# Properties

 Label 1840.2.e.h Level $1840$ Weight $2$ Character orbit 1840.e Analytic conductor $14.692$ Analytic rank $0$ Dimension $16$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(369,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("1840.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.e (of order $$2$$, degree $$1$$, not 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: $$14.6924739719$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024$$ x^16 - 2*x^15 + 2*x^14 + 6*x^13 + 100*x^12 - 196*x^11 + 210*x^10 + 702*x^9 + 1572*x^8 - 1594*x^7 + 2464*x^6 + 9568*x^5 + 15457*x^4 + 4336*x^3 + 128*x^2 - 512*x + 1024 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{3} - \beta_{10} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{9} + \cdots - 1) q^{9}+O(q^{10})$$ q - b5 * q^3 - b10 * q^5 + (b13 - b2) * q^7 + (-b12 - b10 + b9 - b7 + b1 - 1) * q^9 $$q - \beta_{5} q^{3} - \beta_{10} q^{5} + (\beta_{13} - \beta_{2}) q^{7} + ( - \beta_{12} - \beta_{10} + \beta_{9} + \cdots - 1) q^{9}+ \cdots + (3 \beta_{14} - \beta_{12} + \beta_{11} + \cdots + 6) q^{99}+O(q^{100})$$ q - b5 * q^3 - b10 * q^5 + (b13 - b2) * q^7 + (-b12 - b10 + b9 - b7 + b1 - 1) * q^9 + (-b12 - 1) * q^11 + (b15 - b14 - b13 - b5 + b4 + 2*b2 - 1) * q^13 + (-b14 - b13 + b7 + 2*b2 - 1) * q^15 + (b10 - b8 - b7 + b5 - b3 + b2) * q^17 + (b11 + b9 - b8 + 1) * q^19 + (b14 + b11 - b10 - b9 - b8 - b7 - b6 + b4 - 2*b1 - 1) * q^21 - b2 * q^23 + (-b15 + b8 - b7 - b6 - b5 + b1 + 1) * q^25 + (b15 + b13 - b8 + 2*b5 - b3 - b2) * q^27 + (b11 - b8 + b6 - b1 - 3) * q^29 + (-b12 + b11 + b9 - b8 - b1 - 2) * q^31 + (b13 + 2*b10 - 2*b7 + 2*b5 - b2) * q^33 + (-b15 + b14 + b12 + b11 - b9 + b8 + b7 - b6 + 2*b5 - b4 - b3 - 2*b2 + b1) * q^35 + (b15 - b14 - b11 + b10 - b7 - b5 + b4 + b3 - 1) * q^37 + (-b14 + b12 + b11 + 2*b10 + b9 - b8 + 2*b7 + b6 - b4 + b1) * q^39 + (-b14 + b10 + b7 + 2*b6 - b4 - b1 + 1) * q^41 + (b15 + b11 - b5 - b3 + 2*b2) * q^43 + (-b15 - b14 + b13 + b12 + 2*b11 + 2*b10 + b9 + b6 - 2*b4 - b2 + 2*b1 + 2) * q^45 + (-b14 - b11 - b10 + b7 + b4 + b3 + b2 - 1) * q^47 + (-b14 + 2*b12 + 2*b11 + 2*b9 - 2*b8 - b6 - b4 + 2*b1 - 5) * q^49 + (b14 + b12 + b11 - b9 - b8 - b6 + b4 + 2) * q^51 + (-2*b15 + b14 + 2*b11 + 2*b8 + 2*b5 - b4 + 1) * q^53 + (-b15 + 2*b11 + b10 + b6 + b5 + b3 - b2 - 2) * q^55 + (2*b14 + b13 + 2*b11 - b10 + b8 + b7 + b5 - 2*b4 - b3 - 2*b2 + 2) * q^57 + (-b11 + b10 + b8 + b7 + 4) * q^59 + (-b14 + b12 + b10 + b7 - b4 + 2) * q^61 + (-b15 - b14 - 3*b13 + b11 - 3*b10 + 3*b8 + 3*b7 - 4*b5 + b4 + 2*b3 + 6*b2 - 1) * q^63 + (-2*b13 - b12 + 2*b11 - b8 - b7 + b6 - 3*b5 + b4 - b3 - 2*b2) * q^65 + (-b15 + b14 + b11 - b10 + 2*b8 + b7 + b5 - b4 + b3 + 1) * q^67 - b1 * q^69 + (-b14 + 2*b12 - b11 + 2*b10 + b8 + 2*b7 - b4 - 3*b1 - 3) * q^71 + (b14 - 2*b10 + b8 + 2*b7 + 2*b5 - b4 + b3 + b2 + 1) * q^73 + (-b13 - b12 + b11 - b10 + b9 - b8 - 2*b7 - 2*b5 + 2*b4 + b3 + 2*b1 - 4) * q^75 + (-b15 - b14 - b13 - b11 - b10 + 3*b8 + b7 - 4*b5 + b4 + 4*b3 - 2*b2 - 1) * q^77 + (-b14 + b12 - b10 + 2*b9 - b7 - b6 - b4 + 4*b1 + 1) * q^79 + (b14 - b12 - 2*b10 - b9 - 2*b7 - 2*b6 + b4 - 2*b1 + 2) * q^81 + (-b10 + 2*b8 + b7 - 2*b5 + 2*b3 + 4*b2) * q^83 + (2*b15 + b14 + b13 - b12 + b11 - b10 - b9 - 2*b8 + b4 + b2 + 3) * q^85 + (b14 + b11 - b10 + b7 + 4*b5 - b4 - b3 + b2 + 1) * q^87 + (-b14 + b12 - 2*b11 + b10 + 2*b8 + b7 + b6 - b4 - 5) * q^89 + (b14 - b12 - 4*b9 + 2*b6 + b4 - 4*b1 + 4) * q^91 + (-b15 + 2*b14 + b13 + 3*b11 + b10 + 2*b8 - b7 + 4*b5 - 2*b4 - b3 + b2 + 2) * q^93 + (b15 - b14 + b13 - 2*b11 - b10 + b7 + b6 - b4 - 3*b2 + 2*b1 - 1) * q^95 + (b15 - b14 - b11 - b10 - b8 + b7 + b4 + b2 - 1) * q^97 + (3*b14 - b12 + b11 - b10 - 3*b9 - b8 - b7 - b6 + 3*b4 - 4*b1 + 6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 2 q^{5} - 22 q^{9}+O(q^{10})$$ 16 * q - 2 * q^5 - 22 * q^9 $$16 q - 2 q^{5} - 22 q^{9} - 14 q^{11} - 6 q^{15} + 22 q^{19} + 12 q^{25} - 44 q^{29} - 18 q^{31} - 20 q^{35} + 14 q^{41} + 14 q^{45} - 78 q^{49} + 38 q^{51} - 30 q^{55} + 64 q^{59} + 34 q^{61} + 6 q^{65} + 6 q^{69} - 30 q^{71} - 56 q^{75} - 4 q^{79} + 48 q^{81} + 52 q^{85} - 92 q^{89} + 70 q^{91} - 38 q^{95} + 122 q^{99}+O(q^{100})$$ 16 * q - 2 * q^5 - 22 * q^9 - 14 * q^11 - 6 * q^15 + 22 * q^19 + 12 * q^25 - 44 * q^29 - 18 * q^31 - 20 * q^35 + 14 * q^41 + 14 * q^45 - 78 * q^49 + 38 * q^51 - 30 * q^55 + 64 * q^59 + 34 * q^61 + 6 * q^65 + 6 * q^69 - 30 * q^71 - 56 * q^75 - 4 * q^79 + 48 * q^81 + 52 * q^85 - 92 * q^89 + 70 * q^91 - 38 * q^95 + 122 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{15} + 2 x^{14} + 6 x^{13} + 100 x^{12} - 196 x^{11} + 210 x^{10} + 702 x^{9} + 1572 x^{8} + \cdots + 1024$$ :

 $$\beta_{1}$$ $$=$$ $$( - 56\!\cdots\!89 \nu^{15} + \cdots + 95\!\cdots\!24 ) / 20\!\cdots\!20$$ (-564796151969688089*v^15 + 241867657278618280682*v^14 - 1533252977882876294850*v^13 + 5615701405841246795546*v^12 - 5798966345385152663476*v^11 + 22923942005700239322180*v^10 - 135549389895600500067810*v^9 + 523491302527906903267042*v^8 - 537061025633670808973620*v^7 + 169048474594930667975626*v^6 - 84234101814344138888912*v^5 + 3918219876554594510372320*v^4 - 2028057361772854751575673*v^3 - 304907534037070840065976*v^2 + 556650317363414170826464*v + 953326853515704207873024) / 2039997464548851718174720 $$\beta_{2}$$ $$=$$ $$( - 66\!\cdots\!57 \nu^{15} + \cdots - 43\!\cdots\!08 ) / 12\!