LMFDB - Newform orbit 690.2.w.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(7,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([0, 11, 38]))

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

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

chi := DirichletCharacter("690.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.w (of order $44$, degree $20$, 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:	$5.50967773947$
Analytic rank:	$0$
Dimension:	$240$
Relative dimension:	$12$ over $\Q(\zeta_{44})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

$q$-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) = $ $ 240 q + 24 q^{6}+O(q^{10}) $

$\operatorname{Tr}(f)(q) = $ $ 240 q + 24 q^{6} - 44 q^{10} + 24 q^{16} - 44 q^{21} + 96 q^{23} + 16 q^{25} + 16 q^{26} + 44 q^{28} - 16 q^{31} + 44 q^{33} + 16 q^{35} - 24 q^{36} + 44 q^{37} - 88 q^{43} + 8 q^{46} + 96 q^{47} - 24 q^{50} - 24 q^{55} + 44 q^{57} - 16 q^{58} + 88 q^{61} + 56 q^{62} - 88 q^{65} + 132 q^{67} - 56 q^{70} + 16 q^{71} + 48 q^{73} + 24 q^{81} - 24 q^{82} + 44 q^{85} - 16 q^{87} + 44 q^{88} - 124 q^{92} + 32 q^{93} + 20 q^{95} - 24 q^{96} - 56 q^{98}+O(q^{100}) $

