# Properties

 Label 189.2.p.c Level $189$ Weight $2$ Character orbit 189.p 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(26,189)]

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([3, 5]))

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

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

chi := DirichletCharacter("189.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.p (of order $$6$$, 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 $$\Q(\zeta_{6})$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (2 \beta_{2} - 3) q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + 3*b2 * q^4 + (2*b2 - 3) * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + 3 \beta_{2} q^{4} + (2 \beta_{2} - 3) q^{7} + \beta_{3} q^{8} + ( - 2 \beta_{3} + 2 \beta_1) q^{11} + (2 \beta_{2} - 1) q^{13} + (2 \beta_{3} - 3 \beta_1) q^{14} + ( - \beta_{2} + 1) q^{16} + ( - 2 \beta_{3} - 2 \beta_1) q^{17} + ( - 2 \beta_{2} - 2) q^{19} + 10 q^{22} - 2 \beta_1 q^{23} + 5 \beta_{2} q^{25} + (2 \beta_{3} - \beta_1) q^{26} + ( - 3 \beta_{2} - 6) q^{28} + 2 \beta_{3} q^{29} + (\beta_{2} - 2) q^{31} + ( - 3 \beta_{3} + 3 \beta_1) q^{32} + ( - 20 \beta_{2} + 10) q^{34} + (5 \beta_{2} - 5) q^{37} + ( - 2 \beta_{3} - 2 \beta_1) q^{38} + ( - 2 \beta_{3} + 4 \beta_1) q^{41} - 7 q^{43} + 6 \beta_1 q^{44} - 10 \beta_{2} q^{46} + (4 \beta_{3} - 2 \beta_1) q^{47} + ( - 8 \beta_{2} + 5) q^{49} + 5 \beta_{3} q^{50} + (3 \beta_{2} - 6) q^{52} + ( - 2 \beta_{3} + 2 \beta_1) q^{53} + ( - \beta_{3} - 2 \beta_1) q^{56} + (10 \beta_{2} - 10) q^{58} + (2 \beta_{3} + 2 \beta_1) q^{59} + (5 \beta_{2} + 5) q^{61} + (\beta_{3} - 2 \beta_1) q^{62} + 13 q^{64} + \beta_{2} q^{67} + ( - 12 \beta_{3} + 6 \beta_1) q^{68} - 4 \beta_{3} q^{71} + ( - 4 \beta_{2} + 8) q^{73} + (5 \beta_{3} - 5 \beta_1) q^{74} + ( - 12 \beta_{2} + 6) q^{76} + (6 \beta_{3} - 2 \beta_1) q^{77} + (11 \beta_{2} - 11) q^{79} + (10 \beta_{2} + 10) q^{82} + (2 \beta_{3} - 4 \beta_1) q^{83} - 7 \beta_1 q^{86} + 10 \beta_{2} q^{88} + (8 \beta_{3} - 4 \beta_1) q^{89} + ( - 4 \beta_{2} - 1) q^{91} - 6 \beta_{3} q^{92} + (10 \beta_{2} - 20) q^{94} + ( - 2 \beta_{2} + 1) q^{97} + ( - 8 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + 3*b2 * q^4 + (2*b2 - 3) * q^7 + b3 * q^8 + (-2*b3 + 2*b1) * q^11 + (2*b2 - 1) * q^13 + (2*b3 - 3*b1) * q^14 + (-b2 + 1) * q^16 + (-2*b3 - 2*b1) * q^17 + (-2*b2 - 2) * q^19 + 10 * q^22 - 2*b1 * q^23 + 5*b2 * q^25 + (2*b3 - b1) * q^26 + (-3*b2 - 6) * q^28 + 2*b3 * q^29 + (b2 - 2) * q^31 + (-3*b3 + 3*b1) * q^32 + (-20*b2 + 10) * q^34 + (5*b2 - 5) * q^37 + (-2*b3 - 2*b1) * q^38 + (-2*b3 + 4*b1) * q^41 - 7 * q^43 + 6*b1 * q^44 - 10*b2 * q^46 + (4*b3 - 2*b1) * q^47 + (-8*b2 + 5) * q^49 + 5*b3 * q^50 + (3*b2 - 6) * q^52 + (-2*b3 + 2*b1) * q^53 + (-b3 - 2*b1) * q^56 + (10*b2 - 10) * q^58 + (2*b3 + 2*b1) * q^59 + (5*b2 + 5) * q^61 + (b3 - 2*b1) * q^62 + 13 * q^64 + b2 * q^67 + (-12*b3 + 6*b1) * q^68 - 4*b3 * q^71 + (-4*b2 + 8) * q^73 + (5*b3 - 5*b1) * q^74 + (-12*b2 + 6) * q^76 + (6*b3 - 2*b1) * q^77 + (11*b2 - 11) * q^79 + (10*b2 + 10) * q^82 + (2*b3 - 4*b1) * q^83 - 7*b1 * q^86 + 10*b2 * q^88 + (8*b3 - 4*b1) * q^89 + (-4*b2 - 1) * q^91 - 6*b3 * q^92 + (10*b2 - 20) * q^94 + (-2*b2 + 1) * q^97 + (-8*b3 + 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{4} - 8 q^{7}+O(q^{10})$$ 4 * q + 6 * q^4 - 8 * q^7 $$4 q + 6 q^{4} - 8 q^{7} + 2 q^{16} - 12 q^{19} + 40 q^{22} + 10 q^{25} - 30 q^{28} - 6 q^{31} - 10 q^{37} - 28 q^{43} - 20 q^{46} + 4 q^{49} - 18 q^{52} - 20 q^{58} + 30 q^{61} + 52 q^{64} + 2 q^{67} + 24 q^{73} - 22 q^{79} + 60 q^{82} + 20 q^{88} - 12 q^{91} - 60 q^{94}+O(q^{100})$$ 4 * q + 6 * q^4 - 8 * q^7 + 2 * q^16 - 12 * q^19 + 40 * q^22 + 10 * q^25 - 30 * q^28 - 6 * q^31 - 10 * q^37 - 28 * q^43 - 20 * q^46 + 4 * q^49 - 18 * q^52 - 20 * q^58 + 30 * q^61 + 52 * q^64 + 2 * q^67 + 24 * q^73 - 22 * q^79 + 60 * q^82 + 20 * q^88 - 12 * q^91 - 60 * q^94

