
// Make newform 1200.3.c.h in Magma, downloaded from the LMFDB on 25 April 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_h();".
// 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_h();".
// 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, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 2, 0, 2], [0, 2, 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_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, 0, 0], [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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, -1, 0, 1], [0, 0, 0, 0], [0, 7, 0, 0], [0, 0, -3, 0], [0, 25, 0, 0], [0, 0, 0, 9], [-7, 0, 0, 0], [0, 0, 0, -9], [0, 0, 15, 0], [7, 0, 0, 0], [0, 2, 0, 0], [0, 0, -3, 0], [0, 41, 0, 0], [0, 0, 0, 0], [0, 0, 0, 21], [0, 0, 12, 0], [-1, 0, 0, 0], [0, -17, 0, 0], [0, 0, 15, 0], [0, 70, 0, 0], [-58, 0, 0, 0], [0, 0, 0, 42], [0, 0, 48, 0], [0, -49, 0, 0], [0, 0, -21, 0], [0, -154, 0, 0], [0, 0, 0, 63], [25, 0, 0, 0], [0, 0, 0, -6], [0, 34, 0, 0], [0, 0, -69, 0], [0, 0, 0, -42], [-154, 0, 0, 0], [0, 0, -54, 0], [199, 0, 0, 0], [0, -145, 0, 0], [0, 161, 0, 0], [0, 0, 0, -39], [0, 0, 0, 63], [0, 0, 42, 0], [-217, 0, 0, 0], [0, 0, -21, 0], [0, 25, 0, 0], [0, 0, 0, 48], [-103, 0, 0, 0], [7, 0, 0, 0], [0, 161, 0, 0], [0, 0, 0, 21], [97, 0, 0, 0], [0, 0, 0, -93], [0, 0, -21, 0], [119, 0, 0, 0], [0, 0, 102, 0], [0, 0, 0, 42], [0, 0, 0, -3], [0, 0, 21, 0], [-470, 0, 0, 0], [0, -217, 0, 0], [0, 0, -183, 0], [0, 65, 0, 0], [0, 0, 0, 60], [0, -521, 0, 0], [0, 0, -12, 0], [0, -119, 0, 0], [0, 0, 0, 54], [418, 0, 0, 0], [0, -553, 0, 0], [0, 0, 0, -228], [-266, 0, 0, 0], [0, 0, 0, -147], [0, 0, -117, 0], [0, 103, 0, 0], [0, -359, 0, 0], [377, 0, 0, 0], [0, 0, 0, -216], [0, 0, -123, 0], [0, 239, 0, 0], [0, 0, 33, 0], [-455, 0, 0, 0], [0, 0, 105, 0], [-526, 0, 0, 0], [0, 0, 99, 0], [0, -119, 0, 0], [-727, 0, 0, 0], [0, 0, 0, 186], [0, 0, -90, 0], [0, -310, 0, 0], [0, 0, 255, 0], [0, -730, 0, 0], [0, 0, 0, -69], [0, 0, -138, 0], [0, -473, 0, 0], [0, 0, -288, 0], [-175, 0, 0, 0], [0, 0, 0, 123], [0, 0, -258, 0], [0, 0, -201, 0], [0, -175, 0, 0], [863, 0, 0, 0], [0, 778, 0, 0], [0, 0, 0, 150], [0, 0, 0, 135], [0, 0, -135, 0], [535, 0, 0, 0], [0, -49, 0, 0], [0, 0, 0, 147], [0, 0, 0, -273], [0, 0, 228, 0], [455, 0, 0, 0], [0, -566, 0, 0], [0, -578, 0, 0], [0, 0, 0, 225], [593, 0, 0, 0], [559, 0, 0, 0], [0, 0, -192, 0], [0, 854, 0, 0], [0, 0, 0, 114], [0, 0, 0, 126], [0, 0, 315, 0], [-910, 0, 0, 0], [0, 742, 0, 0], [0, 0, 0, -153], [0, 0, 0, -234], [-302, 0, 0, 0], [0, 0, -63, 0], [-95, 0, 0, 0], [0, 0, -309, 0], [0, 871, 0, 0], [0, 406, 0, 0], [830, 0, 0, 0], [0, 0, 0, 462], [-350, 0, 0, 0], [0, -265, 0, 0], [0, 0, -378, 0], [529, 0, 0, 0], [0, 0, 0, 357], [0, 1519, 0, 0], [0, 0, 0, 186], [0, 0, 291, 0], [319, 0, 0, 0], [0, 0, -42, 0], [0, -1375, 0, 0], [0, 0, 0, -99], [142, 0, 0, 0], [0, 0, 441, 0], [0, 1057, 0, 0], [0, 0, 0, 