Display 100 coefficients 10 coefficients

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} $
7.1	−0.997452	+	0.0713392i	−0.212565	+	0.977147i	0.989821	−	0.142315i	−1.95274	+	1.08941i	0.142315	−	0.989821i	1.32241	−	2.42181i	−0.977147	+	0.212565i	−0.909632	−	0.415415i	1.87005	−	1.22594i
7.2	−0.997452	+	0.0713392i	−0.212565	+	0.977147i	0.989821	−	0.142315i	−1.89440	−	1.18796i	0.142315	−	0.989821i	0.244741	−	0.448210i	−0.977147	+	0.212565i	−0.909632	−	0.415415i	1.97432	+	1.04979i
7.3	−0.997452	+	0.0713392i	−0.212565	+	0.977147i	0.989821	−	0.142315i	−0.619349	−	2.14858i	0.142315	−	0.989821i	−1.99507	+	3.65370i	−0.977147	+	0.212565i	−0.909632	−	0.415415i	0.771049	+	2.09892i
7.4	−0.997452	+	0.0713392i	−0.212565	+	0.977147i	0.989821	−	0.142315i	1.62532	+	1.53569i	0.142315	−	0.989821i	1.83156	−	3.35425i	−0.977147	+	0.212565i	−0.909632	−	0.415415i	−1.73074	−	1.41582i
7.5	−0.997452	+	0.0713392i	−0.212565	+	0.977147i	0.989821	−	0.142315i	1.87213	−	1.22275i	0.142315	−	0.989821i	−0.241465	+	0.442209i	−0.977147	+	0.212565i	−0.909632	−	0.415415i	−1.78013	+	1.35319i
7.6	−0.997452	+	0.0713392i	−0.212565	+	0.977147i	0.989821	−	0.142315i	1.98312	+	1.03307i	0.142315	−	0.989821i	−0.653533	+	1.19686i	−0.977147	+	0.212565i	−0.909632	−	0.415415i	−2.05177	−	0.888964i
7.7	0.997452	−	0.0713392i	0.212565	−	0.977147i	0.989821	−	0.142315i	−2.18918	+	0.455502i	0.142315	−	0.989821i	0.172948	−	0.316731i	0.977147	−	0.212565i	−0.909632	−	0.415415i	−2.15111	+	0.610516i
7.8	0.997452	−	0.0713392i	0.212565	−	0.977147i	0.989821	−	0.142315i	−1.40926	−	1.73608i	0.142315	−	0.989821i	1.26240	−	2.31192i	0.977147	−	0.212565i	−0.909632	−	0.415415i	−1.52952	−	1.63113i
7.9	0.997452	−	0.0713392i	0.212565	−	0.977147i	0.989821	−	0.142315i	0.0285953	+	2.23589i	0.142315	−	0.989821i	−1.81898	+	3.33121i	0.977147	−	0.212565i	−0.909632	−	0.415415i	0.188029	+	2.22815i
7.10	0.997452	−	0.0713392i	0.212565	−	0.977147i	0.989821	−	0.142315i	0.855176	+	2.06608i	0.142315	−	0.989821i	1.43032	−	2.61943i	0.977147	−	0.212565i	−0.909632	−	0.415415i	1.00039	+	1.99981i
7.11	0.997452	−	0.0713392i	0.212565	−	0.977147i	0.989821	−	0.142315i	1.40256	−	1.74150i	0.142315	−	0.989821i	1.87882	−	3.44079i	0.977147	−	0.212565i	−0.909632	−	0.415415i	1.27475	−	1.83712i
7.12	0.997452	−	0.0713392i	0.212565	−	0.977147i	0.989821	−	0.142315i	1.93237	−	1.12513i	0.142315	−	0.989821i	−0.968092	+	1.77293i	0.977147	−	0.212565i	−0.909632	−	0.415415i	1.84719	−	1.26012i
37.1	−0.877679	−	0.479249i	0.997452	+	0.0713392i	0.540641	+	0.841254i	−2.21203	−	0.326962i	−0.841254	−	0.540641i	0.672927	+	1.80419i	−0.0713392	−	0.997452i	0.989821	+	0.142315i	1.78476	+	1.34708i
37.2	−0.877679	−	0.479249i	0.997452	+	0.0713392i	0.540641	+	0.841254i	−1.79792	−	1.32947i	−0.841254	−	0.540641i	−1.02771	−	2.75539i	−0.0713392	−	0.997452i	0.989821	+	0.142315i	0.940853	+	2.02850i
37.3	−0.877679	−	0.479249i	0.997452	+	0.0713392i	0.540641	+	0.841254i	0.217321	−	2.22548i	−0.841254	−	0.540641i	0.154967	+	0.415484i	−0.0713392	−	0.997452i	0.989821	+	0.142315i	−1.25730	+	1.84911i
37.4	−0.877679	−	0.479249i	0.997452	+	0.0713392i	0.540641	+	0.841254i	1.06201	+	1.96778i	−0.841254	−	0.540641i	−0.702461	−	1.88337i	−0.0713392	−	0.997452i	0.989821	+	0.142315i	0.0109547	−	2.23604i
37.5	−0.877679	−	0.479249i	0.997452	+	0.0713392i	0.540641	+	0.841254i	2.06085	+	0.867694i	−0.841254	−	0.540641i	−1.60657	−	4.30737i	−0.0713392	−	0.997452i	0.989821	+	0.142315i	−1.39292	−	1.74922i
37.6	−0.877679	−	0.479249i	0.997452	+	0.0713392i	0.540641	+	0.841254i	2.18586	−	0.471182i	−0.841254	−	0.540641i	1.33355	+	3.57540i	−0.0713392	−	0.997452i	0.989821	+	0.142315i	−2.14430	−	0.634025i
37.7	0.877679	+	0.479249i	−0.997452	−	0.0713392i	0.540641	+	0.841254i	−2.09572	−	0.779713i	−0.841254	−	0.540641i	−0.448960	−	1.20371i	0.0713392	+	0.997452i	0.989821	+	0.142315i	−1.46569	−	1.68871i
37.8	0.877679	+	0.479249i	−0.997452	−	0.0713392i	0.540641	+	0.841254i	−1.84527	−	1.26292i	−0.841254	−	0.540641i	1.33335	+	3.57484i	0.0713392	+	0.997452i	0.989821	+	0.142315i	−1.01431	−	1.99278i

See next 80 embeddings (of 240 total)

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
23.d	odd	22	1	inner
115.l	even	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	690.2.w.b	✓	240
5.c	odd	4	1	inner	690.2.w.b	✓	240
23.d	odd	22	1	inner	690.2.w.b	✓	240
115.l	even	44	1	inner	690.2.w.b	✓	240

