LMFDB - Newform orbit 189.4.e.h

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 + 54x^14 - 12x^13 + 2361x^12 - 966x^11 + 29570x^10 - 65952x^9 + 300096x^8 - 356610x^7 + 531858x^6 + 80070x^5 + 77197x^4 - 216x^3 + 4878x^2 + 486x + 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.

$f(q)$	$=$	$q + ( - \beta_{5} - \beta_{2}) q^{2} + (\beta_{4} - \beta_{3} + 6 \beta_1) q^{4} + ( - \beta_{12} - \beta_{11}) q^{5} + (\beta_{7} + \beta_{4} - \beta_{3} + \cdots - 5) q^{7} + ( - \beta_{9} + 5 \beta_{2}) q^{8}+ \cdots + (62 \beta_{12} + 64 \beta_{11} + \cdots - 251 \beta_{2}) q^{98}+O(q^{100})$ q + (-b5 - b2) * q^2 + (b4 - b3 + 6b1) q^4 + (-b12 - b11) * q^5 + (b7 + b4 - b3 - 3b1 - 5) q^7 + (-b9 + 5b2) q^8 + (-b15 + b10 + 6b1) q^10 + (-b14 - b12 + b8 + b6 + 3b5) q^11 + (-b13 - b10 - 3b4 - b1 - 11) q^13 + (b12 - b11 - b9 - b6 + 4b5 + 6b2) * q^14 + (-b15 - 4b10 - b7 + b4 + 12b3 - 22b1 - 20) q^16 + (b12 + 2b8 + 2b6 - 8b5) q^17 + (b15 + 2b13 - 4b10 - b7 + b4 - 3b1 - 1) q^19 + (b11 + b9 + b8 + 3b2) q^20 + (-b15 + 2b13 + 2b10 + 4b7 - 8b4 - b3 + 4b1 + 31) q^22 + (b14 + 2b12 + 2b11 + b9 - b6 - 15b5 - 15b2) * q^23 + (-4b15 + 4b10 - 10b4 + 10b3 + 53b1) q^25 + (2b14 - 9b12 - 9b11 + 2b9 + b6 - 8b5 - 8b2) * q^26 + (-5b10 - 3b7 - 12b4 + 24b3 - 70b1 - 44) q^28 + (7b11 - 4b9 - 2b8 + 20b2) * q^29 + (b15 + b13 - 4b10 - 5b7 + 13b4 - 12b3 + 96b1 - 2) q^31 + (-5b14 - 6b12 - 4b8 - 4b6 + 57b5) q^32 + (-4b15 - b13 - b10 + 16b7 + 30b4 - 4b3 + 7b1 - 111) q^34 + (-b14 + 9b12 - 5b11 - 4b9 - b8 + 3b6 + 11b5 - 8b2) * q^35 + (-b15 - b13 - 15b3 - 87b1 - 87) * q^37 + (b14 + 12b12 - 6b8 - 6b6 - 10b5) * q^38 + (-6b15 - 5b13 - 4b10 - b7 + b4 - 3b3 - 17b1 - 15) q^40 + (5b11 + 4b9 - 2b8 + 28b2) * q^41 + (-2b15 - 6b13 - 6b10 + 8b7 - 16b4 - 2b3 - 2b1 + 183) q^43 + (3b14 + 13b12 + 13b11 + 3b9 + b6 - 81b5 - 81b2) * q^44 + (4b15 + b13 - 7b10 - 5b7 + 21b4 - 20b3 + 185b1 - 2) * q^46 + (8b14 - 15b12 - 15b11 + 8b9 - 2b6 + 8b5 + 8b2) q^47 + (7b13 + 7b10 - b7 - 8b4 + 15b3 - 102b1 + 75) q^49 + (36b11 + 14b9 + 4b8 - 29b2) * q^50 + (-3b15 - 4b13 + 15b10 + 20b7 + 23b4 - 27b3 + 262b1 + 8) q^52 + (9b14 - 2b12 + 3b8 + 3b6 + 37b5) q^53 + (6b15 + 