
// Make newform 616.2.r.f in Magma, downloaded from the LMFDB on 23 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_616_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_616_r_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_616_2_r_f();".
// 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_616_2_r_f();".
// 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 := [10000, -3000, 72900, -179670, 212971, 166407, 191109, 13430, 85487, -14174, 29289, -5753, 6557, -755, 1296, -348, 225, -58, 21, -4, 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, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1783076289543920291440878202489416150266305614961000, 13236687730948644939134821309052808423680122934742700, -23585087514321117937515010600407090612184494400833210, -21081759418369782765196352467160404447136266390351277, -25152060303860130754446861719603191366061957047446509, -5618227724087166678700556947594847475164662325680683, -9399738001633894959794191625594323270867524128110410, -733424827251535299177335007020074163444350020681969, -2541942148839484894924181036938661343951383152960662, -245990707904268698777713225549845955066795576520843, -513677935359061164862839579155176626629076488056089, -91365977834847082232697972415726927199607743085559, -105839972730725981139715584231581748548167601122315, -1294440809575935460602061016335148798592222151552, -11644973493713756475882111663051294613422464464824, 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14444000045504617651064463616074164410802201824299717, 13299927392065125988605572156025763768715294329547452, 2346069176576058144340217392066442474296419218485218, 5564479915643009249802584836530765284126050097827495, -476386298488193663753866090720698862965012515653401, 1895730615927451104079457450106089806101177622874385, -281746662217313777609059086007927277106531360316196, 429612913417282392266076175158735610783550415375816, -30131096626966562704677058524495440322292589392571, 81759510996684390613844247626414918029369887705251, -19915263477167394900108733547278533979783935383924, 14244577722673561911824026616851343908900139748798, -3578709905222524676816195719389821571656617642950, 1352806453429920366570947328827280154822234550507, -253910999274503376562323345229650458611424321078, 67216138265765641349547124933114531351703988958], [416476403000858533497764020926396895044999952605000, 50037035376777365894110234710095248179568951405971150, -113647196426794515399436978641901397321631996573281875, 