# Properties

 Label 189.2.ba.a Level $189$ Weight $2$ Character orbit 189.ba 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(5,189)]

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([5, 15]))

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

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

chi := DirichletCharacter("189.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.ba (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.

 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9}+O(q^{10})$$ 132 * q - 3 * q^2 - 9 * q^3 - 3 * q^4 - 9 * q^5 - 18 * q^6 - 6 * q^7 - 18 * q^8 + 3 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$132 q - 3 q^{2} - 9 q^{3} - 3 q^{4} - 9 q^{5} - 18 q^{6} - 6 q^{7} - 18 q^{8} + 3 q^{9} - 9 q^{11} - 9 q^{12} + 3 q^{14} - 24 q^{15} + 3 q^{16} - 18 q^{17} - 3 q^{18} + 18 q^{20} - 21 q^{21} - 12 q^{22} - 6 q^{23} - 9 q^{24} - 3 q^{25} - 12 q^{28} + 6 q^{29} + 51 q^{30} - 9 q^{31} + 3 q^{32} - 9 q^{33} - 18 q^{34} + 18 q^{35} + 3 q^{37} - 99 q^{38} - 36 q^{39} - 54 q^{40} - 45 q^{42} - 12 q^{43} - 9 q^{44} - 9 q^{45} + 3 q^{46} + 45 q^{47} - 24 q^{49} - 9 q^{50} - 48 q^{51} - 9 q^{52} - 45 q^{53} + 171 q^{54} + 3 q^{56} - 3 q^{58} + 36 q^{59} + 57 q^{60} - 9 q^{61} - 99 q^{62} - 33 q^{63} + 18 q^{64} + 69 q^{65} - 9 q^{66} - 3 q^{67} + 36 q^{68} + 108 q^{69} + 66 q^{70} + 18 q^{71} - 129 q^{72} - 9 q^{73} + 75 q^{74} + 36 q^{75} + 36 q^{76} + 15 q^{77} + 66 q^{78} - 21 q^{79} + 72 q^{80} - 33 q^{81} - 18 q^{82} - 90 q^{83} - 120 q^{84} + 9 q^{85} - 105 q^{86} - 54 q^{87} - 63 q^{88} - 18 q^{89} + 81 q^{90} + 6 q^{91} + 150 q^{92} + 21 q^{93} - 9 q^{94} + 45 q^{95} - 81 q^{96} + 27 q^{98} + 96 q^{99}+O(q^{100})$$ 132 * q - 3 * q^2 - 9 * q^3 - 3 * q^4 - 9 * q^5 - 18 * q^6 - 6 * q^7 - 18 * q^8 + 3 * q^9 - 9 * q^11 - 9 * q^12 + 3 * q^14 - 24 * q^15 + 3 * q^16 - 18 * q^17 - 3 * q^18 + 18 * q^20 - 21 * q^21 - 12 * q^22 - 6 * q^23 - 9 * q^24 - 3 * q^25 - 12 * q^28 + 6 * q^29 + 51 * q^30 - 9 * q^31 + 3 * q^32 - 9 * q^33 - 18 * q^34 + 18 * q^35 + 3 * q^37 - 99 * q^38 - 36 * q^39 - 54 * q^40 - 45 * q^42 - 12 * q^43 - 9 * q^44 - 9 * q^45 + 3 * q^46 + 45 * q^47 - 24 * q^49 - 9 * q^50 - 48 * q^51 - 9 * q^52 - 45 * q^53 + 171 * q^54 + 3 * q^56 - 3 * q^58 + 36 * q^59 + 57 * q^60 - 9 * q^61 - 99 * q^62 - 33 * q^63 + 18 * q^64 + 69 * q^65 - 9 * q^66 - 3 * q^67 + 36 * q^68 + 108 * q^69 + 66 * q^70 + 18 * q^71 - 129 * q^72 - 9 * q^73 + 75 * q^74 + 36 * q^75 + 36 * q^76 + 15 * q^77 + 66 * q^78 - 21 * q^79 + 72 * q^80 - 33 * q^81 - 18 * q^82 - 90 * q^83 - 120 * q^84 + 9 * q^85 - 105 * q^86 - 54 * q^87 - 63 * q^88 - 18 * q^89 + 81 * q^90 + 6 * q^91 + 150 * q^92 + 21 * q^93 - 9 * q^94 + 45 * q^95 - 81 * q^96 + 27 * q^98 + 96 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −2.51634 + 0.443699i −1.35749 + 1.07574i 4.25573 1.54896i −0.181827 + 1.03119i 2.93860 3.30926i 2.00134 1.73050i −5.59594 + 3.23082i 0.685547 2.92062i 2.67551i
5.2 −2.44999 + 0.431999i 1.58275 0.703494i 3.93642 1.43274i 0.403704 2.28952i −3.57380 + 2.40730i −0.412545 2.61339i −4.71627 + 2.72294i 2.01019 2.22691i 5.78368i
5.3 −2.31815 + 0.408753i −0.111866 1.72843i 3.32736 1.21106i −0.175778 + 0.996886i 0.965825 + 3.96105i 1.63463 + 2.08038i −3.14120 + 1.81358i −2.97497 + 0.386707i 2.38278i
5.4 −2.27453 + 0.401061i 1.22609 + 1.22340i 3.13325 1.14041i −0.595188 + 3.37548i −3.27944 2.29092i −2.64569 + 0.0174348i −2.66893 + 1.54091i 0.00659757 + 2.99999i 7.91633i
