
// Make newform 768.5.b.c in Magma, downloaded from the LMFDB on 01 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_768_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_768_b_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_768_5_b_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_768_5_b_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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, -1], [-2, 0, 4, 0], [0, 0, 0, 2]];
    Rf_basisdens := [1, 1, 1, 1];
    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_768_b();"
function MakeCharacter_768_b()
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    v := [1, 1, 2];
    // 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_768_b_Hecke();"
function MakeCharacter_768_b_Hecke(Kf)
    N := 768;
    order := 2;
    char_gens := [511, 517, 257];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0], [1, 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+1)]];
    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 := 5;
    raw_aps := [[0, 0, 0, 0], [0, 3, 0, 0], [0, 0, 0, 21], [0, 0, -22, 0], [0, -12, 0, 0], [0, 0, 0, -91], [-246, 0, 0, 0], [0, 68, 0, 0], [0, 0, 216, 0], [0, 0, 0, 39], [0, 0, 426, 0], [0, 0, 0, -265], [918, 0, 0, 0], [0, -492, 0, 0], [0, 0, 1092, 0], [0, 0, 0, 2313], [0, -132, 0, 0], [0, 0, 0, 673], [0, -628, 0, 0], [0, 0, 528, 0], [926, 0, 0, 0], [0, 0, -1270, 0], [0, -6924, 0, 0], [-11586, 0, 0, 0], [-13118, 0, 0, 0], [0, 0, 0, 2745], [0, 0, 1646, 0], [0, 5820, 0, 0], [0, 0, 0, -8083], [1842, 0, 0, 0], [0, 0, -114, 0], [0, 204, 0, 0], [13254, 0, 0, 0], [0, 7628, 0, 0], [0, 0, 0, -219], [0, 0, 8098, 0], [0, 0, 0, 9673], [0, 20932, 0, 0], [0, 0, 5412, 0], [0, 0, 0, -17205], [0, -9732, 0, 0], [0, 0, 0, -7853], [0, 0, 768, 0], [-26782, 0, 0, 0], [0, 0, 0, 26241], [0, 0, 6662, 0], [0, -13796, 0, 0], [0, 0, 246, 0], [0, -43980, 0, 0], [0, 0, 0, 24235], [-48738, 0, 0, 0], [0, 0, 20496, 0], [73138, 0, 0, 0], [0, -53052, 0, 0], [-48894, 0, 0, 0], [0, 0, -22572, 0], [0, 0, 0, -35769], [0, 0, -31234, 0], [0, 0, 0, 60259], [3054, 0, 0, 0], [0, 76764, 0, 0], [0, 0, 0, -75831], [0, 3132, 0, 0], [0, 0, 40740, 0], [128686, 0, 0, 0], [0, 0, 0, -36993], [0, 32924, 0, 0], [98674, 0, 0, 0], [0, 32868, 0, 0], [0, 0, 0, 90673], [-4302, 0, 0, 0], [0, 0, 53472, 0], [0, 0, 52738, 0], [0, 0, 0, -75889], [0, 20788, 0, 0], [0, 0, -18864, 0], [0, 0, 0, -52875], [0, 0, 0, -13967], [237882, 0, 0, 0], [20270, 0, 0, 0], [0, 13908, 0, 0], [0, 0, 0, -58349], [0, 0, 102636, 0], [-199726, 0, 0, 0], [0, 0, -42282, 0], [0, 29124, 0, 0], [149994, 0, 0, 0], [-284338, 0, 0, 0], [0, 0, 0, -91701], [0, 0, 49774, 0], [0, 39684, 0, 0], [0, 0, -72156, 0], [0, 0, -78262, 0], [0, -131556, 0, 0], [0, 143676, 0, 0], [0, 0, -128904, 0], [0, 0, 0, -19665], [464598, 0, 0, 0], [0, -78236, 0, 0], [0, 0, 0, 180221], [0, 151060, 0, 0], [0, 0, 0, -116637], [0, -242316, 0, 0], [-470058, 0, 0, 0], [0, 184972, 0, 0], [-341038, 0, 0, 0], [0, -65892, 0, 0], [-96846, 0, 0, 0], [0, 0, 150048, 0], [627742, 0, 0, 0], [0, 0, -38450, 0], [0, 0, 0, -123601], [31758, 0, 0, 0], [0, 378796, 0, 0], [0, 0, -120590, 0], [-152214, 0, 0, 0], [0, 412308, 0, 0], [0, 0, 74928, 0], [0, 0, 0, -165357], [0, -146580, 0, 0], [0, 0, 0, 361079], [-552910, 0, 0, 0], [0, 0, 0, -304515], [0, -13692, 0, 0], [0, -249324, 0, 0], [0, 0, 0, 22479], [0, 0, 0, -266501], [0, 0, 84324, 0], [0, 0, 218178, 0], [0, 0, 0, -416491], [0, -403044, 0, 0], [0, 0, -133092, 0], [0, 0, 270506, 0], [0, 0, 0, -147893], [1026150, 0, 0, 0], [362306, 0, 0, 0], [0, 0, 0, -513039], [0, 490772, 0, 0], [0, 0, 0, 380739], [-247674, 0, 0, 0], [0, -531276, 0, 0], [0, 0, 0, 125121], [0, 0, 115690, 0], [0, -9972, 0, 0], [0, 0, 0, -7843], [0, 0, 33336, 0], [0, 0, 0, -172969], [267990, 0, 0, 0], [0, -301612, 0, 0], [0, 0, 238704, 0], [0, 0, 0, 558145], [19170, 0, 0, 0], [0, -328460, 0, 0], [0, 0, 314700, 0], [0, 528964, 0, 0], [0, 0, -287376, 0], [0, 0, 28026, 0], [-1278822, 0, 0, 0], [981262, 0, 0, 0], [0, 0, 0, 142203], [0, 573372, 0, 0], [-602922, 0, 0, 0], [0, 0, 166222, 0], [0, -715452, 0, 0], [-1040742, 0, 0, 0], [0, 0, -273876, 0], [0, 0, 177826, 0], [0, 0, 0, 267935]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_768_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_768_5_b_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_768_5_b_c(:prec:=4)
    chi := MakeCharacter_768_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_768_5_b_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_768_5_b_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_768_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<5,R![1764, 0, 1]>,<11,R![-432, 0, 1]>],Snew);
    return Vf;
end function;