Properties

 Label 189.2.e.d Level $189$ Weight $2$ Character orbit 189.e 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(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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{2} + 4$$ x^4 - 2*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} - \beta_{2} q^{5} + ( - 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8}+O(q^{10})$$ q - b2 * q^2 + (4*b1 - 4) * q^4 - b2 * q^5 + (-3*b1 + 1) * q^7 + 2*b3 * q^8 $$q - \beta_{2} q^{2} + (4 \beta_1 - 4) q^{4} - \beta_{2} q^{5} + ( - 3 \beta_1 + 1) q^{7} + 2 \beta_{3} q^{8} + (6 \beta_1 - 6) q^{10} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{11} - 4 q^{13} + ( - 3 \beta_{3} + 2 \beta_{2}) q^{14} - 4 \beta_1 q^{16} + (\beta_{3} - \beta_{2}) q^{17} + \beta_1 q^{19} + 4 \beta_{3} q^{20} + 12 q^{22} - \beta_{2} q^{23} + (\beta_1 - 1) q^{25} + 4 \beta_{2} q^{26} + (4 \beta_1 + 8) q^{28} - 3 \beta_{3} q^{29} + ( - 7 \beta_1 + 7) q^{31} - 6 q^{34} + ( - 3 \beta_{3} + 2 \beta_{2}) q^{35} - 8 \beta_1 q^{37} + (\beta_{3} - \beta_{2}) q^{38} - 12 \beta_1 q^{40} + 3 \beta_{3} q^{41} - q^{43} - 8 \beta_{2} q^{44} + (6 \beta_1 - 6) q^{46} - \beta_{2} q^{47} + (3 \beta_1 - 8) q^{49} + \beta_{3} q^{50} + ( - 16 \beta_1 + 16) q^{52} + (\beta_{3} - \beta_{2}) q^{53} + 12 q^{55} + (2 \beta_{3} - 6 \beta_{2}) q^{56} + 18 \beta_1 q^{58} + (4 \beta_{3} - 4 \beta_{2}) q^{59} - 5 \beta_1 q^{61} - 7 \beta_{3} q^{62} - 8 q^{64} + 4 \beta_{2} q^{65} + (2 \beta_1 - 2) q^{67} + 4 \beta_{2} q^{68} + (6 \beta_1 + 12) q^{70} + ( - \beta_1 + 1) q^{73} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{74} - 4 q^{76} + (4 \beta_{3} + 2 \beta_{2}) q^{77} + 4 \beta_1 q^{79} + ( - 4 \beta_{3} + 4 \beta_{2}) q^{80} - 18 \beta_1 q^{82} + 6 \beta_{3} q^{83} - 6 q^{85} + \beta_{2} q^{86} + (24 \beta_1 - 24) q^{88} - \beta_{2} q^{89} + (12 \beta_1 - 4) q^{91} + 4 \beta_{3} q^{92} + (6 \beta_1 - 6) q^{94} + (\beta_{3} - \beta_{2}) q^{95} - q^{97} + (3 \beta_{3} + 5 \beta_{2}) q^{98}+O(q^{100})$$ q - b2 * q^2 + (4*b1 - 4) * q^4 - b2 * q^5 + (-3*b1 + 1) * q^7 + 2*b3 * q^8 + (6*b1 - 6) * q^10 + (-2*b3 + 2*b2) * q^11 - 4 * q^13 + (-3*b3 + 2*b2) * q^14 - 4*b1 * q^16 + (b3 - b2) * q^17 + b1 * q^19 + 4*b3 * q^20 + 12 * q^22 - b2 * q^23 + (b1 - 1) * q^25 + 4*b2 * q^26 + (4*b1 + 8) * q^28 - 3*b3 * q^29 + (-7*b1 + 7) * q^31 - 6 * q^34 + (-3*b3 + 2*b2) * q^35 - 8*b1 * q^37 + (b3 - b2) * q^38 - 12*b1 * q^40 + 3*b3 * q^41 - q^43 - 8*b2 * q^44 + (6*b1 - 6) * q^46 - b2 * q^47 + (3*b1 - 8) * q^49 + b3 * q^50 + (-16*b1 + 16) * q^52 + (b3 - b2) * q^53 + 12 * q^55 + (2*b3 - 6*b2) * q^56 + 18*b1 * q^58 + (4*b3 - 4*b2) * q^59 - 5*b1 * q^61 - 7*b3 * q^62 - 8 * q^64 + 4*b2 * q^65 + (2*b1 - 2) * q^67 + 4*b2 * q^68 + (6*b1 + 12) * q^70 + (-b1 + 1) * q^73 + (-8*b3 + 8*b2) * q^74 - 4 * q^76 + (4*b3 + 2*b2) * q^77 + 4*b1 * q^79 + (-4*b3 + 4*b2) * q^80 - 18*b1 * q^82 + 6*b3 * q^83 - 6 * q^85 + b2 * q^86 + (24*b1 - 24) * q^88 - b2 * q^89 + (12*b1 - 4) * q^91 + 4*b3 * q^92 + (6*b1 - 6) * q^94 + (b3 - b2) * q^95 - q^97 + (3*b3 + 5*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4} - 2 q^{7}+O(q^{10})$$ 4 * q - 8 * q^4 - 2 * q^7 $$4 q - 8 q^{4} - 2 q^{7} - 12 q^{10} - 16 q^{13} - 8 q^{16} + 2 q^{19} + 48 q^{22} - 2 q^{25} + 40 q^{28} + 14 q^{31} - 24 q^{34} - 16 q^{37} - 24 q^{40} - 4 q^{43} - 12 q^{46} - 26 q^{49} + 32 q^{52} + 48 q^{55} + 36 q^{58} - 10 q^{61} - 32 q^{64} - 4 q^{67} + 60 q^{70} + 2 q^{73} - 16 q^{76} + 8 q^{79} - 36 q^{82} - 24 q^{85} - 48 q^{88} + 8 q^{91} - 12 q^{94} - 4 q^{97}+O(q^{100})$$ 4 * q - 8 * q^4 - 2 * q^7 - 12 * q^10 - 16 * q^13 - 8 * q^16 + 2 * q^19 + 48 * q^22 - 2 * q^25 + 40 * q^28 + 14 * q^31 - 24 * q^34 - 16 * q^37 - 24 * q^40 - 4 * q^43 - 12 * q^46 - 26 * q^49 + 32 * q^52 + 48 * q^55 + 36 * q^58 - 10 * q^61 - 32 * q^64 - 4 * q^67 + 60 * q^70 + 2 * q^73 - 16 * q^76 + 8 * q^79 - 36 * q^82 - 24 * q^85 - 48 * q^88 + 8 * q^91 - 12 * q^94 - 4 * q^97

