
// Make newform 63.3.m.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_63_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_63_m_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_63_3_m_b();".
// 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_63_3_m_b();".
// 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, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_63_m();"
function MakeCharacter_63_m()
    N := 63;
    order := 6;
    char_gens := [29, 10];
    v := [6, 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_63_m_Hecke();"
function MakeCharacter_63_m_Hecke(Kf)
    N := 63;
    order := 6;
    char_gens := [29, 10];
    char_values := [[1, 0], [0, 1]];
    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 := 3;
    raw_aps := [[0, 0], [0, 0], [0, 0], [8, -3], [0, 0], [-7, 14], [0, 0], [-42, 21], [0, 0], [0, 0], [35, 35], [0, -73], [0, 0], [61, 0], [0, 0], [0, 0], [0, 0], [-112, 56], [13, -13], [0, 0], [-63, -63], [0, -11], [0, 0], [0, 0], [112, -224], [0, 0], [154, -77], [0, 0], [71, -71], [0, 0], [107, 0], [0, 0], [0, 0], [91, -182], [0, 0], [-286, 286], [-168, -168], [0, 262], [0, 0], [0, 0], [0, 0], [-105, 210], [0, 0], [239, -239], [0, 0], [-56, -56], [166, 0], [168, -336], [0, 0], [-238, 119], [0, 0], [0, 0], [224, 224], [0, 0], [0, 0], [0, 0], [0, 0], [-560, 280], [-407, 407], [0, 0], [133, 133], [0, 0], [35, -70], [0, 0], [210, -105], [0, 0], [0, -661], [649, 0], [0, 0], [-280, 560], [0, 0], [0, 0], [315, 315], [0, 577], [-611, 0], [0, 0], [0, 0], [-770, 385], [0, 0], [-161, -161], [0, 0], [-481, 0], [0, 0], [-455, 910], [1008, -504], [0, 0], [0, 0], [0, 767], [0, 0], [-397, 0], [0, 0], [0, 0], [-349, 349], [0, 0], [0, 851], [0, 0], [0, 0], [0, 0], [1190, -595], [0, 241], [506, 0], [0, 0], [0, 0], [0, 0], [181, -181], [-665, -665], [0, 0], [0, 0], [0, 0], [-49, 98], [-294, 147], [1126, -1126], [0, 0], [245, 245], [674, 0], [0, 0], [-203, 406], [0, 0], [0, 0], [0, 0], [-441, -441], [-23, 0], [0, 0], [0, 0], [1078, -539], [0, 0], [0, -934], [0, 0], [637, -1274], [-434, 217], [-1331, 1331], [0, 0], [0, 179], [838, 0], [0, 0], [-735, 1470], [0, 0], [112, 112], [0, 0], [0, 0], [-336, 672], [0, 0], [-1058, 1058], [0, 0], [231, 231], [0, 0], [-945, 1890], [0, 0], [1120, -560], [0, 0], [0, 598], [0, 0], [-443, 0], [0, 0], [1667, -1667], [0, 0], [0, -1837], [0, 0], [1015, -2030], [0, 0], [0, 0], [0, 0], [1787, 0], [0, 0], [0, 0], [0, 0], [-1693, 1693], [-1127, -1127]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_63_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_63_3_m_b();".
// 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_63_3_m_b(:prec:=2)
    chi := MakeCharacter_63_m();
    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_63_3_m_b();".
// 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_63_3_m_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_63_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;