# Properties

 Label 189.2.u.a Level $189$ Weight $2$ Character orbit 189.u Analytic conductor $1.509$ Analytic rank $0$ Dimension $132$ 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(4,189)]

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([2, 12]))

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

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

chi := DirichletCharacter("189.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.u (of order $$9$$, 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 $$\Q(\zeta_{9})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9}+O(q^{10})$$ 132 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 3 * q^5 - 18 * q^6 - 6 * q^7 - 6 * q^8 - 15 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 3 q^{2} - 3 q^{3} - 3 q^{4} - 3 q^{5} - 18 q^{6} - 6 q^{7} - 6 q^{8} - 15 q^{9} + 3 q^{10} - 15 q^{11} - 3 q^{12} - 12 q^{13} - 30 q^{14} + 9 q^{16} + 27 q^{17} - 3 q^{18} + 3 q^{19} - 18 q^{20} + 15 q^{21} - 12 q^{22} - 36 q^{23} - 72 q^{24} - 3 q^{25} + 30 q^{26} - 12 q^{27} - 12 q^{28} - 30 q^{29} - 3 q^{30} - 3 q^{31} - 75 q^{32} + 15 q^{33} - 18 q^{34} + 15 q^{35} - 60 q^{36} - 6 q^{37} + 69 q^{38} + 51 q^{39} + 51 q^{40} - 39 q^{42} - 12 q^{43} - 6 q^{44} - 21 q^{45} - 6 q^{46} - 21 q^{47} + 90 q^{48} - 42 q^{49} - 39 q^{50} + 33 q^{51} + 9 q^{52} + 9 q^{53} - 9 q^{54} - 24 q^{55} + 111 q^{56} - 18 q^{57} - 3 q^{58} + 27 q^{59} - 63 q^{60} - 21 q^{61} + 75 q^{62} + 63 q^{63} - 30 q^{64} - 90 q^{65} - 3 q^{66} - 3 q^{67} - 30 q^{68} - 6 q^{69} + 39 q^{70} - 18 q^{71} + 183 q^{72} - 42 q^{73} + 51 q^{74} - 45 q^{75} - 24 q^{76} + 15 q^{77} - 30 q^{78} + 15 q^{79} + 102 q^{80} - 87 q^{81} - 6 q^{82} - 42 q^{83} + 135 q^{84} - 63 q^{85} - 93 q^{86} + 75 q^{87} - 51 q^{88} + 75 q^{89} - 39 q^{90} - 21 q^{91} - 66 q^{92} + 81 q^{93} + 33 q^{94} + 15 q^{95} - 171 q^{96} - 12 q^{97} - 36 q^{98} + 24 q^{99}+O(q^{100})$$ 132 * q - 3 * q^2 - 3 * q^3 - 3 * q^4 - 3 * q^5 - 18 * q^6 - 6 * q^7 - 6 * q^8 - 15 * q^9 + 3 * q^10 - 15 * q^11 - 3 * q^12 - 12 * q^13 - 30 * q^14 + 9 * q^16 + 27 * q^17 - 3 * q^18 + 3 * q^19 - 18 * q^20 + 15 * q^21 - 12 * q^22 - 36 * q^23 - 72 * q^24 - 3 * q^25 + 30 * q^26 - 12 * q^27 - 12 * q^28 - 30 * q^29 - 3 * q^30 - 3 * q^31 - 75 * q^32 + 15 * q^33 - 18 * q^34 + 15 * q^35 - 60 * q^36 - 6 * q^37 + 69 * q^38 + 51 * q^39 + 51 * q^40 - 39 * q^42 - 12 * q^43 - 6 * q^44 - 21 * q^45 - 6 * q^46 - 21 * q^47 + 90 * q^48 - 42 * q^49 - 39 * q^50 + 33 * q^51 + 9 * q^52 + 9 * q^53 - 9 * q^54 - 24 * q^55 + 111 * q^56 - 18 * q^57 - 3 * q^58 + 27 * q^59 - 63 * q^60 - 21 * q^61 + 75 * q^62 + 63 * q^63 - 30 * q^64 - 90 * q^65 - 3 * q^66 - 3 * q^67 - 30 * q^68 - 6 * q^69 + 39 * q^70 - 18 * q^71 + 183 * q^72 - 42 * q^73 + 51 * q^74 - 45 * q^75 - 24 * q^76 + 15 * q^77 - 30 * q^78 + 15 * q^79 + 102 * q^80 - 87 * q^81 - 6 * q^82 - 42 * q^83 + 135 * q^84 - 63 * q^85 - 93 * q^86 + 75 * q^87 - 51 * q^88 + 75 * q^89 - 39 * q^90 - 21 * q^91 - 66 * q^92 + 81 * q^93 + 33 * q^94 + 15 * q^95 - 171 * q^96 - 12 * q^97 - 36 * q^98 + 24 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −0.467294 2.65016i 1.66541 + 0.475808i −4.92558 + 1.79277i 2.75387 1.00233i 0.482727 4.63595i −0.609446 2.57460i 4.36176 + 7.55480i 2.54721 + 1.58484i −3.94319 6.82980i
4.2 −0.460894 2.61386i −1.36647 + 1.06432i −4.74046 + 1.72539i −0.528985 + 0.192535i 3.41177 + 3.08122i 0.317729 + 2.62660i 4.04059 + 6.99852i 0.734460 2.90871i 0.747066 + 1.29396i
4.3 −0.422628 2.39684i −0.118761 1.72797i −3.68685 + 1.34190i −2.38847 + 0.869333i −4.09149 + 1.01494i 2.35689 1.20212i 2.34068 + 4.05418i −2.97179 + 0.410433i 3.09309 + 5.35738i