82804342832243163592154684785247924087795179843782232, 56577635131807009888881928116881292669998650193721769, 88934155378666237544825170205159321343538403470407033, -2762937395384201355821650559736492345027985445991410, 41772485298179714349839402459812949166190368139723809, -9808891856389838262084159474574822676515944642177853, 14471799642725031104026660553541968523937464301232423, -3444471970778751392879925564929460561004361375044281, 3184559981036015275670904867371252338353463077458809, -491281139068469463910462393517722772319425181401645, 649530647623249932606711819276924653096804291253832, -188206923784360757404777718567978237933719862459576, 112803260555402873041038656259657071988840612732080, -29740299407773405857241924634698539729166567554631, 10400419717950898126863715649345380358729761808667, -2010976286639818399320252272286798676796959754083, 476510198789462892340586485950346317205130340162]];
    Rf_basisdens := [1, 1, 8707082140645302621003645575223925946567069737049000, 870708214064530262100364557522392594656706973704900, 2176770535161325655250911393805981486641767434262250, 435354107032265131050182278761196297328353486852450, 8707082140645302621003645575223925946567069737049000, 870708214064530262100364557522392594656706973704900, 4353541070322651310501822787611962973283534868524500, 2176770535161325655250911393805981486641767434262250, 870708214064530262100364557522392594656706973704900, 4353541070322651310501822787611962973283534868524500, 16616568970697142406495506822946423562150896444750, 1741416428129060524200729115044785189313413947409800, 4353541070322651310501822787611962973283534868524500, 8707082140645302621003645575223925946567069737049000, 4353541070322651310501822787611962973283534868524500, 1741416428129060524200729115044785189313413947409800, 870708214064530262100364557522392594656706973704900, 4353541070322651310501822787611962973283534868524500];
    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_616_r();"
function MakeCharacter_616_r()
    N := 616;
    order := 5;
    char_gens := [463, 309, 353, 57];
    v := [5, 5, 5, 4];
    // 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_616_r_Hecke();"
function MakeCharacter_616_r_Hecke(Kf)
    N := 616;
    order := 5;
    char_gens := [463, 309, 353, 57];
    char_values := [[1, 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, 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, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, 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 := 2;