\cdots\!20$$ (-669081183175634857*v^15 + 31617792777529048366*v^14 - 64646323960557883050*v^13 + 63600734450499465858*v^12 + 101144014888243416932*v^11 + 3167278369797318418580*v^10 - 6371259042835319510930*v^9 + 6577290238793402720266*v^8 + 18861957671154867372740*v^7 + 49351663880097415453818*v^6 - 56255878203811954909816*v^5 + 73974887245017554157920*v^4 + 267645466920360925996151*v^3 + 493638074924085942890092*v^2 + 69896317461757032781472*v - 4340717231201691735808) / 127499841534303232385920 $$\beta_{3}$$ $$=$$ $$( - 66\!\cdots\!57 \nu^{15} + \cdots - 43\!\cdots\!08 ) / 31\!\cdots\!80$$ (-669081183175634857*v^15 + 31617792777529048366*v^14 - 64646323960557883050*v^13 + 63600734450499465858*v^12 + 101144014888243416932*v^11 + 3167278369797318418580*v^10 - 6371259042835319510930*v^9 + 6577290238793402720266*v^8 + 18861957671154867372740*v^7 + 49351663880097415453818*v^6 - 56255878203811954909816*v^5 + 73974887245017554157920*v^4 + 267645466920360925996151*v^3 + 461763114540510134793612*v^2 + 101771277845332840877952*v - 4340717231201691735808) / 31874960383575808096480 $$\beta_{4}$$ $$=$$ $$( - 36\!\cdots\!99 \nu^{15} + \cdots - 93\!\cdots\!76 ) / 50\!\cdots\!80$$ (-36082429475973024899*v^15 + 51817197715454225402*v^14 + 98141784974047919250*v^13 + 80508305370487005526*v^12 - 4150958833832665209716*v^11 + 4424452632901492664220*v^10 + 5800880410569448661690*v^9 + 2054109647928134982382*v^8 - 114373918633476906704980*v^7 - 9697761543566197765634*v^6 - 52052886724240319553832*v^5 + 255183413268314405913440*v^4 - 1232478353488625770263843*v^3 - 2575592145277412776789596*v^2 - 3335136930759889767274816*v - 93942894839672333928576) / 509999366137212929543680 $$\beta_{5}$$ $$=$$ $$( - 31\!\cdots\!61 \nu^{15} + \cdots - 14\!\cdots\!44 ) / 20\!\cdots\!20$$ (-318969229217745937361*v^15 + 1841229998511317961538*v^14 - 3285807383750098948290*v^13 + 726871944630147495194*v^12 - 23161774996655402727844*v^11 + 167555889184928866487620*v^10 - 316526821134642417546610*v^9 + 52212436728277648932258*v^8 + 458943318236379704058780*v^7 + 952700810705768643736474*v^6 - 2054926237083438692768608*v^5 + 115121855831410995632480*v^4 + 4286478475719333680330703*v^3 + 2635056706542329997920176*v^2 + 1194224413274039153395616*v - 140707114255972724633344) / 2039997464548851718174720 $$\beta_{6}$$ $$=$$ $$( - 32\!\cdots\!63 \nu^{15} + \cdots - 12\!\cdots\!80 ) / 40\!\cdots\!44$$ (-328086612481608009763*v^15 + 639944811506344667566*v^14 - 508649117447187324278*v^13 - 2037151664803358920002*v^12 - 32676821886335924461052*v^11 + 63267592666342922762540*v^10 - 53565452553247896896982*v^9 - 239291997595419620454186*v^8 - 501938626773392038340220*v^7 + 568205842884924476879182*v^6 - 519643278737243550826480*v^5 - 3264034951805109084713312*v^4 - 4552141809370651565356611*v^3 - 653589136267567895383144*v^2 + 1227514889489791068838304*v - 1299074755592200570567680) / 407999492909770343634944 $$\beta_{7}$$ $$=$$ $$( 84\!