
// Make newform 612.2.w.c in Magma, downloaded from the LMFDB on 30 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_612_w();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_612_w_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_612_2_w_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_612_2_w_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 := [4, 0, 462752, 0, 715372, 0, 414440, 0, 116445, 0, 17508, 0, 1434, 0, 60, 0, 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], [1177839845590, 0, -4765884821288, 0, -5118882005523, 0, -1728005430420, 0, -264194956239, 0, -20013985161, 0, -730091944, 0, -10183231, 0], [834139859368, 0, -17202998438278, 0, -18805602643828, 0, -6973443020659, 0, -1225172767250, 0, -110704440654, 0, -4947624218, 0, -86362753, 0], [-37457113132, -12739898076102, -154574621844, -19981915899780, -121753921784, -11706031006803, -46130941900, -3313465288556, -8866998107, -500634663986, -872350943, -41140786928, -41712081, -1725343305, -766785, -28803044], [37457113132, -12739898076102, 154574621844, -19981915899780, 121753921784, -11706031006803, 46130941900, -3313465288556, 8866998107, -500634663986, 872350943, -41140786928, 41712081, -1725343305, 766785, -28803044], [691188026456, 235059249358272, 12105874045474, 353089832963568, 14685750256246, 199956544935140, 6081895983335, 55492341655432, 1159917430273, 8290208940598, 111434060969, 676673874128, 5215141977, 28257173258, 94265158, 470397100], [-691188026456, 235059249358272, -12105874045474, 353089832963568, -14685750256246, 199956544935140, -6081895983335, 55492341655432, -1159917430273, 8290208940598, -111434060969, 676673874128, -5215141977, 28257173258, -94265158, 470397100], [0, 874905524, 0, 1331873950, 0, 761904098, 0, 212104547, 0, 31676582, 0, 2581674, 0, 107628, 0, 1789], [-345529433242, 283805077784822, 10340297968278, 441876241326876, 9591550356055, 257208904350375, 3092284932373, 72530523488424, 489648271146, 10941952531944, 41289910746, 899051581736, 1769619153, 37723759683, 30183059, 630241980], [345529433242, 283805077784822, -10340297968278, 441876241326876, -9591550356055, 257208904350375, -3092284932373, 72530523488424, -489648271146, 10941952531944, -41289910746, 899051581736, -1769619153, 37723759683, -30183059, 630241980], [487357836308, -165761866021268, 1371589038938, -264225212267472, -99090963372, -156317046040216, -297914865297, -44397804203222, -73093876203, -6709646651825, -7315411723, -550744767037, -333947021, -23059716203, -5768030, -384330901], [487357836308, 