    raw_aps := [[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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -1, 0, -1, 0, -1, 0, 0, 0, 0, -1, 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, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0], [-1, -1, -2, 0, -2, 0, -2, 0, 0, 0, 0, -1, 1, 1, 0, 1, 0, 0, -1, 1], [1, 0, 0, 0, 1, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, 1, -1, 0, 0, 1], [2, 0, 1, 0, 2, 0, 1, 0, -1, 0, 0, 0, -1, -2, 0, -1, 0, -1, -1, 0], [-2, 0, -2, 0, -1, 0, -2, 0, 0, -1, 1, -2, 0, 1, -1, 1, 1, 1, 0, 1], [-1, -1, -1, 1, -1, 0, -2, 0, 1, -1, 0, -1, 1, 1, 0, 0, -1, -1, 0, 1], [-1, 1, 0, 0, -1, 0, 0, 0, 0, 1, -1, -1, 0, 0, -1, 0, 1, 0, 0, 0], [-1, -1, -1, 1, -2, -1, -3, 0, 1, -1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1], [-2, 0, 0, 1, -2, 0, -2, 0, 0, 0, 0, 0, -1, 2, -1, 1, 0, 1, 1, 0], [-3, 0, -1, 1, -1, -1, -2, 0, 1, 0, 1, -1, 1, 0, 0, 1, 0, 1, -1, 1], [0, 0, 0, -2, 0, 0, -2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, -1, 1, 2, 5, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, -1], [1, 1, 1, -1, -2, 0, -1, 0, 0, 1, -2, -1, -1, -1, 0, 0, -1, -1, -1, -1], [5, 1, 2, 0, 5, 1, 5, 0, 0, 0, -1, 5, 0, 0, 0, -1, -1, 0, 0, 1], [-1, 1, -1, -2, 1, 2, -1, 1, 0, -2, -2, -1, 0, -2, 0, -1, 0, -2, -1, -1], [2, 2, 0, 0, 0, 2, 2, 0, 2, -2, -2, 0, 0, 0, 0, 1, -2, 0, 0, 1], [0, -1, 0, 1, 0, -1, 1, 0, 0, 1, -1, -3, -1, 0, 0, 0, -1, -1, -2, 0], [2, -4, -2, 2, -2, 0, -2, -2, 0, 0, 2, 2, 0, -1, 2, -1, -1, 0, -2, 2], [5, -1, 0, 0, 5, -1, 5, -3, 0, 0, -1, 5, 0, 0, 0, 3, -1, 0, 0, -1], [-3, -1, 0, -2, -1, 2, -2, -1, 1, 1, 1, -1, 1, 2, 2, 1, -2, 1, 1, 1], [4, -2, 1, 2, 2, 0, 1, -2, 0, 1, 0, 4, 1, 1, 0, 1, 3, 0, 2, -2], [4, -1, 1, -2, 2, 0, 1, 2, 0, 1, 2, 4, 3, -1, 2, -1, 1, 0, 4, -4], [1, 0, 1, 0, 1, 2, 1, 0, 2, 0, 0, 1, 0, -1, 0, 0, -1, -1, 1, -1], [-4, 0, -1, 1, -4, 1, -5, 1, 1, 0, 0, 0, 1, 2, -1, 1, 0, 1, 1, 0], [1, 1, -1, 0, -1, 0, 6, 1, -2, 0, -2, 4, -2, -1, -1, 1, 1, -2, -2, 1], [6, 0, 2, 2, 6, -3, 2, -3, -2, 0, 0, 0, 0, -3, 1, 2, 0, 2, 2, 0], [2, 1, -2, 0, 3, 1, 2, -3, 0, 0, 1, 3, 0, 0, 1, -1, 0, 1, 0, 0], [-5, 1, -1, 0, -2, 0, -1, 1, -4, 1, 1, -5, -4, -3, 4, 2, -4, 1, -5, 2], [-1, -2, 1, 0, -2, -2, -1, -3, 1, -1, -2, -2, 0, 0, 1, 1, -3, 1, 0, 0], [3, -1, 3, 1, 6, -2, 4, 0, -1, -2, -1, 3, 2, -3, 0, 0, 0, 0, 1, -3], [-3, 1, -2, 0, -4, 1, -3, 4, -1, 1, 1, -4, 0, 0, 2, 3, -1, 2, 0, -1], [-5, 0, -1, -3, -5, 3, -8, 3, 1, 0, 0, 0, 5, 4, 0, 1, 0, 0, 1, 0], [-2, 