\cdots\!19 \nu^{15} + \cdots - 10\!\cdots\!44 ) / 10\!\cdots\!60$$ (841925339302843628619*v^15 - 1535375885944134270142*v^14 + 614001498766608317510*v^13 + 7835326530712091726514*v^12 + 81160127013367967422236*v^11 - 152213113407290106228940*v^10 + 76879314495638174174630*v^9 + 854491089510550619162458*v^8 + 1037048304461043182970780*v^7 - 1382131543294845713179486*v^6 + 1350815537979474984914032*v^5 + 9891141608439985398920800*v^4 + 10856809983417508211138923*v^3 + 1549991709567331298025256*v^2 - 3939967276533955427877024*v - 107593881030981543276544) / 1019998732274425859087360 $$\beta_{8}$$ $$=$$ $$( - 20\!\cdots\!05 \nu^{15} + \cdots - 23\!\cdots\!28 ) / 20\!\cdots\!72$$ (-207962376041506841905*v^15 + 531602863276948713442*v^14 - 748886722375901601090*v^13 - 858852135854606012262*v^12 - 20011902708376888146916*v^11 + 50823127396663014338116*v^10 - 75638304146095329018290*v^9 - 105935593052591979557342*v^8 - 239794915024620709995940*v^7 + 359794110659287610999514*v^6 - 773356052025354301260832*v^5 - 1555862406023457302487712*v^4 - 2205874733214161441488273*v^3 - 606086541272980490786928*v^2 - 733530723301828313458784*v - 23061364425557027753728) / 203999746454885171817472 $$\beta_{9}$$ $$=$$ $$( 21\!\cdots\!49 \nu^{15} + \cdots + 44\!\cdots\!56 ) / 20\!\cdots\!20$$ (2118185236141746471449*v^15 - 3953155420784877654762*v^14 + 2132038220718324369090*v^13 + 18550348604166644455654*v^12 + 206585678634484606502836*v^11 - 391518309605053145121860*v^10 + 242799571396014564554210*v^9 + 2055560822793859187884318*v^8 + 2808490629040326766600500*v^7 - 3540845704199600616790986*v^6 + 3423207863830156165574352*v^5 + 25958054272212434904396320*v^4 + 28059114061089377308872313*v^3 + 4011928424236009986551736*v^2 - 7551143469775208630023904*v + 4455281824367152369273856) / 2039997464548851718174720 $$\beta_{10}$$ $$=$$ $$( 21\!\cdots\!67 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 20\!\cdots\!72$$ (216832476518453171567*v^15 - 408368235739557435366*v^14 + 230984654232606903502*v^13 + 1835948719271709389514*v^12 + 21089846935305423863020*v^11 - 40411825872447519151324*v^10 + 26651039266139952258734*v^9 + 202760223167881268421202*v^8 + 284665898450793362180972*v^7 - 363797928128516647631366*v^6 + 398765757204922242534320*v^5 + 2423008249317927822153440*v^4 + 2965824734178438435235855*v^3 + 421969477678591882074120*v^2 - 591486967727086100610080*v - 20422553595681348505600) / 203999746454885171817472 $$\beta_{11}$$ $$=$$ $$( - 12\!