165761866021268, 1371589038938, 264225212267472, -99090963372, 156317046040216, -297914865297, 44397804203222, -73093876203, 6709646651825, -7315411723, 550744767037, -333947021, 23059716203, -5768030, 384330901], [1180083186368, 401308473215848, 25565490162696, 618686634269978, 25301012335574, 356174500011984, 8749033688262, 99592230993357, 1442431117793, 14926761716220, 123337412129, 1220103229442, 5266939953, 50982942440, 88634985, 848959971], [-487357836308, -400184344456296, -1371589038938, -614687276659692, 99090963372, -354203774305028, 297914865297, -99105919846354, 73093876203, -14849116624604, 7315411723, -1212588675134, 333947021, -50607784084, 5768030, -841692656], [2719728082012, -925127497705982, -21061790265862, -1424112015933552, -24727944611256, -823930529271495, -9899692593497, -231543839462682, -1870905126755, -34847511079170, -180044740671, -2857862201528, -8466451041, -119731104237, -153784274, -1997867746], [2719728082012, 925127497705982, -21061790265862, 1424112015933552, -24727944611256, 823930529271495, -9899692593497, 231543839462682, -1870905126755, 34847511079170, -180044740671, 2857862201528, -8466451041, 119731104237, -153784274, 1997867746]];
    Rf_basisdens := [1, 450248976368, 450248976368, 52970467808, 52970467808, 900497952736, 900497952736, 2572256, 900497952736, 900497952736, 450248976368, 450248976368, 900497952736, 450248976368, 900497952736, 900497952736];
    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_612_w();"
function MakeCharacter_612_w()
    N := 612;
    order := 8;
    char_gens := [307, 137, 37];
    v := [8, 8, 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_612_w_Hecke();"
function MakeCharacter_612_w_Hecke(Kf)
    N := 612;
    order := 8;
    char_gens := [307, 137, 37];
    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], [0, 0, 0, 1, 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 := 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, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, -2, 0, -1, 1, 0, 0, 0, 1, 1, 1, -1, 0, 1, 1], [1, 0, 1, -1, 0, -1, 1, 0, 0, 0, -1, -1, 1, 0, -1, -1], [0, 0, 1, -1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0], [1, 0, 1, -1, 0, 0, 1, 2, 0, 0, -1, 0, 1, 0, 0, -1], [-1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, -1, 0], [0, 1, -1, 0, -3, 1, 1, 0, 1, 1, 0, 1, -1, 1, 0, 0], [0, 1, 