Properties

 Label 189.2.c.c Level $189$ Weight $2$ Character orbit 189.c Analytic conductor $1.509$ Analytic rank $0$ Dimension $4$ 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: $$\Q(\sqrt{-2}, \sqrt{3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 4x^{2} + 1$$ x^4 + 4*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - \beta_{2} q^{5} + ( - \beta_{3} + 1) q^{7} - 2 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - b2 * q^5 + (-b3 + 1) * q^7 - 2*b1 * q^8 $$q - \beta_1 q^{2} - \beta_{2} q^{5} + ( - \beta_{3} + 1) q^{7} - 2 \beta_1 q^{8} + \beta_{3} q^{10} - \beta_1 q^{11} - \beta_{3} q^{13} + ( - 2 \beta_{2} - \beta_1) q^{14} - 4 q^{16} + 3 \beta_{2} q^{17} + 3 \beta_{3} q^{19} - 2 q^{22} + 2 \beta_1 q^{23} - 2 q^{25} - 2 \beta_{2} q^{26} + 5 \beta_1 q^{29} - \beta_{3} q^{31} - 3 \beta_{3} q^{34} + ( - \beta_{2} + 3 \beta_1) q^{35} + 5 q^{37} + 6 \beta_{2} q^{38} + 2 \beta_{3} q^{40} + 5 \beta_{2} q^{41} + 5 q^{43} + 4 q^{46} - 5 \beta_{2} q^{47} + ( - 2 \beta_{3} - 5) q^{49} + 2 \beta_1 q^{50} + 8 \beta_1 q^{53} + \beta_{3} q^{55} + ( - 4 \beta_{2} - 2 \beta_1) q^{56} + 10 q^{58} + 5 \beta_{2} q^{59} + \beta_{3} q^{61} - 2 \beta_{2} q^{62} - 8 q^{64} + 3 \beta_1 q^{65} + 2 q^{67} + (\beta_{3} + 6) q^{70} - 10 \beta_1 q^{71} - 5 \beta_1 q^{74} + ( - 2 \beta_{2} - \beta_1) q^{77} - 13 q^{79} + 4 \beta_{2} q^{80} - 5 \beta_{3} q^{82} + \beta_{2} q^{83} - 9 q^{85} - 5 \beta_1 q^{86} - 4 q^{88} - 6 \beta_{2} q^{89} + ( - \beta_{3} - 6) q^{91} + 5 \beta_{3} q^{94} - 9 \beta_1 q^{95} + 7 \beta_{3} q^{97} + ( - 4 \beta_{2} + 5 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 - b2 * q^5 + (-b3 + 1) * q^7 - 2*b1 * q^8 + b3 * q^10 - b1 * q^11 - b3 * q^13 + (-2*b2 - b1) * q^14 - 4 * q^16 + 3*b2 * q^17 + 3*b3 * q^19 - 2 * q^22 + 2*b1 * q^23 - 2 * q^25 - 2*b2 * q^26 + 5*b1 * q^29 - b3 * q^31 - 3*b3 * q^34 + (-b2 + 3*b1) * q^35 + 5 * q^37 + 6*b2 * q^38 + 2*b3 * q^40 + 5*b2 * q^41 + 5 * q^43 + 4 * q^46 - 5*b2 * q^47 + (-2*b3 - 5) * q^49 + 2*b1 * q^50 + 8*b1 * q^53 + b3 * q^55 + (-4*b2 - 2*b1) * q^56 + 10 * q^58 + 5*b2 * q^59 + b3 * q^61 - 2*b2 * q^62 - 8 * q^64 + 3*b1 * q^65 + 2 * q^67 + (b3 + 6) * q^70 - 10*b1 * q^71 - 5*b1 * q^74 + (-2*b2 - b1) * q^77 - 13 * q^79 + 4*b2 * q^80 - 5*b3 * q^82 + b2 * q^83 - 9 * q^85 - 5*b1 * q^86 - 4 * q^88 - 6*b2 * q^89 + (-b3 - 6) * q^91 + 5*b3 * q^94 - 9*b1 * q^95 + 7*b3 * q^97 + (-4*b2 + 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{7}+O(q^{10})$$ 4 * q + 4 * q^7 $$4 q + 4 q^{7} - 16 q^{16} - 8 q^{22} - 8 q^{25} + 20 q^{37} + 20 q^{43} + 16 q^{46} - 20 q^{49} + 40 q^{58} - 32 q^{64} + 8 q^{67} + 24 q^{70} - 52 q^{79} - 36 q^{85} - 16 q^{88} - 24 q^{91}+O(q^{100})$$ 4 * q + 4 * q^7 - 16 * q^16 - 8 * q^22 - 8 * q^25 + 20 * q^37 + 20 * q^43 + 16 * q^46 - 20 * q^49 + 40 * q^58 - 32 * q^64 + 8 * q^67 + 24 * q^70 - 52 * q^79 - 36 * q^85 - 16 * q^88 - 24 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 4x^{2} + 1$$ :