12b13 + 12b10 - 24b7 - 4b4 + 6b3 - 166) * q^55 + (-4b14 + 8b12 + b11 + 12b9 + 3b8 - 2b6 + 100b5 - 74b2) q^56 + (3b15 + 3b13 + 62b3 - 295b1 - 295) * q^58 + (5b14 + 27b12 - b8 - b6 - 91b5) q^59 + (2b15 + b13 + 4b10 + b7 - b4 - 23b3 - 24b1 - 26) * q^61 + (-3b11 - 13b9 - 5b8 + 200b2) * q^62 + (5b15 + 3b13 + 3b10 - 20b7 - 77b4 + 5b3 - 7b1 + 612) q^64 + (-17b14 - 5b12 - 5b11 - 17b9 + 7b6 - 105b5 - 105b2) q^65 + (14b15 + b13 - 17b10 - 5b7 - 2b4 + 3b3 + 75b1 - 2) * q^67 + (-27b14 + 11b12 + 11b11 - 27b9 - 3b6 + 203b5 + 203b2) q^68 + (-14b15 - 21b13 + 8b10 + 24b7 + 2b4 + 11b3 + 179b1 + 358) q^70 + (-45b11 + 3b9 + 5b8 - 21b2) * q^71 + (14b15 + 7b13 - 35b10 - 35b7 - 6b4 + 13b3 - 602b1 - 14) q^73 + (14b14 - 9b12 + b8 + b6 - 16b5) q^74 + (3b15 - 5b13 - 5b10 - 12b7 - 32b4 + 3b3 - 11b1 - 56) q^76 + (13b14 - 47b12 - 26b11 - 4b9 - b8 + 3b6 + 25b5 - 64b2) q^77 + (b15 - 3b13 + 16b10 + 4b7 - 4b4 + 25b3 - 61b1 - 69) * q^79 + (-11b14 - 59b12 + 9b8 + 9b6 - 27b5) q^80 + (9b15 + b13 + 32b10 + 8b7 - 8b4 - 50b3 - 397b1 - 413) * q^82 + (10b11 - 2b9 + 8b8 + 98b2) * q^83 + (10b15 + 4b13 + 4b10 - 40b7 + 18b4 + 10b3 - 16b1 + 144) q^85 + (12b14 - 48b12 - 48b11 + 12b9 - 4b6 - 311b5 - 311b2) q^86 + (-13b15 - 13b13 + 52b10 + 65b7 + 63b4 - 76b3 + 889b1 + 26) q^88 + (-4b14 - 17b12 - 17b11 - 4b9 + 4b5 + 4b2) * q^89 + (21b15 + 7b13 + b10 - 34b7 + 49b4 - 15b3 - 123b1 - 80) * q^91 + (-14b11 - 16b9 + 234b2) q^92 + (-11b15 - 4b13 + 23b10 + 20b7 + 94b4 - 98b3 - 42b1 + 8) q^94 + (5b14 + 44b12 + 13b8 + 13b6 + 129b5) q^95 + (2b15 - 8b13 - 8b10 - 8b7 - 50b4 + 2b3 - 12b1 + 65) q^97 + (62b12 + 64b11 + 15b9 - 6b6 - 39b5 - 251b2) * q^98
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 48 q^{4} - 60 q^{7} - 44 q^{10} - 168 q^{13} - 156 q^{16} - 12 q^{19} + 448 q^{22} - 408 q^{25} - 152 q^{28} - 800 q^{31} - 1896 q^{34} - 692 q^{37} - 96 q^{40} + 2912 q^{43} - 1524 q^{46} + 2020 q^{49}+ \cdots + 1168 q^{97}+O(q^{100})$ 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 $x^{16} + 54 x^{14} - 12 x^{13} + 2361 x^{12} - 966 x^{11} + 29570 x^{10} - 65952 x^{9} + 300096 x^{8} + \cdots + 81$ 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 :

