# Properties

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

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(17,189)]

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([5, 1]))

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

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

chi := DirichletCharacter("189.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.s (of order $$6$$, degree $$2$$, not 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: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 3 q^{5} + (\zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8}+O(q^{10})$$ q + (z + 1) * q^2 + z * q^4 + 3 * q^5 + (z - 3) * q^7 + (-2*z + 1) * q^8 $$q + (\zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 3 q^{5} + (\zeta_{6} - 3) q^{7} + ( - 2 \zeta_{6} + 1) q^{8} + (3 \zeta_{6} + 3) q^{10} + (2 \zeta_{6} - 1) q^{11} + ( - \zeta_{6} - 1) q^{13} + ( - \zeta_{6} - 4) q^{14} + ( - 5 \zeta_{6} + 5) q^{16} + (3 \zeta_{6} - 3) q^{17} + (3 \zeta_{6} - 6) q^{19} + 3 \zeta_{6} q^{20} + (3 \zeta_{6} - 3) q^{22} + ( - 6 \zeta_{6} + 3) q^{23} + 4 q^{25} - 3 \zeta_{6} q^{26} + ( - 2 \zeta_{6} - 1) q^{28} + ( - 3 \zeta_{6} + 6) q^{29} + (2 \zeta_{6} - 4) q^{31} + ( - 3 \zeta_{6} + 6) q^{32} + (3 \zeta_{6} - 6) q^{34} + (3 \zeta_{6} - 9) q^{35} - 7 \zeta_{6} q^{37} - 9 q^{38} + ( - 6 \zeta_{6} + 3) q^{40} + (3 \zeta_{6} - 3) q^{41} - \zeta_{6} q^{43} + (\zeta_{6} - 2) q^{44} + ( - 9 \zeta_{6} + 9) q^{46} + ( - 5 \zeta_{6} + 8) q^{49} + (4 \zeta_{6} + 4) q^{50} + ( - 2 \zeta_{6} + 1) q^{52} + (5 \zeta_{6} + 5) q^{53} + (6 \zeta_{6} - 3) q^{55} + (5 \zeta_{6} - 1) q^{56} + 9 q^{58} + (8 \zeta_{6} + 8) q^{61} - 6 q^{62} - q^{64} + ( - 3 \zeta_{6} - 3) q^{65} + 4 \zeta_{6} q^{67} - 3 q^{68} + ( - 3 \zeta_{6} - 12) q^{70} + (4 \zeta_{6} - 2) q^{71} + ( - 3 \zeta_{6} - 3) q^{73} + ( - 14 \zeta_{6} + 7) q^{74} + ( - 3 \zeta_{6} - 3) q^{76} + ( - 5 \zeta_{6} + 1) q^{77} + (8 \zeta_{6} - 8) q^{79} + ( - 15 \zeta_{6} + 15) q^{80} + (3 \zeta_{6} - 6) q^{82} + 15 \zeta_{6} q^{83} + (9 \zeta_{6} - 9) q^{85} + ( - 2 \zeta_{6} + 1) q^{86} + 3 q^{88} - 3 \zeta_{6} q^{89} + (\zeta_{6} + 4) q^{91} + ( - 3 \zeta_{6} + 6) q^{92} + (9 \zeta_{6} - 18) q^{95} + (\zeta_{6} - 2) q^{97} + ( - 2 \zeta_{6} + 13) q^{98} +O(q^{100})$$ q + (z + 1) * q^2 + z * q^4 + 3 * q^5 + (z - 3) * q^7 + (-2*z + 1) * q^8 + (3*z + 3) * q^10 + (2*z - 1) * q^11 + (-z - 1) * q^13 + (-z - 4) * q^14 + (-5*z + 5) * q^16 + (3*z - 3) * q^17 + (3*z - 6) * q^19 + 3*z * q^20 + (3*z - 3) * q^22 + (-6*z + 3) * q^23 + 4 * q^25 - 3*z * q^26 + (-2*z - 1) * q^28 + (-3*z + 6) * q^29 + (2*z - 4) * q^31 + (-3*z + 6) * q^32 + (3*z - 6) * q^34 + (3*z - 9) * q^35 - 7*z * q^37 - 9 * q^38 + (-6*z + 3) * q^40 + (3*z - 3) * q^41 - z * q^43 + (z - 2) * q^44 + (-9*z + 9) * q^46 + (-5*z + 8) * q^49 + (4*z + 4) * q^50 + (-2*z + 1) * q^52 + (5*z + 5) * q^53 + (6*z - 3) * q^55 + (5*z - 1) * q^56 + 9 * q^58 + (8*z + 8) * q^61 - 6 * q^62 - q^64 + (-3*z - 3) * q^65 + 4*z * q^67 - 3 * q^68 + (-3*z - 12) * q^70 + (4*z - 2) * q^71 + (-3*z - 3) * q^73 + (-14*z + 7) * q^74 + (-3*z - 3) * q^76 + (-5*z + 1) * q^77 + (8*z - 8) * q^79 + (-15*z + 15) * q^80 + (3*z - 6) * q^82 + 15*z * q^83 + (9*z - 9) * q^85 + (-2*z + 1) * q^86 + 3 * q^88 - 3*z * q^89 + (z + 4) * q^91 + (-3*z + 6) * q^92 + (9*z - 18) * q^95 + (z - 2) * q^97 + (-2*z + 13) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + q^{4} + 6 q^{5} - 5 q^{7}+O(q^{10})$$ 2 * q + 3 * q^2 + q^4 + 6 * q^5 - 5 * q^7 $$2 q + 3 q^{2} + q^{4} + 6 q^{5} - 5 q^{7} + 9 q^{10} - 3 q^{13} - 9 q^{14} + 5 q^{16} - 3 q^{17} - 9 q^{19} + 3 q^{20} - 3 q^{22} + 8 q^{25} - 3 q^{26} - 4 q^{28} + 9 q^{29} - 6 q^{31} + 9 q^{32} - 9 q^{34} - 15 q^{35} - 7 q^{37} - 18 q^{38} - 3 q^{41} - q^{43} - 3 q^{44} + 9 q^{46} + 11 q^{49} + 12 q^{50} + 15 q^{53} + 3 q^{56} + 18 q^{58} + 24 q^{61} - 12 q^{62} - 2 q^{64} - 9 q^{65} + 4 q^{67} - 6 q^{68} - 27 q^{70} - 9 q^{73} - 9 q^{76} - 3 q^{77} - 8 q^{79} + 15 q^{80} - 9 q^{82} + 15 q^{83} - 9 q^{85} + 6 q^{88} - 3 q^{89} + 9 q^{91} + 9 q^{92} - 27 q^{95} - 3 q^{97} + 24 q^{98}+O(q^{100})$$ 2 * q + 3 * q^2 + q^4 + 6 * q^5 - 5 * q^7 + 9 * q^10 - 3 * q^13 - 9 * q^14 + 5 * q^16 - 3 * q^17 - 9 * q^19 + 3 * q^20 - 3 * q^22 + 8 * q^25 - 3 * q^26 - 4 * q^28 + 9 * q^29 - 6 * q^31 + 9 * q^32 - 9 * q^34 - 15 * q^35 - 7 * q^37 - 18 * q^38 - 3 * q^41 - q^43 - 3 * q^44 + 9 * q^46 + 11 * q^49 + 12 * q^50 + 15 * q^53 + 3 * q^56 + 18 * q^58 + 24 * q^61 - 12 * q^62 - 2 * q^64 - 9 * q^65 + 4 * q^67 - 6 * q^68 - 27 * q^70 - 9 * q^73 - 9 * q^76 - 3 * q^77 - 8 * q^79 + 15 * q^80 - 9 * q^82 + 15 * q^83 - 9 * q^85 + 6 * q^88 - 3 * q^89 + 9 * q^91 + 9 * q^92 - 27 * q^95 - 3 * q^97 + 24 * q^98