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

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
 0.517638i − 1.93185i − 0.517638i 1.93185i
1.41421i 0 0 −1.73205 0 1.00000 2.44949i 2.82843i 0 2.44949i
188.2 1.41421i 0 0 1.73205 0 1.00000 + 2.44949i 2.82843i 0 2.44949i
188.3 1.41421i 0 0 −1.73205 0 1.00000 + 2.44949i 2.82843i 0 2.44949i
188.4 1.41421i 0 0 1.73205 0 1.00000 2.44949i 2.82843i 0 2.44949i
 $$n$$: 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.c 4
3.b odd 2 1 inner 189.2.c.c 4
4.b odd 2 1 3024.2.k.g 4
7.b odd 2 1 inner 189.2.c.c 4
9.c even 3 2 567.2.o.e 8
9.d odd 6 2 567.2.o.e 8
12.b even 2 1 3024.2.k.g 4
21.c even 2 1 inner 189.2.c.c 4
28.d even 2 1 3024.2.k.g 4
63.l odd 6 2 567.2.o.e 8
63.o even 6 2 567.2.o.e 8
84.h odd 2 1 3024.2.k.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.c 4 1.a even 1 1 trivial
189.2.c.c 4 3.b odd 2 1 inner
189.2.c.c 4 7.b odd 2 1 inner
189.2.c.c 4 21.c even 2 1 inner
567.2.o.e 8 9.c even 3 2
567.2.o.e 8 9.d odd 6 2
567.2.o.e 8 63.l odd 6 2
567.2.o.e 8 63.o even 6 2
3024.2.k.g 4 4.b odd 2 1
3024.2.k.g 4 12.b even 2 1
3024.2.k.g 4 28.d even 2 1
3024.2.k.g 4 84.h odd 2 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 3)^{2}$$
$7$ $$(T^{2} - 2 T + 7)^{2}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$(T^{2} + 6)^{2}$$
$17$ $$(T^{2} - 27)^{2}$$
$19$ $$(T^{2} + 54)^{2}$$
$23$ $$(T^{2} + 8)^{2}$$
$29$ $$(T^{2} + 50)^{2}$$
$31$ $$(T^{2} + 6)^{2}$$
$37$ $$(T - 5)^{4}$$
$41$ $$(T^{2} - 75)^{2}$$
$43$ $$(T - 5)^{4}$$
$47$ $$(T^{2} - 75)^{2}$$
$53$ $$(T^{2} + 128)^{2}$$
$59$ $$(T^{2} - 75)^{2}$$
$61$ $$(T^{2} + 6)^{2}$$
$67$ $$(T - 2)^{4}$$
$71$ $$(T^{2} + 200)^{2}$$
$73$ $$T^{4}$$
$79$ $$(T + 13)^{4}$$
$83$ $$(T^{2} - 3)^{2}$$
$89$ $$(T^{2} - 108)^{2}$$
$97$ $$(T^{2} + 294)^{2}$$