$\beta_{1}$	$=$	$( 51\!\cdots\!86 \nu^{15} + \cdots - 15\!\cdots\!87 ) / 34\!\cdots\!04$ (5150955099921168548586v^15 - 399066979643989434973v^14 + 278014912120055287156056v^13 - 83387557077442080040203v^12 + 12158828863833005482946292v^11 - 5917812735287788738645344v^10 + 152377692231961500414711582v^9 - 351447040288173944394280142v^8 + 1568119463587256938331080650v^7 - 1948431641273295864938623590v^6 + 2843045574249633988600391868v^5 + 239921202700739614544036220v^4 + 306862261345572305282838288v^3 - 62802406547786806787941369v^2 + 18876629001889791000696798*v - 1570364393273348796566487) / 3417839928129147250678704
$\beta_{2}$	$=$	$( 44\!\cdots\!71 \nu^{15} + \cdots + 10\!\cdots\!79 ) / 29\!\cdots\!40$ (442764159594379869520971v^15 + 92045143513852266048383v^14 + 23875233760018211350286395v^13 - 334521476496697879981549v^12 + 1042378325146991278463590872v^11 - 209637242658343244713775850v^10 + 12920716480808433551935581884v^9 - 26431419694369820486902515736v^8 + 125681615744961907464618798700v^7 - 127946844118204810042200678910v^6 + 190708322691323970606768128508v^5 + 100146641482197993232718041374v^4 + 14936832517605486267172602005v^3 + 6897374060046734350153592973v^2 + 724005707439463626563830881*v + 1045588544089657175339954379) / 290516393890977516307689840
$\beta_{3}$	$=$	$( 95\!\cdots\!24 \nu^{15} + \cdots + 13\!\cdots\!43 ) / 29\!\cdots\!84$ (95380685141312083531524v^15 - 19999835154833760133025v^14 + 5143920844575013833434298v^13 - 2223805481128098770493153v^12 + 225076637537986043826828356v^11 - 139256162798744285266167836v^10 + 2824074367489539682128456470v^9 - 6874798095428693104070498202v^8 + 29746512388308646051695896130v^7 - 39600997957091737754309283690v^6 + 55789433302351489421009172356v^5 - 991471541688804280620645008v^4 + 2823503042864611244771923334v^3 - 4527163588678219491999721417v^2 + 152717063340804932034735444*v + 13553422352698653807833043) / 29051639389097751630768984
$\beta_{4}$	$=$	$( - 78\!\cdots\!07 \nu^{15} + \cdots + 45\!\cdots\!11 ) / 17\!\cdots\!08$ (-78587806494897782607v^15 - 18514612371959954123v^14 - 4238897266583391539275v^13 - 56958150753926795849v^12 - 185067326029031479134312v^11 + 32121640103510367075210v^10 - 2294800844682407360497616v^9 + 4629860906669748741870256v^8 - 22231547944601577747330100v^7 + 22124852259119506579511074v^6 - 33866059298945644669248108v^5 - 17791507215158118971325294v^4 - 5560894533930725148743453v^3 - 1225257977546479953252873v^2 - 128607778453102121256201*v + 45149992925452738096311) / 17364996646203079277208
$\beta_{5}$	$=$	$( 66\!\cdots\!74 \nu^{15} + \cdots - 22\!\cdots\!33 ) / 14\!\cdots\!20$ (665522250423697530353874v^15 - 125243857984949990751497v^14 + 35941861102546041760672534v^13 - 14738278130936968798845517v^12 + 1573020323036532483998365408v^11 - 938037836291855080686051628v^10 + 19810054007942904483154236558v^9 - 47573713680789795812173517278v^8 + 208123772376420301381650915250v^7 - 274861520920805544623772742090v^6 + 399390063142360978998688646512v^5 - 13098061260473914994913455176v^4 + 43133473362055627998543680712v^3 - 8839371024656621237947489541v^2 + 2733811532980690542634041264*v - 220930559608896963997451433) / 145258196945488758153844920