\cdots\!41 \nu^{15} + \cdots - 12\!\cdots\!84 ) / 10\!\cdots\!60$$ (-1282048923496956438741*v^15 + 3164479609138396473898*v^14 - 4285355384275934205450*v^13 - 5638677744919485282366*v^12 - 124348621205043592627444*v^11 + 303961652938262561218260*v^10 - 434567402565538232210490*v^9 - 688987991591815620730262*v^8 - 1585255762916027177913780*v^7 + 2235828650644175579974914*v^6 - 4509793508171907734061728*v^5 - 10003211668266940224284960*v^4 - 15001687353545491172481717*v^3 - 3590288385190530566279984*v^2 - 2610188589858814773945824*v - 120787935180359939517184) / 1019998732274425859087360 $$\beta_{12}$$ $$=$$ $$( - 25\!\cdots\!67 \nu^{15} + \cdots - 31\!\cdots\!68 ) / 20\!\cdots\!20$$ (-2588229413723518163967*v^15 + 4615436144052771597446*v^14 - 1300938577750539720430*v^13 - 23832307778719830030122*v^12 - 247870842045704795997228*v^11 + 458174412566717236259100*v^10 - 184409721653063663205070*v^9 - 2618657823981625025068594*v^8 - 3075848072628037146480300*v^7 + 4187584486409972800336038*v^6 - 3641530117296416951306416*v^5 - 29807609026592396802595040*v^4 - 31880221946092013629388959*v^3 - 4568188273249911843048328*v^2 + 8597925159288225369863712*v - 3181384674636323243527168) / 2039997464548851718174720 $$\beta_{13}$$ $$=$$ $$( - 26\!\cdots\!23 \nu^{15} + \cdots - 44\!\cdots\!52 ) / 20\!\cdots\!20$$ (-2642888945664910864623*v^15 + 7849910266256606593854*v^14 - 11457146342780235799230*v^13 - 8707977648300977704858*v^12 - 250389486494265538272092*v^11 + 762176261837051384990140*v^10 - 1161038922935840085830670*v^9 - 1117119434130211646290466*v^8 - 2503967682918694073411740*v^7 + 6950964135040340249863142*v^6 - 12018735453826353009247904*v^5 - 17183680768812435879037280*v^4 - 18490637950705990608063631*v^3 + 10409412107504918728814928*v^2 - 4246082060596735800332192*v - 448532962164852865847552) / 2039997464548851718174720 $$\beta_{14}$$ $$=$$ $$( 18\!\cdots\!37 \nu^{15} + \cdots - 15\!\cdots\!52 ) / 12\!\cdots\!20$$ (189621980412060243137*v^15 - 390372336275624352541*v^14 + 382730328923720987360*v^13 + 1201904523584497153952*v^12 + 19224867526049022000338*v^11 - 38258112235964529343520*v^10 + 40301528362365775227230*v^9 + 139489491216171912857464*v^8 + 318883474884996061259370*v^7 - 323581731844098202246038*v^6 + 478381537503435627193246*v^5 + 1984719931099756217426160*v^4 + 3248825928763831075053889*v^3 + 1058982307105449261342133*v^2 + 50175033137633848711128*v - 150605332666768364582752) / 127499841534303232385920 $$\beta_{15}$$ $$=$$ $$( - 32\!\cdots\!01 \nu^{15} + \cdots - 49\!\cdots\!24 ) / 20\!