# Properties

 Label 189.4.e.h Level $189$ Weight $4$ Character orbit 189.e Analytic conductor $11.151$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,4,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, 4, names="a")

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

chi := DirichletCharacter("189.109");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$11.1513609911$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 54 x^{14} - 12 x^{13} + 2361 x^{12} - 966 x^{11} + 29570 x^{10} - 65952 x^{9} + 300096 x^{8} + \cdots + 81$$ x^16 + 54*x^14 - 12*x^13 + 2361*x^12 - 966*x^11 + 29570*x^10 - 65952*x^9 + 300096*x^8 - 356610*x^7 + 531858*x^6 + 80070*x^5 + 77197*x^4 - 216*x^3 + 4878*x^2 + 486*x + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{9}\cdot 7^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{5} - \beta_{2}) q^{2} + (\beta_{4} - \beta_{3} + 6 \beta_1) q^{4} + ( - \beta_{12} - \beta_{11}) q^{5} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 5) q^{7}+ \cdots + ( - \beta_{9} + 5 \beta_{2}) q^{8}+O(q^{10})$$ q + (-b5 - b2) * q^2 + (b4 - b3 + 6*b1) * q^4 + (-b12 - b11) * q^5 + (b7 + b4 - b3 - 3*b1 - 5) * q^7 + (-b9 + 5*b2) * q^8 $$q + ( - \beta_{5} - \beta_{2}) q^{2} + (\beta_{4} - \beta_{3} + 6 \beta_1) q^{4} + ( - \beta_{12} - \beta_{11}) q^{5} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 5) q^{7}+ \cdots + (62 \beta_{12} + 64 \beta_{11} + \cdots - 251 \beta_{2}) q^{98}+O(q^{100})$$ q + (-b5 - b2) * q^2 + (b4 - b3 + 6*b1) * q^4 + (-b12 - b11) * q^5 + (b7 + b4 - b3 - 3*b1 - 5) * q^7 + (-b9 + 5*b2) * q^8 + (-b15 + b10 + 6*b1) * q^10 + (-b14 - b12 + b8 + b6 + 3*b5) * q^11 + (-b13 - b10 - 3*b4 - b1 - 11) * q^13 + (b12 - b11 - b9 - b6 + 4*b5 + 6*b2) * q^14 + (-b15 - 4*b10 - b7 + b4 + 12*b3 - 22*b1 - 20) * q^16 + (b12 + 2*b8 + 2*b6 - 8*b5) * q^17 + (b15 + 2*b13 - 4*b10 - b7 + b4 - 3*b1 - 1) * q^19 + (b11 + b9 + b8 + 3*b2) * q^20 + (-b15 + 2*b13 + 2*b10 + 4*b7 - 8*b4 - b3 + 4*b1 + 31) * q^22 + (b14 + 2*b12 + 2*b11 + b9 - b6 - 15*b5 - 15*b2) * q^23 + (-4*b15 + 4*b10 - 10*b4 + 10*b3 + 53*b1) * q^25 + (2*b14 - 9*b12 - 9*b11 + 2*b9 + b6 - 8*b5 - 8*b2) * q^26 + (-5*b10 - 3*b7 - 12*b4 + 24*b3 - 70*b1 - 44) * q^28 + (7*b11 - 4*b9 - 2*b8 + 20*b2) * q^29 + (b15 + b13 - 4*b10 - 5*b7 + 13*b4 - 12*b3 + 96*b1 - 2) * q^31 + (-5*b14 - 6*b12 - 4*b8 - 4*b6 + 57*b5) * q^32 + (-4*b15 - b13 - b10 + 16*b7 + 30*b4 - 4*b3 + 7*b1 - 111) * q^34 + (-b14 + 9*b12 - 5*b11 - 4*b9 - b8 + 3*b6 + 11*b5 - 8*b2) * q^35 + (-b15 - b13 - 15*b3 - 87*b1 - 87) * q^37 + (b14 + 12*b12 - 6*b8 - 6*b6 - 10*b5) * q^38 + (-6*b15 - 5*b13 - 4*b10 - b7 + b4 - 3*b3 - 17*b1 - 15) * q^40 + (5*b11 + 4*b9 - 2*b8 + 28*b2) * q^41 + (-2*b15 - 6*b13 - 6*b10 + 8*b7 - 16*b4 - 2*b3 - 2*b1 + 183) * q^43 + (3*b14 + 13*b12 + 13*b11 + 3*b9 + b6 - 81*b5 - 81*b2) * q^44 + (4*b15 + b13 - 7*b10 - 5*b7 + 21*b4 - 20*b3 + 185*b1 - 2) * q^46 + (8*b14 - 15*b12 - 15*b11 + 8*b9 - 2*b6 + 8*b5 + 8*b2) * q^47 + (7*b13 + 7*b10 - b7 - 8*b4 + 15*b3 - 102*b1 + 75) * q^49 + (36*b11 + 14*b9 + 4*b8 - 29*b2) * q^50 + (-3*b15 - 4*b13 + 15*b10 + 20*b7 + 23*b4 - 27*b3 + 262*b1 + 8) * q^52 + (9*b14 - 2*b12 + 3*b8 + 3*b6 + 37*b5) * q^53 + (6*b15 + 12*b13 + 12*b10 - 24*b7 - 4*b4 + 6*b3 - 166) * q^55 + (-4*b14 + 8*b12 + b11 + 12*b9 + 3*b8 - 2*b6 + 100*b5 - 74*b2) * q^56 + (3*b15 + 3*b13 + 62*b3 - 295*b1 - 295) * q^58 + (5*b14 + 27*b12 - b8 - b6 - 91*b5) * q^59 + (2*b15 + b13 + 4*b10 + b7 - b4 - 23*b3 - 24*b1 - 26) * q^61 + (-3*b11 - 13*b9 - 5*b8 + 200*b2) * q^62 + (5*b15 + 3*b13 + 3*b10 - 20*b7 - 77*b4 + 5*b3 - 7*b1 + 612) * q^64 + (-17*b14 - 5*b12 - 5*b11 - 17*b9 + 7*b6 - 105*b5 - 105*b2) * q^65 + (14*b15 + b13 - 17*b10 - 5*b7 - 2*b4 + 3*b3 + 75*b1 - 2) * q^67 + (-27*b14 + 11*b12 + 11*b11 - 27*b9 - 3*b6 + 203*b5 + 203*b2) * q^68 + (-14*b15 - 21*b13 + 8*b10 + 24*b7 + 2*b4 + 11*b3 + 179*b1 + 358) * q^70 + (-45*b11 + 3*b9 + 5*b8 - 21*b2) * q^71 + (14*b15 + 7*b13 - 35*b10 - 35*b7 - 6*b4 + 13*b3 - 602*b1 - 14) * q^73 + (14*b14 - 9*b12 + b8 + b6 - 16*b5) * q^74 + (3*b15 - 5*b13 - 5*b10 - 12*b7 - 32*b4 + 3*b3 - 11*b1 - 56) * q^76 + (13*b14 - 47*b12 - 26*b11 - 4*b9 - b8 + 3*b6 + 25*b5 - 64*b2) * q^77 + (b15 - 3*b13 + 16*b10 + 4*b7 - 4*b4 + 25*b3 - 61*b1 - 69) * q^79 + (-11*b14 - 59*b12 + 9*b8 + 9*b6 - 27*b5) * q^80 + (9*b15 + b13 + 32*b10 + 8*b7 - 8*b4 - 50*b3 - 397*b1 - 413) * q^82 + (10*b11 - 2*b9 + 8*b8 + 98*b2) * q^83 + (10*b15 + 4*b13 + 4*b10 - 40*b7 + 18*b4 + 10*b3 - 16*b1 + 144) * q^85 + (12*b14 - 48*b12 - 48*b11 + 12*b9 - 4*b6 - 311*b5 - 311*b2) * q^86 + (-13*b15 - 13*b13 + 52*b10 + 65*b7 + 63*b4 - 76*b3 + 889*b1 + 26) * q^88 + (-4*b14 - 17*b12 - 17*b11 - 4*b9 + 4*b5 + 4*b2) * q^89 + (21*b15 + 7*b13 + b10 - 34*b7 + 49*b4 - 15*b3 - 123*b1 - 80) * q^91 + (-14*b11 - 16*b9 + 234*b2) * q^92 + (-11*b15 - 4*b13 + 23*b10 + 20*b7 + 94*b4 - 98*b3 - 42*b1 + 8) * q^94 + (5*b14 + 44*b12 + 13*b8 + 13*b6 + 129*b5) * q^95 + (2*b15 - 8*b13 - 8*b10 - 8*b7 - 50*b4 + 2*b3 - 12*b1 + 65) * q^97 + (62*b12 + 64*b11 + 15*b9 - 6*b6 - 39*b5 - 251*b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 48 q^{4} - 60 q^{7}+O(q^{10})$$ 16 * q - 48 * q^4 - 60 * q^7 $$16 q - 48 q^{4} - 60 q^{7} - 44 q^{10} - 168 q^{13} - 156 q^{16} - 12 q^{19} + 448 q^{22} - 408 q^{25} - 152 q^{28} - 800 q^{31} - 1896 q^{34} - 692 q^{37} - 96 q^{40} + 2912 q^{43} - 1524 q^{46} + 2020 q^{49} - 1972 q^{52} - 2560 q^{55} - 2372 q^{58} - 216 q^{61} + 9928 q^{64} - 684 q^{67} + 4316 q^{70} + 4564 q^{73} - 760 q^{76} - 556 q^{79} - 3340 q^{82} + 2592 q^{85} - 6696 q^{88} - 184 q^{91} + 492 q^{94} + 1168 q^{97}+O(q^{100})$$ 16 * q - 48 * q^4 - 60 * q^7 - 44 * q^10 - 168 * q^13 - 156 * q^16 - 12 * q^19 + 448 * q^22 - 408 * q^25 - 152 * q^28 - 800 * q^31 - 1896 * q^34 - 692 * q^37 - 96 * q^40 + 2912 * q^43 - 1524 * q^46 + 2020 * q^49 - 1972 * q^52 - 2560 * q^55 - 2372 * q^58 - 216 * q^61 + 9928 * q^64 - 684 * q^67 + 4316 * q^70 + 4564 * q^73 - 760 * q^76 - 556 * q^79 - 3340 * q^82 + 2592 * q^85 - 6696 * q^88 - 184 * q^91 + 492 * q^94 + 1168 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 54 x^{14} - 12 x^{13} + 2361 x^{12} - 966 x^{11} + 29570 x^{10} - 65952 x^{9} + 300096 x^{8} + \cdots + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( 51\!\cdots\!86 \nu^{15} + \cdots - 15\!\cdots\!87 ) / 34\!