
// Make newform 275.3.d.c in Magma, downloaded from the LMFDB on 08 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_275_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_275_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_275_3_d_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_275_3_d_c();".
// 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 ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [256, 1664, 5408, 7088, 2916, -5016, 3920, 724, 749, -1222, 770, -118, 51, -54, 32, -8, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1143599264828719472144, -238225223087026630432, -981515064677011835976, -2138634042363623579676, 1824016623936130884408, -1091316693169880830788, 32460364067993845518, -446611853807793021913, 442893480642803976402, -231724362207142322226, 44606399946990703620, -24080867698733495687, 19522254988205748198, -9838419486534991212, 2311820544100620666, -274223940304429169], [15067874601472290320, -2655401805418001056, -10911220375354635944, -23670368447056532012, 24145142415053732880, -11754397194954160332, 372665734690933846, -5172598096612164777, 5222225978340301914, -2591726624411708810, 497544489463531036, -277319409845813415, 226089326193995086, -111088409910698964, 25763923212930370, -2994297419615393], [-9674090539850689904, 14987333603488307680, 61469789341420665976, 132901095665974881860, -133054620565628886640, 64029287248153279140, -2575157522712897378, 31127112543300196683, -30695843189741642558, 15015697307741378334, -2910576203629004852, 1652801081022864581, -1333404800467108858, 648824338305365020, -149805677524499814, 17340994225293299], [-4872988745652174469056, -25368080605396316832960, -29839855554800633877600, -23091501917291921085660, 31536015769415375895480, -21363760129957091427432, -2135327360214752406440, -5828769097721764864275, 7546246586390613117330, -4349373910690396323390, 735507757400761455542, -354346635350256900525, 331861741308128808330, -182585393798321797320, 44431423666697545340, -5416815988184978583], [47783920905553656384, 249590578858194428224, 257755082545867795584, 241572809822491720644, -296428108894350377352, 196404785516310036264, 15309052701853259728, 58131423762999518717, -71128400831769687198, 40276880607015004674, -6976590512891024962, 3440229817902220483, -3129395124129403062, 1694191387362303528, -409761113306107084, 49722990453455689], [20826719246177544522048, 107666046752342388406848, 194236293786131749570704, 84666176747761702235508, -147749005570357275072104, 100004730692355388775888, 21396760451854628255612, 21593961615290065133709, -35125548703612766659686, 21111982673537727750098, -3176101215200978060062, 1444292805281714785491, -1543252200504249975974, 894683743467714619296, -221604450921266449408, 27334116808089729793], [-12368343399849277528, 30224368028777734880, 123857067187115899576, 267343069629621435684, -264595169535503537840, 126574190669530792516, -5904728358813298322, 65089274801965665523, -63264265218595253662, 30705752950663967902, -6028019317204030532, 3489553021990883469, -2785646211061360546, 1343406444343004108, -308941912993891862, 35643483073044507], [240019216623817752256, 1244092425554284615104, 2281817519997410601680, 1034799811680295021836, -1654673955324068538840, 1095709922680987041584, 262574320587229276780, 249155661399020519411, -393980791616304365354, 234136233148448915070, -34453858188763086890, 