\cdots\!20$$ (-3214385228459778752201*v^15 + 8944116696865206225138*v^14 - 13659830033743327363890*v^13 - 11093928427081377174326*v^12 - 303190775204524701949444*v^11 + 860187450071408174746660*v^10 - 1367315172251677890979810*v^9 - 1409871110534423896856302*v^8 - 3059708196081617915726020*v^7 + 6701984866603385326334154*v^6 - 13689050286567661049021408*v^5 - 21446083076286990646032800*v^4 - 25100415896991282501011497*v^3 + 4428663247793606577665776*v^2 - 5841096957539534050870624*v - 495178165772403862599424) / 2039997464548851718174720
 $$\nu$$ $$=$$ $$( \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} ) / 2$$ (b11 + b10 - b8 - b7) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{11} + \beta_{10} - \beta_{8} - \beta_{7} - 2\beta_{3} + 8\beta_{2} ) / 2$$ (b11 + b10 - b8 - b7 - 2*b3 + 8*b2) / 2 $$\nu^{3}$$ $$=$$ $$( \beta_{15} - 2 \beta_{14} - 2 \beta_{13} + \beta_{12} - 6 \beta_{11} + 6 \beta_{10} + 2 \beta_{9} + \cdots - 2 ) / 2$$ (b15 - 2*b14 - 2*b13 + b12 - 6*b11 + 6*b10 + 2*b9 + 7*b8 - 6*b7 + b6 - b5 + b3 + 2*b2 - 2) / 2 $$\nu^{4}$$ $$=$$ $$( 2\beta_{12} - 7\beta_{11} - 5\beta_{10} - 4\beta_{9} + 7\beta_{8} - 5\beta_{7} - 18\beta_{6} + 16\beta _1 - 54 ) / 2$$ (2*b12 - 7*b11 - 5*b10 - 4*b9 + 7*b8 - 5*b7 - 18*b6 + 16*b1 - 54) / 2 $$\nu^{5}$$ $$=$$ $$( - 11 \beta_{15} + 22 \beta_{13} + 11 \beta_{12} - 39 \beta_{11} - 39 \beta_{10} + 22 \beta_{9} + \cdots + 10 ) / 2$$ (-11*b15 + 22*b13 + 11*b12 - 39*b11 - 39*b10 + 22*b9 + 32*b8 + 45*b7 + 13*b6 + 7*b5 - 20*b4 - 13*b3 - 10*b2 - 6*b1 + 10) / 2 $$\nu^{6}$$ $$=$$ $$- 13 \beta_{15} - 25 \beta_{13} - 13 \beta_{11} - 18 \beta_{10} + 65 \beta_{8} + 18 \beta_{7} + \cdots - 207 \beta_{2}$$ -13*b15 - 25*b13 - 13*b11 - 18*b10 + 65*b8 + 18*b7 - 12*b5 + 78*b3 - 207*b2 $$\nu^{7}$$ $$=$$ $$( - 106 \beta_{15} + 180 \beta_{14} + 206 \beta_{13} - 106 \beta_{12} + 363 \beta_{11} - 363 \beta_{10} + \cdots - 12 ) / 2$$ (-106*b15 + 180*b14 + 206*b13 - 106*b12 + 363*b11 - 363*b10 - 206*b9 - 413*b8 + 269*b7 - 144*b6 + 52*b5 - 144*b3 + 12*b2 + 92*b1 - 12) / 2 $$\nu^{8}$$ $$=$$ $$( - 16 \beta_{14} - 250 \beta_{12} + 113 \beta_{11} + 153 \beta_{10} + 520 \beta_{9} - 113 \beta_{8} + \cdots + 3352 ) / 2$$ (-16*b14 - 250*b12 + 113*b11 + 153*b10 + 520*b9 - 113*b8 + 153*b7 + 1364*b6 - 16*b4 - 1122*b1 + 3352) / 2 $$\nu^{9}$$ $$=$$ $$( 957 \beta_{15} - 1868 \beta_{13} - 957 \beta_{12} + 1948 \beta_{11} + 1948 \beta_{10} - 1868 \beta_{9} + \cdots - 2514 ) / 2$$ (957*b15 - 1868*b13 - 957*b12 + 1948*b11 + 1948*b10 - 1868*b9 - 1513*b8 - 3018*b7 - 1505*b6 - 435*b5 + 1574*b4 + 1505*b3 - 940*b2 + 1070*b1 - 2514) / 2 $$\nu^{10}$$ $$=$$ $$( 2136 \beta_{15} + 366 \beta_{14} + 5172 \beta_{13} + 1063 \beta_{11} - 497 \beta_{10} - 10989 \beta_{8} + \cdots + 366 ) / 2$$ (2136*b15 + 366*b14 + 5172*b13 + 1063*b11 - 497*b10 - 10989*b8 + 497*b7 + 2304*b5 - 366*b4 - 12052*b3 + 27654*b2 + 366) / 2 $$\nu^{11}$$ $$=$$ $$( 8319 \beta_{15} - 13624 \beta_{14} - 16858 \beta_{13} + 8319 \beta_{12} - 25513 \beta_{11} + \cdots + 14772 ) / 2$$ (8319*b15 - 13624*b14 - 16858*b13 + 8319*b12 - 25513*b11 + 25513*b10 + 16858*b9 + 29978*b8 - 14759*b7 + 15219*b6 - 3931*b5 + 15219*b3 - 14772*b2 - 11288*b1 + 14772) / 2 $$\nu^{12}$$ $$=$$ $$2763 \beta_{14} + 8523 \beta_{12} + 7616 \beta_{11} - 4366 \beta_{10} - 25265 \beta_{9} - 7616 \beta_{8} + \cdots - 119772$$ 2763*b14 + 8523*b12 + 7616*b11 - 4366*b10 - 25265*b9 - 7616*b8 - 4366*b7 - 53720*b6 + 2763*b4 + 43038*b1 - 119772 $$\nu^{13}$$ $$=$$ $$( - 70750 \beta_{15} + 152444 \beta_{13} + 70750 \beta_{12} - 116569 \beta_{11} - 116569 \beta_{10} + \cdots + 298240 ) / 2$$ (-70750*b15 + 152444*b13 + 70750*b12 - 116569*b11 - 116569*b10 + 152444*b9 + 67759*b8 + 218389*b7 + 150630*b6 + 36830*b5 - 117752*b4 - 150630*b3 + 180488*b2 - 113800*b1 + 298240) / 2 $$\nu^{14}$$ $$=$$ $$( - 129196 \beta_{15} - 69708 \beta_{14} - 488396 \beta_{13} - 80835 \beta_{11} + 214549 \beta_{10} + \cdots - 69708 ) / 2$$ (-129196*b15 - 69708*b14 - 488396*b13 - 80835*b11 + 214549*b10 + 884019*b8 - 214549*b7 - 195824*b5 + 69708*b4 + 964854*b3 - 2013820*b2 - 69708) / 2 $$\nu^{15}$$ $$=$$ $$( - 594527 \beta_{15} + 1021262 \beta_{14} + 1383542 \beta_{13} - 594527 \beta_{12} + 1890076 \beta_{11} + \cdots - 1986426 ) / 2$$ (-594527*b15 + 1021262*b14 + 1383542*b13 - 594527*b12 + 1890076*b11 - 1890076*b10 - 1383542*b9 - 2423779*b8 + 954884*b7 - 1468895*b6 + 349447*b5 - 1468895*b3 + 1986426*b2 + 1119448*b1 - 1986426) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$$.

 $$n$$ $$737$$ $$1151$$ $$1201$$ $$1381$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
369.1
 −0.945903 + 0.945903i 1.65386 − 1.65386i 1.81400 + 1.81400i −1.13508 − 1.13508i −2.15699 + 2.15699i 0.303680 + 0.303680i 1.95202 − 1.95202i −0.485591 + 0.485591i −0.485591 − 0.485591i 1.95202 + 1.95202i 0.303680 − 0.303680i −2.15699 − 2.15699i −1.13508 + 1.13508i 1.81400 − 1.81400i 1.65386 + 1.65386i −0.945903 − 0.945903i
0 3.20935i 0 2.02615 0.945903i 0 4.54713i 0 −7.29996 0
369.2 0 2.89996i 0 −1.50491 + 1.65386i 0 0.580879i 0 −5.40975 0
369.3 0 2.54092i 0 −1.30744 1.81400i 0 0.780573i 0 −3.45626 0
369.4 0 2.51561i 0 −1.92655 + 1.13508i 0 4.64022i 0 −3.32827 0
369.5 0 1.69755i 0 0.589417 2.15699i 0 4.22860i 0 0.118308 0
369.6 0 0.540724i 0 2.21535 0.303680i 0 1.15693i 0 2.70762 0
369.7 0 0.493532i 0 1.09069 + 1.95202i 0 4.54439i 0 2.75643 0
369.8 0 0.296848i 0 −2.18271 0.485591i 0 3.46037i 0 2.91188 0
369.9 0 0.296848i 0 −2.18271 + 0.485591i 0 3.46037i 0 2.91188 0