0, -2, 3, -2, -1, -1, -1, 2, 0, 0, 0, 0, 2, -1, -1, 0, -2, -1, 0], [-1, 3, 3, 0, -1, 0, 3, 0, 0, 3, 1, -1, -3, -2, 1, -2, -1, 0, 0, 0], [6, 0, 4, 0, 1, 0, 4, 0, 0, 3, -3, 6, 0, 1, -3, 1, 2, 0, 4, -4], [-5, 0, -2, 3, -5, 0, -7, 0, 2, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0], [2, 0, -1, -3, 2, 5, -4, 5, 1, 0, 0, 0, 2, 2, 2, 0, 0, -1, 0, 0], [4, -1, 5, 0, 1, -1, 4, 0, 1, -1, -5, 1, 0, 0, -4, -3, -1, -4, 0, -5], [2, 2, 2, -2, 3, 2, 2, 0, -3, -1, -3, -5, 1, -2, 0, 0, 1, 1, 7, -2], [6, 1, 5, 0, 1, 1, 6, 2, -1, 1, 1, 1, 0, 0, 0, -2, 1, 0, 0, -6], [5, -4, 3, 4, 1, -4, 5, -4, -4, 2, 3, 1, -4, -1, 1, -1, -1, 3, 0, -1], [-8, 0, 4, 4, 2, -4, -2, 0, -5, 2, 3, -7, -5, -1, -2, -2, 2, 3, 1, -2], [-3, 4, -2, -2, -8, 0, -2, 2, 0, 0, 2, -3, 1, 0, 2, 0, 0, 0, -1, 1], [7, 0, -2, -3, 7, -2, 3, -2, 2, 0, 0, 0, 0, 0, 1, 2, 0, -1, 2, 0], [-1, -2, -1, 2, -2, 0, 0, 0, 2, 2, -4, 2, -2, 1, 0, 0, -2, -2, 2, 1], [-3, -1, -4, 1, 3, 0, -4, -1, 0, -2, 0, -3, 5, 1, 0, 1, 1, 0, 1, -1], [-3, 1, -3, -1, -2, 0, -3, 1, 0, 2, 1, -3, 0, -1, 1, -1, -2, 0, -3, 3], [-1, 0, -1, -5, -1, 5, -6, 5, 1, 0, 0, 0, 0, -2, 0, 1, 0, -1, 1, 0], [-4, -5, 1, 1, -2, -1, 13, -5, -4, 3, 2, 9, -4, 0, 1, 2, -1, 2, -2, 2], [-3, 5, 2, 0, 4, 0, 2, 0, 0, 2, -1, -3, -1, 2, -1, 2, 0, 0, 1, -1], [-2, -6, -2, 6, -10, -2, -9, 0, 0, 1, 2, -8, -1, 2, 0, 0, 1, 1, 0, 2], [3, -1, 0, 3, 1, -3, 1, -1, 1, -1, -1, 2, 1, 4, -3, -1, 3, -1, 5, -1], [-9, -1, -4, 0, -4, -1, -9, -4, 3, -3, -1, -4, 0, 0, 0, 3, -1, 0, 0, 3], [-4, 0, -4, 0, -5, 4, -6, 0, 3, -1, 1, -1, 1, 4, 0, 0, -2, -2, -3, 4], [-4, -2, -2, 1, -3, 0, -2, -1, 0, 0, 2, -4, -3, 1, 2, 1, 0, 0, -5, 5], [-6, -3, -8, 0, -5, -3, -6, -2, 2, -2, 2, -5, 0, 0, 1, 2, 1, 1, 0, 5], [3, 0, 0, -1, 3, 0, 5, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0], [-1, 7, -1, -7, -3, 6, -4, 0, 5, -1, -5, -5, 1, 1, 0, 0, -2, -2, 1, 1], [-2, 0, 2, -1, -1, 1, 1, 0, -3, 3, 2, -2, -3, 2, -1, 1, 1, 2, 1, 1], [-3, 0, -4, 1, -3, 0, -15, 0, 4, 0, 0, 0, 7, 4, -1, 2, 0, 1, 2, 0], [3, -3, -2, 0, 2, -3, 3, 2, 2, -2, -1, 2, 0, 0, -4, -7, 3, -4, 0, 0], [9, 1, 4, 0, 12, 0, 4, 0, 0, 4, -1, 9, -5, -3, -1, -3, 1, 0, -1, 1], [-8, 4, 3, 0, 3, 0, -6, 4, -2, 0, 2, -8, -2, -3, -4, -3, 4, 2, 0, -3], [-2, 3, -2, -3, 0, 1, 1, 0, 2, 1, 1, 0, -1, 2, 0, 0, 4, 4, 2, 2], [6, -3, 1, 0, 12, -3, 6, -2, -1, 1, -3, 12, 0, 0, 0, -2, -3, 0, 0, 1], [4, 0, 0, 1, 4, -5, -5, -5, 0, 0, 0, 0, 1, 0, -4, -2, 0, -3, -2, 0], [-6, 3, -3, 1, -1, -1, -5, 3, 4, -2, 0, -1, 4, 1, 0, 1, 0, 0, 0, 1], [-2, 2, -2, -2, -7, 5, -5, 0, 1, 2, 