16208287985411428493, -17314281466129378746, 9992309789873830656, -2467682657602190520, 304056114626190127], [41848303621051479975744, 218173940823097434197056, 220568211611603028825392, 204396705416507215688916, -264570949218479893314088, 181626844036325797915552, 9017188416239806962836, 52863016091889954615533, -63435607815570953360502, 36071662280813543621746, -6407304989257831059606, 3131266206442145890867, -2790456097998261369478, 1504097057924230683792, -363446351706889254696, 44113723981629103665], [-102980796015921868336, 261966241461684096352, 1071377634913537901784, 2305146553607721205844, -2391605660213682264792, 1071226833041605883084, -49620855511184306410, 571399021030932352747, -564724215695594374454, 264496682741149751654, -51716859033558523644, 30694431509086347621, -24672010279833872194, 11667880961402859668, -2653134620407099054, 300281369975679315], [-16971713695243920586496, 14199543777096822981792, 58056760013298412459432, 124869642993712492103532, -134400242932010120233056, 58066978075596732372692, -2555118391959813965622, 30729690985913188588021, -30711298320872339045586, 14251643929143973739802, -2770551325651266243572, 1644677763340852169835, -1329606320506763213510, 627204321188355031484, -142373215842824628642, 16046402982557990365], [-81322996863684295808960, -419984921446004792935104, -785741364760782575428976, -322978418287719386068252, 583168674266171951445176, -397691113822270303948624, -85107251926187094846212, -84196728550733010396039, 138561748032173246836690, -83625381897508233679862, 12601133117466693445866, -5689891053346850028313, 6087152964706522075634, -3535822025821810783616, 876501451954526123120, -108195488251500798403], [-10407210472764190571680, 27745791726376950490336, 113500678537875924558520, 244314816164036742208356, -253951853188812724088176, 114051506511339794206812, -5109617645655150243058, 60015336959822514773815, -59506879529989506455174, 27927783008207710423022, -5445450772897847414636, 3217266996377624081049, -2591861428321274274674, 1228411769335929932132, -279622940481143165302, 31681426257220138223], [-855029220790410392512, -4421896578817262813888, -8851110390182841182384, -3506725691883236205404, 6041464736065852934392, -4073690111361356177312, -986524970164723772820, -885539892819063646783, 1436560773489552860450, -861762699571889234614, 127091670641734345706, -58880885215686803521, 63118046860550334658, -36594534393627143632, 9054509015984957984, -1117650522215340667], [-813802282047589552448, -4230223033816736176704, -6238298761162618732080, -3737732722476812208116, 5380455426259980853800, -3628413529926065056848, -533222219906166017620, -936581288862058778381, 1285363546893930274774, -748630663146153874050, 121369986843041505302, -58114010484886922803, 56513396384645377926, -31563688407854514496, 7716841179719169800, -944272070273722513]];
    Rf_basisdens := [1, 796923041584880842832, 3041690998415575736, 6083381996831151472, 1593846083169761685664, 12166763993662302944, 3187692166339523371328, 3041690998415575736, 24333527987324605888, 3187692166339523371328, 16955809395422996656, 796923041584880842832, 3187692166339523371328, 796923041584880842832, 24333527987324605888, 12166763993662302944];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_275_d();"
function MakeCharacter_275_d()
    N := 275;
    order := 2;
    char_gens := [177, 101];
    v := [1, 1];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_275_d_Hecke();"
