# Properties

 Label 189.2.e.f Level $189$ Weight $2$ Character orbit 189.e Analytic conductor $1.509$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ x^6 - 3*x^5 + 10*x^4 - 15*x^3 + 19*x^2 - 12*x + 3 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{5} - \beta_{4} + 1) q^{2} + (\beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - 2 \beta_{3} - \beta_1 - 4) q^{8}+O(q^{10})$$ q + (b5 - b4 + 1) * q^2 + (b5 - 2*b4 - b3 - b2 - b1) * q^4 + b2 * q^5 + (b5 - b4 + b3 - b1 + 1) * q^7 + (-2*b3 - b1 - 4) * q^8 $$q + (\beta_{5} - \beta_{4} + 1) q^{2} + (\beta_{5} - 2 \beta_{4} + \cdots - \beta_1) q^{4}+ \cdots + ( - 4 \beta_{3} + \beta_{2} + \cdots - 10) q^{98}+O(q^{100})$$ q + (b5 - b4 + 1) * q^2 + (b5 - 2*b4 - b3 - b2 - b1) * q^4 + b2 * q^5 + (b5 - b4 + b3 - b1 + 1) * q^7 + (-2*b3 - b1 - 4) * q^8 + (-b5 + b4 + b3 + b2 + b1) * q^10 + (b5 + 2*b4 - b1) * q^11 + (-b3 + 2*b1 + 1) * q^13 + (2*b5 - 2*b4 - b3 - b2 - b1 - 2) * q^14 + (-4*b5 + 5*b4 + b2 - 5) * q^16 + (-b5 + b4 + 2*b3 + 2*b2 + b1) * q^17 + (b5 + b4 - b2 - 1) * q^19 + (2*b1 + 5) * q^20 + (-b3 + 2*b1 - 1) * q^22 + (-b5 - 2*b4 - b2 + 2) * q^23 + (b5 + b4 + 2*b3 + 2*b2 - b1) * q^25 + (-6*b4 - b2 + 6) * q^26 + (-b4 - b3 + b2 - 3*b1 - 6) * q^28 + (2*b3 - b1 - 4) * q^29 + (-2*b5 + 3*b4 - b3 - b2 + 2*b1) * q^31 + (-4*b5 + 10*b4 + b3 + b2 + 4*b1) * q^32 + (3*b3 + 3*b1 + 6) * q^34 + (-b5 + 4*b4 + b3 - b2 + b1 - 3) * q^35 + (-b5 - 3*b4 - 2*b2 + 3) * q^37 + (-3*b4 - 2*b3 - 2*b2) * q^38 + (3*b5 - 9*b4 + 9) * q^40 + (b3 - 5*b1 - 2) * q^41 + (-3*b1 - 4) * q^43 - b2 * q^44 + (3*b5 - 3*b1) * q^46 + (-4*b5 - 2*b4 - b2 + 2) * q^47 + (3*b5 - 2*b3 - b2 - 3*b1 + 1) * q^49 + (b3 + 3*b1) * q^50 + (3*b5 - 5*b4 - 3*b3 - 3*b2 - 3*b1) * q^52 + (2*b5 + 7*b4 - b3 - b2 - 2*b1) * q^53 + (-2*b3 + b1 + 1) * q^55 + (-6*b5 + 9*b4 + b3 + 3*b2 + 2*b1 - 13) * q^56 + (-2*b5 + 5*b4 - b2 - 5) * q^58 + (-b5 - 5*b4 - b3 - b2 + b1) * q^59 + (-2*b5 + b4 + 2*b2 - 1) * q^61 + (b3 + 2*b1 + 8) * q^62 + (3*b3 + 3*b1 + 13) * q^64 + (-b5 - 2*b4 + 3*b2 + 2) * q^65 + (-b5 - 4*b4 + b3 + b2 + b1) * q^67 + (7*b5 - 16*b4 - 2*b2 + 16) * q^68 + (-b5 + b4 - 2*b2 + 3*b1 + 5) * q^70 + (3*b3 + 3*b1 + 3) * q^71 + (-5*b5 + 2*b3 + 2*b2 + 5*b1) * q^73 + (5*b5 - 2*b4 - b3 - b2 - 5*b1) * q^74 + (-3*b1 - 7) * q^76 + (4*b5 + 2*b4 + 2*b3 + 3*b2 - 2*b1 - 1) * q^77 + (3*b5 + 2*b4 + 3*b2 - 2) * q^79 + (5*b5 - 8*b4 - 3*b3 - 3*b2 - 5*b1) * q^80 + (-b5 + 16*b4 + 4*b2 - 16) * q^82 + (-3*b3 + 6*b1) * q^83 + (3*b3 - 3*b1 - 9) * q^85 + (-4*b5 + 13*b4 + 3*b2 - 13) * q^86 + (-3*b5 - 3*b4 - 3*b3 - 3*b2 + 3*b1) * q^88 + (3*b4 + 4*b2 - 3) * q^89 + (-6*b4 + b3 - b2 + 3*b1 - 1) * q^91 + (-b3 - 2*b1 - 5) * q^92 + (3*b5 + 9*b4 + 3*b3 + 3*b2 - 3*b1) * q^94 + (-2*b5 + 5*b4 - 3*b3 - 3*b2 + 2*b1) * q^95 + (-2*b3 - 2*b1 - 6) * q^97 + (-4*b3 + b2 - b1 - 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} - 4 