# Properties

 Label 189.2.c.b Level $189$ Weight $2$ Character orbit 189.c Analytic conductor $1.509$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 5x^{2} + 25$$ x^4 - 5*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 3 q^{4} - \beta_{3} q^{5} + (\beta_{2} - 2) q^{7} + \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 3 * q^4 - b3 * q^5 + (b2 - 2) * q^7 + b1 * q^8 $$q - \beta_1 q^{2} - 3 q^{4} - \beta_{3} q^{5} + (\beta_{2} - 2) q^{7} + \beta_1 q^{8} + 5 \beta_{2} q^{10} - \beta_1 q^{11} - 2 \beta_{2} q^{13} + (\beta_{3} + 2 \beta_1) q^{14} - q^{16} - 3 \beta_{2} q^{19} + 3 \beta_{3} q^{20} - 5 q^{22} - \beta_1 q^{23} + 10 q^{25} - 2 \beta_{3} q^{26} + ( - 3 \beta_{2} + 6) q^{28} + 2 \beta_1 q^{29} + \beta_{2} q^{31} + 3 \beta_1 q^{32} + (2 \beta_{3} - 3 \beta_1) q^{35} - q^{37} - 3 \beta_{3} q^{38} - 5 \beta_{2} q^{40} - \beta_{3} q^{41} + 2 q^{43} + 3 \beta_1 q^{44} - 5 q^{46} - 2 \beta_{3} q^{47} + ( - 4 \beta_{2} + 1) q^{49} - 10 \beta_1 q^{50} + 6 \beta_{2} q^{52} - 4 \beta_1 q^{53} + 5 \beta_{2} q^{55} + ( - \beta_{3} - 2 \beta_1) q^{56} + 10 q^{58} + 2 \beta_{3} q^{59} - 4 \beta_{2} q^{61} + \beta_{3} q^{62} + 13 q^{64} + 6 \beta_1 q^{65} - 10 q^{67} + ( - 10 \beta_{2} - 15) q^{70} + 5 \beta_1 q^{71} - 6 \beta_{2} q^{73} + \beta_1 q^{74} + 9 \beta_{2} q^{76} + (\beta_{3} + 2 \beta_1) q^{77} + 2 q^{79} + \beta_{3} q^{80} + 5 \beta_{2} q^{82} - 2 \beta_{3} q^{83} - 2 \beta_1 q^{86} + 5 q^{88} + 3 \beta_{3} q^{89} + (4 \beta_{2} + 6) q^{91} + 3 \beta_1 q^{92} + 10 \beta_{2} q^{94} + 9 \beta_1 q^{95} + 8 \beta_{2} q^{97} + ( - 4 \beta_{3} - \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - 3 * q^4 - b3 * q^5 + (b2 - 2) * q^7 + b1 * q^8 + 5*b2 * q^10 - b1 * q^11 - 2*b2 * q^13 + (b3 + 2*b1) * q^14 - q^16 - 3*b2 * q^19 + 3*b3 * q^20 - 5 * q^22 - b1 * q^23 + 10 * q^25 - 2*b3 * q^26 + (-3*b2 + 6) * q^28 + 2*b1 * q^29 + b2 * q^31 + 3*b1 * q^32 + (2*b3 - 3*b1) * q^35 - q^37 - 3*b3 * q^38 - 5*b2 * q^40 - b3 * q^41 + 2 * q^43 + 3*b1 * q^44 - 5 * q^46 - 2*b3 * q^47 + (-4*b2 + 1) * q^49 - 10*b1 * q^50 + 6*b2 * q^52 - 4*b1 * q^53 + 5*b2 * q^55 + (-b3 - 2*b1) * q^56 + 10 * q^58 + 2*b3 * q^59 - 4*b2 * q^61 + b3 * q^62 + 13 * q^64 + 6*b1 * q^65 - 10 * q^67 + (-10*b2 - 15) * q^70 + 5*b1 * q^71 - 6*b2 * q^73 + b1 * q^74 + 9*b2 * q^76 + (b3 + 2*b1) * q^77 + 2 * q^79 + b3 * q^80 + 5*b2 * q^82 - 2*b3 * q^83 - 2*b1 * q^86 + 5 * q^88 + 3*b3 * q^89 + (4*b2 + 6) * q^91 + 3*b1 * q^92 + 10*b2 * q^94 + 9*b1 * q^95 + 8*b2 * q^97 + (-4*b3 - b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{4} - 