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 2\nu ) / 2$$ (v^3 + 2*v) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{3} + 4\nu ) / 2$$ (-v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 3$$ (b3 + b2) / 3 $$\nu^{2}$$ $$=$$ $$2\beta_1$$ 2*b1 $$\nu^{3}$$ $$=$$ $$( -2\beta_{3} + 4\beta_{2} ) / 3$$ (-2*b3 + 4*b2) / 3

Character values

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

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

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 − 0.707107i −1.22474 + 0.707107i
−1.22474 2.12132i 0 −2.00000 + 3.46410i −1.22474 2.12132i 0 −0.500000 2.59808i 4.89898 0 −3.00000 + 5.19615i
109.2 1.22474 + 2.12132i 0 −2.00000 + 3.46410i 1.22474 + 2.12132i 0 −0.500000 2.59808i −4.89898 0 −3.00000 + 5.19615i
163.1 −1.22474 + 2.12132i 0 −2.00000 3.46410i −1.22474 + 2.12132i 0 −0.500000 + 2.59808i 4.89898 0 −3.00000 5.19615i
163.2 1.22474 2.12132i 0 −2.00000 3.46410i 1.22474 2.12132i 0 −0.500000 + 2.59808i −4.89898 0 −3.00000 5.19615i
 $$n$$: 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.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 189.2.e.d 4
3.b odd 2 1 inner 189.2.e.d 4
7.c even 3 1 inner 189.2.e.d 4
7.c even 3 1 1323.2.a.v 2
7.d odd 6 1 1323.2.a.u 2
9.c even 3 1 567.2.g.g 4
9.c even 3 1 567.2.h.g 4
9.d odd 6 1 567.2.g.g 4
9.d odd 6 1 567.2.h.g 4
21.g even 6 1 1323.2.a.u 2
21.h odd 6 1 inner 189.2.e.d 4
21.h odd 6 1 1323.2.a.v 2
63.g even 3 1 567.2.h.g 4
63.h even 3 1 567.2.g.g 4
63.j odd 6 1 567.2.g.g 4
63.n odd 6 1 567.2.h.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.d 4 1.a even 1 1 trivial
189.2.e.d 4 3.b odd 2 1 inner
189.2.e.d 4 7.c even 3 1 inner
189.2.e.d 4 21.h odd 6 1 inner
567.2.g.g 4 9.c even 3 1
567.2.g.g 4 9.d odd 6 1
567.2.g.g 4 63.h even 3 1
567.2.g.g 4 63.j odd 6 1
567.2.h.g 4 9.c even 3 1
567.2.h.g 4 9.d odd 6 1
567.2.h.g 4 63.g even 3 1
567.2.h.g 4 63.n odd 6 1
1323.2.a.u 2 7.d odd 6 1
1323.2.a.u 2 21.g even 6 1
1323.2.a.v 2 7.c even 3 1
1323.2.a.v 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6T_{2}^{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 36$$
$3$ $$T^{4}$$
$5$ $$T^{4} + 6T^{2} + 36$$
$7$ $$(T^{2} + T + 7)^{2}$$
$11$ $$T^{4} + 24T^{2} + 576$$
$13$ $$(T + 4)^{4}$$
$17$ $$T^{4} + 6T^{2} + 36$$
$19$ $$(T^{2} - T + 1)^{2}$$
$23$ $$T^{4} + 6T^{2} + 36$$
$29$ $$(T^{2} - 54)^{2}$$
$31$ $$(T^{2} - 7 T + 49)^{2}$$
$37$ $$(T^{2} + 8 T + 64)^{2}$$
$41$ $$(T^{2} - 54)^{2}$$
$43$ $$(T + 1)^{4}$$
$47$ $$T^{4} + 6T^{2} + 36$$
$53$ $$T^{4} + 6T^{2} + 36$$
$59$ $$T^{4} + 96T^{2} + 9216$$
$61$ $$(T^{2} + 5 T + 25)^{2}$$
$67$ $$(T^{2} + 2 T + 4)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} - T + 1)^{2}$$
$79$ $$(T^{2} - 4 T + 16)^{2}$$
$83$ $$(T^{2} - 216)^{2}$$
$89$ $$T^{4} + 6T^{2} + 36$$
$97$ $$(T + 1)^{4}$$