\cdots\!04$$ (5150955099921168548586*v^15 - 399066979643989434973*v^14 + 278014912120055287156056*v^13 - 83387557077442080040203*v^12 + 12158828863833005482946292*v^11 - 5917812735287788738645344*v^10 + 152377692231961500414711582*v^9 - 351447040288173944394280142*v^8 + 1568119463587256938331080650*v^7 - 1948431641273295864938623590*v^6 + 2843045574249633988600391868*v^5 + 239921202700739614544036220*v^4 + 306862261345572305282838288*v^3 - 62802406547786806787941369*v^2 + 18876629001889791000696798*v - 1570364393273348796566487) / 3417839928129147250678704 $$\beta_{2}$$ $$=$$ $$( 44\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!79 ) / 29\!\cdots\!40$$ (442764159594379869520971*v^15 + 92045143513852266048383*v^14 + 23875233760018211350286395*v^13 - 334521476496697879981549*v^12 + 1042378325146991278463590872*v^11 - 209637242658343244713775850*v^10 + 12920716480808433551935581884*v^9 - 26431419694369820486902515736*v^8 + 125681615744961907464618798700*v^7 - 127946844118204810042200678910*v^6 + 190708322691323970606768128508*v^5 + 100146641482197993232718041374*v^4 + 14936832517605486267172602005*v^3 + 6897374060046734350153592973*v^2 + 724005707439463626563830881*v + 1045588544089657175339954379) / 290516393890977516307689840 $$\beta_{3}$$ $$=$$ $$( 95\!\cdots\!24 \nu^{15} + \cdots + 13\!\cdots\!43 ) / 29\!\cdots\!84$$ (95380685141312083531524*v^15 - 19999835154833760133025*v^14 + 5143920844575013833434298*v^13 - 2223805481128098770493153*v^12 + 225076637537986043826828356*v^11 - 139256162798744285266167836*v^10 + 2824074367489539682128456470*v^9 - 6874798095428693104070498202*v^8 + 29746512388308646051695896130*v^7 - 39600997957091737754309283690*v^6 + 55789433302351489421009172356*v^5 - 991471541688804280620645008*v^4 + 2823503042864611244771923334*v^3 - 4527163588678219491999721417*v^2 + 152717063340804932034735444*v + 13553422352698653807833043) / 29051639389097751630768984 $$\beta_{4}$$ $$=$$ $$( - 78\!\cdots\!07 \nu^{15} + \cdots + 45\!\cdots\!11 ) / 17\!\cdots\!08$$ (-78587806494897782607*v^15 - 18514612371959954123*v^14 - 4238897266583391539275*v^13 - 56958150753926795849*v^12 - 185067326029031479134312*v^11 + 32121640103510367075210*v^10 - 2294800844682407360497616*v^9 + 4629860906669748741870256*v^8 - 22231547944601577747330100*v^7 + 22124852259119506579511074*v^6 - 33866059298945644669248108*v^5 - 17791507215158118971325294*v^4 - 5560894533930725148743453*v^3 - 1225257977546479953252873*v^2 - 128607778453102121256201*v + 45149992925452738096311) / 17364996646203079277208 $$\beta_{5}$$ $$=$$ $$( 66\!\cdots\!74 \nu^{15} + \cdots - 22\!\cdots\!33 ) / 14\!\cdots\!20$$ (665522250423697530353874*v^15 - 125243857984949990751497*v^14 + 35941861102546041760672534*v^13 - 14738278130936968798845517*v^12 + 1573020323036532483998365408*v^11 - 938037836291855080686051628*v^10 + 19810054007942904483154236558*v^9 - 47573713680789795812173517278*v^8 + 208123772376420301381650915250*v^7 - 274861520920805544623772742090*v^6 + 399390063142360978998688646512*v^5 - 13098061260473914994913455176*v^4 + 43133473362055627998543680712*v^3 - 8839371024656621237947489541*v^2 + 2733811532980690542634041264*v - 220930559608896963997451433) / 145258196945488758153844920 $$\beta_{6}$$ $$=$$ $$( - 73\!\cdots\!09 \nu^{15} + \cdots + 89\!\cdots\!53 ) / 11\!