
// Make newform 1200.3.l.l in Magma, downloaded from the LMFDB on 11 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_l_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_l_l();".
// 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_l_l();".
// 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_l();"
function MakeCharacter_1200_l()
    N := 1200;
    order := 2;
    char_gens := [751, 901, 401, 577];
    v := [2, 2, 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_1200_l_Hecke();"
function MakeCharacter_1200_l_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]];
    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, 0], [0, 0], [0, 0], [0, 14], [-22, 0], [0, -34], [0, 0], [-2, 0], [0, 0], [0, 0], [0, 0], [0, -14], [0, -86], [0, 0], [-118, 0], [0, 0], [0, 0], [0, 0], [98, 0], [0, -154], [0, 0], [0, 0], [0, 0], [0, 0], [0, 106], [22, 0], [0, -206], [0, 0], [0, 0], [0, -226], [-262, 0], [0, 0], [238, 0], [0, 0], [0, 0], [0, -254], [0, 154], [0, 0], [122, 0], [0, 0], [0, 0], [0, 374], [-142, 0], [-362, 0], [0, 0], [0, -134], [-218, 0], [0, 34], [0, 0], [-478, 0], [0, 0], [0, -466], [0, 446], [0, 0], [-482, 0], [0, 0], [0, 0], [0, 0], [0, 394], [0, 0], [0, 0], [0, 0], [0, 134], [-122, 0], [0, 0], [0, 586], [-458, 0], [0, 274], [0, 0], [0, 0], [0, 0], [-742, 0], [0, 686], [0, 0], [0, 0], [0, 0], [142, 0], [0, 0], [602, 0], [0, 0], [0, 0], [-622, 0], [0, 566], [0, 0], [0, 0], [0, 0], [0, 0], [0, 346], [0, 0], [0, 0], [0, 0], [938, 0], [0, -994], [0, 0], [0, 0], [0, 0], [-1078, 0], [0, 0], [0, 614], [0, -154], [0, 0], [358, 0], [0, 0], [0, -854], [0, -1166], [0, 0], [242, 0], [0, 0], [0, 0], [0, -1186], [698, 0], [238, 0], [0, 0], [0, 0], [0, 706], [0, 1114], [0, 0], [-838, 0], [0, 0], [0, 374], [0, 86], [-1322, 0], [0, 0], [742, 0], [0, 0], [0, 0], [0, 0], [-1462, 0], [0, -514], [1438, 0], [0, 0], [0, 0], [-578, 0], [0, -1526], [0, 0], [0, -826], [0, 0], [-1082, 0], [0, 0], [0, 0], [0, -374], [502, 0], [0, 0], [0, 0], [0, -1666], [218, 0], [0, -274], [0, 0], [0, 0], [0, 0], [0, -1694], [0, 0], [0, 0], [1298, 0], [0, 0], [0, 0], [0, 0], [0, -1574], [0, 1474], [0, 0], [0, 0], [0, 1934], [0, -1954], [958, 0], [0, 0], [1822, 0], [0, -1046], [0, 0], [1802, 0], [0, 0], [0, 0], [2018, 0], [0, 0], [-602, 0], [0, 0], [0, 0], [22, 0], [0, 0], [0, 0], [0, 0], [0, 1694], [0, 2126], [0, 0], [0, 0], [0, 0], [1582, 0], [0, 0], [0, 0], [0, 2006], [-2282, 0], [0, 0], [0, -1094], [0, 34], [1442, 0], [0, 0], [0, 1454], [0, -1474], [0, 0], [2398, 0], [0, 0], [1342, 0], [0, 0], [0, -826], [-2302, 0], [0, -2554], [0, 0], [-2042, 0], [0, 0], [0, 0], [0, 0], [0, 2506], [0, 0], [-1198, 0], [0, 0], [0, 0], [0, -734], [0, -326], [602, 0], [1298, 0], [0, 0], [0, 0], [0, 2266], [-2618, 0], [0, 514], [0, 0], [0, 0], [0, 0], [0, 0], [2858, 0], [1918, 0], [0, 0], [0, 0], [0, 946], [862, 0], [0, 2794], [0, 0], [0, 0], [0, -2074], [-1562, 0], [0, 0], [2902, 0], [0, 2674], [0, 0], [0, 0], [0, 0], [2618, 0], [0, 1166], [0, 0], [0, 0], [0, -3134], [-2258, 0], [0, 154], [0, 0], [-2758, 0], [0, 0], [0, -2506], [0, 0], [0, 0], [0, -3014], [2662, 0], [0, 0], [0, 2414], [-1462, 0], [0, 0], [0, 0], [0, 0], [0, -3446], [3242, 0], [0, 0], [0, 0], [578, 0], [0, 0], [0, 0], [0, 1546], [2422, 0], [-238, 0], [0, 0], [0, 1646], [3598, 0], [0, 226], [-2278, 0], [0, 0], [0, 0], [0, 0], [0, -2026], [-3502, 0], [0, 0], [0, 0], [0, -2534], [0, 3394], [0, 0], [0, 0], [0, 0], [-3842, 0], [0, -2966], [0, 0], [0, 0], [0, 0], [0, -3226], [-862, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_3_l_l();".
// 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_l_l(:prec:=2)
    chi := MakeCharacter_1200_l();
    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_l_l();".
// 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_l_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![0, 1]>,<11,R![0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;