0, -1, 1, -1, 0, 1, -1, 0, -1, -1, 0, 0, 0, -1], [2, 1, 0, -2, -2, 1, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, 1, -1, 0, -2, 1, 0, -1, -1, 0, 0, 1, 1, -1, 0], [0, 0, 0, -1, 1, 0, 1, -1, -2, 0, -1, 0, 1, 0, -1, 0], [4, 1, 1, -2, 2, -1, 1, 1, 0, 1, 0, 0, 2, 1, -1, -2], [0, 1, -3, 0, -1, 0, 0, 2, -1, 3, 1, 0, -1, -1, 1, 1], [2, 0, 2, 0, 2, -2, 0, 0, 0, 0, 0, -2, 2, 0, 0, -2], [-3, -1, -1, 1, 0, 1, 0, 1, 1, -1, 0, 0, -1, -1, 0, 1], [2, 1, -1, -1, 0, 0, 0, 2, -1, 1, 0, 2, 0, -1, 0, 1], [-2, -2, 2, -1, -2, 2, 0, -1, 4, -2, -2, 0, -2, 0, 0, 0], [-2, -2, 0, 4, -4, 1, -1, 0, 2, -2, 1, 1, 0, 0, 1, 1], [0, 0, 0, -1, -4, 1, 1, 4, -1, 1, 1, 2, 0, 2, -1, 0], [-2, -1, -2, 1, 2, 0, -1, 2, -1, 1, 0, 2, 0, 1, 1, 2], [3, 2, 1, -1, 0, -1, 1, 0, -2, 0, -3, -1, 1, -2, -1, -1], [-2, -2, -2, 2, 2, 2, -2, -2, 2, -2, 2, 0, -2, -2, 4, 2], [-2, 0, -2, 1, -1, 0, -2, -2, 1, 1, 2, 2, -2, 2, 2, 0], [-1, -1, 2, 2, -1, 0, -1, -3, 1, -3, 0, -2, 0, -1, -1, 0], [-4, 2, -2, 0, 0, 1, -1, 0, -2, 2, 1, 1, 0, 0, 2, 2], [1, 4, -1, 1, -1, -1, 1, 0, -4, 4, 1, 1, 0, 0, -2, -2], [-2, 4, -5, 3, 6, -1, -2, 3, -6, 3, 2, 2, 0, -3, -1, 1], [5, -3, 6, -3, 4, -3, -2, -1, 3, -5, -3, -3, 4, -1, 0, -1], [1, -2, 2, -1, 2, -1, -2, 0, -1, -4, 1, -1, 1, 0, -1, -1], [4, -1, 3, 4, 1, -2, 0, 2, 1, -3, -1, -4, -1, 1, -1, -1], [-2, -1, 0, 2, -2, -2, -1, -3, -2, 1, 0, 0, 2, 0, 0, 1], [8, -2, 8, 4, -4, 0, 0, 0, 5, -5, -3, -3, 0, 0, -1, -1], [-1, 0, 1, 1, 0, 0, -2, 0, 2, 0, 0, -2, -3, -2, 2, 0], [0, 0, 0, 6, 6, 1, 1, 2, -1, -1, 0, 0, 0, 0, -1, 1], [-4, 0, -6, 3, 0, 1, -2, -2, -4, 0, 2, 3, -1, 0, 2, 3], [-1, 0, -1, 2, 1, 0, -1, -2, 0, 0, -1, -2, -1, -4, 0, 1], [-4, 1, 1, 3, -4, -2, -1, 2, -1, -1, 0, 2, 1, 1, -1, 0], [4, -2, -1, 6, -5, 1, -2, 5, 2, -1, -1, -1, 0, -3, 2, 0], [0, 0, 2, 0, 1, -4, 0, -1, -1, -4, -2, -2, 2, -2, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2], [-6, 0, -6, 1, 0, 2, -2, 1, 0, 0, 2, 2, -2, -2, 2, 2], [4, 0, 4, 1, 5, -1, 3, -2, 1, 1, -5, 1, 4, 0, -2, -2], [-4, -3, 0, -6, 1, 3, 0, 6, 3, 1, 3, 3, 0, 3, 0, 3], [-6, 1, -3, 0, -6, 1, 1, 1, -2, 3, 2, 2, 0, 1, 1, 0], [2, -1, -1, -5, -6, 4, 1, -2, 3, 1, 2, 2, -1, 1, 3, 2], [0, 1, -1, -2, 2, 0, 0, -8, -1, -1, -1, -4, -4, -3, -2, 2], [0, 5, 3, -4, -1, -1, 5, 8, 1, 5, -2, 1, 1, 5, -2, -4], [0, 3, 4, 1, 5, -1, 4, -3, -1, 0, -5, -1, 2, 2, -2, -1], [2, 2, -2, -5, 0, 1, 0, 2, -2, 2, 0, -3, 1, -2, 0, 1], [1, 1, -3, -1, -5, 0, 1, -4, 0, 1, -1, 1, -3, 1, 0, -1], [4, 