# Properties

 Label 189.2.p.d Level $189$ Weight $2$ Character orbit 189.p Analytic conductor $1.509$ Analytic rank $0$ Dimension $12$ 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(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: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81$$ x^12 - 9*x^10 + 59*x^8 - 180*x^6 + 403*x^4 - 198*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{9} - \beta_{7} - \beta_{3} + 1) q^{4} + (\beta_{6} + \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{4} - 1) q^{7} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b9 - b7 - b3 + 1) * q^4 + (b6 + b5) * q^5 + (-b9 + b4 - 1) * q^7 + (b11 - b10 + b8) * q^8 $$q - \beta_1 q^{2} + (\beta_{9} - \beta_{7} - \beta_{3} + 1) q^{4} + (\beta_{6} + \beta_{5}) q^{5} + ( - \beta_{9} + \beta_{4} - 1) q^{7} + (\beta_{11} - \beta_{10} + \beta_{8}) q^{8} + (\beta_{9} + \beta_{7} + \cdots - 2 \beta_{2}) q^{10}+ \cdots + (7 \beta_{11} - \beta_{10} + \cdots + 4 \beta_1) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b9 - b7 - b3 + 1) * q^4 + (b6 + b5) * q^5 + (-b9 + b4 - 1) * q^7 + (b11 - b10 + b8) * q^8 + (b9 + b7 - 2*b4 - 2*b2) * q^10 + (b11 - b10 + b1) * q^11 + (-b9 + 2*b3 + b2 - 1) * q^13 + (-b11 + b10 - b8 + b6) * q^14 + (-b7 + b4 - b3 + 2*b2) * q^16 + (b11 - b1) * q^17 + (-b9 + 2*b7) * q^19 + (b10 + b8) * q^20 + (-b9 + 2*b7 + 2*b4 + b2 - 3) * q^22 + (-b8 - b6 + b5 - b1) * q^23 + (2*b9 - 2*b7 - b4 + 4*b3 + b2 - 4) * q^25 + (-4*b11 - b6 - b5 - 2*b1) * q^26 + (-2*b9 + 2*b7 - b4 + b3 - 2*b2 + 2) * q^28 + (-b11 - b10 + b8) * q^29 + (-b9 - b7 - b4 - b3 - b2 - 1) * q^31 + (-b6 - 2*b5) * q^32 + (2*b9 - b7 - 6*b3 + 3) * q^34 + (3*b11 + 2*b10 - b8 - b6 + 3*b1) * q^35 + (b9 + b7 - b4 + 2*b3 - 2*b2) * q^37 + (-2*b11 + b10 - 2*b8 + 2*b1) * q^38 + (b9 - 3*b7 + b4) * q^40 + (b11 + b10 + b8 - b5 + 2*b1) * q^41 + (-b9 - b7 + 2*b4 + b2 + 1) * q^43 + (-2*b8 + b6 - b5 + 6*b1) * q^44 + (-b4 - 3*b3 + b2 + 3) * q^46 + (4*b11 - b6 - b5 + 2*b1) * q^47 + (2*b9 + b7 - 3*b3 - b2 + 4) * q^49 + (-b11 - 3*b10 + 3*b8 - 2*b6 - b5) * q^50 + (-b9 - b7 + 4*b3 + 4) * q^52 + (-2*b11 - b10 + b6 + 2*b5 - 2*b1) * q^53 + (3*b9 - b7 - b2) * q^55 + (-4*b11 + 2*b10 - b8 - b6 - 4*b1) * q^56 + (-b7 + b4 + 3*b3 + 2*b2) * q^58 + (-b11 - b6 + b1) * q^59 + (-2*b9 + 2*b7 + 2*b4 - b3 + 2) * q^61 + (-2*b11 + 2*b10 + 2*b8 + b5 - 4*b1) * q^62 + (-b7 + 2) * q^64 + (-3*b8 + b6 - b5 - 3*b1) * q^65 + (b9 - b7 - 2*b4 - 7*b3 + 2*b2 + 7) * q^67 + (6*b11 - 2*b10 + b8 + 3*b1) * q^68 + (2*b7 - 3*b4 - b2 - 9) * q^70 + (-b11 + 2*b6 + b5) * q^71 + (-2*b9 - 2*b7 + 3*b4 - 3*b3 + 3*b2 - 3) * q^73 + (-b11 + b10 + b6 + 2*b5 - b1) * q^74 + (-3*b9 + 3*b7 + 12*b3 - 3*b2 - 6) * q^76 + (b10 + b8 - b6 - b5 - 4*b1) * q^77 + (b9 + 2*b7 - 2*b4 - 5*b3 - 4*b2) * q^79 + (3*b11 + b10 - 2*b8 + b6 - 3*b1) * q^80 + (b9 - b7 - b4 + 3*b3 - 6) * q^82 + (2*b11 + b5 + 4*b1) * q^83 + (2*b9 + b7 - 4*b4 - 2*b2) * q^85 + (-b8 + b6 - b5 - 2*b1) * q^86 + (-5*b9 + 5*b7 + 2*b4 + 12*b3 - 2*b2 - 12) * q^88 + (-4*b11 - 2*b10 + b8 - 2*b1) * q^89 + (-2*b7 + 3*b4 + 4*b3 + 4*b2 - 2) * q^91 + (-3*b10 + 3*b8 + 2*b6 + b5) * q^92 + (b9 + b7 + 2*b4 - 6*b3 + 2*b2 - 6) * q^94 + (-3*b10 + b6 + 2*b5) * q^95 + (-2*b9 - 8*b3 + 2*b2 + 4) * q^97 + (7*b11 - b10 - b8 + b6 + b5 + 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 8 q^{4} - 8 q^{7}+O(q^{10})$$ 12 * q + 8 * q^4 - 8 * q^7 $$12 q + 8 q^{4} - 8 q^{7} - 6 q^{10} - 4 q^{16} - 6 q^{19} - 40 q^{22} - 24 q^{25} + 28 q^{28} - 12 q^{31} + 8 q^{37} + 12 q^{40} + 20 q^{43} + 14 q^{46} + 24 q^{49} + 78 q^{52} + 20 q^{58} + 18 q^{61} + 28 q^{64} + 36 q^{67} - 120 q^{70} - 42 q^{73} - 36 q^{79} - 54 q^{82} - 12 q^{85} - 74 q^{88} + 6 q^{91} - 114 q^{94}+O(q^{100})$$ 12 * q + 8 * q^4 - 8 * q^7 - 6 * q^10 - 4 * q^16 - 6 * q^19 - 40 * q^22 - 24 * q^25 + 28 * q^28 - 12 * q^31 + 8 * q^37 + 12 * q^40 + 20 * q^43 + 14 * q^46 + 24 * q^49 + 78 * q^52 + 20 * q^58 + 18 * q^61 + 28 * q^64 + 36 * q^67 - 120 * q^70 - 42 * q^73 - 36 * q^79 - 54 * q^82 - 12 * q^85 - 74 * q^88 + 6 * q^91 - 114 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 9x^{10} + 59x^{8} - 180x^{6} + 403x^{4} - 198x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( 81\nu^{11} - 531\nu^{9} + 3481\nu^{7} - 3627\nu^{5} + 1782\nu^{3} + 76298\nu ) / 21995$$ (81*v^11 - 531*v^9 + 3481*v^7 - 3627*v^5 + 1782*v^3 + 76298*v) / 21995 $$\beta_{2}$$ $$=$$ $$( 117\nu^{10} - 767\nu^{8} + 7472\nu^{6} - 27234\nu^{4} + 90554\nu^{2} - 60864 ) / 21995$$ (117*v^10 - 767*v^8 + 7472*v^6 - 27234*v^4 + 90554*v^2 - 60864) / 21995 $$\beta_{3}$$ $$=$$ $$( -1298\nu^{10} + 10953\nu^{8} - 71803\nu^{6} + 202311\nu^{4} - 490451\nu^{2} + 240966 ) / 197955$$ (-1298*v^10 + 10953*v^8 - 71803*v^6 + 202311*v^4 - 490451*v^2 + 240966) / 197955 $$\beta_{4}$$ $$=$$ $$( 461\nu^{10} - 5466\nu^{8} + 28501\nu^{6} - 98847\nu^{4} + 142112\nu^{2} - 186237 ) / 65985$$ (461*v^10 - 5466*v^8 + 28501*v^6 - 98847*v^4 + 142112*v^2 - 186237) / 65985 $$\beta_{5}$$ $$=$$ $$( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} - 438921\nu ) / 197955$$ (1298*v^11 - 10953*v^9 + 71803*v^7 - 202311*v^5 + 490451*v^3 - 438921*v) / 197955 $$\beta_{6}$$ $$=$$ $$( 1298\nu^{11} - 10953\nu^{9} + 71803\nu^{7} - 202311\nu^{5} + 490451\nu^{3} + 154944\nu ) / 197955$$ (1298*v^11 - 10953*v^9 + 71803*v^7 - 202311*v^5 + 490451*v^3 + 154944*v) / 197955 $$\beta_{7}$$ $$=$$ $$( -288\nu^{10} + 1888\nu^{8} - 9933\nu^{6} + 12896\nu^{4} - 6336\nu^{2} - 68439 ) / 21995$$ (-288*v^10 + 1888*v^8 - 9933*v^6 + 12896*v^4 - 6336*v^2 - 68439) / 21995 $$\beta_{8}$$ $$=$$ $$( -288\nu^{11} + 1888\nu^{9} - 9933\nu^{7} + 12896\nu^{5} - 6336\nu^{3} - 90434\nu ) / 21995$$ (-288*v^11 + 1888*v^9 - 9933*v^7 + 12896*v^5 - 6336*v^3 - 90434*v) / 21995 $$\beta_{9}$$ $$=$$ $$( 1138\nu^{10} - 12348\nu^{8} + 80948\nu^{6} - 273351\nu^{4} + 552916\nu^{2} - 271656 ) / 65985$$ (1138*v^10 - 12348*v^8 + 80948*v^6 - 273351*v^4 + 552916*v^2 - 271656) / 65985 $$\beta_{10}$$ $$=$$ $$( 4712\nu^{11} - 47997\nu^{9} + 314647\nu^{7} - 1022364\nu^{5} + 2149199\nu^{3} - 1055934\nu ) / 197955$$ (4712*v^11 - 47997*v^9 + 314647*v^7 - 1022364*v^5 + 2149199*v^3 - 1055934*v) / 197955 $$\beta_{11}$$ $$=$$ $$( -5192\nu^{11} + 43812\nu^{9} - 287212\nu^{7} + 809244\nu^{5} - 1763849\nu^{3} + 172044\nu ) / 197955$$ (-5192*v^11 + 43812*v^9 - 287212*v^7 + 809244*v^5 - 1763849*v^3 + 172044*v) / 197955
 $$\nu$$ $$=$$ $$( \beta_{6} - \beta_{5} ) / 3$$ (b6 - b5) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{9} + 2\beta_{7} + \beta_{4} - 9\beta_{3} - \beta_{2} + 9 ) / 3$$ (-2*b9 + 2*b7 + b4 - 9*b3 - b2 + 9) / 3 $$\nu^{3}$$ $$=$$ $$( 3\beta_{11} + 8\beta_{6} + 4\beta_{5} ) / 3$$ (3*b11 + 8*b6 + 4*b5) / 3 $$\nu^{4}$$ $$=$$ $$-\beta_{9} + 2\beta_{7} - 2\beta_{4} - 12\beta_{3} - 4\beta_{2}$$ -b9 + 2*b7 - 2*b4 - 12*b3 - 4*b2 $$\nu^{5}$$ $$=$$ $$( 18\beta_{11} + 3\beta_{10} + 17\beta_{6} + 34\beta_{5} + 18\beta_1 ) / 3$$ (18*b11 + 3*b10 + 17*b6 + 34*b5 + 18*b1) / 3 $$\nu^{6}$$ $$=$$ $$( 32\beta_{9} - 5\beta_{7} - 64\beta_{4} - 32\beta_{2} - 153 ) / 3$$ (32*b9 - 5*b7 - 64*b4 - 32*b2 - 153) / 3 $$\nu^{7}$$ $$=$$ $$( 27\beta_{8} - 74\beta_{6} + 74\beta_{5} + 96\beta_1 ) / 3$$ (27*b8 - 74*b6 + 74*b5 + 96*b1) / 3 $$\nu^{8}$$ $$=$$ $$51\beta_{9} - 51\beta_{7} - 55\beta_{4} + 222\beta_{3} + 55\beta_{2} - 222$$ 51*b9 - 51*b7 - 55*b4 + 222*b3 + 55*b2 - 222 $$\nu^{9}$$ $$=$$ $$( -495\beta_{11} - 177\beta_{10} + 177\beta_{8} - 656\beta_{6} - 328\beta_{5} ) / 3$$ (-495*b11 - 177*b10 + 177*b8 - 656*b6 - 328*b5) / 3 $$\nu^{10}$$ $$=$$ $$( -191\beta_{9} - 835\beta_{7} + 835\beta_{4} + 2952\beta_{3} + 1670\beta_{2} ) / 3$$ (-191*b9 - 835*b7 + 835*b4 + 2952*b3 + 1670*b2) / 3 $$\nu^{11}$$ $$=$$ $$( -2505\beta_{11} - 1026\beta_{10} - 1477\beta_{6} - 2954\beta_{5} - 2505\beta_1 ) / 3$$ (-2505*b11 - 1026*b10 - 1477*b6 - 2954*b5 - 2505*b1) / 3

## Character values

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

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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
 0.617942 + 0.356769i −1.90412 − 1.09935i 1.65604 + 0.956115i −1.65604 − 0.956115i 1.90412 + 1.09935i −0.617942 − 0.356769i 0.617942 − 0.356769i −1.90412 + 1.09935i 1.65604 − 0.956115i −1.65604 + 0.956115i 1.90412 − 1.09935i −0.617942 + 0.356769i
−2.15715 1.24543i 0 2.10220 + 3.64112i −0.617942 + 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
26.2 −1.58850 0.917122i 0 0.682224 + 1.18165i 1.90412 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.3 −0.568650 0.328310i 0 −0.784425 1.35866i −1.65604 + 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.4 0.568650 + 0.328310i 0 −0.784425 1.35866i 1.65604 2.86834i 0 −2.58392 + 0.568650i 2.34338i 0 1.88341 1.08739i
26.5 1.58850 + 0.917122i 0 0.682224 + 1.18165i −1.90412 + 3.29804i 0 2.11581 + 1.58850i 1.16576i 0 −6.04940 + 3.49262i
26.6 2.15715 + 1.24543i 0 2.10220 + 3.64112i 0.617942 1.07031i 0 −1.53189 2.15715i 5.49086i 0 2.66599 1.53921i
80.1 −2.15715 + 1.24543i 0 2.10220 3.64112i −0.617942 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
80.2 −1.58850 + 0.917122i 0 0.682224 1.18165i 1.90412 + 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.3 −0.568650 + 0.328310i 0 −0.784425 + 1.35866i −1.65604 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.4 0.568650 0.328310i 0 −0.784425 + 1.35866i 1.65604 + 2.86834i 0 −2.58392 0.568650i 2.34338i 0 1.88341 + 1.08739i
80.5 1.58850 0.917122i 0 0.682224 1.18165i −1.90412 3.29804i 0 2.11581 1.58850i 1.16576i 0 −6.04940 3.49262i