function MakeCharacter_275_d_Hecke( : Kf := false)
    N := 275;
    order := 2;
    char_gens := [177, 101];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 3;
    raw_aps := [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 1, 0, -2, 0, 0, 0, 0, 0, 0, -1, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 0, 2, 0, -1, 0, 0, 1, 0, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, -3, 0, 0, 0, 0, 0, 2, -1], [4, 0, -1, 5, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, 0, -2, 0, 0, 2, 0, 0, -7, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 0, -4, 0, 0, 0, 0, 0, 0, 1], [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 0, -2, 0, 0], [0, 0, 0, 0, 4, 0, 7, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, 0, 6, 0, 0, -4, 0, 0, -9, 0, 0, 0], [4, 0, 4, 7, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -12, 0, 0, 2, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, -7, 0, -18, 0, 0, -5, 0, 0, 7, 0, 0, 0], [32, 0, -5, 9, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 11, 0, 0, 0, 0, 0, 0, 0, 0, -7, 1, 0, 3, 0, 0], [0, 0, 0, 0, 0, 9, 0, 0, -2, 0, 0, 0, 0, 0, -10, 1], [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 10, -1, 0, -6, 0, 0], [-58, 0, -5, -4, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -19, 0, 16, 0, 0, 5, 0, 0, 15, 0, 0, 0], [0, 0, 0, 0, 0, -7, 0, 0, 9, 0, 0, 0, 0, 0, 11, 3], [0, 0, 0, 0, 32, 0, 11, 0, 0, 2, 0, 0, 3, 0, 0, 0], [0, 39, 0, 0, 0, 0, 0, 0, 0, 0, -3, 5, 0, -1, 0, 0], [0, 0, 0, 0, 0, 12, 0, 0, 2, 0, 0, 0, 0, 0, 8, -2], [0, 0, 0, 0, -9, 0, 18, 0, 0, 8, 0, 0, 2, 0, 0, 0], [0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 19, 1, 0, -3, 0, 0], [0, 0, 0, 0, 0, 16, 0, 0, 1, 0, 0, 0, 0, 0, 10, -4], [0, 0, 0, 0, 53, 0, 36, 0, 0, -11, 0, 0, 11, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 7, 0, 0, 0, 0, 0, -11, 3], [0, 0, 0, 0, 0, -17, 0, 0, -1, 0, 0, 0, 0, 0, 6, 3], [0, 0, 0, 0, 0, -3, 0, 0, -13, 0, 0, 0, 0, 0, -11, 5], [0, 0, 0, 0, 2, 0, 26, 0, 0, 4, 0, 0, -27, 0, 0, 0], [0, 0, 0, 0, 23, 0, -1, 0, 0, 10, 0, 0, 4, 0, 0, 0], [0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -9, 4, 0, -4, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -35, -1, 0, 1, 0, 0], [96, 0, -2, -1, 0, 0, 0, 17, 0, 0, 0, 0, 0, 0, 0, 0], [-10, 0, 6, -8, 0, 0, 0, 44, 0, 0, 0, 0, 0, 0, 0, 0], [42, 0, -30, 2, 0, 0, 0, -10, 0, 0, 0, 0, 0, 0, 0, 0], [0, -33, 0, 0, 0, 0, 0, 0, 0, 0, 28, 4, 0, 1, 0, 0], [0, -2, 0, 0, 0, 0, 0, 0, 0, 0, -32, -9, 0, -4, 0, 0], [-28, 0, -45, 21, 0, 0, 0, 20, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -18, 0, 0, 13, 0, 0, 0, 0, 0, -22, -6], [0, 0, 0, 0, -80, 0, 5, 0, 0, -14, 0, 0, 51, 0, 0, 0], [0, 6, 0, 0, 0, 0, 0, 0, 0, 0, -8, -5, 0, 12, 0, 0], [140, 0, 6, -25, 0, 0, 0, -49, 0, 0, 0, 0, 0, 0, 0, 0], [0, -31, 0, 0, 0, 0, 0, 0, 0, 0, 4, -14, 0, 3, 0, 0], [0, 0, 0, 0, 0, 29, 0, 0, 6, 0, 0, 0, 0, 0, -22, 5], [0, 0, 0, 0, 0, 37, 0, 0, -2, 0, 0, 0, 0, 0, -12, 1], [-66, 0, -16, -30, 0, 0, 0, -50, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, -22, 0, 0, 0], [0, -76, 0, 0, 0, 0, 0, 0, 0, 0, 11, 16, 0, 4, 0, 0], [-66, 0, -56, 16, 0, 0, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 18, 0, 0, -9, 0, 0, 0, 0, 0, -3, 10], [0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 25, -11, 0, 7, 0, 0], [0, 0, 0, 0, 0, 10, 0, 0, 10, 0, 0, 0, 0, 0, -30, 2], [0, 38, 0, 0, 0, 0, 0, 0, 0, 0, -4, 11, 0, 12, 0, 0], [0, 