0, -4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2], [-1, 0, -2, 0, -4, -2, -1, 4, -4, -2, -1, -2, 3, -2, 1, 0], [-2, 0, -2, 4, 2, -1, -3, 2, -1, -1, 2, 2, -2, 2, -1, 3], [4, 1, -5, 2, -1, -2, -2, -8, -9, 1, 3, 3, 2, 1, 1, 2], [10, 3, 1, -5, 2, -2, 0, 6, -3, -1, -2, -2, 0, -3, -2, -1], [-1, -1, 1, -2, -2, 2, 1, 2, 0, 1, -2, 0, -1, -1, -1, 0], [3, 2, 3, -3, 3, -4, 4, 0, 0, 0, -6, -6, 0, 0, -5, -5], [10, 3, 4, -6, 0, -2, 0, -4, -1, -3, -3, -2, 1, -1, -4, -4], [1, -2, 8, 3, 3, -3, 0, -5, 2, -8, -3, -5, 5, 2, -3, -1], [7, 0, 3, -9, 0, -1, -1, 6, -4, 2, -3, -1, 3, -2, -3, -1], [-4, 0, -4, -1, -5, 5, 1, -12, -1, -1, 3, 5, -4, 4, 0, 4], [-2, 2, -8, 2, -2, 3, -3, 0, -2, 2, 5, 5, 0, 0, 3, 3], [-4, 0, -6, -1, -2, 3, 3, 6, 1, 7, 3, 4, -2, 2, 5, 2], [0, -1, -2, 0, -1, 5, 2, 4, 3, 3, 1, 5, -2, 3, 2, -1], [-2, 3, -6, -4, -8, 2, 3, 8, 4, 6, 1, 2, -4, 0, 4, 2], [6, -5, 3, -5, -1, 1, 0, 8, 5, -3, 1, 3, 2, 5, 1, 0], [5, 2, 6, 0, -2, -4, 5, 2, -2, -2, -7, -4, 1, -2, -5, 0], [2, 0, -2, -10, 1, 7, 1, -6, 5, 3, 2, 7, -4, 4, 2, -1], [1, 1, 5, 11, 3, -5, 0, -5, -1, -5, -3, -7, 2, -1, -3, -2], [-6, 0, -6, 8, 2, 7, 1, 12, 5, 5, 2, 2, -6, -2, 3, 3], [12, -3, 13, -5, 2, -6, 4, -2, 1, -3, -8, -8, 6, -3, -4, -5], [-1, 2, -7, -9, 0, 1, 1, -8, -6, 6, 5, 5, 1, 2, 3, 1], [-12, -4, -8, 1, -1, 4, -4, 0, -4, 4, 6, 6, 0, 0, 6, 6], [-3, -2, 7, 5, 6, 2, -2, 2, 4, -8, -2, -2, 1, 0, -2, 0], [-2, 0, -4, -8, -2, 2, 0, 2, 0, 4, 2, 2, 2, 0, 2, 0], [12, -1, 5, -2, 9, 0, 4, -8, 5, -1, -5, -3, 0, -1, -1, -2], [-5, 3, -2, -5, 4, -3, 0, 7, -1, 3, -1, -3, -2, -1, 2, -3], [-4, 5, 1, -6, -6, -1, 1, -7, 0, 1, 0, 0, -2, 1, -5, -2], [11, 0, 5, 5, -5, -1, 1, 0, 4, -4, -3, -3, 0, 0, 0, 0], [8, 2, 10, -6, 7, -7, 1, 2, -5, -5, -6, -7, 4, -2, -4, -7], [-5, 0, -5, -1, -6, 7, 2, 6, 2, 2, 2, 1, -5, -2, 1, 4], [0, -4, 0, -6, 1, 3, -3, 4, -1, -3, 2, 3, 2, 0, -2, 3], [2, 1, -3, 7, -1, -3, 0, 4, -1, 3, 1, -1, -4, -1, 1, -4], [2, 1, 1, 1, 6, 0, 1, -6, 3, 1, -4, -4, -1, -3, -1, 0], [2, 0, -2, 6, -6, 4, -4, 0, 0, 0, 2, 2, 0, 0, 2, 2], [3, 1, -2, -4, 6, 1, -3, -9, -1, -1, 2, -1, -3, -2, 1, 3], [2, -1, 9, -4, 13, -3, 1, 6, 3, -1, -2, -5, 3, -1, -4, -4], [3, 4, 2, -3, 3, -1, 2, 3, 0, 4, -3, 1, 1, 4, -3, -3], [0, 0, 0, 2, 2, 1, 1, -4, -4, -4, 1, 5, 0, 6, -3, 3], [12, -1, 1, 1, 0, 0, 0, 12, 1, -1, 0, -6, 0, 1, 0, -1], [6, 2, -3, 8, -9, -1, -2, 9, 2, 1, 3, -1, 0, 1, 2, 0], [-7, 0, -7, -5, -2, 3, -1, 8, 2, 4, 3, 3, -3, 2, 3, 3], [-4, 2, -4, 1, 5, 0, -8, 2, -4, 2, 7, 3, 0, 2, -1, 1], [-12, 1, -5, 13, 8, 