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
690.2.w.b	✓	240	1.a	even	1	1	trivial
690.2.w.b	✓	240	5.c	odd	4	1	inner
690.2.w.b	✓	240	23.d	odd	22	1	inner
690.2.w.b	✓	240	115.l	even	44	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{240} + 88 T_{7}^{237} - 1408 T_{7}^{236} + 924 T_{7}^{235} + 3872 T_{7}^{234} + \cdots + 55\!\cdots\!96 $ T7^240 + 88*T7^237 - 1408*T7^236 + 924*T7^235 + 3872*T7^234 - 121308*T7^233 + 966606*T7^232 - 1414952*T7^231 - 4796440*T7^230 + 77828432*T7^229 - 406627817*T7^228 + 974520052*T7^227 + 2403215848*T7^226 - 32808880192*T7^225 + 151113664523*T7^224 - 273487541228*T7^223 - 832433272704*T7^222 + 10850368527856*T7^221 - 71696129569354*T7^220 + 81017116115276*T7^219 + 301335956315944*T7^218 - 5777739420618836*T7^217 + 41880648144599981*T7^216 - 69128116042789540*T7^215 - 232919575512182328*T7^214 + 4047759587673052592*T7^213 - 24415346852341745456*T7^212 + 51909842221008946504*T7^211 + 167817296671174242488*T7^210 - 2532010328734671086544*T7^209 + 13891873973752509248963*T7^208 - 29145363651079540110840*T7^207 - 79907836444352332407816*T7^206 + 1134680983056484359351944*T7^205 - 5720842789301531436315546*T7^204 + 10659916497344163461887376*T7^203 + 34360505982796100765215448*T7^202 - 439630391191015672954551828*T7^201 + 2118545384642200629832214520*T7^200 - 3914995403741647465056276064*T7^199 - 12044052817639515968072156544*T7^198 + 146186477016049555755262614516*T7^197 - 648392703244181958306345545590*T7^196 + 985677712296582650570258347512*T7^195 + 4593139130774673680687010600328*T7^194 - 45211628000279134596769148663444*T7^193 + 192475701572363783225234232118179*T7^192 - 253064158039796915330149753350800*T7^191 - 1326764225937378603831043431392032*T7^190 + 11522156888850202531747057004164924*T7^189 - 43802617120268940925703381509740984*T7^188 + 28624327301908926968152504502081696*T7^187 + 387365736588699354373384627500717512*T7^186 - 2624583456541506832125515735677438372*T7^185 + 9184397667789339669115589358384922147*T7^184 - 3750710039308115715017998676645324808*T7^183 - 79991169597873865305365328373838682016*T7^182 + 525708665443484283678271733170006760292*T7^181 - 1786451657835590273419760732080733735528*T7^180 - 676825949735223073085773524869519801308*T7^179 + 17950855744925256774028961190010424117968*T7^178 - 90026434265702337449671912633373515652252*T7^177 + 242052888490426307511977753586583668937458*T7^176 + 313805404931197562159428440581749304691376*T7^175 - 2695377328433841561301885213008560474084920*T7^174 + 14318124071570652838333558751993891886112344*T7^173 - 37257239097779400636308921513483166193276631*T7^172 - 77752881855954298138398910139014458929885432*T7^171 + 461706708126428215664171431454108583722195336*T7^170 - 1625778885355798001469107255271141173423614552*T7^169 + 3433066562479392419240641851510307428346693693*T7^168 + 17578451139078929959743523957737439198448739536*T7^167 - 41517428997622238982352681387915634540370296504*T7^166 + 214603595458879547894833283676392971020957123852*T7^165 - 15510145352858069019510020624924127913043969617*T7^164 - 2429135318159989441639230346100938924418571873848*T7^163 + 4684979314142671880868069269333891227216023045608*T7^162 - 3786169263354407797435360191594317248141029099052*T7^161 + 22447757663878548537093211692907526113797624366691*T7^160 + 274430002060589061552696600894214251580035600579904*T7^159 + 155856336537423511154552609862751724469312967349520*T7^158 + 2349130173808065773795851306849285869281473683126624*T7^157 + 7646428108271112664920494081877547164143294139747580*T7^156 - 15345959616105372825001036696836166808056952453967928*T7^155 + 4796981398647757512158152172841772946528942084269232*T7^154 + 84382543309210061804836633483567614271542159138249584*T7^153 + 215467480666215364021857820524358510514813127629426966*T7^152 + 1257067877862609504870267872546310664147600074914935064*T7^151 + 3297805118713360789338057037653539425554255169936253416*T7^150 + 14858876468939065819143881323528799635922280993844043492*T7^149 + 72399453339669469563598786413464422278057690151504297824*T7^148 + 32799176335329998184616927680430309346944530927970728916*T7^147 - 118899175137544969529376525978865430027457355852504615520*T7^146 + 317868980780465690545170416483319137120577173899917796972*T7^145 + 3135786616826786703169890472929690397168356101706069209115*T7^144 + 7681437927833876195011172306548153062542325279450029531548*T7^143 + 1715324463309682004807764612471624131005161678621074985816*T7^142 - 79248068213270625149428640870461492987134491888650835335376*T7^141 - 354423226976887404771456208266630045409661638340822711894178*T7^140 - 1516166485201312539153676008802518524901627801865011317813988*T7^139 - 5445325601271283316723001860464561708359906081415180980453760*T7^138 - 12358634483206189304684678663231681194005310202656828435448012*T7^137 + 746382359954261379330891890455079098657266973695186961956603*T7^136 + 131537271873593943698141169286380900673808695161619043422366396*T7^135 + 620561213836527352030747235420275996816578604270098750624342600*T7^134 + 1767093177516967288032589019475735076735502684550057492916349492*T7^133 + 3239325400511660875711960497280628773904955229385279937135434765*T7^132 + 2384268436845106770849205010887502229609647399153510981922541328*T7^131 - 11487498391492768884535993381423817897713675935231343964668037584*T7^130 - 75188021759504529132298783550443401635413870341571751017615225608*T7^129 - 278229701497827682392890401661875268046831795444707834212713028745*T7^128 - 668619781111810796033426085630242150337737320235335460840519047112*T7^127 - 889835258606960975139494799098231596286279897485608308352030889304*T7^126 + 305384052358894299813591435923313992276588428499130209905300611624*T7^125 + 4379457292558591810898064963699981961779430829047372690232609498404*T7^124 + 11758752728820240069365007816502105538749419206511747131557495353016*T7^123 + 24456376417576093935965303225294229715266898644518493660508195842480*T7^122 + 58835101749714338380871923498279408043454642311911175368744320741252*T7^121 + 133843224957231853568411376435560289685135821404792654008146774588682*T7^120 + 188645249390736873668412547532985847575729431100863579795895149918448*T7^119 + 197330915018218119261423341702191581083715261559929786154809211744936*T7^118 + 1115506527937507671540411194091428622010790311335099367831481535440884*T7^117 + 5931058517695184632725065812143552466648309220430647339883848707154630*T7^116 + 13903593737988561802156402864374656547233701180927486705754561213409828*T7^115 + 9243928212951635408643427976777314786871289757579834816544057329453144*T7^114 - 29026983896648313670202855847860412747468424623701800394978974310651884*T7^113 - 40370400604921129493677787854493971437845889579166144108631682990441718*T7^112 + 121048045177435043705707700237300300461089925309495350664674184116536640*T7^111 + 408283914054064351817053484879871808014908936841206751527294291456766752*T7^110 + 115135484473821685238429346769056219640343937542292866924149386161524728*T7^109 - 855975167484942221411454817255520929238376220081588537873823759989877842*T7^108 - 99780300840846713129718067839639213589323598272115349674954797318026320*T7^107 + 6002601650329685399732831660445389307910836093384632758437026172699471168*T7^106 + 8664677204371892193791710558516188276790753827032940659650247703426905648*T7^105 - 12169163697804331276442418615988757192378385937362885794585217607243578775*T7^104 - 60016221067332462080113711388923074180218081205234093108074622348687371500*T7^103 - 54974798664470599211939548754894414211277869250947347145398032587849414864*T7^102 + 89971680290272399730300966515556362095779043948957632736706182898856044176*T7^101 + 391498113364446315942568962053724079253210367159245087047152101713480370711*T7^100 + 607782353779737201764983293990354924695891719931858029411060114352012960040*T7^99 + 449979832659801676810872466801898649938970746783618724230983875425935238656*T7^98 - 1287655524189407562606278809912477782984561728953995071441423883653279490276*T7^97 - 5818365892537991430663486650968536322492972616712506210029423412673819138043*T7^96 - 11361616763512076349491249557012860982470904750692137813986715721006741713376*T7^95 - 3149873538550294826888255481160191427375840707473071277810186571185474972632*T7^94 + 35256292517683875060627168380727122535102840052404872397107427742570993319428*T7^93 + 