222], [-946, 0, 0, 0], [0, 0, 0, 138], [0, 1463, 0, 0], [0, 0, 378, 0], [0, 1289, 0, 0], [0, 0, 0, 456], [0, -14, 0, 0], [0, 0, -246, 0], [-1423, 0, 0, 0], [0, 0, 147, 0], [0, 1655, 0, 0], [0, 0, -588, 0], [0, 0, 0, 147], [0, 0, 0, 594], [0, 1162, 0, 0], [0, 0, -252, 0], [0, 0, 0, -318], [0, 0, 0, 0], [535, 0, 0, 0], [0, 1274, 0, 0], [265, 0, 0, 0], [0, 0, 0, 78], [0, 0, -333, 0], [-553, 0, 0, 0], [0, 0, 138, 0], [0, -1127, 0, 0], [1505, 0, 0, 0], [0, 0, -648, 0], [631, 0, 0, 0], [0, 0, 21, 0], [0, -535, 0, 0], [1561, 0, 0, 0], [0, 1447, 0, 0], [0, 0, -546, 0], [0, -743, 0, 0], [0, 0, 0, 246], [0, 0, 0, 402], [0, 0, -114, 0], [0, -745, 0, 0], [0, 569, 0, 0], [-215, 0, 0, 0], [0, 0, 105, 0], [0, 1609, 0, 0], [0, 0, 0, -351], [919, 0, 0, 0], [0, 0, -153, 0], [0, 0, 0, 375], [0, 0, 0, -642], [1754, 0, 0, 0], [0, -1706, 0, 0], [0, 0, 0, 546], [0, 0, 0, -399], [0, 0, 441, 0], [-374, 0, 0, 0], [0, -910, 0, 0], [310, 0, 0, 0], [0, 0, -528, 0], [0, 0, 0, -621], [1217, 0, 0, 0], [0, 0, 0, 273], [0, 0, -246, 0], [1255, 0, 0, 0], [0, 1367, 0, 0], [0, 0, -402, 0], [0, 113, 0, 0], [0, 0, 0, 84], [0, 0, -420, 0], [986, 0, 0, 0], [0, 1495, 0, 0], [0, 0, -63, 0], [0, 0, 0, 75], [0, 0, 0, -951], [1367, 0, 0, 0], [-823, 0, 0, 0], [0, 0, -450, 0], [0, 1622, 0, 0], [0, 0, 0, 417], [217, 0, 0, 0], [0, 0, 0, 315], [0, 0, -648, 0], [0, -1217, 0, 0], [0, 0, 705, 0], [0, -1823, 0, 0], [-322, 0, 0, 0], [-2681, 0, 0, 0], [0, 0, 528, 0], [0, 158, 0, 0], [0, 0, 0, -294], [-2906, 0, 0, 0], [0, 0, 0, -285], [0, 0, -846, 0], [0, 0, -609, 0], [0, 0, 0, -633], [-281, 0, 0, 0], [0, -1519, 0, 0], [1825, 0, 0, 0], [0, 0, 0, 570], [0, 0, -594, 0], [0, 1375, 0, 0], [0, 0, 387, 0], [-2023, 0, 0, 0], [0, 0, 0, -1008], [0, 530, 0, 0], [0, 0, 393, 0], [0, 0, 0, 42], [1201, 0, 0, 0], [0, 0, 0, 483], [0, 0, 990, 0], [1823, 0, 0, 0], [0, -590, 0, 0], [0, 0, 0, 777], [0, -1582, 0, 0], [0, -2407, 0, 0], [0, 0, 0, -153], [-98, 0, 0, 0], [0, 1225, 0, 0], [0, 0, 0, -504], [14, 0, 0, 0], [0, 0, 174, 0], [0, 0, 504, 0], [0, -1954, 0, 0], [0, 0, 0, -612], [-1561, 0, 0, 0], [0, 2338, 0, 0], [0, -266, 0, 0], [-2791, 0, 0, 0], [0, -25, 0, 0], [0, -463, 0, 0], [0, 0, 0, 348], [-791, 0, 0, 0], [-3505, 0, 0, 0], [0, 0, -258, 0], [0, 0, 0, -525], [1162, 0, 0, 0], [0, 0, 0, 879], [-1417, 0, 0, 0], [0, 847, 0, 0], [0, 0, 492, 0], [0, -95, 0, 0], [0, 0, 0, 267], [14, 0, 0, 0], [0, 0, -795, 0], [0, 0, -228, 0], [0, 0, 0, 1176], [0, 0, 0, -180], [0, 0, 798, 0], [0, -2927, 0, 0], [0, 0, 471, 0], [463, 0, 0, 0], [0, 0, 0, -879], [0, 0, -867, 0], [0, 2146, 0, 0], [0, -1106, 0, 0], [0, 0, 0, -1302], [641, 0, 0, 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_h();".
// 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_h(:prec:=4)
    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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![49, 0, 1]>,<11,R![72, 0, 1]>],Snew);
    return Vf;
end function;