2, -1, -12, -1, -1, 0, 2, -1, -1, 1, 2], [-10, -7, 4, -2, -8, -1, 0, -5, 3, -6, -6, -2, 0, -2, 1, 2], [-12, -4, 0, 7, -7, 6, -6, 0, 8, -8, 1, 1, 0, 0, 3, 3], [-14, 2, -6, -5, 6, 0, 0, 9, 2, 7, 2, 0, -4, -2, 4, 0], [-8, 0, -8, 16, 8, 3, -5, 0, 0, 0, 3, -1, -8, -6, 5, 3], [-8, 0, 4, -7, -9, 0, -1, 9, 6, 0, -1, -2, -5, -2, 1, 0], [-7, -4, 1, -5, 2, 4, -2, -2, 8, 0, 0, 2, -3, -2, 2, 0], [0, 0, 0, 4, 4, 0, 0, 6, -1, -1, -3, -1, 0, -4, -1, 1], [-2, 0, -10, 5, 7, 0, -4, -8, -10, 0, 5, 5, 0, 0, 1, 5], [-3, -3, 1, -4, -1, 9, 0, -1, 3, -1, 1, 1, -5, 3, 1, 4], [2, 1, 3, 5, 0, -6, 1, 6, -3, -1, -2, 0, 5, 1, -3, -2], [7, -2, 3, 10, -10, -4, 4, 0, 0, 0, -2, -2, 0, 0, -3, -3], [-12, 6, -4, -1, 1, -4, 0, -14, -6, 4, -1, -1, -2, -6, -1, 1], [1, 5, -5, 0, 9, 1, -2, -1, -3, 5, 3, 5, -1, 5, 5, 0], [-12, 0, -12, 17, 5, 11, -1, -6, 5, 5, 9, 3, -12, 0, 8, 4], [0, 1, -6, 2, 1, -1, -6, -2, -7, 3, -1, -1, 0, -1, 0, -1], [-13, 2, -7, 9, -2, 6, 2, 4, 2, 4, 2, 6, -5, 2, 4, 2], [-5, -8, 3, 3, -3, 1, -1, 0, 6, -6, -3, -3, 0, 0, 0, 0], [-3, -10, -1, -2, 5, 7, -5, 5, 7, -7, 4, 1, -3, 1, 5, 10], [1, 4, -3, -2, 2, 7, -4, -2, 4, 2, 6, 1, -6, 0, 2, 1], [12, -1, 7, -3, 11, -4, 1, -6, -1, -1, -8, -4, 4, -1, -10, -3], [-11, -9, -7, 5, -11, 6, -3, -8, 6, -3, 5, 1, -7, 1, 6, 9], [-4, -5, -5, 1, -2, 6, -5, 2, -1, 1, -2, -2, -1, -7, 5, 0], [16, 6, 8, -9, 9, -1, 1, 0, -1, 1, -3, -3, 0, 0, -6, -6], [10, -2, 6, -2, 7, -8, 4, -1, -2, -3, -7, -15, 6, -6, -5, -3], [4, 0, 4, 0, 4, -5, -1, -6, 1, 1, -4, -2, 4, -2, -3, -1], [-3, -3, -7, 5, 13, 0, -11, -7, -7, -3, 8, 2, 0, -3, 4, 5], [9, 3, 4, 4, 10, -11, 1, -7, -7, -5, -6, -9, 1, -5, -9, 1], [4, -2, -4, 6, 3, 0, -2, -3, -5, -2, 8, 8, 4, 6, 2, 0], [-2, 5, 4, -7, 1, -6, 5, -1, -4, -1, -5, -5, 2, 0, -5, 0], [8, -3, 7, -7, 0, -2, -1, 4, -1, -3, 0, -2, 3, -1, -3, -2], [4, 0, 4, -4, 0, -1, 3, 0, 2, 2, -3, -3, 4, -2, 2, -6], [-2, -3, 4, -2, -20, 6, 4, -12, 9, -1, -1, 2, -1, 5, 4, 0], [2, 2, -5, -5, -2, 5, -2, -1, -2, 3, 0, 10, 0, 3, 3, 5], [2, 0, 2, -8, -6, 2, 4, 0, 4, 4, 6, 4, 2, 12, 2, -4], [-13, 2, -2, 0, -6, 4, 6, 13, 6, 2, 4, 2, -4, 2, 0, 0], [-2, 3, -5, 10, -1, 0, 0, 0, -3, 5, 1, 8, -5, -3, 1, 3], [12, -1, 15, -6, 0, -10, 2, -4, -1, -3, -7, -8, 10, 1, -6, -8], [-10, -1, -1, -6, -2, 2, -4, 2, -3, 1, 3, 0, -4, -1, 4, 0], [2, -5, 3, -7, -1, 7, 0, 4, 5, -3, 1, 5, -2, 5, 1, 8], [2, 0, 2, -3, -1, -7, -5, -8, -8, -8, 4, -6, 2, 0, -1, -1], [1, 0, 1, -9, -8, -2, -1, -10, -2, -2, -5, 0, 1, -4, 0, -1], [-4, -6, 0, -5, -3, 