
// Make newform 1200.3.c.a in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_c_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_1200_3_c_a();".
// 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_1200_3_c_a();".
// 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];
        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_1200_c();"
function MakeCharacter_1200_c()
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    v := [2, 2, 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_1200_c_Hecke();"
function MakeCharacter_1200_c_Hecke(Kf)
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    char_values := [[1, 0], [1, 0], [-1, 0], [-1, 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 := 3;
    raw_aps := [[0, 0], [0, 3], [0, 0], [0, 11], [0, 0], [0, -1], [0, 0], [-37, 0], [0, 0], [0, 0], [13, 0], [0, -26], [0, 0], [0, 61], [0, 0], [0, 0], [0, 0], [47, 0], [0, -109], [0, 0], [0, -46], [-142, 0], [0, 0], [0, 0], [0, 169], [0, 0], [0, -194], [0, 0], [-143, 0], [0, 0], [0, 146], [0, 0], [0, 0], [-22, 0], [0, 0], [-227, 0], [0, -311], [0, -299], [0, 0], [0, 0], [0, 0], [-313, 0], [0, 0], [0, 239], [0, 0], [-277, 0], [253, 0], [0, 421], [0, 0], [-383, 0], [0, 0], [0, 0], [-193, 0], [0, 0], [0, 0], [0, 0], [0, 0], [-242, 0], [0, 529], [0, 0], [0, -59], [0, 0], [0, 611], [0, 0], [0, 599], [0, 0], [-362, 0], [0, 649], [0, 0], [502, 0], [0, 0], [0, 0], [0, 491], [0, -121], [83, 0], [0, 0], [0, 0], [0, -431], [0, 0], [-143, 0], [0, 0], [-358, 0], [0, 0], [0, 359], [803, 0], [0, 0], [0, 0], [0, 814], [0, 0], [0, 526], [0, 0], [0, 0], [0, -349], [0, 0], [-877, 0], [0, 0], [0, 0], [0, 0], [0, -179], [-793, 0], [0, 506], [0, 0], [0, 0], [0, 0], [-1067, 0], [0, -71], [0, 0], [0, 0], [0, 0], [-673, 0], [0, -814], [0, -1126], [0, 0], [1163, 0], [-587, 0], [0, 0], [0, -314], [0, 0], [0, 0], [0, 0], [122, 0], [0, 1154], [0, 0], [0, 0], [1318, 0], [0, 0], [457, 0], [0, 0], [0, -1429], [0, 1034], [-1222, 0], [0, 0], [-1202, 0], [0, -1511], [0, 0], [-863, 0], [0, 0], [0, -949], [0, 0], [0, 0], [253, 0], [0, 0], [0, 1621], [0, 0], [-458, 0], [0, 0], [0, -481], [0, 0], [1418, 0], [0, 0], [0, 1129], [0, 0], [0, -1259], [0, 0], [0, -214], [0, 0], [-1837, 0], [0, 0], [0, 649], [0, 0], [0, 0], [0, 0], [0, -1534], [0, 0], [0, 0], [0, 0], [1693, 0], [0, 1894], [1657, 0], [0, 0], [0, 0], [167, 0], [0, 0], [0, -2041], [1403, 0], [0, 0], [1573, 0], [0, 0], [0, -2099], [-263, 0], [0, 851], [0, 0], [0, 2039], [0, 0], [0, 0], [0, 0], [0, -1991], [0, 1021], [1417, 0], [0, 0], [0, 719], [0, 0], [-467, 0], [0, 0], [0, 0], [0, 0], [-2398, 0], [0, 74], [0, 0], [0, 0], [0, 0], [-2162, 0], [0, -2426], [2302, 0], [0, 0], [0, 0], [2483, 0], [0, 0], [0, 0], [-1907, 0], [0, 1969], [0, 0], [0, -419], [0, 0], [0, 0], [-2158, 0], [0, 1571], [0, 0], [0, 0], [0, 0], [-913, 0], [923, 0], [0, 0], [0, 2446], [0, 0], [-2183, 0], [0, 0], [0, 0], [0, -1669], [0, 0], [0, 1319], [218, 0], [733, 0], [0, 0], [0, 1366], [0, 0], [1822, 0], [0, 0], [0, 0], [0, 0], [0, 0], [3013, 0], [0, 2461], [2977, 0], [0, 0], [0, 0], [0, -1909], [0, 0], [3083, 0], [0, 0], [0, 694], [0, 0], [0, 0], [2857, 0], [0, 0], [0, 0], [-433, 0], [0, 3146], [0, 0], [0, -1586], [0, 3301], [0, 0], [-2138, 0], [0, -2161], [0, 0], [698, 0], [0, 0], [0, 0], [0, 1846], [0, 0], [3407, 0], [0, 2906], [0, -3406], [-2557, 0], [0, -671], [0, 541], [0, 0], [97, 0], [1727, 0], [0, 0], [0, 0], [-962, 0], [0, 0], [-2353, 0], [0, 3491], [0, 0], [0, 479], [0, 0], [-3742, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 3359], [0, 0], [-2027, 0], [0, 0], [0, 0], [0, -2374], [0, 914], [0, 0], [3923, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_c_a();".
// 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_1200_3_c_a(:prec:=2)
    chi := MakeCharacter_1200_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1200_3_c_a();".
// 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_1200_3_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![121, 0, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;