\cdots\!32$$ (-7378249582352633616809*v^15 + 7640516173576034579001*v^14 - 396486260130533734438365*v^13 + 501010878979267442799633*v^12 - 17406995501606988686312928*v^11 + 25135602796127952251865918*v^10 - 220977529076860573592357800*v^9 + 710311663408836311201832420*v^8 - 2660715630374200481681211576*v^7 + 4789207986003342252715895190*v^6 - 6063461101389177663116968668*v^5 + 2705051621646512950567970286*v^4 + 1162352296016671188979227565*v^3 + 467098110845212066943802927*v^2 + 83276969081471319038266269*v + 8932657899805938939981753) / 1185781199555010270643632 $$\beta_{7}$$ $$=$$ $$( - 18\!\cdots\!03 \nu^{15} + \cdots + 17\!\cdots\!42 ) / 29\!\cdots\!84$$ (-181202429822774059445203*v^15 + 173977431935443638961454*v^14 - 9811685959379603724793333*v^13 + 11565395949374069055808906*v^12 - 431363096763830035881827228*v^11 + 585885681184940118029980022*v^10 - 5589979039086480130957442826*v^9 + 17110561464270230737000486850*v^8 - 66667094487687553817090945818*v^7 + 118423275991500249424147295008*v^6 - 166515257410212088424334640544*v^5 + 86533867862175077271211405370*v^4 - 14103155587023330050892289055*v^3 + 7760418160366236295272417264*v^2 - 2675857350338389303631635689*v + 17606182425241232279099142) / 29051639389097751630768984 $$\beta_{8}$$ $$=$$ $$( - 46\!\cdots\!69 \nu^{15} + \cdots - 84\!\cdots\!78 ) / 41\!\cdots\!12$$ (-46137478859790502534169*v^15 - 18003661639799948801112*v^14 - 2497338363865019696115297*v^13 - 416038060561248460663806*v^12 - 109030841460406320382014300*v^11 + 2273497446134516638901454*v^10 - 1360721802377728113197714806*v^9 + 2522331908486871096491201670*v^8 - 12828449250602365675826380110*v^7 + 11521798797354065797958395260*v^6 - 19974983405575959441115234416*v^5 - 10518093913416689404508414166*v^4 - 8515955152820740899495849605*v^3 - 724055489241304417393132566*v^2 - 75981746281782619859736105*v - 8494466193943117728132078) / 4150234198442535947252712 $$\beta_{9}$$ $$=$$ $$( 35\!\cdots\!57 \nu^{15} + \cdots + 10\!\cdots\!43 ) / 29\!\cdots\!40$$ (3581213557777451280388457*v^15 + 497101835269179788970811*v^14 + 192780893717033095789404005*v^13 - 15980579473944711138709193*v^12 + 8416740229132048443443408064*v^11 - 2270254578144792137065853430*v^10 + 103992904622903652639078628208*v^9 - 220543126481258728893468067532*v^8 + 1024155956141169744925415357600*v^7 - 1083369088036026586115709143390*v^6 + 1539084062908760277548679251916*v^5 + 807377687758217219205149874858*v^4 - 39477049489181380817607242085*v^3 + 55616734255955182638475383621*v^2 + 5838615735956173206944959347*v + 10771434886564083995036979543) / 290516393890977516307689840 $$\beta_{10}$$ $$=$$ $$( - 37\!\cdots\!34 \nu^{15} + \cdots - 45\!\cdots\!93 ) / 29\!\cdots\!84$$ (-370727423561987712791834*v^15 - 138943322352577418084677*v^14 - 20010099818056432848765736*v^13 - 3066464048607306731551049*v^12 - 873124319747606545549609124*v^11 + 29333329071129918628681280*v^10 - 10806301579935763307328500414*v^9 + 20305374462049276803890790674*v^8 - 101803954946404128926376815122*v^7 + 89605900266476711030909858062*v^6 - 143950743942351235848030195932*v^5 - 109762217468378527910055229084*v^4 - 31002241094379135143472273216*v^3 - 12836668733018897803488941265*v^2 - 639346001539411016632446822*v - 459002540690788617114448293) / 29051639389097751630768984 $$\beta_{11}$$ $$=$$ $$( 22\!\cdots\!31 \nu^{15} + \cdots + 33\!