$\beta_{6}$	$=$	$( - 73\!\cdots\!09 \nu^{15} + \cdots + 89\!\cdots\!53 ) / 11\!\cdots\!32$ (-7378249582352633616809v^15 + 7640516173576034579001v^14 - 396486260130533734438365v^13 + 501010878979267442799633v^12 - 17406995501606988686312928v^11 + 25135602796127952251865918v^10 - 220977529076860573592357800v^9 + 710311663408836311201832420v^8 - 2660715630374200481681211576v^7 + 4789207986003342252715895190v^6 - 6063461101389177663116968668v^5 + 2705051621646512950567970286v^4 + 1162352296016671188979227565v^3 + 467098110845212066943802927v^2 + 83276969081471319038266269*v + 8932657899805938939981753) / 1185781199555010270643632
$\beta_{7}$	$=$	$( - 18\!\cdots\!03 \nu^{15} + \cdots + 17\!\cdots\!42 ) / 29\!\cdots\!84$ (-181202429822774059445203v^15 + 173977431935443638961454v^14 - 9811685959379603724793333v^13 + 11565395949374069055808906v^12 - 431363096763830035881827228v^11 + 585885681184940118029980022v^10 - 5589979039086480130957442826v^9 + 17110561464270230737000486850v^8 - 66667094487687553817090945818v^7 + 118423275991500249424147295008v^6 - 166515257410212088424334640544v^5 + 86533867862175077271211405370v^4 - 14103155587023330050892289055v^3 + 7760418160366236295272417264v^2 - 2675857350338389303631635689*v + 17606182425241232279099142) / 29051639389097751630768984
$\beta_{8}$	$=$	$( - 46\!\cdots\!69 \nu^{15} + \cdots - 84\!\cdots\!78 ) / 41\!\cdots\!12$ (-46137478859790502534169v^15 - 18003661639799948801112v^14 - 2497338363865019696115297v^13 - 416038060561248460663806v^12 - 109030841460406320382014300v^11 + 2273497446134516638901454v^10 - 1360721802377728113197714806v^9 + 2522331908486871096491201670v^8 - 12828449250602365675826380110v^7 + 11521798797354065797958395260v^6 - 19974983405575959441115234416v^5 - 10518093913416689404508414166v^4 - 8515955152820740899495849605v^3 - 724055489241304417393132566v^2 - 75981746281782619859736105*v - 8494466193943117728132078) / 4150234198442535947252712
$\beta_{9}$	$=$	$( 35\!\cdots\!57 \nu^{15} + \cdots + 10\!\cdots\!43 ) / 29\!\cdots\!40$ (3581213557777451280388457v^15 + 497101835269179788970811v^14 + 192780893717033095789404005v^13 - 15980579473944711138709193v^12 + 8416740229132048443443408064v^11 - 2270254578144792137065853430v^10 + 103992904622903652639078628208v^9 - 220543126481258728893468067532v^8 + 1024155956141169744925415357600v^7 - 1083369088036026586115709143390v^6 + 1539084062908760277548679251916v^5 + 807377687758217219205149874858v^4 - 39477049489181380817607242085v^3 + 55616734255955182638475383621v^2 + 5838615735956173206944959347*v + 10771434886564083995036979543) / 290516393890977516307689840
$\beta_{10}$	$=$	$( - 37\!\cdots\!34 \nu^{15} + \cdots - 45\!\cdots\!93 ) / 29\!\cdots\!84$ (-370727423561987712791834v^15 - 138943322352577418084677v^14 - 20010099818056432848765736v^13 - 3066464048607306731551049v^12 - 873124319747606545549609124v^11 + 29333329071129918628681280v^10 - 10806301579935763307328500414v^9 + 20305374462049276803890790674v^8 - 101803954946404128926376815122v^7 + 89605900266476711030909858062v^6 - 143950743942351235848030195932v^5 - 109762217468378527910055229084v^4 - 31002241094379135143472273216v^3 - 12836668733018897803488941265v^2 - 639346001539411016632446822*v - 459002540690788617114448293) / 29051639389097751630768984