q^{4} + q^{5} + 2 q^{7} - 18 q^{8}+O(q^{10})$$ 6 * q + 2 * q^2 - 4 * q^4 + q^5 + 2 * q^7 - 18 * q^8 $$6 q + 2 q^{2} - 4 q^{4} + q^{5} + 2 q^{7} - 18 q^{8} + q^{10} + 7 q^{11} + 4 q^{13} - 17 q^{14} - 10 q^{16} - 5 q^{19} + 26 q^{20} - 8 q^{22} + 6 q^{23} + 2 q^{25} + 17 q^{26} - 30 q^{28} - 26 q^{29} + 8 q^{31} + 25 q^{32} + 24 q^{34} - 10 q^{35} + 8 q^{37} - 7 q^{38} + 24 q^{40} - 4 q^{41} - 18 q^{43} - q^{44} + 3 q^{46} + 9 q^{47} + 12 q^{49} - 8 q^{50} - 9 q^{52} + 24 q^{53} + 8 q^{55} - 48 q^{56} - 14 q^{58} - 15 q^{59} + q^{61} + 42 q^{62} + 66 q^{64} + 10 q^{65} - 14 q^{67} + 39 q^{68} + 26 q^{70} + 6 q^{71} - 7 q^{73} - 36 q^{76} - q^{77} - 6 q^{79} - 16 q^{80} - 43 q^{82} - 6 q^{83} - 54 q^{85} - 32 q^{86} - 9 q^{88} - 5 q^{89} - 33 q^{91} - 24 q^{92} + 27 q^{94} + 16 q^{95} - 28 q^{97} - 49 q^{98}+O(q^{100})$$ 6 * q + 2 * q^2 - 4 * q^4 + q^5 + 2 * q^7 - 18 * q^8 + q^10 + 7 * q^11 + 4 * q^13 - 17 * q^14 - 10 * q^16 - 5 * q^19 + 26 * q^20 - 8 * q^22 + 6 * q^23 + 2 * q^25 + 17 * q^26 - 30 * q^28 - 26 * q^29 + 8 * q^31 + 25 * q^32 + 24 * q^34 - 10 * q^35 + 8 * q^37 - 7 * q^38 + 24 * q^40 - 4 * q^41 - 18 * q^43 - q^44 + 3 * q^46 + 9 * q^47 + 12 * q^49 - 8 * q^50 - 9 * q^52 + 24 * q^53 + 8 * q^55 - 48 * q^56 - 14 * q^58 - 15 * q^59 + q^61 + 42 * q^62 + 66 * q^64 + 10 * q^65 - 14 * q^67 + 39 * q^68 + 26 * q^70 + 6 * q^71 - 7 * q^73 - 36 * q^76 - q^77 - 6 * q^79 - 16 * q^80 - 43 * q^82 - 6 * q^83 - 54 * q^85 - 32 * q^86 - 9 * q^88 - 5 * q^89 - 33 * q^91 - 24 * q^92 + 27 * q^94 + 16 * q^95 - 28 * q^97 - 49 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ v^2 - v + 2 $$\beta_{2}$$ $$=$$ $$( -\nu^{5} + \nu^{4} - 8\nu^{3} + 5\nu^{2} - 18\nu + 6 ) / 3$$ (-v^5 + v^4 - 8*v^3 + 5*v^2 - 18*v + 6) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2\nu^{3} + 6\nu^{2} - 5\nu + 3$$ v^4 - 2*v^3 + 6*v^2 - 5*v + 3 $$\beta_{4}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 9 ) / 3$$ (-2*v^5 + 5*v^4 - 16*v^3 + 19*v^2 - 21*v + 9) / 3 $$\beta_{5}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 30\nu - 9 ) / 3$$ (2*v^5 - 5*v^4 + 19*v^3 - 22*v^2 + 30*v - 9) / 3
 $$\nu$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + \beta _1 + 2 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} + 4\beta _1 - 4 ) / 3$$ (-2*b5 - b4 - b3 - 2*b2 + 4*b1 - 4) / 3 $$\nu^{3}$$ $$=$$ $$( 7\beta_{5} + 5\beta_{4} + 2\beta_{3} + 4\beta_{2} + \beta _1 - 10 ) / 3$$ (7*b5 + 5*b4 + 2*b3 + 4*b2 + b1 - 10) / 3 $$\nu^{4}$$ $$=$$ $$( 16\beta_{5} + 11\beta_{4} + 8\beta_{3} + 10\beta_{2} - 17\beta _1 + 5 ) / 3$$ (16*b5 + 11*b4 + 8*b3 + 10*b2 - 17*b1 + 5) / 3 $$\nu^{5}$$ $$=$$ $$( -14\beta_{5} - 16\beta_{4} + 5\beta_{3} - 5\beta_{2} - 23\beta _1 + 47 ) / 3$$ (-14*b5 - 16*b4 + 5*b3 - 5*b2 - 23*b1 + 47) / 3

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{4}$$