\cdots\!79 ) / 14\!\cdots\!20$$ (2289167883872891291150831*v^15 + 627420074432135434000673*v^14 + 123588042832718453901704065*v^13 + 6386433259237099110525131*v^12 + 5395753424776406429835162552*v^11 - 730936881254907217767908190*v^10 + 67021974193589028530144363484*v^9 - 132451463917252456080785597776*v^8 + 644852294661348734077554382300*v^7 - 626908185570784693395088158490*v^6 + 987671105811000701345435576388*v^5 + 519171675475924636810246102794*v^4 + 222428943273666648886756094785*v^3 + 35750388942425214750207298203*v^2 + 3752277027081566743984640571*v + 3310396194125585240714736279) / 145258196945488758153844920 $$\beta_{12}$$ $$=$$ $$( 49\!\cdots\!31 \nu^{15} + \cdots - 16\!\cdots\!07 ) / 29\!\cdots\!40$$ (4998542427516925336278531*v^15 - 1090288236405243753967463*v^14 + 269998200596936769336168811*v^13 - 118845324104131662579154303*v^12 + 11818767259407872899010172832*v^11 - 7402999434033410305883790322*v^10 + 149040219219300537152985478512*v^9 - 361947314427688630409111620012*v^8 + 1574154920688246687446763207760*v^7 - 2114379398528963194371454132450*v^6 + 3069157212268333177811738925028*v^5 - 200867138436006223737091483954*v^4 + 331513256607379064367614787033*v^3 - 67959586376872875103251681569*v^2 + 41764693992637524946085799981*v - 1698392289271548563809157007) / 290516393890977516307689840 $$\beta_{13}$$ $$=$$ $$( - 25\!\cdots\!02 \nu^{15} + \cdots - 33\!\cdots\!25 ) / 58\!\cdots\!68$$ (-2571831061897469598114502*v^15 - 493877469490488980635899*v^14 - 138735270869336879994709404*v^13 + 4210350941760129617515551*v^12 - 6058443645835285290299979476*v^11 + 1317463601632217043187740056*v^10 - 75234957599634968389557350878*v^9 + 154911001630630906925272891286*v^8 - 735074006714935417538144034426*v^7 + 759756008244781386304974139518*v^6 - 1150871009667652044360949647404*v^5 - 515084123499938412097060102564*v^4 - 171459431213353795023532713060*v^3 - 23568065621853474074904047959*v^2 - 4322794507103219074613130762*v - 3317453458629006220339166325) / 58103278778195503261537968 $$\beta_{14}$$ $$=$$ $$( 39\!\cdots\!04 \nu^{15} + \cdots - 12\!\cdots\!68 ) / 72\!\cdots\!60$$ (3911264155408570583939304*v^15 - 694118293076704457345747*v^14 + 211221676579589381568989329*v^13 - 84431993902682656026315592*v^12 + 9243517985344549650308741938*v^11 - 5418030775292864868093348118*v^10 + 116356897336999386253673581698*v^9 - 278527017467178868822677710938*v^8 + 1219877326870322355058022530300*v^7 - 1604361396434478753759038607160*v^6 + 2332068617501933062829815627702*v^5 - 62652997359749764024278509146*v^4 + 251847313221794279320194469182*v^3 - 51605525411741198441767997291*v^2 + 13097877297782791499527921839*v - 1289871806587074023992553268) / 72629098472744379076922460 $$\beta_{15}$$ $$=$$ $$( - 10\!\cdots\!87 \nu^{15} + \cdots + 86\!\cdots\!95 ) / 14\!\cdots\!92$$ (-1073865354571902572408887*v^15 + 179196365203365095644700*v^14 - 58014230781407180494288473*v^13 + 22553910299694760921607847*v^12 - 2538913821945140369206816736*v^11 + 1460265363455016395184416846*v^10 - 31986879836697318747160200580*v^9 + 76126132490064599383326439026*v^8 - 334803809171548014454396381296*v^7 + 438158821213752374865905238138*v^6 - 641836824574537975441293452396*v^5 + 15301213626900151314291998654*v^4 - 77071655551893286686923333217*v^3 + 6211379408222858418994419562*v^2 - 6426318911590096700217830559*v + 86065612803115055061832395) / 14525819694548875815384492