$\beta_{11}$	$=$	$( 22\!\cdots\!31 \nu^{15} + \cdots + 33\!\cdots\!79 ) / 14\!\cdots\!20$ (2289167883872891291150831v^15 + 627420074432135434000673v^14 + 123588042832718453901704065v^13 + 6386433259237099110525131v^12 + 5395753424776406429835162552v^11 - 730936881254907217767908190v^10 + 67021974193589028530144363484v^9 - 132451463917252456080785597776v^8 + 644852294661348734077554382300v^7 - 626908185570784693395088158490v^6 + 987671105811000701345435576388v^5 + 519171675475924636810246102794v^4 + 222428943273666648886756094785v^3 + 35750388942425214750207298203v^2 + 3752277027081566743984640571*v + 3310396194125585240714736279) / 145258196945488758153844920
$\beta_{12}$	$=$	$( 49\!\cdots\!31 \nu^{15} + \cdots - 16\!\cdots\!07 ) / 29\!\cdots\!40$ (4998542427516925336278531v^15 - 1090288236405243753967463v^14 + 269998200596936769336168811v^13 - 118845324104131662579154303v^12 + 11818767259407872899010172832v^11 - 7402999434033410305883790322v^10 + 149040219219300537152985478512v^9 - 361947314427688630409111620012v^8 + 1574154920688246687446763207760v^7 - 2114379398528963194371454132450v^6 + 3069157212268333177811738925028v^5 - 200867138436006223737091483954v^4 + 331513256607379064367614787033v^3 - 67959586376872875103251681569v^2 + 41764693992637524946085799981*v - 1698392289271548563809157007) / 290516393890977516307689840
$\beta_{13}$	$=$	$( - 25\!\cdots\!02 \nu^{15} + \cdots - 33\!\cdots\!25 ) / 58\!\cdots\!68$ (-2571831061897469598114502v^15 - 493877469490488980635899v^14 - 138735270869336879994709404v^13 + 4210350941760129617515551v^12 - 6058443645835285290299979476v^11 + 1317463601632217043187740056v^10 - 75234957599634968389557350878v^9 + 154911001630630906925272891286v^8 - 735074006714935417538144034426v^7 + 759756008244781386304974139518v^6 - 1150871009667652044360949647404v^5 - 515084123499938412097060102564v^4 - 171459431213353795023532713060v^3 - 23568065621853474074904047959v^2 - 4322794507103219074613130762*v - 3317453458629006220339166325) / 58103278778195503261537968
$\beta_{14}$	$=$	$( 39\!\cdots\!04 \nu^{15} + \cdots - 12\!\cdots\!68 ) / 72\!\cdots\!60$ (3911264155408570583939304v^15 - 694118293076704457345747v^14 + 211221676579589381568989329v^13 - 84431993902682656026315592v^12 + 9243517985344549650308741938v^11 - 5418030775292864868093348118v^10 + 116356897336999386253673581698v^9 - 278527017467178868822677710938v^8 + 1219877326870322355058022530300v^7 - 1604361396434478753759038607160v^6 + 2332068617501933062829815627702v^5 - 62652997359749764024278509146v^4 + 251847313221794279320194469182v^3 - 51605525411741198441767997291v^2 + 13097877297782791499527921839*v - 1289871806587074023992553268) / 72629098472744379076922460
$\beta_{15}$	$=$	$( - 10\!\cdots\!87 \nu^{15} + \cdots + 86\!\cdots\!95 ) / 14\!\cdots\!92$ (-1073865354571902572408887v^15 + 179196365203365095644700v^14 - 58014230781407180494288473v^13 + 22553910299694760921607847v^12 - 2538913821945140369206816736v^11 + 1460265363455016395184416846v^10 - 31986879836697318747160200580v^9 + 76126132490064599383326439026v^8 - 334803809171548014454396381296v^7 + 438158821213752374865905238138v^6 - 641836824574537975441293452396v^5 + 15301213626900151314291998654v^4 - 77071655551893286686923333217v^3 + 6211379408222858418994419562v^2 - 6426318911590096700217830559*v + 86065612803115055061832395) / 14525819694548875815384492

