# Properties

 Label 189.2.be.a Level $189$ Weight $2$ Character orbit 189.be Analytic conductor $1.509$ Analytic rank $0$ Dimension $132$ 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(20,189)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([7, 9]))

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

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

chi := DirichletCharacter("189.20");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.be (of order $$18$$, degree $$6$$, 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: $$132$$ Relative dimension: $$22$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 12 q^{2} - 12 q^{4} - 6 q^{7} - 18 q^{8} - 6 q^{9}+O(q^{10})$$ 132 * q - 12 * q^2 - 12 * q^4 - 6 * q^7 - 18 * q^8 - 6 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 12 q^{2} - 12 q^{4} - 6 q^{7} - 18 q^{8} - 6 q^{9} - 18 q^{11} + 3 q^{14} - 24 q^{15} - 24 q^{16} - 12 q^{18} - 12 q^{21} - 12 q^{22} + 12 q^{23} - 12 q^{25} - 12 q^{28} - 48 q^{29} + 42 q^{30} - 6 q^{32} - 36 q^{35} - 36 q^{36} - 6 q^{37} - 18 q^{39} - 12 q^{43} - 18 q^{44} - 6 q^{46} - 24 q^{49} + 18 q^{50} + 24 q^{51} + 57 q^{56} - 12 q^{58} - 6 q^{60} + 21 q^{63} + 18 q^{64} + 78 q^{65} - 12 q^{67} - 69 q^{70} + 18 q^{71} + 114 q^{72} - 6 q^{74} - 57 q^{77} + 12 q^{78} + 24 q^{79} - 42 q^{81} - 48 q^{84} + 54 q^{85} - 42 q^{86} - 72 q^{88} + 6 q^{91} - 120 q^{92} - 60 q^{93} + 126 q^{95} + 126 q^{98} - 192 q^{99}+O(q^{100})$$ 132 * q - 12 * q^2 - 12 * q^4 - 6 * q^7 - 18 * q^8 - 6 * q^9 - 18 * q^11 + 3 * q^14 - 24 * q^15 - 24 * q^16 - 12 * q^18 - 12 * q^21 - 12 * q^22 + 12 * q^23 - 12 * q^25 - 12 * q^28 - 48 * q^29 + 42 * q^30 - 6 * q^32 - 36 * q^35 - 36 * q^36 - 6 * q^37 - 18 * q^39 - 12 * q^43 - 18 * q^44 - 6 * q^46 - 24 * q^49 + 18 * q^50 + 24 * q^51 + 57 * q^56 - 12 * q^58 - 6 * q^60 + 21 * q^63 + 18 * q^64 + 78 * q^65 - 12 * q^67 - 69 * q^70 + 18 * q^71 + 114 * q^72 - 6 * q^74 - 57 * q^77 + 12 * q^78 + 24 * q^79 - 42 * q^81 - 48 * q^84 + 54 * q^85 - 42 * q^86 - 72 * q^88 + 6 * q^91 - 120 * q^92 - 60 * q^93 + 126 * q^95 + 126 * q^98 - 192 * q^99