85523931357875078225878445195361090927090921640220968785029690984861422141254*T7^92 + 47808763892933423854935329481205637982946489158500833363704695143904504968480*T7^91 - 145138157003792529306002516379884008517262424660465298783495136604968078118016*T7^90 - 297507662073200687193951477877112802177755662245564131625286476256720291743872*T7^89 + 91340063862495417060825344442717092645786402130628903488317497062477499611080*T7^88 + 936783731612681020769960763337160508665995988453439567742152812922731094364068*T7^87 + 604995004335274946177919244440280441970686975038139293149502505889771971316384*T7^86 - 2679566023083594200764794031577190477887626844249962713522284812673500673593516*T7^85 - 5265146961841460681564595897687548947718000333025541647714376115762485515999743*T7^84 + 3083119258815160277330533973032858550844657728530245653743873291637475337932948*T7^83 + 22715536185800946934242378037961804180034005168774116200094163284443040473538072*T7^82 + 15181670477282758440240134971546619193302029943839994224648911661779149233411288*T7^81 - 79356935913486635542696061183987698257007228489889868347870351213148756260409593*T7^80 - 225598573309831472967459422283348500822553445798015803442595754108793326024807852*T7^79 - 155354965187957262067832233812798315669545788300504490511030809834573329659773872*T7^78 + 496598391927876095825564781748261698412120218533302620667836906887874928884737552*T7^77 + 1601871483352190638090878935016268468774907821040372867999207372682460137079989630*T7^76 + 1899396712635105798970473146523330306138469371367508778663928196649330200424196696*T7^75 - 533524454530563784733160413008574639482777138025170370568564631637250487060922176*T7^74 - 5998940505245760605879940610098858919136333678702361465591093342430976838372234860*T7^73 - 10438995183482187702487802683404106017673985669982862418697100945416657703198093629*T7^72 - 6233665750171759526451532685945357177859806829369773075949065876197005944707470784*T7^71 + 10567771464570420587067250258148260069194910710488542654448244585728009597389814728*T7^70 + 30487199182825132749331574936681223671902196690239529065123426065123514342889750912*T7^69 + 31034295158728608988511548792914410201303653934780289653626569486887659335729232800*T7^68 - 3726633953113425687996196852193962009161177123995371672568550248624964294681423520*T7^67 - 56433953906436322713578087478561267034933734584558691143554594166614244579891438344*T7^66 - 76202318387695114384838633596377533636944754888405950929874585313734748056077302124*T7^65 - 21499921875311001303311102746457979827013459942094550609444173406147598550031199382*T7^64 + 87363926959731190217835338833924530089154615271542207782195585075510621140816598628*T7^63 + 170050572119529372420834982489652858795748847530752849429902466790113249667065145640*T7^62 + 155768625188131836259004977184807291333347869677187629990215231863771793796772390184*T7^61 + 53731377588321550144468599787772657566205555458656242559564886021032706221385895253*T7^60 - 55140280280178899036174199502318508499869201387462352592894779395043743279815996940*T7^59 - 96696765437218947741926062116586790250355794673216504640827447195417083553611340032*T7^58 - 67613216825356853939135756095639032621810963043509148326562337315829298930555939888*T7^57 - 20045888339947139148359637556082709345853641953688523946223270059408463449930992536*T7^56 + 787255693589321993074531238056426235519621947768274563156594814223701156045629416*T7^55 - 5074705440223386207007517096757446454983040296997624658338581969438717933952933944*T7^54 - 11399204762581892704231478690569570142033041395703799540973409474655223069254862120*T7^53 - 4189101948869459944922233293647004104141391406898956015633912765696294358440396235*T7^52 + 7748323010278606261420106768622250519698721762019756934551273957604266331561317888*T7^51 + 11453701224427095600438065030175430277718761209802897433279356536954492722636779416*T7^50 + 7048637443444192763136930668128452076507437818212555681166371374801720172843961720*T7^49 + 1258937905906714920168237539433086337985440939670045152628394664983761993142702105*T7^48 + 178082008368397075011933281921657267904978073103462703558274042374076559657751632*T7^47 + 1831724664133788344496986678100905757372612008792091578749133001352814976329027680*T7^46 + 3072284599569413116244060438830113939672647178169246081194330414281136809491200756*T7^45 + 1075157568756284578552356232795975753518586317147626517829035120972734017692104758*T7^44 - 2097059544751819515074278247453178072786946624697802818845918834284410421842563676*T7^43 - 1954551947924111290544331963522758192223325942017152443839717655951001050261029336*T7^42 - 1872439644398326103789077798015179837045995203013553166024640283488014855067641332*T7^41 - 541530307124802632312082478576914674850485859681841708494211176997457772856070135*T7^40 + 650687945403967727120164770070611243653013527797023700422210154778712242208418076*T7^39 + 223214849397732881712357660720037735996494073340000503930378163840033725666146920*T7^38 + 378745343235274250549631222413802965941126326551549545530204452974465279988005600*T7^37 + 186523191932278290267975637801929407093574816827887832841424530104809016996039188*T7^36 - 45253959315639161524286076572064040737233115693281498157799308782406335495332768*T7^35 + 40662409708122027602625574512799369956876320011104968390920867381152732335201376*T7^34 - 17510085171992430191443825775730453471929641220048402650787790741822481213833920*T7^33 - 16760626912832914646849934285539089837343041086975977884730484877767826406890176*T7^32 + 9942272299714645639968329025149758579863182524648237240290587895377825953278144*T7^31 - 714471455963850504659870106681181886228546882981572808697096988922789094310016*T7^30 + 435718607341824087794302437194876402495056125095841061703826087094477572864*T7^29 + 1548052606891251255825386800424366726163237878084764828619986544626460153323072*T7^28 - 212699944921475174195235407811577192029786231760158807566387040517384621688064*T7^27 - 71751356157125824571576531490639453867768302828464355264313995221242324188160*T7^26 - 73497892044515912610663319522734216506511584567728700233862820247902813348864*T7^25 - 54785036238955617272071807879794198475577625747295321720308743728186827029248*T7^24 + 17351448108469591109608547992572848212879475303286797742612339673042074988544*T7^23 - 11385100642245536315139425876090532951432757797441800680957119223395471722496*T7^22 + 1777951831633490004527761848744120178532647342699098789957537064449699887104*T7^21 + 5802873568369223445080601229394050052933120872149365900638551345548887195648*T7^20 - 2045414552988254601229168620370579614106347937269621709167281226456469610496*T7^19 + 255886990252969479196991775175499948360778828512617729464715411492961460224*T7^18 + 215552056542582094955597030780129913968077366399839960103609922511748464640*T7^17 - 199688251769431793942219974506398775949714813748988860450586776518656106496*T7^16 - 3115132962853904034271530484275634864670415769968919355662458272394379264*T7^15 + 16136858520787034084058821221795023184173384732548816934439811331717758976*T7^14 - 6850255320068062529441344940967949179415209818344061605398494933262139392*T7^13 + 1123821085312457711955451085067564920060584275497224025694482891648319488*T7^12 + 985529753923056201257928662613065780052321541092343227055726223850668032*T7^11 - 53886815617939318621005306857123612764522171129428456311640526143225856*T7^10 + 44254705181917138390328954376554901233559002346864982245521369048416256*T7^9 + 16482491252124012027337815360693570654519606944518765775641982471569408*T7^8 - 1814502098698644968552130174643134307032246905808184595025150764646400*T7^7 + 90527010218862718592769369612585436773731550967216496430585825722368*T7^6 - 3560174252194349268518374293816062743317425875312401866131118751744*T7^5 + 114834989876026090425408247119671112935362604477610429457587765248*T7^4 - 2718873494071157519314113022150971119750646733885011954228002816*T7^3 + 51100157762351921333718553040586245359352925838826013225648128*T7^2 - 750031460280840396027495318089704390788471272150349408894976*T7 + 5504358656065313755056355870627693124119489667155908100096 acting on $S_{2}^{\mathrm{new}}(690, [\chi])$.

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.w (of order \(44\), degree \(20\), minimal)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 7.12
Significant digits:
Format:

Introduction

L-functions

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