 $$\nu$$ $$=$$ $$( - 3 \beta_{15} - 2 \beta_{14} - \beta_{13} - 5 \beta_{12} + 6 \beta_{10} + \beta_{8} + 5 \beta_{7} + \cdots + 2 ) / 42$$ (-3*b15 - 2*b14 - b13 - 5*b12 + 6*b10 + b8 + 5*b7 + b6 + 14*b5 + 2*b4 - 3*b3 + 5*b1 + 2) / 42 $$\nu^{2}$$ $$=$$ $$( 4 \beta_{15} + 9 \beta_{14} - \beta_{13} + 12 \beta_{12} + 12 \beta_{11} + 20 \beta_{10} + 9 \beta_{9} + \cdots - 565 ) / 42$$ (4*b15 + 9*b14 - b13 + 12*b12 + 12*b11 + 20*b10 + 9*b9 + 5*b7 + 6*b6 + 63*b5 - 5*b4 + 4*b3 + 63*b2 - 555*b1 - 565) / 42 $$\nu^{3}$$ $$=$$ $$( 65 \beta_{15} - 144 \beta_{13} - 449 \beta_{11} - 144 \beta_{10} - 83 \beta_{9} - 115 \beta_{8} + \cdots - 13 ) / 84$$ (65*b15 - 144*b13 - 449*b11 - 144*b10 - 83*b9 - 115*b8 - 260*b7 - 188*b4 + 65*b3 + 749*b2 - 274*b1 - 13) / 84 $$\nu^{4}$$ $$=$$ $$( 52 \beta_{15} - 312 \beta_{14} + 218 \beta_{13} - 339 \beta_{12} - 706 \beta_{10} - 285 \beta_{8} + \cdots - 436 ) / 42$$ (52*b15 - 312*b14 + 218*b13 - 339*b12 - 706*b10 - 285*b8 - 1090*b7 - 285*b6 - 3318*b5 - 317*b4 + 535*b3 + 18174*b1 - 436) / 42 $$\nu^{5}$$ $$=$$ $$( 5672 \beta_{15} + 2377 \beta_{14} + 8249 \beta_{13} + 17713 \beta_{12} + 17713 \beta_{11} + \cdots + 19531 ) / 84$$ (5672*b15 + 2377*b14 + 8249*b13 + 17713*b12 + 17713*b11 - 10308*b10 + 2377*b9 - 2577*b7 - 5063*b6 - 25375*b5 + 2577*b4 + 2753*b3 - 25375*b2 + 14377*b1 + 19531) / 84 $$\nu^{6}$$ $$=$$ $$( - 1577 \beta_{15} - 983 \beta_{13} - 1614 \beta_{11} - 983 \beta_{10} - 1823 \beta_{9} + 2046 \beta_{8} + \cdots + 122121 ) / 7$$ (-1577*b15 - 983*b13 - 1614*b11 - 983*b10 - 1823*b9 + 2046*b8 + 6308*b7 + 4927*b4 - 1577*b3 - 25221*b2 + 2171*b1 + 122121) / 7 $$\nu^{7}$$ $$=$$ $$( - 165693 \beta_{15} - 37672 \beta_{14} - 53663 \beta_{13} - 349141 \beta_{12} + 326682 \beta_{10} + \cdots + 107326 ) / 42$$ (-165693*b15 - 37672*b14 - 53663*b13 - 349141*b12 + 326682*b10 + 107141*b8 + 268315*b7 + 107141*b6 + 500836*b5 + 36178*b4 - 89841*b3 - 480797*b1 + 107326) / 42 $$\nu^{8}$$ $$=$$ $$( 198134 \beta_{15} + 400050 \beta_{14} - 205994 \beta_{13} + 246687 \beta_{12} + 246687 \beta_{11} + \cdots - 28093496 ) / 42$$ (198134*b15 + 400050*b14 - 205994*b13 + 246687*b12 + 246687*b11 + 1616512*b10 + 400050*b9 + 404128*b7 + 527709*b6 + 6507396*b5 - 404128*b4 - 939709*b3 + 6507396*b2 - 27285240*b1 - 28093496) / 42 $$\nu^{9}$$ $$=$$ $$( 4545349 \beta_{15} - 8840700 \beta_{13} - 27656533 \beta_{11} - 8840700 \beta_{10} - 2499019 \beta_{9} + \cdots - 101048477 ) / 84$$ (4545349*b15 - 8840700*b13 - 27656533*b11 - 8840700*b10 - 2499019*b9 - 9004295*b8 - 18181396*b7 - 7117696*b4 + 4545349*b3 + 42607789*b2 - 17931398*b1 - 101048477) / 84 $$\nu^{10}$$ $$=$$ $$( 10805111 \beta_{15} - 15065301 \beta_{14} + 17122465 \beta_{13} - 4441368 \beta_{12} - 62172506 \beta_{10} + \cdots - 34244930 ) / 42$$ (10805111*b15 - 15065301*b14 + 17122465*b13 - 4441368*b12 - 62172506*b10 - 22724454*b8 - 85612325*b7 - 22724454*b6 - 273256347*b5 - 40321252*b4 + 57443717*b3 + 1064254659*b1 - 34244930) / 42 $$\nu^{11}$$ $$=$$ $$( 349102204 \beta_{15} + 84438449 \beta_{14} + 542822449 \beta_{13} + 1100326265 \beta_{12} + \cdots + 4651037963 ) / 84$$ (349102204*b15 + 84438449*b14 + 542822449*b13 + 1100326265*b12 + 1100326265*b11 - 774880980*b10 + 84438449*b9 - 193720245*b7 - 377807215*b6 - 1866273143*b5 + 193720245*b4 + 113845861*b3 - 1866273143*b2 + 4263597473*b1 + 4651037963) / 84 $$\nu^{12}$$ $$=$$ $$( - 120521474 \beta_{15} - 31201500 \beta_{13} + 5945004 \beta_{11} - 31201500 \beta_{10} + \cdots + 7846479013 ) / 7$$ (-120521474*b15 - 31201500*b13 + 5945004*b11 - 31201500*b10 - 96209332*b9 + 163091760*b8 + 482085896*b7 + 399791968*b4 - 120521474*b3 - 1895037564*b2 + 209841448*b1 + 7846479013) / 7 $$\nu^{13}$$ $$=$$ $$( - 11044886505 \beta_{15} - 1428712508 \beta_{14} - 4137003643 \beta_{13} - 21978952367 \beta_{12} + \cdots + 8274007286 ) / 42$$ (-11044886505*b15 - 1428712508*b14 - 4137003643*b13 - 21978952367*b12 + 23455897434*b10 + 7927216447*b8 + 20685018215*b7 + 7927216447*b6 + 41204820020*b5 + 2728166636*b4 - 6865170279*b3 - 88018702561*b1 + 8274007286) / 42 $$\nu^{14}$$ $$=$$ $$( 4861786192 \beta_{15} + 22350858507 \beta_{14} - 25647216547 \beta_{13} - 9895364610 \beta_{12} + \cdots - 1860633895699 ) / 42$$ (4861786192*b15 + 22350858507*b14 - 25647216547*b13 - 9895364610*b12 - 9895364610*b11 + 122036010956*b10 + 22350858507*b9 + 30509002739*b7 + 42095759784*b6 + 471803384241*b5 - 30509002739*b4 - 68792253050*b3 + 471803384241*b2 - 1799615890221*b1 - 1860633895699) / 42 $$\nu^{15}$$ $$=$$ $$( 353535630725 \beta_{15} - 548381305872 \beta_{13} - 1762964503613 \beta_{11} - 548381305872 \beta_{10} + \cdots - 11080741238257 ) / 84$$ (353535630725*b15 - 548381305872*b13 - 1762964503613*b11 - 548381305872*b10 - 95608150535*b9 - 665795979367*b8 - 1414142522900*b7 - 617116818428*b4 + 353535630725*b3 + 3637334096921*b2 - 1255452567322*b1 - 11080741238257) / 84

## Character values

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

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

## 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
 −2.17666 + 3.77009i 1.03883 − 1.79930i −0.0640761 + 0.110983i 0.128480 − 0.222533i −0.178435 + 0.309059i −3.06544 + 5.30949i 3.25730 − 5.64181i 1.06000 − 1.83598i −2.17666 − 3.77009i 1.03883 + 1.79930i −0.0640761 − 0.110983i 0.128480 + 0.222533i −0.178435 − 0.309059i −3.06544 − 5.30949i 3.25730 + 5.64181i 1.06000 + 1.83598i
−2.73089 4.73004i 0 −10.9155 + 18.9063i −0.0995681 0.172457i 0 −16.0061 + 9.31685i 75.5424 0 −0.543819 + 0.941923i
109.2 −1.74618 3.02447i 0 −2.09829 + 3.63435i 3.93765 + 6.82022i 0 18.4135 1.98614i −13.2829 0 13.7517 23.8187i
109.3 −1.64637 2.85159i 0 −1.42105 + 2.46133i −10.8582 18.8070i 0 1.08769 18.4883i −16.9836 0 −35.7533 + 61.9265i
109.4 −0.884622 1.53221i 0 2.43489 4.21735i 6.52561 + 11.3027i 0 −18.4950 0.966772i −22.7698 0 11.5454 19.9972i
109.5 0.884622 + 1.53221i 0 2.43489 4.21735i −6.52561 11.3027i 0 −18.4950 0.966772i 22.7698 0 11.5454 19.9972i
109.6 1.64637 + 2.85159i 0 −1.42105 + 2.46133i 10.8582 + 18.8070i 0 1.08769 18.4883i 16.9836 0 −35.7533 + 61.9265i
109.7 1.74618 + 3.02447i 0 −2.09829 + 3.63435i −3.93765 6.82022i 0 18.4135 1.98614i 13.2829 0 13.7517 23.8187i
109.8 2.73089 + 4.73004i 0 −10.9155 + 18.9063i 0.0995681 + 0.172457i 0 −16.0061 + 9.31685i −75.5424 0 −0.543819 + 0.941923i