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

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

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1$$ $$1 - \beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 −1.93649 − 1.11803i 1.93649 + 1.11803i −1.93649 + 1.11803i 1.93649 − 1.11803i
−1.93649 1.11803i 0 1.50000 + 2.59808i 0 0 −2.00000 + 1.73205i 2.23607i 0 0
26.2 1.93649 + 1.11803i 0 1.50000 + 2.59808i 0 0 −2.00000 + 1.73205i 2.23607i 0 0
80.1 −1.93649 + 1.11803i 0 1.50000 2.59808i 0 0 −2.00000 1.73205i 2.23607i 0 0
80.2 1.93649 1.11803i 0 1.50000 2.59808i 0 0 −2.00000 1.73205i 2.23607i 0 0
 $$n$$: 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.d odd 6 1 inner
21.g 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.p.c 4
3.b odd 2 1 inner 189.2.p.c 4
7.c even 3 1 1323.2.c.b 4
7.d odd 6 1 inner 189.2.p.c 4
7.d odd 6 1 1323.2.c.b 4
9.c even 3 1 567.2.i.c 4
9.c even 3 1 567.2.s.e 4
9.d odd 6 1 567.2.i.c 4
9.d odd 6 1 567.2.s.e 4
21.g even 6 1 inner 189.2.p.c 4
21.g even 6 1 1323.2.c.b 4
21.h odd 6 1 1323.2.c.b 4
63.i even 6 1 567.2.s.e 4
63.k odd 6 1 567.2.i.c 4
63.s even 6 1 567.2.i.c 4
63.t odd 6 1 567.2.s.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 5T_{2}^{2} + 25$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 5T^{2} + 25$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4 T + 7)^{2}$$
$11$ $$T^{4} - 20T^{2} + 400$$
$13$ $$(T^{2} + 3)^{2}$$
$17$ $$T^{4} + 60T^{2} + 3600$$
$19$ $$(T^{2} + 6 T + 12)^{2}$$
$23$ $$T^{4} - 20T^{2} + 400$$
$29$ $$(T^{2} + 20)^{2}$$
$31$ $$(T^{2} + 3 T + 3)^{2}$$
$37$ $$(T^{2} + 5 T + 25)^{2}$$
$41$ $$(T^{2} - 60)^{2}$$
$43$ $$(T + 7)^{4}$$
$47$ $$T^{4} + 60T^{2} + 3600$$
$53$ $$T^{4} - 20T^{2} + 400$$
$59$ $$T^{4} + 60T^{2} + 3600$$
$61$ $$(T^{2} - 15 T + 75)^{2}$$
$67$ $$(T^{2} - T + 1)^{2}$$
$71$ $$(T^{2} + 80)^{2}$$
$73$ $$(T^{2} - 12 T + 48)^{2}$$
$79$ $$(T^{2} + 11 T + 121)^{2}$$
$83$ $$(T^{2} - 60)^{2}$$
$89$ $$T^{4} + 240 T^{2} + 57600$$
$97$ $$(T^{2} + 3)^{2}$$