86, 0, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, -4, 0, 0], [0, 56, 0, 0, 0, 0, 0, 0, 0, 0, 66, -1, 0, 10, 0, 0], [196, 0, 3, -53, 0, 0, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 0, -102, 0, 0, -5, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, -2, 0, 82, 0, 0, 26, 0, 0, 27, 0, 0, 0], [-132, 0, 22, 47, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 4, 28, 0, -3, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, 0, -27, 1, 0, 35, 0, 0], [0, 0, 0, 0, 0, 16, 0, 0, -22, 0, 0, 0, 0, 0, -4, -2], [0, 0, 0, 0, 57, 0, 64, 0, 0, 5, 0, 0, -19, 0, 0, 0], [0, 0, 0, 0, 0, -40, 0, 0, -8, 0, 0, 0, 0, 0, 14, 4], [0, 0, 0, 0, -52, 0, -121, 0, 0, -20, 0, 0, -1, 0, 0, 0], [0, -80, 0, 0, 0, 0, 0, 0, 0, 0, 2, -21, 0, 10, 0, 0], [-230, 0, -14, -62, 0, 0, 0, 60, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -24, 0, -37, 0, 0, 20, 0, 0, 37, 0, 0, 0], [-30, 0, -34, 86, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -42, 0, -29, 0, 0, -20, 0, 0, 11, 0, 0, 0], [42, 0, -35, -80, 0, 0, 0, -39, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 15, 0, 0, -15, 0, 0, 0, 0, 0, 33, 15], [-222, 0, 42, -44, 0, 0, 0, -120, 0, 0, 0, 0, 0, 0, 0, 0], [162, 0, 60, -54, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 69, 0, 0, 11, 0, 0, 0, 0, 0, -43, 5], [0, 0, 0, 0, 57, 0, -94, 0, 0, 20, 0, 0, -12, 0, 0, 0], [0, 0, 0, 0, 0, 79, 0, 0, 2, 0, 0, 0, 0, 0, -26, -9], [0, 0, 0, 0, -143, 0, -80, 0, 0, -5, 0, 0, -43, 0, 0, 0], [270, 0, -14, -26, 0, 0, 0, -106, 0, 0, 0, 0, 0, 0, 0, 0], [0, 75, 0, 0, 0, 0, 0, 0, 0, 0, -38, -50, 0, 25, 0, 0], [0, 0, 0, 0, 0, -50, 0, 0, 34, 0, 0, 0, 0, 0, -25, 6], [0, 0, 0, 0, -112, 0, -149, 0, 0, -11, 0, 0, -34, 0, 0, 0], [0, 0, 0, 0, 25, 0, 119, 0, 0, -26, 0, 0, 60, 0, 0, 0], [0, 0, 0, 0, 0, -53, 0, 0, 32, 0, 0, 0, 0, 0, -34, -1], [0, 0, 0, 0, 144, 0, -53, 0, 0, 25, 0, 0, -80, 0, 0, 0], [0, 0, 0, 0, 0, -76, 0, 0, 57, 0, 0, 0, 0, 0, -24, 4], [14, 0, 34, -116, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0], [0, -61, 0, 0, 0, 0, 0, 0, 0, 0, -23, 5, 0, 11, 0, 0], [318, 0, -58, -4, 0, 0, 0, -174, 0, 0, 0, 0, 0, 0, 0, 0], [-246, 0, 34, -6, 0, 0, 0, -62, 0, 0, 0, 0, 0, 0, 0, 0], [0, 47, 0, 0, 0, 0, 0, 0, 0, 0, -97, -25, 0, 43, 0, 0], [0, 0, 0, 0, 0, -4, 0, 0, -44, 0, 0, 0, 0, 0, 33, 4], [0, -35, 0, 0, 0, 0, 0, 0, 0, 0, -17, -7, 0, 25, 0, 0], [0, -52, 0, 0, 0, 0, 0, 0, 0, 0, 64, -19, 0, -46, 0, 0], [0, 16, 0, 0, 0, 0, 0, 0, 0, 0, -28, -51, 0, 14, 0, 0], [0, 0, 0, 0, 0, 49, 0, 0, -3, 0, 0, 0, 0, 0, 15, -3], [0, 0, 0, 0, 0, 16, 0, 0, -5, 0, 0, 0, 0, 0, 54, 16], [0, 0, 0, 0, -111, 0, -102, 0, 0, -12, 0, 0, -114, 0, 0, 0], [0, 0, 0, 0, 61, 0, -51, 0, 0, -6, 0, 0, 28, 0, 0, 0], [0, -163, 0, 0, 0, 0, 0, 0, 0, 0, -9, 33, 0, -15, 0, 0], [212, 0, 5, -69, 0, 0, 0, 22, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -66, 0, 0, -2, 0, 0, 0, 0, 0, 8, 2], [0, -8, 0, 0, 0, 0, 0, 0, 0, 0, -7, 22, 0, 4, 0, 0], [0, -13, 0, 0, 0, 0, 0, 0, 0, 0, 29, 9, 0, -45, 0, 0], [0, 0, 0, 0, 73, 0, -98, 0, 0, 61, 0, 0, -113, 0, 0, 0], [-250, 0, -64, 18, 0, 0, 0, 76, 0, 0, 0, 0, 0, 0, 0, 0], [44, 0, -21, 95, 0, 0, 0, 214, 0, 0, 0, 0, 0, 0, 0, 0], [330, 0, 1, 48, 0, 0, 0, 109, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 115, 0, 75, 0, 0, 44, 0, 0, -34, 0, 0, 0], [0, 0, 0, 0, 266, 0, -21, 0, 0, 19, 0, 0, -72, 0, 0, 0], [0, 0, 0, 0, 328, 0, 26, 0, 0, 16, 0, 0, -79, 0, 0, 0], [0, 0, 0, 0, 0, 21, 0, 0, 48, 0, 0, 0, 0, 0, 17, 17], [136, 