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.50000 + 0.866025i 0 0.500000 + 0.866025i 3.00000 0 −2.50000 + 0.866025i 1.73205i 0 4.50000 + 2.59808i
89.1 1.50000 0.866025i 0 0.500000 0.866025i 3.00000 0 −2.50000 0.866025i 1.73205i 0 4.50000 2.59808i
 $$n$$: 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
63.s 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.s.a 2
3.b odd 2 1 63.2.s.a yes 2
4.b odd 2 1 3024.2.df.a 2
7.b odd 2 1 1323.2.s.a 2
7.c even 3 1 1323.2.i.a 2
7.c even 3 1 1323.2.o.a 2
7.d odd 6 1 189.2.i.a 2
7.d odd 6 1 1323.2.o.b 2
9.c even 3 1 63.2.i.a 2
9.c even 3 1 567.2.p.a 2
9.d odd 6 1 189.2.i.a 2
9.d odd 6 1 567.2.p.b 2
12.b even 2 1 1008.2.df.a 2
21.c even 2 1 441.2.s.a 2
21.g even 6 1 63.2.i.a 2
21.g even 6 1 441.2.o.a 2
21.h odd 6 1 441.2.i.a 2
21.h odd 6 1 441.2.o.b 2
28.f even 6 1 3024.2.ca.a 2
36.f odd 6 1 1008.2.ca.a 2
36.h even 6 1 3024.2.ca.a 2
63.g even 3 1 441.2.s.a 2
63.h even 3 1 441.2.o.a 2
63.i even 6 1 567.2.p.a 2
63.i even 6 1 1323.2.o.a 2
63.j odd 6 1 1323.2.o.b 2
63.k odd 6 1 63.2.s.a yes 2
63.l odd 6 1 441.2.i.a 2
63.n odd 6 1 1323.2.s.a 2
63.o even 6 1 1323.2.i.a 2
63.s even 6 1 inner 189.2.s.a 2
63.t odd 6 1 441.2.o.b 2
63.t odd 6 1 567.2.p.b 2
84.j odd 6 1 1008.2.ca.a 2
252.n even 6 1 1008.2.df.a 2
252.bn odd 6 1 3024.2.df.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.i.a 2 9.c even 3 1
63.2.i.a 2 21.g even 6 1
63.2.s.a yes 2 3.b odd 2 1
63.2.s.a yes 2 63.k odd 6 1
189.2.i.a 2 7.d odd 6 1
189.2.i.a 2 9.d odd 6 1
189.2.s.a 2 1.a even 1 1 trivial
189.2.s.a 2 63.s even 6 1 inner
441.2.i.a 2 21.h odd 6 1
441.2.i.a 2 63.l odd 6 1
441.2.o.a 2 21.g even 6 1
441.2.o.a 2 63.h even 3 1
441.2.o.b 2 21.h odd 6 1
441.2.o.b 2 63.t odd 6 1
441.2.s.a 2 21.c even 2 1
441.2.s.a 2 63.g even 3 1
567.2.p.a 2 9.c even 3 1
567.2.p.a 2 63.i even 6 1
567.2.p.b 2 9.d odd 6 1
567.2.p.b 2 63.t odd 6 1
1008.2.ca.a 2 36.f odd 6 1
1008.2.ca.a 2 84.j odd 6 1
1008.2.df.a 2 12.b even 2 1
1008.2.df.a 2 252.n even 6 1
1323.2.i.a 2 7.c even 3 1
1323.2.i.a 2 63.o even 6 1
1323.2.o.a 2 7.c even 3 1
1323.2.o.a 2 63.i even 6 1
1323.2.o.b 2 7.d odd 6 1
1323.2.o.b 2 63.j odd 6 1
1323.2.s.a 2 7.b odd 2 1
1323.2.s.a 2 63.n odd 6 1
3024.2.ca.a 2 28.f even 6 1
3024.2.ca.a 2 36.h even 6 1
3024.2.df.a 2 4.b odd 2 1
3024.2.df.a 2 252.bn odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 3T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 3T + 3$$
$3$ $$T^{2}$$
$5$ $$(T - 3)^{2}$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + 3$$
$13$ $$T^{2} + 3T + 3$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 9T + 27$$
$23$ $$T^{2} + 27$$
$29$ $$T^{2} - 9T + 27$$
$31$ $$T^{2} + 6T + 12$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 15T + 75$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 24T + 192$$
$67$ $$T^{2} - 4T + 16$$
$71$ $$T^{2} + 12$$
$73$ $$T^{2} + 9T + 27$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} - 15T + 225$$
$89$ $$T^{2} + 3T + 9$$
$97$ $$T^{2} + 3T + 3$$
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