0, -5, -2, 2, 0, 0, 0, 0, -1, 2], [-3, 2, -3, -2, -13, 0, -15, 0, 4, -2, -1, -5, 2, 3, 0, 0, 1, 1, 0, 3], [4, 1, 9, -1, 4, 1, 16, 1, -4, 5, -4, 12, -4, -7, 1, -4, -1, -4, -3, -4], [-3, 5, 0, 0, -8, 5, -3, -3, 1, -1, 1, -8, 0, 0, 1, 2, 0, 1, 0, 3], [-13, 1, 0, -2, -8, 0, 0, 2, 0, 0, 4, -13, 0, 3, 4, 3, -3, 0, 0, 0], [-1, -3, -1, 3, 4, 0, 4, 0, -8, 0, 4, 0, 0, 1, 0, 0, -1, -1, -3, 1], [2, 1, 1, 1, 4, -1, -6, 1, 5, -3, -3, -1, 5, 4, -5, -4, 5, -3, 8, -4], [4, 0, 3, 0, 2, 0, 4, 0, -1, 1, 1, 2, 0, 0, 1, -2, 0, 1, 0, 2], [8, 5, 3, -3, -1, 0, 3, 3, 0, 1, 1, 8, -4, 4, 1, 4, 1, 0, -1, 1], [-5, 0, 2, 0, 1, 0, 3, 0, 1, 1, -2, 4, 1, -2, 0, -1, 0, -2, -1, -1], [1, 0, 2, 6, 1, -3, 6, -3, -2, 0, 0, 0, -7, -1, 2, 0, 0, 5, 0, 0], [4, 3, 1, -1, 4, 0, 1, 1, 0, -2, -3, 4, 3, -1, -3, -1, 0, 0, 4, -4], [7, -4, 7, 4, 3, -1, 5, 0, -2, 2, -6, 8, -2, -7, 0, 0, -1, -1, 2, -7], [-8, -1, 3, 5, 1, -5, 0, -1, -3, 2, 3, -3, -3, -3, 0, -1, 0, 3, -2, -1], [6, 0, 6, 5, 6, -4, 0, -4, -6, 0, 0, 0, -1, 4, -3, 4, 0, 3, 4, 0], [6, -2, 3, -5, 7, 0, 3, 5, 0, 8, 1, 6, -2, 3, 1, 3, 0, 0, 1, -1], [-6, 2, -5, 0, 0, 2, -6, 8, 3, -3, 1, 0, 0, 0, -2, -3, 3, -2, 0, 3], [-5, -2, -4, 3, -5, -3, -6, -2, 3, 1, 2, -3, 3, 3, 0, 5, 0, 2, -2, 5], [3, 6, 2, 3, -2, -3, 8, 6, -4, 4, 2, 4, -4, 1, 0, 2, 0, 2, -1, 2], [1, -2, -4, 0, -4, -2, 1, -3, 0, 0, 3, -4, 0, 0, 2, 5, 1, 2, 0, 0], [9, 1, 9, 0, 11, 1, 9, 5, -5, 5, 2, 11, 0, 0, 1, -3, 1, 1, 0, -6], [15, 0, 5, 5, 15, -1, 14, -1, -5, 0, 0, 0, -2, 4, -3, -2, 0, 0, -2, 0], [4, -4, 4, 4, -8, -5, -4, 0, -11, 4, 2, -9, -4, -4, 0, 0, 0, 0, -6, -4], [2, -6, 2, 6, 2, -5, 0, 0, -1, -2, 3, 3, 2, -2, 0, 0, 3, 3, 1, -2], [-4, 0, 2, -3, -4, 1, 3, 1, -2, 0, 0, 0, -7, -5, 3, -1, 0, -1, -1, 0], [-10, 0, -1, -2, -10, 3, -5, 3, 1, 0, 0, 0, -1, 0, 1, 0, 0, -2, 0, 0], [-1, -1, -1, 1, -9, 0, -7, 0, 5, 2, 8, -8, -2, 1, 0, 0, 5, 5, 7, 1], [12, -5, -3, 0, -2, -5, 12, 2, -7, 7, 7, -2, 0, 0, 3, 4, 4, 3, 0, -3], [-1, 2, -5, -1, -5, 0, -5, 1, 0, -1, 0, -1, 7, 7, 0, 7, -1, 0, 2, -2], [1, 0, -1, -3, 1, 6, -12, 6, 1, 0, 0, 0, -1, -2, 2, 2, 0, 1, 2, 0], [1, 1, 1, -1, -7, 1, -7, 0, -6, 0, -1, -12, 0, -1, 0, 0, 0, 0, 5, -1], [13, 3, -3, -1, 7, 0, -3, 1, 0, 3, 2, 13, -5, 2, 2, 2, -3, 0, -8, 8], [-8, 0, -1, -3, -8, -5, -2, -5, 1, 0, 0, 0, -2, 0, 1, 0, 0, -3, 0, 0], [2, -5, 7, 1, 2, -1, -4, -5, -1, 5, 0, -5, -1, -4, 3, -2, -3, 0, -2, -2], [6, -4, 2, 0, 12, -4, 6, -8, 1, -1, -7, 12, 0, 0, 0, 2, -7, 0, 0, -6], [-1, -9, -1, 9, -7, -3, -6, 0, 5, 1, 1, -2, -1, 1, 0, 0, -2, -2, 0, 1], [-3, 4, -8, -9, -1, 9, -1, 