0, -12, -45, 0, 0, 0, -199, 0, 0, 0, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0, 0, 0, 0, 0, -24, 20, 0, 33, 0, 0], [0, -131, 0, 0, 0, 0, 0, 0, 0, 0, -13, 41, 0, 1, 0, 0], [0, 0, 0, 0, -309, 0, -75, 0, 0, 34, 0, 0, 4, 0, 0, 0], [-372, 0, 22, -55, 0, 0, 0, -215, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -78, 0, 0, 80, 0, 0, 0, 0, 0, 13, 6], [318, 0, 52, 126, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0], [436, 0, 83, -201, 0, 0, 0, -28, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -168, 0, 49, 0, 0, -8, 0, 0, 31, 0, 0, 0], [0, -48, 0, 0, 0, 0, 0, 0, 0, 0, -40, -31, 0, -26, 0, 0], [0, 0, 0, 0, 0, -9, 0, 0, -15, 0, 0, 0, 0, 0, -105, 5], [0, 36, 0, 0, 0, 0, 0, 0, 0, 0, 27, 48, 0, -36, 0, 0], [568, 0, -23, 9, 0, 0, 0, -250, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 138, 0, -25, 0, 0, -62, 0, 0, 65, 0, 0, 0], [0, 0, 0, 0, 0, -9, 0, 0, 24, 0, 0, 0, 0, 0, -8, -17], [0, 0, 0, 0, 0, 83, 0, 0, 65, 0, 0, 0, 0, 0, 9, -13], [0, 0, 0, 0, -362, 0, 94, 0, 0, -30, 0, 0, -111, 0, 0, 0], [0, -60, 0, 0, 0, 0, 0, 0, 0, 0, -68, 73, 0, -38, 0, 0], [0, 0, 0, 0, -228, 0, -35, 0, 0, 72, 0, 0, -63, 0, 0, 0], [0, 0, 0, 0, 0, -77, 0, 0, -59, 0, 0, 0, 0, 0, 103, 11], [0, 0, 0, 0, 0, -64, 0, 0, -71, 0, 0, 0, 0, 0, 36, 0], [0, 0, 0, 0, 0, -61, 0, 0, 29, 0, 0, 0, 0, 0, 135, 9], [0, 0, 0, 0, 330, 0, 9, 0, 0, -39, 0, 0, -20, 0, 0, 0], [0, 30, 0, 0, 0, 0, 0, 0, 0, 0, -66, -67, 0, 24, 0, 0], [-334, 0, 158, -60, 0, 0, 0, -20, 0, 0, 0, 0, 0, 0, 0, 0], [-478, 0, -20, 82, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 0, 0], [0, -42, 0, 0, 0, 0, 0, 0, 0, 0, -110, -71, 0, 64, 0, 0], [0, -153, 0, 0, 0, 0, 0, 0, 0, 0, 50, -30, 0, 37, 0, 0], [330, 0, 56, -68, 0, 0, 0, -262, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 188, 0, -13, 0, 0, 15, 0, 0, -168, 0, 0, 0], [0, 321, 0, 0, 0, 0, 0, 0, 0, 0, -87, -5, 0, 17, 0, 0], [162, 0, 140, -136, 0, 0, 0, 196, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -167, 0, 17, 0, 0, 6, 0, 0, 22, 0, 0, 0], [0, -175, 0, 0, 0, 0, 0, 0, 0, 0, -33, 45, 0, -91, 0, 0], [0, 0, 0, 0, -155, 0, -107, 0, 0, 126, 0, 0, -36, 0, 0, 0], [-308, 0, -5, 73, 0, 0, 0, 46, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 131, 0, 0, -60, 0, 0, 0, 0, 0, 4, -5], [222, 0, 75, 118, 0, 0, 0, 63, 0, 0, 0, 0, 0, 0, 0, 0], [0, 325, 0, 0, 0, 0, 0, 0, 0, 0, 83, -23, 0, 1, 0, 0], [0, 0, 0, 0, 0, -86, 0, 0, -76, 0, 0, 0, 0, 0, 101, 10], [0, 0, 0, 0, -193, 0, 349, 0, 0, 68, 0, 0, 116, 0, 0, 0], [0, 191, 0, 0, 0, 0, 0, 0, 0, 0, -22, 30, 0, 53, 0, 0], [0, -128, 0, 0, 0, 0, 0, 0, 0, 0, 35, -80, 0, 0, 0, 0], [506, 0, 48, 94, 0, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -39, 0, -56, 0, 0, -69, 0, 0, 133, 0, 0, 0], [0, 0, 0, 0, 270, 0, 79, 0, 0, -64, 0, 0, -25, 0, 0, 0], [-18, 0, -10, 100, 0, 0, 0, 280, 0, 0, 0, 0, 0, 0, 0, 0], [0, 166, 0, 0, 0, 0, 0, 0, 0, 0, -110, -75, 0, 88, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_275_d_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_275_3_d_c();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_275_3_d_c(:prec:=16)
    chi := MakeCharacter_275_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_275_3_d_c();".
// 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_275_3_d_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_275_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![895, 0, -890, 0, 280, 0, -30, 0, 1]>],Snew);
    return Vf;
end function;