4.4 −0.324131 1.83824i 0.743955 1.56414i −1.39467 + 0.507617i 1.37479 0.500384i −3.11640 0.860580i −2.42382 + 1.06071i −0.481419 0.833843i −1.89306 2.32730i −1.36544 2.36501i
4.5 −0.310937 1.76341i 0.625951 + 1.61499i −1.13356 + 0.412582i −0.00172074 0.000626297i 2.65326 1.60597i 2.59651 + 0.508047i −0.710598 1.23079i −2.21637 + 2.02181i 0.00163946 + 0.00283963i
4.6 −0.279530 1.58529i −1.44281 + 0.958286i −0.555633 + 0.202234i 0.230811 0.0840085i 1.92247 + 2.01940i −1.31413 2.29631i −1.13383 1.96386i 1.16338 2.76524i −0.197697 0.342421i
4.7 −0.266701 1.51254i −1.44858 0.949540i −0.337258 + 0.122752i 3.71318 1.35149i −1.04988 + 2.44427i 2.56740 + 0.639091i −1.26026 2.18283i 1.19675 + 2.75096i −3.03449 5.25589i
4.8 −0.203963 1.15673i −1.51129 0.846167i 0.582964 0.212181i −3.41115 + 1.24156i −0.670539 + 1.92074i −1.99948 + 1.73265i −1.53891 2.66548i 1.56800 + 2.55761i 2.13189 + 3.69255i
4.9 −0.165988 0.941364i 1.70872 0.283332i 1.02077 0.371530i −2.75749 + 1.00364i −0.550345 1.56150i 0.0940141 2.64408i −1.47507 2.55489i 2.83945 0.968269i 1.40250 + 2.42921i
4.10 −0.134854 0.764795i 1.54799 + 0.776991i 1.31266 0.477769i 1.10529 0.402293i 0.385486 1.28868i −0.975813 + 2.45923i −1.31901 2.28459i 1.79257 + 2.40555i −0.456725 0.791071i
4.11 −0.0305699 0.173370i −0.226706 + 1.71715i 1.85026 0.673440i 2.52551 0.919210i 0.304634 0.0131890i −1.78991 1.94839i −0.349362 0.605113i −2.89721 0.778577i −0.236568 0.409748i
4.12 0.0290601 + 0.164808i 0.733837 1.56891i 1.85307 0.674462i −0.999212 + 0.363683i 0.279894 + 0.0753494i 1.97858 + 1.75648i 0.332357 + 0.575660i −1.92297 2.30265i −0.0889751 0.154109i
4.13 0.0570934 + 0.323793i −0.691412 1.58806i 1.77780 0.647067i 1.85642 0.675680i 0.474729 0.314543i −2.33644 1.24140i 0.639805 + 1.10817i −2.04390 + 2.19601i 0.324770 + 0.562517i
4.14 0.0954712 + 0.541444i −1.72856 0.109886i 1.59534 0.580656i −0.969531 + 0.352880i −0.105531 0.946411i 2.17686 1.50376i 1.01650 + 1.76063i 2.97585 + 0.379890i −0.283627 0.491257i
4.15 0.101938 + 0.578121i −0.121715 + 1.72777i 1.55555 0.566175i −3.06658 + 1.11614i −1.01127 + 0.105760i −1.48906 + 2.18694i 1.07293 + 1.85836i −2.97037 0.420592i −0.957867 1.65908i
4.16 0.241593 + 1.37014i 1.22928 + 1.22020i 0.0604707 0.0220095i −1.40760 + 0.512324i −1.37486 + 1.97907i 1.17416 2.37094i 1.43604 + 2.48730i 0.0222396 + 2.99992i −1.04202 1.80483i
4.17 0.246187 + 1.39619i −1.04527 + 1.38109i −0.00936568 + 0.00340883i 3.04103 1.10685i −2.18560 1.11939i 1.78778 + 1.95034i 1.41067 + 2.44335i −0.814825 2.88722i 2.29403 + 3.97338i
4.18 0.340870 + 1.93317i −0.913186 1.47176i −1.74157 + 0.633879i 1.35943 0.494793i 2.53389 2.26702i −0.147009 + 2.64166i 0.143948 + 0.249325i −1.33218 + 2.68799i 1.41991 + 2.45935i
4.19 0.371090 + 2.10456i 0.871315 1.49693i −2.41207 + 0.877922i 2.18602 0.795646i 3.47372 + 1.27824i 1.27150 2.32019i −0.605711 1.04912i −1.48162 2.60860i 2.48569 + 4.30535i
4.20 0.373934 + 2.12069i −1.62126 + 0.609530i −2.47810 + 0.901956i −1.18205 + 0.430229i −1.89887 3.21026i −2.62839 0.302569i −0.686013 1.18821i 2.25695 1.97641i −1.35439 2.34587i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.u even 9 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.u.a 132
3.b odd 2 1 567.2.u.a 132
7.c even 3 1 189.2.w.a yes 132
21.h odd 6 1 567.2.w.a 132
27.e even 9 1 189.2.w.a yes 132
27.f odd 18 1 567.2.w.a 132
189.u even 9 1 inner 189.2.u.a 132
189.bc odd 18 1 567.2.u.a 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.u.a 132 1.a even 1 1 trivial
189.2.u.a 132 189.u even 9 1 inner
189.2.w.a yes 132 7.c even 3 1
189.2.w.a yes 132 27.e even 9 1
567.2.u.a 132 3.b odd 2 1
567.2.u.a 132 189.bc odd 18 1
567.2.w.a 132 21.h odd 6 1
567.2.w.a 132 27.f odd 18 1

## Hecke kernels

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