369.10 0 0.493532i 0 1.09069 1.95202i 0 4.54439i 0 2.75643 0
369.11 0 0.540724i 0 2.21535 + 0.303680i 0 1.15693i 0 2.70762 0
369.12 0 1.69755i 0 0.589417 + 2.15699i 0 4.22860i 0 0.118308 0
369.13 0 2.51561i 0 −1.92655 1.13508i 0 4.64022i 0 −3.32827 0
369.14 0 2.54092i 0 −1.30744 + 1.81400i 0 0.780573i 0 −3.45626 0
369.15 0 2.89996i 0 −1.50491 1.65386i 0 0.580879i 0 −5.40975 0
369.16 0 3.20935i 0 2.02615 + 0.945903i 0 4.54713i 0 −7.29996 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 369.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.2.e.h 16
4.b odd 2 1 920.2.e.c 16
5.b even 2 1 inner 1840.2.e.h 16
5.c odd 4 1 9200.2.a.dd 8
5.c odd 4 1 9200.2.a.de 8
20.d odd 2 1 920.2.e.c 16
20.e even 4 1 4600.2.a.bj 8
20.e even 4 1 4600.2.a.bk 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.c 16 4.b odd 2 1
920.2.e.c 16 20.d odd 2 1
1840.2.e.h 16 1.a even 1 1 trivial
1840.2.e.h 16 5.b even 2 1 inner
4600.2.a.bj 8 20.e even 4 1
4600.2.a.bk 8 20.e even 4 1
9200.2.a.dd 8 5.c odd 4 1
9200.2.a.de 8 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1840, [\chi])$$:

 $$T_{3}^{16} + 35T_{3}^{14} + 479T_{3}^{12} + 3218T_{3}^{10} + 10815T_{3}^{8} + 16123T_{3}^{6} + 7441T_{3}^{4} + 1264T_{3}^{2} + 64$$ T3^16 + 35*T3^14 + 479*T3^12 + 3218*T3^10 + 10815*T3^8 + 16123*T3^6 + 7441*T3^4 + 1264*T3^2 + 64 $$T_{7}^{16} + 95 T_{7}^{14} + 3621 T_{7}^{12} + 69892 T_{7}^{10} + 703096 T_{7}^{8} + 3332320 T_{7}^{6} + \cdots + 541696$$ T7^16 + 95*T7^14 + 3621*T7^12 + 69892*T7^10 + 703096*T7^8 + 3332320*T7^6 + 5335056*T7^4 + 3052288*T7^2 + 541696

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} + 35 T^{14} + \cdots + 64$$
$5$ $$T^{16} + 2 T^{15} + \cdots + 390625$$
$7$ $$T^{16} + 95 T^{14} + \cdots + 541696$$
$11$ $$(T^{8} + 7 T^{7} + \cdots + 440)^{2}$$
$13$ $$T^{16} + \cdots + 196560400$$
$17$ $$T^{16} + 147 T^{14} + \cdots + 30976$$
$19$ $$(T^{8} - 11 T^{7} + \cdots + 11192)^{2}$$
$23$ $$(T^{2} + 1)^{8}$$
$29$ $$(T^{8} + 22 T^{7} + \cdots - 400)^{2}$$
$31$ $$(T^{8} + 9 T^{7} + \cdots - 287276)^{2}$$
$37$ $$T^{16} + \cdots + 259081216$$
$41$ $$(T^{8} - 7 T^{7} + \cdots + 1197584)^{2}$$
$43$ $$T^{16} + \cdots + 59754824704$$
$47$ $$T^{16} + \cdots + 5405190400$$
$53$ $$T^{16} + \cdots + 41160294400$$
$59$ $$(T^{8} - 32 T^{7} + \cdots + 29696)^{2}$$
$61$ $$(T^{8} - 17 T^{7} + \cdots + 200)^{2}$$
$67$ $$T^{16} + \cdots + 463227249664$$
$71$ $$(T^{8} + 15 T^{7} + \cdots - 8900000)^{2}$$
$73$ $$T^{16} + \cdots + 311006982400$$
$79$ $$(T^{8} + 2 T^{7} + \cdots - 5248)^{2}$$
$83$ $$T^{16} + \cdots + 17501839590400$$
$89$ $$(T^{8} + 46 T^{7} + \cdots + 12804160)^{2}$$
$97$ $$T^{16} + 255 T^{14} + \cdots + 8761600$$