## 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
 0.5 + 2.05195i 0.5 − 1.41036i 0.5 + 0.224437i 0.5 − 2.05195i 0.5 + 1.41036i 0.5 − 0.224437i
−0.730252 1.26483i 0 −0.0665372 + 0.115246i −0.296790 0.514055i 0 2.32383 1.26483i −2.72665 0 −0.433463 + 0.750780i
109.2 0.380438 + 0.658939i 0 0.710533 1.23068i 1.59097 + 2.75564i 0 −2.56238 + 0.658939i 2.60301 0 −1.21053 + 2.09671i
109.3 1.34981 + 2.33795i 0 −2.64400 + 4.57954i −0.794182 1.37556i 0 1.23855 + 2.33795i −8.87636 0 2.14400 3.71351i
163.1 −0.730252 + 1.26483i 0 −0.0665372 0.115246i −0.296790 + 0.514055i 0 2.32383 + 1.26483i −2.72665 0 −0.433463 0.750780i
163.2 0.380438 0.658939i 0 0.710533 + 1.23068i 1.59097 2.75564i 0 −2.56238 0.658939i 2.60301 0 −1.21053 2.09671i
163.3 1.34981 2.33795i 0 −2.64400 4.57954i −0.794182 + 1.37556i 0 1.23855 2.33795i −8.87636 0 2.14400 + 3.71351i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.e.f yes 6
3.b odd 2 1 189.2.e.e 6
7.c even 3 1 inner 189.2.e.f yes 6
7.c even 3 1 1323.2.a.x 3
7.d odd 6 1 1323.2.a.y 3
9.c even 3 1 567.2.g.i 6
9.c even 3 1 567.2.h.h 6
9.d odd 6 1 567.2.g.h 6
9.d odd 6 1 567.2.h.i 6
21.g even 6 1 1323.2.a.z 3
21.h odd 6 1 189.2.e.e 6
21.h odd 6 1 1323.2.a.ba 3
63.g even 3 1 567.2.h.h 6
63.h even 3 1 567.2.g.i 6
63.j odd 6 1 567.2.g.h 6
63.n odd 6 1 567.2.h.i 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.e.e 6 3.b odd 2 1
189.2.e.e 6 21.h odd 6 1
189.2.e.f yes 6 1.a even 1 1 trivial
189.2.e.f yes 6 7.c even 3 1 inner
567.2.g.h 6 9.d odd 6 1
567.2.g.h 6 63.j odd 6 1
567.2.g.i 6 9.c even 3 1
567.2.g.i 6 63.h even 3 1
567.2.h.h 6 9.c even 3 1
567.2.h.h 6 63.g even 3 1
567.2.h.i 6 9.d odd 6 1
567.2.h.i 6 63.n odd 6 1
1323.2.a.x 3 7.c even 3 1
1323.2.a.y 3 7.d odd 6 1
1323.2.a.z 3 21.g even 6 1
1323.2.a.ba 3 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 2T_{2}^{5} + 7T_{2}^{4} + 15T_{2}^{2} - 9T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} - T^{5} + 7 T^{4} + \cdots + 9$$
$7$ $$T^{6} - 2 T^{5} + \cdots + 343$$
$11$ $$T^{6} - 7 T^{5} + \cdots + 9$$
$13$ $$(T^{3} - 2 T^{2} - 19 T + 47)^{2}$$
$17$ $$T^{6} + 33 T^{4} + \cdots + 81$$
$19$ $$T^{6} + 5 T^{5} + \cdots + 841$$
$23$ $$T^{6} - 6 T^{5} + \cdots + 81$$
$29$ $$(T^{3} + 13 T^{2} + 30 T + 9)^{2}$$
$31$ $$T^{6} - 8 T^{5} + \cdots + 4761$$
$37$ $$T^{6} - 8 T^{5} + \cdots + 8649$$
$41$ $$(T^{3} + 2 T^{2} + \cdots - 387)^{2}$$
$43$ $$(T^{3} + 9 T^{2} + \cdots - 101)^{2}$$
$47$ $$T^{6} - 9 T^{5} + \cdots + 81$$
$53$ $$T^{6} - 24 T^{5} + \cdots + 59049$$
$59$ $$T^{6} + 15 T^{5} + \cdots + 6561$$
$61$ $$T^{6} - T^{5} + \cdots + 14641$$
$67$ $$T^{6} + 14 T^{5} + \cdots + 961$$
$71$ $$(T^{3} - 3 T^{2} + \cdots - 243)^{2}$$
$73$ $$T^{6} + 7 T^{5} + \cdots + 962361$$
$79$ $$T^{6} + 6 T^{5} + \cdots + 16129$$
$83$ $$(T^{3} + 3 T^{2} + \cdots + 729)^{2}$$
$89$ $$T^{6} + 5 T^{5} + \cdots + 239121$$
$97$ $$(T^{3} + 14 T^{2} + \cdots - 24)^{2}$$