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1 −0.878867 2.41467i −1.25404 1.19473i −3.52612 + 2.95877i −0.588152 3.33557i −1.78274 + 4.07810i −0.566293 2.58444i 5.79269 + 3.34441i 0.145238 + 2.99648i −7.53740 + 4.35172i
20.2 −0.878867 2.41467i 1.25404 + 1.19473i −3.52612 + 2.95877i 0.588152 + 3.33557i 1.78274 4.07810i −2.64351 0.108907i 5.79269 + 3.34441i 0.145238 + 2.99648i 7.53740 4.35172i
20.3 −0.752422 2.06726i −1.63905 0.559928i −2.17534 + 1.82533i 0.562637 + 3.19087i 0.0757386 + 3.80964i 1.24057 + 2.33688i 1.59981 + 0.923650i 2.37296 + 1.83550i 6.17302 3.56400i
20.4 −0.752422 2.06726i 1.63905 + 0.559928i −2.17534 + 1.82533i −0.562637 3.19087i −0.0757386 3.80964i 2.51680 + 0.815925i 1.59981 + 0.923650i 2.37296 + 1.83550i −6.17302 + 3.56400i
20.5 −0.681310 1.87188i −0.917364 + 1.46916i −1.50768 + 1.26509i −0.291735 1.65451i 3.37511 + 0.716242i −2.47296 + 0.940471i −0.0549773 0.0317412i −1.31689 2.69552i −2.89829 + 1.67333i
20.6 −0.681310 1.87188i 0.917364 1.46916i −1.50768 + 1.26509i 0.291735 + 1.65451i −3.37511 0.716242i 0.496759 2.59870i −0.0549773 0.0317412i −1.31689 2.69552i 2.89829 1.67333i
20.7 −0.422526 1.16088i −1.51095 + 0.846773i 0.362974 0.304571i 0.253991 + 1.44045i 1.62142 + 1.39625i 1.07486 2.41758i −2.64668 1.52806i 1.56595 2.55887i 1.56488 0.903482i
20.8 −0.422526 1.16088i 1.51095 0.846773i 0.362974 0.304571i −0.253991 1.44045i −1.62142 1.39625i −2.19420 + 1.47834i −2.64668 1.52806i 1.56595 2.55887i −1.56488 + 0.903482i
20.9 −0.234777 0.645045i −0.440723 1.67504i 1.17113 0.982692i −0.231854 1.31491i −0.977005 + 0.677547i 2.46223 + 0.968204i −2.09779 1.21116i −2.61153 + 1.47646i −0.793741 + 0.458267i
20.10 −0.234777 0.645045i 0.440723 + 1.67504i 1.17113 0.982692i 0.231854 + 1.31491i 0.977005 0.677547i 1.38106 + 2.25670i −2.09779 1.21116i −2.61153 + 1.47646i 0.793741 0.458267i
20.11 −0.0448460 0.123213i −1.62941 0.587383i 1.51892 1.27452i −0.231324 1.31190i 0.000699206 0.227107i −2.62614 0.321557i −0.452264 0.261115i 2.30996 + 1.91418i −0.151270 + 0.0873359i
20.12 −0.0448460 0.123213i 1.62941 + 0.587383i 1.51892 1.27452i 0.231324 + 1.31190i −0.000699206 0.227107i −0.772696 2.53040i −0.452264 0.261115i 2.30996 + 1.91418i 0.151270 0.0873359i
20.13 0.157445 + 0.432576i −0.888001 + 1.48710i 1.36976 1.14936i −0.725040 4.11191i −0.783093 0.149993i 2.17886 1.50085i 1.51017 + 0.871900i −1.42291 2.64109i 1.66456 0.961033i
20.14 0.157445 + 0.432576i 0.888001 1.48710i 1.36976 1.14936i 0.725040 + 4.11191i 0.783093 + 0.149993i −1.09969 + 2.40638i 1.51017 + 0.871900i −1.42291 2.64109i −1.66456 + 0.961033i
20.15 0.471430 + 1.29524i −1.37746 1.05005i 0.0766803 0.0643424i 0.459149 + 2.60396i 0.710690 2.27917i 1.90985 1.83097i 2.50689 + 1.44736i 0.794802 + 2.89280i −3.15631 + 1.82230i
20.16 0.471430 + 1.29524i 1.37746 + 1.05005i 0.0766803 0.0643424i −0.459149 2.60396i −0.710690 + 2.27917i −1.47152 + 2.19878i 2.50689 + 1.44736i 0.794802 + 2.89280i 3.15631 1.82230i
20.17 0.492726 + 1.35375i −0.333137 + 1.69971i −0.0577819 + 0.0484848i 0.440273 + 2.49691i −2.46514 + 0.386507i −1.60923 2.10009i 2.40115 + 1.38630i −2.77804 1.13247i −3.16327 + 1.82631i
20.18 0.492726 + 1.35375i 0.333137 1.69971i −0.0577819 + 0.0484848i −0.440273 2.49691i 2.46514 0.386507i −2.34762 1.22011i 2.40115 + 1.38630i −2.77804 1.13247i 3.16327 1.82631i
20.19 0.735125 + 2.01974i −1.28358 + 1.16294i −2.00685 + 1.68395i 0.0677816 + 0.384409i −3.29242 1.73760i 0.618456 + 2.57245i −1.15362 0.666045i 0.295160 2.98544i −0.726578 + 0.419490i
20.20 0.735125 + 2.01974i 1.28358 1.16294i −2.00685 + 1.68395i −0.0677816 0.384409i 3.29242 + 1.73760i 2.64076 + 0.162359i −1.15362 0.666045i 0.295160 2.98544i 0.726578 0.419490i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 20.22 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.b odd 2 1 inner
27.f odd 18 1 inner
189.be even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.be.a 132
3.b odd 2 1 567.2.be.a 132
7.b odd 2 1 inner 189.2.be.a 132
21.c even 2 1 567.2.be.a 132
27.e even 9 1 567.2.be.a 132
27.f odd 18 1 inner 189.2.be.a 132
189.y odd 18 1 567.2.be.a 132
189.be even 18 1 inner 189.2.be.a 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.be.a 132 1.a even 1 1 trivial
189.2.be.a 132 7.b odd 2 1 inner
189.2.be.a 132 27.f odd 18 1 inner
189.2.be.a 132 189.be even 18 1 inner
567.2.be.a 132 3.b odd 2 1
567.2.be.a 132 21.c even 2 1
567.2.be.a 132 27.e even 9 1
567.2.be.a 132 189.y odd 18 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(189, [\chi])$$.