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$( - 3 \beta_{15} - 2 \beta_{14} - \beta_{13} - 5 \beta_{12} + 6 \beta_{10} + \beta_{8} + 5 \beta_{7} + \cdots + 2 ) / 42$ (-3b15 - 2b14 - b13 - 5b12 + 6b10 + b8 + 5b7 + b6 + 14b5 + 2b4 - 3b3 + 5*b1 + 2) / 42
$\nu^{2}$	$=$	$( 4 \beta_{15} + 9 \beta_{14} - \beta_{13} + 12 \beta_{12} + 12 \beta_{11} + 20 \beta_{10} + 9 \beta_{9} + \cdots - 565 ) / 42$ (4b15 + 9b14 - b13 + 12b12 + 12b11 + 20b10 + 9b9 + 5b7 + 6b6 + 63b5 - 5b4 + 4b3 + 63b2 - 555*b1 - 565) / 42
$\nu^{3}$	$=$	$( 65 \beta_{15} - 144 \beta_{13} - 449 \beta_{11} - 144 \beta_{10} - 83 \beta_{9} - 115 \beta_{8} + \cdots - 13 ) / 84$ (65b15 - 144b13 - 449b11 - 144b10 - 83b9 - 115b8 - 260b7 - 188b4 + 65b3 + 749b2 - 274*b1 - 13) / 84
$\nu^{4}$	$=$	$( 52 \beta_{15} - 312 \beta_{14} + 218 \beta_{13} - 339 \beta_{12} - 706 \beta_{10} - 285 \beta_{8} + \cdots - 436 ) / 42$ (52b15 - 312b14 + 218b13 - 339b12 - 706b10 - 285b8 - 1090b7 - 285b6 - 3318b5 - 317b4 + 535b3 + 18174b1 - 436) / 42
$\nu^{5}$	$=$	$( 5672 \beta_{15} + 2377 \beta_{14} + 8249 \beta_{13} + 17713 \beta_{12} + 17713 \beta_{11} + \cdots + 19531 ) / 84$ (5672b15 + 2377b14 + 8249b13 + 17713b12 + 17713b11 - 10308b10 + 2377b9 - 2577b7 - 5063b6 - 25375b5 + 2577b4 + 2753b3 - 25375b2 + 14377b1 + 19531) / 84
$\nu^{6}$	$=$	$( - 1577 \beta_{15} - 983 \beta_{13} - 1614 \beta_{11} - 983 \beta_{10} - 1823 \beta_{9} + 2046 \beta_{8} + \cdots + 122121 ) / 7$ (-1577b15 - 983b13 - 1614b11 - 983b10 - 1823b9 + 2046b8 + 6308b7 + 4927b4 - 1577b3 - 25221b2 + 2171*b1 + 122121) / 7
$\nu^{7}$	$=$	$( - 165693 \beta_{15} - 37672 \beta_{14} - 53663 \beta_{13} - 349141 \beta_{12} + 326682 \beta_{10} + \cdots + 107326 ) / 42$ (-165693b15 - 37672b14 - 53663b13 - 349141b12 + 326682b10 + 107141b8 + 268315b7 + 107141b6 + 500836b5 + 36178b4 - 89841b3 - 480797b1 + 107326) / 42
$\nu^{8}$	$=$	$( 198134 \beta_{15} + 400050 \beta_{14} - 205994 \beta_{13} + 246687 \beta_{12} + 246687 \beta_{11} + \cdots - 28093496 ) / 42$ (198134b15 + 400050b14 - 205994b13 + 246687b12 + 246687b11 + 1616512b10 + 400050b9 + 404128b7 + 527709b6 + 6507396b5 - 404128b4 - 939709b3 + 6507396b2 - 27285240b1 - 28093496) / 42
$\nu^{9}$	$=$	$( 4545349 \beta_{15} - 8840700 \beta_{13} - 27656533 \beta_{11} - 8840700 \beta_{10} - 2499019 \beta_{9} + \cdots - 101048477 ) / 84$ (4545349b15 - 8840700b13 - 27656533b11 - 8840700b10 - 2499019b9 - 9004295b8 - 18181396b7 - 7117696b4 + 4545349b3 + 42607789b2 - 17931398*b1 - 101048477) / 84
$\nu^{10}$	$=$	$( 10805111 \beta_{15} - 15065301 \beta_{14} + 17122465 \beta_{13} - 4441368 \beta_{12} - 62172506 \beta_{10} + \cdots - 34244930 ) / 42$ (10805111b15 - 15065301b14 + 17122465b13 - 4441368b12 - 62172506b10 - 22724454b8 - 85612325b7 - 22724454b6 - 273256347b5 - 40321252b4 + 57443717b3 + 1064254659b1 - 34244930) / 42
$\nu^{11}$	$=$	$( 349102204 \beta_{15} + 84438449 \beta_{14} + 542822449 \beta_{13} + 1100326265 \beta_{12} + \cdots + 4651037963 ) / 84$ (349102204b15 + 84438449b14 + 542822449b13 + 1100326265b12 + 1100326265b11 - 774880980b10 + 84438449b9 - 193720245b7 - 377807215b6 - 1866273143b5 + 193720245b4 + 113845861b3 - 1866273143b2 + 4263597473b1 + 4651037963) / 84
$\nu^{12}$	$=$	$( - 120521474 \beta_{15} - 31201500 \beta_{13} + 5945004 \beta_{11} - 31201500 \beta_{10} + \cdots + 7846479013 ) / 7$ (-120521474b15 - 31201500b13 + 5945004b11 - 31201500b10 - 96209332b9 + 163091760b8 + 482085896b7 + 399791968b4 - 120521474b3 - 1895037564b2 + 209841448*b1 + 7846479013) / 7
$\nu^{13}$	$=$	$( - 11044886505 \beta_{15} - 1428712508 \beta_{14} - 4137003643 \beta_{13} - 21978952367 \beta_{12} + \cdots + 8274007286 ) / 42$ (-11044886505b15 - 1428712508b14 - 4137003643b13 - 21978952367b12 + 23455897434b10 + 7927216447b8 + 20685018215b7 + 7927216447b6 + 41204820020b5 + 2728166636b4 - 6865170279b3 - 88018702561b1 + 8274007286) / 42
$\nu^{14}$	$=$	$( 4861786192 \beta_{15} + 22350858507 \beta_{14} - 25647216547 \beta_{13} - 9895364610 \beta_{12} + \cdots - 1860633895699 ) / 42$ (4861786192b15 + 22350858507b14 - 25647216547b13 - 9895364610b12 - 9895364610b11 + 122036010956b10 + 22350858507b9 + 30509002739b7 + 42095759784b6 + 471803384241b5 - 30509002739b4 - 68792253050b3 + 471803384241b2 - 1799615890221b1 - 1860633895699) / 42
$\nu^{15}$	$=$	$( 353535630725 \beta_{15} - 548381305872 \beta_{13} - 1762964503613 \beta_{11} - 548381305872 \beta_{10} + \cdots - 11080741238257 ) / 84$ (353535630725b15 - 548381305872b13 - 1762964503613b11 - 548381305872b10 - 95608150535b9 - 665795979367b8 - 1414142522900b7 - 617116818428b4 + 353535630725b3 + 3637334096921b2 - 1255452567322*b1 - 11080741238257) / 84