4, 3, -7, 0, 2, 3, -2, 4, 1, -4, 0, -3, 1], [-1, 5, 0, 0, 1, 5, -1, 0, 8, -8, 1, 1, 0, 0, 1, -6, 0, 1, 0, 1], [1, 2, 1, -2, 12, 5, 9, 0, 5, -3, -3, 2, 3, -1, 0, 0, 1, 1, 2, -1], [0, 3, 3, -4, 1, 0, 3, 4, 0, 2, 0, 0, 1, -1, 0, -1, 3, 0, 4, -4], [-3, 0, -1, 1, -3, 7, -10, 7, 1, 0, 0, 0, 1, -1, 4, -3, 0, -4, -3, 0], [9, -5, -8, 7, -6, -7, 0, -5, 2, -2, 4, 2, 2, 5, 0, 6, 0, 4, -1, 6], [-2, 0, 3, 1, -2, 1, -8, 1, -3, 0, 0, 0, -5, -12, 6, -3, 0, 6, -3, 0], [0, 2, 0, -2, 11, 1, 15, 0, -8, 4, -3, 10, -4, 0, 0, 0, -4, -4, -8, 0], [5, 0, -1, 2, 5, 3, 2, 3, 1, 0, 0, 0, -5, -6, 4, -5, 0, -1, -5, 0], [15, -8, 3, 0, 6, -8, 15, -7, -3, 3, 5, 6, 0, 0, 4, 0, 1, 4, 0, 1], [-7, -1, 2, -2, -8, 0, 2, 2, 0, -2, -2, -7, 1, -3, -2, -3, 2, 0, 3, -3], [-4, -4, -4, 4, -12, 7, -13, 0, -2, -1, -1, 0, 1, 4, 0, 0, 0, 0, -4, 4], [-12, -6, -4, 6, 3, -6, -9, -6, 7, -7, -2, -2, 7, 0, -1, -3, 1, -2, 3, -3], [-29, 2, -16, -3, -7, 3, -32, 2, 11, -9, -6, -21, 11, 5, 6, 7, -6, -6, -2, 7], [-4, 9, 12, 0, 8, 0, 12, 0, 0, -1, -3, -4, -7, -6, -3, -6, 1, 0, 5, -5], [-2, 3, 4, 0, -2, 3, -2, 1, -3, 3, 5, -2, 0, 0, 2, 0, 3, 2, 0, -2], [-3, 3, -2, -5, 0, 5, 1, 3, -1, -2, 2, 0, -1, 2, -3, 0, 3, 2, 2, 0], [2, -3, 4, 8, -4, 0, 4, -8, 0, -7, -7, 2, -6, -3, -7, -3, -3, 0, -2, 2], [-7, 0, -5, -6, -7, 6, -7, 6, 5, 0, 0, 0, 7, -3, 8, 1, 0, -2, 1, 0], [-3, 5, 2, 0, 10, 5, -3, 2, -1, 1, -1, 10, 0, 0, 2, 8, -3, 2, 0, -2], [5, 0, 5, 0, 7, -2, 5, 0, 1, -2, 5, -2, 2, -5, 0, 0, 3, 3, -2, -5], [-14, -4, -17, 3, -12, -3, -23, -4, 10, -5, 4, -13, 10, 8, 5, 12, -5, 4, -4, 12], [-3, -1, -3, 1, -8, 5, -5, 0, -4, 3, -3, -8, -3, 3, 0, 0, -1, -1, -5, 3], [-15, 10, 0, -8, -10, 0, 0, 8, 0, -1, -1, -15, 0, -1, -1, -1, 3, 0, 0, 0], [0, 4, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, -2, 3, 2, -2, 0, -3], [1, 0, 3, 5, 1, -7, -2, -7, -3, 0, 0, 0, -4, 1, -2, 1, 0, -5, 1, 0], [20, -3, 7, -1, 23, 0, 7, 1, 0, 1, -3, 20, 1, -1, -3, -1, 5, 0, 8, -8], [-16, 0, -5, 0, -11, 0, -5, 0, 0, 5, -1, -16, 3, 4, -1, 4, 2, 0, -2, 2], [-3, 5, -2, 3, -1, -3, 11, 5, 1, -1, 5, 12, 1, 3, 2, 1, -2, 5, 2, 1], [-6, 0, 5, 4, -6, 1, 7, 1, -5, 0, 0, 0, -3, -2, -7, 0, 0, -2, 0, 0], [-17, -5, -5, 0, -7, -5, -17, 1, 2, -2, 5, -7, 0, 0, 4, -4, 1, 4, 0, 3], [-26, 0, -2, 0, -13, 0, -26, 8, 3, -3, 0, -13, 0, 0, -5, -9, 5, -5, 0, 5], [2, 0, 2, 0, -3, 0, 2, 0, 6, -6, -3, -3, 0, 0, -5, -3, 2, -5, 0, 4], [-11, 0, -4, 5, -11, 3, -16, 3, 4, 0, 0, 0, 2, 5, -5, 2, 0, 3, 2, 0], [-3, 0, -3, 0, -12, 7, -10, 0, -5, 2, 5, 2, -2, 3, 0, 0, 4, 4, -6, 3], [-8, -5, -9, 2, -11, 