163.1 −2.73089 + 4.73004i 0 −10.9155 18.9063i −0.0995681 + 0.172457i 0 −16.0061 9.31685i 75.5424 0 −0.543819 0.941923i
163.2 −1.74618 + 3.02447i 0 −2.09829 3.63435i 3.93765 6.82022i 0 18.4135 + 1.98614i −13.2829 0 13.7517 + 23.8187i
163.3 −1.64637 + 2.85159i 0 −1.42105 2.46133i −10.8582 + 18.8070i 0 1.08769 + 18.4883i −16.9836 0 −35.7533 61.9265i
163.4 −0.884622 + 1.53221i 0 2.43489 + 4.21735i 6.52561 11.3027i 0 −18.4950 + 0.966772i −22.7698 0 11.5454 + 19.9972i
163.5 0.884622 1.53221i 0 2.43489 + 4.21735i −6.52561 + 11.3027i 0 −18.4950 + 0.966772i 22.7698 0 11.5454 + 19.9972i
163.6 1.64637 2.85159i 0 −1.42105 2.46133i 10.8582 18.8070i 0 1.08769 + 18.4883i 16.9836 0 −35.7533 61.9265i
163.7 1.74618 3.02447i 0 −2.09829 3.63435i −3.93765 + 6.82022i 0 18.4135 + 1.98614i 13.2829 0 13.7517 + 23.8187i
163.8 2.73089 4.73004i 0 −10.9155 18.9063i 0.0995681 0.172457i 0 −16.0061 9.31685i −75.5424 0 −0.543819 0.941923i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 109.8 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.c even 3 1 inner
21.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.4.e.h 16
3.b odd 2 1 inner 189.4.e.h 16
7.c even 3 1 inner 189.4.e.h 16
7.c even 3 1 1323.4.a.bo 8
7.d odd 6 1 1323.4.a.bn 8
21.g even 6 1 1323.4.a.bn 8
21.h odd 6 1 inner 189.4.e.h 16
21.h odd 6 1 1323.4.a.bo 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.h 16 1.a even 1 1 trivial
189.4.e.h 16 3.b odd 2 1 inner
189.4.e.h 16 7.c even 3 1 inner
189.4.e.h 16 21.h odd 6 1 inner
1323.4.a.bn 8 7.d odd 6 1
1323.4.a.bn 8 21.g even 6 1
1323.4.a.bo 8 7.c even 3 1
1323.4.a.bo 8 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(189, [\chi])$$:

 $$T_{2}^{16} + 56 T_{2}^{14} + 2151 T_{2}^{12} + 42140 T_{2}^{10} + 593317 T_{2}^{8} + 5029374 T_{2}^{6} + \cdots + 152473104$$ T2^16 + 56*T2^14 + 2151*T2^12 + 42140*T2^10 + 593317*T2^8 + 5029374*T2^6 + 30217320*T2^4 + 80385480*T2^2 + 152473104 $$T_{13}^{4} + 42T_{13}^{3} - 4475T_{13}^{2} - 94812T_{13} + 5259952$$ T13^4 + 42*T13^3 - 4475*T13^2 - 94812*T13 + 5259952

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} + \cdots + 152473104$$
$3$ $$T^{16}$$
$5$ $$T^{16} + \cdots + 39033114624$$
$7$ $$(T^{8} + 30 T^{7} + \cdots + 13841287201)^{2}$$
$11$ $$T^{16} + \cdots + 13\!\cdots\!44$$
$13$ $$(T^{4} + 42 T^{3} + \cdots + 5259952)^{4}$$
$17$ $$T^{16} + \cdots + 33\!\cdots\!24$$
$19$ $$(T^{8} + \cdots + 141394168701184)^{2}$$
$23$ $$T^{16} + \cdots + 37\!\cdots\!84$$
$29$ $$(T^{8} + \cdots + 18\!\cdots\!32)^{2}$$
$31$ $$(T^{8} + \cdots + 11\!\cdots\!81)^{2}$$
$37$ $$(T^{8} + \cdots + 16\!\cdots\!76)^{2}$$
$41$ $$(T^{8} + \cdots + 13\!\cdots\!88)^{2}$$
$43$ $$(T^{4} - 728 T^{3} + \cdots - 12025459739)^{4}$$
$47$ $$T^{16} + \cdots + 47\!\cdots\!00$$
$53$ $$T^{16} + \cdots + 20\!\cdots\!04$$
$59$ $$T^{16} + \cdots + 37\!\cdots\!64$$
$61$ $$(T^{8} + \cdots + 17\!\cdots\!25)^{2}$$
$67$ $$(T^{8} + \cdots + 28\!\cdots\!36)^{2}$$
$71$ $$(T^{8} + \cdots + 87\!\cdots\!00)^{2}$$
$73$ $$(T^{8} + \cdots + 10\!\cdots\!44)^{2}$$
$79$ $$(T^{8} + \cdots + 87\!\cdots\!96)^{2}$$
$83$ $$(T^{8} + \cdots + 16\!\cdots\!72)^{2}$$
$89$ $$T^{16} + \cdots + 48\!\cdots\!44$$
$97$ $$(T^{4} - 292 T^{3} + \cdots + 100316149821)^{4}$$