Display $\beta_i$ in terms of $\nu^j$

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

−10.9155

18.9063i

−0.0995681

−

0.172457i

−16.0061

9.31685i

75.5424

−0.543819

0.941923i

109.2

−1.74618

−

3.02447i

−2.09829

3.63435i

3.93765

6.82022i

18.4135

−

1.98614i

−13.2829

13.7517

−

23.8187i

109.3

−1.64637

−

2.85159i

−1.42105

2.46133i

−10.8582

−

18.8070i

1.08769

−

18.4883i

−16.9836

−35.7533

61.9265i

109.4

−0.884622

−

1.53221i

2.43489

−

4.21735i

6.52561

11.3027i

−18.4950

−

0.966772i

−22.7698

11.5454

−

19.9972i

109.5

0.884622

1.53221i

2.43489

−

4.21735i

−6.52561

−

11.3027i

−18.4950

−

0.966772i

22.7698

11.5454

−

19.9972i

109.6

1.64637

2.85159i

−1.42105

2.46133i

10.8582

18.8070i

1.08769

−

18.4883i

16.9836

−35.7533

61.9265i

109.7

1.74618

3.02447i

−2.09829

3.63435i

−3.93765

−

6.82022i

18.4135

−

1.98614i

13.2829

13.7517

−

23.8187i

109.8

2.73089

4.73004i

−10.9155

18.9063i

0.0995681

0.172457i

−16.0061

9.31685i

−75.5424

−0.543819

0.941923i

163.1

−2.73089

4.73004i

−10.9155

−

18.9063i

−0.0995681

0.172457i

−16.0061

−

9.31685i

75.5424

−0.543819

−

0.941923i

163.2

−1.74618

3.02447i

−2.09829

−

3.63435i

3.93765

−

6.82022i

18.4135

1.98614i

−13.2829

13.7517

23.8187i

163.3

−1.64637

2.85159i

−1.42105

−

2.46133i

−10.8582

18.8070i

1.08769

18.4883i

−16.9836

−35.7533

−

61.9265i

163.4

−0.884622

1.53221i

2.43489

4.21735i

6.52561

−

11.3027i

−18.4950

0.966772i

−22.7698

11.5454

19.9972i

163.5

0.884622

−

1.53221i

2.43489

4.21735i

−6.52561

11.3027i

−18.4950

0.966772i

22.7698

11.5454

19.9972i

163.6

1.64637

−

2.85159i

−1.42105

−

2.46133i

10.8582

−

18.8070i

1.08769

18.4883i

16.9836

−35.7533

−

61.9265i

163.7

1.74618

−

3.02447i

−2.09829

−

3.63435i

−3.93765

6.82022i

18.4135

1.98614i

13.2829

13.7517

23.8187i

163.8

2.73089

−

4.73004i

−10.9155

−

18.9063i

0.0995681

−

0.172457i

−16.0061

−

9.31685i

−75.5424

−0.543819

−

0.941923i

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

T_{13}^{4} + 42T_{13}^{3} - 4475T_{13}^{2} - 94812T_{13} + 5259952

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{16} + \cdots + 152473104$ T^16 + 56T^14 + 2151T^12 + 42140T^10 + 593317T^8 + 5029374T^6 + 30217320T^4 + 80385480*T^2 + 152473104
$3$	$T^{16}$ T^16
$5$	$T^{16} + \cdots + 39033114624$ T^16 + 704T^14 + 375444T^12 + 74627264T^10 + 10930325968T^8 + 599009013120T^6 + 24845549154048T^4 + 985254230016*T^2 + 39033114624
$7$	$(T^{8} + 30 T^{7} + \cdots + 13841287201)^{2}$ (T^8 + 30T^7 - 55T^6 - 9786T^5 - 178164T^4 - 3356598T^3 - 6470695T^2 + 1210608210*T + 13841287201)^2
$11$	$T^{16} + \cdots + 13\!\cdots\!44$ T^16 + 8288T^14 + 44402688T^12 + 143032881152T^10 + 336827252346880T^8 + 514857795463151616T^6 + 566352783635405340672T^4 + 338793666874690089517056*T^2 + 135228298497403593575890944
$13$	$(T^{4} + 42 T^{3} + \cdots + 5259952)^{4}$ (T^4 + 42T^3 - 4475T^2 - 94812*T + 5259952)^4
$17$	$T^{16} + \cdots + 33\!\cdots\!24$ T^16 + 34104T^14 + 779209092T^12 + 10239188889888T^10 + 98713419424723728T^8 + 547092920553330493440T^6 + 2031878038296227655195648T^4 + 8271441785272206058979328*T^2 + 33634827889107993428557824
$19$	$(T^{8} + \cdots + 141394168701184)^{2}$ (T^8 + 6T^7 + 14507T^6 + 1636182T^5 + 226469793T^4 + 12609515520T^3 + 570115522928T^2 + 10244082035712*T + 141394168701184)^2