5.5 −1.98316 + 0.349685i −1.71808 0.219557i 1.93127 0.702926i 0.521105 2.95533i 3.48401 0.165370i −1.49002 + 2.18628i −0.0963007 + 0.0555993i 2.90359 + 0.754431i 6.04313i
5.6 −1.27749 + 0.225257i 1.73184 0.0269248i −0.298135 + 0.108512i 0.194492 1.10302i −2.20635 + 0.424505i 0.0727513 + 2.64475i 2.60324 1.50298i 2.99855 0.0932590i 1.45291i
5.7 −1.25823 + 0.221860i 0.0538771 + 1.73121i −0.345469 + 0.125740i 0.640361 3.63167i −0.451876 2.16631i −1.30830 2.29964i 2.61972 1.51249i −2.99419 + 0.186545i 4.71154i
5.8 −1.21351 + 0.213974i −1.60161 0.659440i −0.452565 + 0.164720i −0.378134 + 2.14451i 2.08467 + 0.457534i −0.671624 2.55909i 2.64823 1.52896i 2.13028 + 2.11232i 2.68329i
5.9 −1.19961 + 0.211524i −0.0787989 + 1.73026i −0.485062 + 0.176548i −0.243175 + 1.37911i −0.271462 2.09230i 2.41247 + 1.08628i 2.65438 1.53251i −2.98758 0.272685i 1.70584i
5.10 −0.408566 + 0.0720413i 1.12900 1.31353i −1.71765 + 0.625173i 0.0312679 0.177329i −0.366645 + 0.617997i 1.82229 1.91814i 1.37531 0.794036i −0.450699 2.96595i 0.0747032i
5.11 0.220121 0.0388132i −0.104309 1.72891i −1.83244 + 0.666953i 0.595276 3.37598i −0.0900651 0.376520i −2.47857 + 0.925586i −0.764613 + 0.441450i −2.97824 + 0.360682i 0.766227i
5.12 0.284285 0.0501271i 1.43787 + 0.965670i −1.80108 + 0.655540i −0.499509 + 2.83286i 0.457172 + 0.202449i 0.989753 2.45365i −0.979151 + 0.565313i 1.13496 + 2.77702i 0.830377i
5.13 0.357086 0.0629638i −1.26279 1.18548i −1.75584 + 0.639073i −0.363271 + 2.06021i −0.525567 0.343807i 1.11830 + 2.39779i −1.21478 + 0.701352i 0.189288 + 2.99402i 0.758545i
5.14 0.491776 0.0867134i −1.45806 + 0.934910i −1.64506 + 0.598753i 9.76251e−5 0 0.000553660i −0.635971 + 0.586200i −2.32328 1.26584i −1.62200 + 0.936464i 1.25189 2.72631i 0 0.000280742i
5.15 0.826512 0.145736i 0.181944 + 1.72247i −1.21750 + 0.443135i −0.133334 + 0.756174i 0.401405 + 1.39712i −1.31947 + 2.29325i −2.39534 + 1.38295i −2.93379 + 0.626786i 0.644419i
5.16 1.10969 0.195669i 1.47374 + 0.909998i −0.686257 + 0.249777i 0.705645 4.00191i 1.81345 + 0.721453i 2.51373 + 0.825312i −2.66435 + 1.53826i 1.34381 + 2.68220i 4.57896i
5.17 1.72891 0.304854i 1.65859 0.499066i 1.01682 0.370091i −0.0612092 + 0.347134i 2.71542 1.36847i −2.64149 + 0.150064i −1.39560 + 0.805749i 2.50187 1.65549i 0.618825i
5.18 1.77439 0.312873i 0.626754 1.61468i 1.17119 0.426279i −0.155940 + 0.884381i 0.606919 3.06116i 2.64257 0.129722i −1.17597 + 0.678945i −2.21436 2.02401i 1.61803i
5.19 1.77697 0.313327i −1.53787 0.796838i 1.18006 0.429505i 0.415679 2.35743i −2.98242 0.934099i 0.444023 2.60823i −1.16293 + 0.671420i 1.73010 + 2.45087i 4.31932i
5.20 1.95754 0.345166i −1.39494 + 1.02672i 1.83342 0.667310i −0.649669 + 3.68445i −2.37625 + 2.49133i 2.58740 0.552593i −0.0842052 + 0.0486159i 0.891690 2.86442i 7.43669i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 5.22 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.ba 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.ba.a 132
3.b odd 2 1 567.2.ba.a 132
7.d odd 6 1 189.2.bd.a yes 132
21.g even 6 1 567.2.bd.a 132
27.e even 9 1 567.2.bd.a 132
27.f odd 18 1 189.2.bd.a yes 132
189.x odd 18 1 567.2.ba.a 132
189.ba even 18 1 inner 189.2.ba.a 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.ba.a 132 1.a even 1 1 trivial
189.2.ba.a 132 189.ba even 18 1 inner
189.2.bd.a yes 132 7.d odd 6 1
189.2.bd.a yes 132 27.f odd 18 1
567.2.ba.a 132 3.b odd 2 1
567.2.ba.a 132 189.x odd 18 1
567.2.bd.a 132 21.g even 6 1
567.2.bd.a 132 27.e even 9 1

## Hecke kernels

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