
// Make newform 690.2.n.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_690_n();"
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_690_2_n_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
// To make the character of type GrpDrchElt, type "MakeCharacter_690_n();"
function MakeCharacter_690_n()
    N := 690;
    order := 22;
    char_gens := [461, 277, 511];
    v := [11, 11, 5];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_690_n_Hecke(Kf)
    return MakeCharacter_690_n();
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_690_2_n_a();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_690_2_n_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_690_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<17,R![23195739069871839618284415396329887294406262784, -390241161791397190014452937834378149135692333056, 5782238217569462418573682056614629986597021417472, -16454566183708492480825472726976188461221010079744, -121814761553785184277656054028628859458097683955712, 3658654344085438247505812529144880644463952872767488, 7614010381435447922208445549173070612152491489345536, -11940557973031155013101506989951328385039684878319616, 208008061790227054293404087600920063951469855688609792, 69639694796622890042999182150170175314681581538981888, -1982815449158674127002397385269906363807919716787139584, -522075434134161849103271451488404580927623460787716096, 12047351388780124799148802295626451915070342505521408000, -8155313107491226002164440391377692754224643488614903808, -26061746406336026947419326070662860442291957120512715776, 17725076402759028353819530209461324479326873778593685760, 111025242866005813984329305135645450682589155432087156992, -277681504932920553570004608920674798121819003146520256384, 341924100098113898862366866121072247014970826760267364672, -349581283582082587890029782000181420510711431191979672320, 436172747850824655250895545541441595561071645281610924064, -544812823735632404587881517731127303142368870686917949952, 515742009419193160247246635445433929053392729774787300160, -392192095479967940852591851793735571334579074905411997808, 366789389123181506632270640446193724487686069820991406352, -476900090746976385318602454104716291043344143975562523960, 581018204346587528681340239779027636163122225114863532168, -592601048728748754856002095589919281448044931531156882576, 555346398062579478490160090070707885192255628709185126710, -513689682154271225652142552913572067027157597339503663410, 446866184429334921092262847613231698140085865318515000293, -334963401018103493491672618492450759822370793181425335242, 207247567604978210080900033884639695857033102189435973663, -108052220701118856493988089068666206228146066522159660721, 52762624406408063785372673910800189263912074531850153797, -28364031332072827077512443785158605270675013008614684732, 17319560331078533610258567788762012411397843061263835435, -10625735543475602894232216267889341342102576813528312558, 6040753237381086245567697112956987509167846344034611865, -3181982631957767977698963597660240939928232827289576257, 1589787243409157367178370367147151920627947696173515872, -762201913138802050851919506153130760818138894735870592, 352652221243636951079209211721197858965590983618833729, -158470193152606488851332508990231588150414816636303098, 68043628765114544445394642390871124804356977499050757, -26426587582473829046769947685472673080414812878547857, 8641092899782746384824983951899735062873370232550997, -2214268639250074664407853851232391729729237978841178, 382237437249337945355105387388078615523263070587496, 10482439514753070471976491874299298307152585361515, -56604163248569477342416711182992774984975652260382, 36774388746468266587906360119968357420231420349739, -15043749659458377565920034337128021484481131637393, 3972662493610280519123107737552445819921105679778, -557746337530048297090624566583231299637241725695, -24402536284334929907055262375920312905807616933, 43520522667090880864081031225212675813541574032, -24088529938097989244123201383419828973808077442, 10941798463942872215865428893307178663170343591, -3190067617045617759214725787900842245574622372, 376004296821692139349403384716438164013506682, 72039603333423136060837273661653760213960450, -31282759720566274026293926887532197663456109, 3589874176410794108695549031753532045479004, -472783506241234249762206634533575940934365, 348986381845656973961035221526761356485422, -123512864262825847620148305553687431014496, 28406149044439609135260431188405432789613, -7292024470355940511015085969738009821525, 1197755129263286390623176401860474432349, 393379251235008752695474551105362742424, -280937402282455326422338343062330968725, 60646352413825230574624557478993806076, -1724336964838846113356058913474688130, -1425758239770304037532256638684011789, 167773960715626212615782507268042607, 7708577562416803221574273425009513, -2310425446971073790608326933183129, 2492792223174868526703777407505216, -1114272435062404105445492479750036, 98023630062243021114139293397441, 40257629787314079017317115517952, -7448302107925437285920332896542, -1242267251082336073641950771703, 395505373978873323050631252978, 14142524360280576476111526488, -12539000831746079894182739208, 167269234099604710479468986, 239340380115139131088612308, 22593968808139498517238898, -9781369352105337220496090, -1125141911442505144564398, 453788432457980954657591, 15470763723061257995725, -12153815707682285284473, -594073622392487735828, 333737120206191692944, 27494847258145139378, -9951744867208942168, -941569306890225068, 291669073896447406, 18406327873076044, -5174998791961364, -610034308428589, 96826931813176, 15687165893927, -1990633582551, -299761009922, 38920871417, 5282672142, -951651623, -87149315, 21313899, 1613953, -329135, -20438, 6090, 176, -83, 0, 1]>],Snew);
    return Vf;
end function;