8 q^{7}+O(q^{10})$$ 4 * q - 12 * q^4 - 8 * q^7 $$4 q - 12 q^{4} - 8 q^{7} - 4 q^{16} - 20 q^{22} + 40 q^{25} + 24 q^{28} - 4 q^{37} + 8 q^{43} - 20 q^{46} + 4 q^{49} + 40 q^{58} + 52 q^{64} - 40 q^{67} - 60 q^{70} + 8 q^{79} + 20 q^{88} + 24 q^{91}+O(q^{100})$$ 4 * q - 12 * q^4 - 8 * q^7 - 4 * q^16 - 20 * q^22 + 40 * q^25 + 24 * q^28 - 4 * q^37 + 8 * q^43 - 20 * q^46 + 4 * q^49 + 40 * q^58 + 52 * q^64 - 40 * q^67 - 60 * q^70 + 8 * q^79 + 20 * q^88 + 24 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 5$$ (v^3) / 5 $$\beta_{2}$$ $$=$$ $$( 2\nu^{2} - 5 ) / 5$$ (2*v^2 - 5) / 5 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 10\nu ) / 5$$ (-v^3 + 10*v) / 5
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_1 ) / 2$$ (b3 + b1) / 2 $$\nu^{2}$$ $$=$$ $$( 5\beta_{2} + 5 ) / 2$$ (5*b2 + 5) / 2 $$\nu^{3}$$ $$=$$ $$5\beta_1$$ 5*b1

## 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
 1.93649 + 1.11803i −1.93649 + 1.11803i 1.93649 − 1.11803i −1.93649 − 1.11803i
2.23607i 0 −3.00000 −3.87298 0 −2.00000 + 1.73205i 2.23607i 0 8.66025i
188.2 2.23607i 0 −3.00000 3.87298 0 −2.00000 1.73205i 2.23607i 0 8.66025i
188.3 2.23607i 0 −3.00000 −3.87298 0 −2.00000 1.73205i 2.23607i 0 8.66025i
188.4 2.23607i 0 −3.00000 3.87298 0 −2.00000 + 1.73205i 2.23607i 0 8.66025i
 $$n$$: 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 inner
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.b 4
3.b odd 2 1 inner 189.2.c.b 4
4.b odd 2 1 3024.2.k.i 4
7.b odd 2 1 inner 189.2.c.b 4
9.c even 3 1 567.2.o.c 4
9.c even 3 1 567.2.o.d 4
9.d odd 6 1 567.2.o.c 4
9.d odd 6 1 567.2.o.d 4
12.b even 2 1 3024.2.k.i 4
21.c even 2 1 inner 189.2.c.b 4
28.d even 2 1 3024.2.k.i 4
63.l odd 6 1 567.2.o.c 4
63.l odd 6 1 567.2.o.d 4
63.o even 6 1 567.2.o.c 4
63.o even 6 1 567.2.o.d 4
84.h odd 2 1 3024.2.k.i 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 5$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 5)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 15)^{2}$$
$7$ $$(T^{2} + 4 T + 7)^{2}$$
$11$ $$(T^{2} + 5)^{2}$$
$13$ $$(T^{2} + 12)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 27)^{2}$$
$23$ $$(T^{2} + 5)^{2}$$
$29$ $$(T^{2} + 20)^{2}$$
$31$ $$(T^{2} + 3)^{2}$$
$37$ $$(T + 1)^{4}$$
$41$ $$(T^{2} - 15)^{2}$$
$43$ $$(T - 2)^{4}$$
$47$ $$(T^{2} - 60)^{2}$$
$53$ $$(T^{2} + 80)^{2}$$
$59$ $$(T^{2} - 60)^{2}$$
$61$ $$(T^{2} + 48)^{2}$$
$67$ $$(T + 10)^{4}$$
$71$ $$(T^{2} + 125)^{2}$$
$73$ $$(T^{2} + 108)^{2}$$
$79$ $$(T - 2)^{4}$$
$83$ $$(T^{2} - 60)^{2}$$
$89$ $$(T^{2} - 135)^{2}$$
$97$ $$(T^{2} + 192)^{2}$$