 4T + 7$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 4T + 16$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 3T + 9$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T + 2)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} - 5T + 25$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 6T + 36$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 4T + 16$$
$97$ $$(T - 9)^{2}$$