80.6 2.15715 1.24543i 0 2.10220 3.64112i 0.617942 + 1.07031i 0 −1.53189 + 2.15715i 5.49086i 0 2.66599 + 1.53921i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.6 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.d 12
3.b odd 2 1 inner 189.2.p.d 12
7.c even 3 1 1323.2.c.d 12
7.d odd 6 1 inner 189.2.p.d 12
7.d odd 6 1 1323.2.c.d 12
9.c even 3 1 567.2.i.f 12
9.c even 3 1 567.2.s.f 12
9.d odd 6 1 567.2.i.f 12
9.d odd 6 1 567.2.s.f 12
21.g even 6 1 inner 189.2.p.d 12
21.g even 6 1 1323.2.c.d 12
21.h odd 6 1 1323.2.c.d 12
63.i even 6 1 567.2.s.f 12
63.k odd 6 1 567.2.i.f 12
63.s even 6 1 567.2.i.f 12
63.t odd 6 1 567.2.s.f 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.d 12 1.a even 1 1 trivial
189.2.p.d 12 3.b odd 2 1 inner
189.2.p.d 12 7.d odd 6 1 inner
189.2.p.d 12 21.g even 6 1 inner
567.2.i.f 12 9.c even 3 1
567.2.i.f 12 9.d odd 6 1
567.2.i.f 12 63.k odd 6 1
567.2.i.f 12 63.s even 6 1
567.2.s.f 12 9.c even 3 1
567.2.s.f 12 9.d odd 6 1
567.2.s.f 12 63.i even 6 1
567.2.s.f 12 63.t odd 6 1
1323.2.c.d 12 7.c even 3 1
1323.2.c.d 12 7.d odd 6 1
1323.2.c.d 12 21.g even 6 1
1323.2.c.d 12 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} - 10T_{2}^{10} + 75T_{2}^{8} - 232T_{2}^{6} + 535T_{2}^{4} - 225T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 10 T^{10} + \cdots + 81$$
$3$ $$T^{12}$$
$5$ $$T^{12} + 27 T^{10} + \cdots + 59049$$
$7$ $$(T^{6} + 4 T^{5} + \cdots + 343)^{2}$$
$11$ $$T^{12} - 37 T^{10} + \cdots + 50625$$
$13$ $$(T^{6} + 42 T^{4} + \cdots + 675)^{2}$$
$17$ $$T^{12} + 30 T^{10} + \cdots + 59049$$
$19$ $$(T^{6} + 3 T^{5} + \cdots + 243)^{2}$$
$23$ $$T^{12} - 94 T^{10} + \cdots + 531441$$
$29$ $$(T^{6} + 37 T^{4} + \cdots + 81)^{2}$$
$31$ $$(T^{6} + 6 T^{5} + \cdots + 64827)^{2}$$
$37$ $$(T^{6} - 4 T^{5} + \cdots + 4489)^{2}$$
$41$ $$(T^{6} - 114 T^{4} + \cdots - 243)^{2}$$
$43$ $$(T^{3} - 5 T^{2} - 16 T - 1)^{4}$$
$47$ $$T^{12} + \cdots + 387420489$$
$53$ $$T^{12} - 118 T^{10} + \cdots + 81$$
$59$ $$T^{12} + 63 T^{10} + \cdots + 4782969$$
$61$ $$(T^{6} - 9 T^{5} + \cdots + 49923)^{2}$$
$67$ $$(T^{6} - 18 T^{5} + \cdots + 458329)^{2}$$
$71$ $$(T^{6} + 85 T^{4} + \cdots + 19881)^{2}$$
$73$ $$(T^{6} + 21 T^{5} + \cdots + 177147)^{2}$$
$79$ $$(T^{6} + 18 T^{5} + \cdots + 49)^{2}$$
$83$ $$(T^{6} - 159 T^{4} + \cdots - 6075)^{2}$$
$89$ $$T^{12} + \cdots + 7695324729$$
$97$ $$(T^{6} + 204 T^{4} + \cdots + 1728)^{2}$$