0, -9, -2, 0, 4, 6, -8, 4, 8, 6, 8, -7, 0, -5, 5], [-7, 0, -8, -2, -17, 0, -8, 2, 0, 2, -1, -7, 3, 4, -1, 4, -1, 0, -5, 5], [1, 2, 1, -2, -3, 2, -3, 0, 5, 0, -10, -2, 0, -1, 0, 0, -11, -11, -7, -1], [2, 0, 1, -5, 2, -6, 10, -6, -1, 0, 0, 0, 1, -5, -2, -1, 0, 3, -1, 0], [5, 2, -1, 0, -15, 2, 5, 2, -6, 6, 3, -15, 0, 0, 4, 2, -1, 4, 0, 7], [14, -4, 5, 4, 3, -4, 12, -4, -4, 2, 2, 8, -4, -4, -4, -3, 4, 2, -1, -3], [5, -7, 2, -1, 4, 1, -2, -7, 6, -2, 1, 4, 6, 1, 1, -4, -1, 1, 5, -4], [3, -13, -6, 0, 1, -13, 3, -2, 0, 0, 1, 1, 0, 0, -3, 2, 4, -3, 0, -2], [3, 0, 8, 0, 3, -1, 20, -1, -8, 0, 0, 0, -8, 6, -6, -4, 0, -3, -4, 0], [16, 1, 1, 0, 5, 0, 15, 1, 6, -4, 0, 21, 6, -7, 2, -5, -2, 0, -2, -5], [-1, 0, 1, 17, -1, -4, -7, -4, -1, 0, 0, 0, 0, 9, -2, -2, 0, 1, -2, 0], [2, -1, 2, 1, 3, 5, 10, 0, -1, 7, 3, 2, -7, -2, 0, 0, -4, -4, -6, -2], [-5, 5, -14, 0, 0, 5, -5, -4, 4, -4, 3, 0, 0, 0, 3, 1, 0, 3, 0, 10], [3, -1, 2, 4, 1, 0, 2, -4, 0, -5, 2, 3, -7, -3, 2, -3, -2, 0, -5, 5], [8, 9, -6, 0, 3, 9, 8, 1, 4, -4, -1, 3, 0, 0, -1, 2, 0, -1, 0, -4], [1, -4, 6, 0, -1, -4, 1, -3, -4, 4, 3, -1, 0, 0, 1, 8, 2, 1, 0, -2], [-10, 0, 9, -2, -3, 0, 9, 2, 0, -5, -2, -10, -3, -7, -2, -7, -2, 0, 6, -6], [-3, 2, -2, 0, 0, 2, -3, -11, 2, -2, 0, 0, 0, 0, -2, -11, 2, -2, 0, 1], [-12, 4, -9, -3, 1, 3, -2, 4, 10, -10, -1, 8, 10, 9, -6, -1, 6, -1, 10, -1], [-6, -7, -6, 7, 2, -5, 3, 0, 3, 1, 7, 12, -1, 6, 0, 0, 0, 0, -7, 6], [-5, -10, -8, 3, -6, -3, -14, -10, 2, -2, -5, -12, 2, 6, 3, 6, -3, -5, 0, 6], [-3, 0, -1, -3, -3, 5, 2, 5, 1, 0, 0, 0, 1, 1, -1, -5, 0, -9, -5, 0], [5, -4, -7, 2, -7, 0, -7, -2, 0, 6, 2, 5, 3, 3, 2, 3, 2, 0, -4, 4], [-8, 1, -8, -1, -18, -3, -20, 0, 6, -2, 6, -25, 2, 8, 0, 0, -2, -2, -3, 8], [-14, 3, -10, -4, -7, 4, -7, 3, 2, -3, 3, -5, 2, 7, 9, 7, -9, 3, 0, 7], [-5, -9, -5, 9, -6, -7, 1, 0, -6, 7, 9, -18, -7, 5, 0, 0, 3, 3, -3, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_616_r_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_616_2_r_f();".
// 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_616_2_r_f(:prec:=20)
    chi := MakeCharacter_616_r();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_616_2_r_f();".
// 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_616_2_r_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_616_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![10000, -43000, 97900, -125070, 446621, -188178, 296139, -12610, 75177, -6414, 22864, 1027, 6377, 395, 1081, 112, 85, -8, 11, 1, 1]>],Snew);
    return Vf;
end function;