0, 0, 2, 6, 0, 3, -3, 0, 6, 3, -3], [6, -1, 3, -3, 8, -6, -1, 4, -3, -1, -2, -4, 3, -1, -3, -2], [11, -1, 7, -3, 12, -3, -2, -5, 1, -1, 2, -4, 3, -1, 2, -3], [5, -5, 1, -14, 4, 4, 1, -8, 5, 0, -1, 1, 1, 1, 4, -1], [4, 3, -3, -9, 6, 2, 5, 16, -1, 11, 0, 8, 3, 5, 5, 0], [-6, -9, 1, 4, 4, 1, -1, 5, 7, -6, -5, 1, -4, 7, 2, 3], [9, 0, 5, -4, 4, -9, 6, 0, -4, 0, -6, -5, 8, 0, -6, -5], [20, 2, 3, -3, -16, 0, 3, 1, -1, -2, -4, 0, 2, -3, -2, 0], [0, 0, 0, 6, 6, 1, 1, 4, 0, 0, 5, 1, 0, 6, -1, 1], [-8, -2, -5, 3, 4, 0, -5, -1, -1, -4, 4, 0, -2, 3, 2, 0], [-10, 2, 0, -12, 7, 2, 2, -7, -4, 4, 8, 8, -2, 10, -2, 0], [6, 1, 4, 6, 4, -8, 7, 0, -4, -2, -9, -6, 0, -2, -10, -2], [-14, -4, -4, -1, -4, 1, 0, -6, 4, 4, 4, -3, -3, 4, 4, 1], [2, 0, 2, 9, 11, 1, 3, -6, 3, 3, -1, 3, 2, 4, -2, 0], [2, 6, 0, -7, -6, 4, 6, 5, 4, 8, 2, -4, 0, 2, -4, -8], [-8, 2, -7, -3, 2, 2, -5, -2, -4, 4, 2, -5, -2, -3, 5, 0], [15, 4, 1, -11, 11, -5, 5, 0, -2, 2, -1, -1, 0, 0, -2, -2], [-6, 2, -5, -6, 5, 7, 0, -5, 2, 9, 5, 3, -4, 5, 0, 0], [11, -2, 7, 6, -4, -3, 0, -8, 0, -6, 0, -3, 2, 6, -2, -3], [-6, 5, -8, 0, -4, -5, 6, -9, -13, 6, 0, -2, -2, -2, -5, 2], [5, 8, 3, -3, 3, -5, 5, 0, 0, 0, -5, -5, 0, 0, -4, -4], [-16, 1, -7, -1, 5, -1, -1, 7, 4, 3, 5, -1, -6, -7, 6, -1], [-1, -8, 10, -1, 6, 3, 2, 3, 16, -8, -6, -9, -3, -8, 0, -1], [0, -7, 6, -8, -2, 2, 1, 8, 8, -6, 7, 2, 0, 4, 0, 2], [-14, 4, -12, 11, -10, 9, -1, -8, 1, 1, 5, 6, -10, -2, 3, 4], [-4, 6, -2, 6, -6, -1, 1, 0, -4, 4, 3, 3, 0, 0, 2, 2], [8, 3, -7, 2, -3, -3, 3, -12, -13, 3, -2, 5, 3, 3, -2, 2], [6, -2, -4, 3, -5, 4, 0, -12, 4, -2, -3, 1, -4, -2, 1, 3], [7, 2, 10, -10, 6, -9, 0, -5, -2, -10, -6, -8, 3, -2, -6, 2], [-12, -1, -4, -11, -13, 8, -1, 13, -2, 7, 9, 9, -4, 8, 1, 0], [6, 3, -1, -4, 22, -6, -2, 18, -9, 5, 1, 0, 6, -3, -2, 0], [-9, -4, 8, -15, 17, 2, 1, 10, 0, -4, -1, 2, 5, 2, -5, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_612_w_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_612_2_w_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_612_2_w_c(:prec:=16)
    chi := MakeCharacter_612_w();
    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_612_2_w_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_612_2_w_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_612_w();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 2512, 461560, -216880, 363976, -123784, 126468, -54720, 6565, -2504, 1344, 296, -120, -32, 28, -8, 1]>],Snew);
    return Vf;
end function;