$23$	$T^{16} + \cdots + 37\!\cdots\!84$ T^16 + 20496T^14 + 306939348T^12 + 2134523204352T^10 + 10891837621949904T^8 + 9650097754668524160T^6 + 6333916466375567048448T^4 + 1776863182844288317556736*T^2 + 370876875037988574452649984
$29$	$(T^{8} + \cdots + 18\!\cdots\!32)^{2}$ (T^8 - 131000T^6 + 4759603708T^4 - 33070541140224*T^2 + 18208268086079232)^2
$31$	$(T^{8} + \cdots + 11\!\cdots\!81)^{2}$ (T^8 + 400T^7 + 146142T^6 + 18947824T^5 + 3216564223T^4 + 181995567504T^3 + 49682529244566T^2 + 2302882627538808*T + 118057702007277081)^2
$37$	$(T^{8} + \cdots + 16\!\cdots\!76)^{2}$ (T^8 + 346T^7 + 122319T^6 + 13962202T^5 + 2983445953T^4 + 299880110868T^3 + 54170749556928T^2 + 3012690414934080*T + 164348368660152576)^2
$41$	$(T^{8} + \cdots + 13\!\cdots\!88)^{2}$ (T^8 - 228440T^6 + 14366890684T^4 - 155523979924992*T^2 + 139421904745068288)^2
$43$	$(T^{4} - 728 T^{3} + \cdots - 12025459739)^{4}$ (T^4 - 728T^3 - 17166T^2 + 89054896*T - 12025459739)^4
$47$	$T^{16} + \cdots + 47\!\cdots\!00$ T^16 + 437304T^14 + 143878920516T^12 + 17621995712781600T^10 + 1565436202969647690000T^8 + 71183421986176742256000000T^6 + 2278933037107875490431600000000T^4 + 3367730030822816455359360000000000*T^2 + 4760858355905995100012304000000000000
$53$	$T^{16} + \cdots + 20\!\cdots\!04$ T^16 + 719040T^14 + 353192089716T^12 + 88304539440188736T^10 + 15788196795332952466128T^8 + 1772341817586836785738339968T^6 + 144149953865993786006294399315712T^4 + 6599125287754855265153118808826136576*T^2 + 200257325981366411028422040902180136751104
$59$	$T^{16} + \cdots + 37\!\cdots\!64$ T^16 + 984576T^14 + 647430432192T^12 + 231694316416253952T^10 + 59737627928135970459648T^8 + 9933990926569804997378703360T^6 + 1198099827522347710897881064931328T^4 + 82248711459125692708640309336994742272*T^2 + 3719013789801954660633902216933841058136064
$61$	$(T^{8} + \cdots + 17\!\cdots\!25)^{2}$ (T^8 + 108T^7 + 115850T^6 + 35423280T^5 + 14689630083T^4 + 2715378470064T^3 + 407701497127466T^2 + 30675929696583660*T + 1727746243726888225)^2
$67$	$(T^{8} + \cdots + 28\!\cdots\!36)^{2}$ (T^8 + 342T^7 + 637247T^6 + 303026814T^5 + 369774865245T^4 + 136634204005704T^3 + 49072173061975652T^2 + 4048676928668080800*T + 283440442826007477136)^2
$71$	$(T^{8} + \cdots + 87\!\cdots\!00)^{2}$ (T^8 - 1647576T^6 + 674817953040T^4 - 6295304638812672*T^2 + 8784101891148595200)^2
$73$	$(T^{8} + \cdots + 10\!\cdots\!44)^{2}$ (T^8 - 2282T^7 + 3961827T^6 - 3571443218T^5 + 2713315823845T^4 - 1052378348663964T^3 + 543899368099097388T^2 - 120259178981926964784*T + 108924162892095829161744)^2
$79$	$(T^{8} + \cdots + 87\!\cdots\!96)^{2}$ (T^8 + 278T^7 + 354127T^6 + 118095806T^5 + 106709565625T^4 + 28642908807656T^3 + 8694006870027952T^2 + 288143301722122880*T + 8728684609283389696)^2
$83$	$(T^{8} + \cdots + 16\!\cdots\!72)^{2}$ (T^8 - 1245384T^6 + 385045815888T^4 - 43313260552611840*T^2 + 1618973493511880604672)^2
$89$	$T^{16} + \cdots + 48\!\cdots\!44$ T^16 + 295712T^14 + 73553216148T^12 + 3865980405090368T^10 + 157123456224977102800T^8 + 1640898147337523740009344T^6 + 13692808578023971000424531712T^4 + 8433201862125679551332606613504*T^2 + 4851085399713824919348686418284544
$97$	$(T^{4} - 292 T^{3} + \cdots + 100316149821)^{4}$ (T^4 - 292T^3 - 678686T^2 + 59319684*T + 100316149821)^4
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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 109.8
Significant digits:
Format:

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$ -expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	189.4.e.h

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