
// Make newform 1200.3.l.g in Magma, downloaded from the LMFDB on 25 April 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_g();".
// 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_g();".
// 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 := [5, 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+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], [-2, 1], [0, 0], [-6, 0], [0, -2], [-16, 0], [0, -2], [2, 0], [0, 6], [0, -14], [18, 0], [16, 0], [0, 28], [16, 0], [0, -22], [0, -2], [0, -2], [82, 0], [24, 0], [0, 56], [74, 0], [-138, 0], [0, -42], [0, 48], [166, 0], [0, 30], [26, 0], [0, 90], [38, 0], [0, -14], [-26, 0], [0, -6], [0, 54], [82, 0], [0, -50], [158, 0], [-164, 0], [236, 0], [0, 42], [0, -6], [0, -86], [2, 0], [0, 92], [214, 0], [0, 42], [242, 0], [-2, 0], [86, 0], [0, -26], [-282, 0], [0, 162], [0, 112], [262, 0], [0, -210], [0, -90], [0, -26], [0, -166], [-82, 0], [-24, 0], [0, -84], [-144, 0], [0, -210], [184, 0], [0, -72], [394, 0], [0, -202], [198, 0], [-394, 0], [0, -82], [-362, 0], [0, 138], [0, -132], [-186, 0], [44, 0], [362, 0], [0, -162], [0, 198], [-124, 0], [0, -120], [458, 0], [0, 266], [562, 0], [0, -156], [-226, 0], [2, 0], [0, 90], [0, 140], [-334, 0], [0, 42], [366, 0], [0, 202], [0, 264], [-886, 0], [0, 182], [2, 0], [0, -98], [0, -358], [0, -236], [376, 0], [-198, 0], [1024, 0], [0, -30], [0, 114], [0, 384], [-962, 0], [886, 0], [0, -294], [0, 50], [0, -100], [2, 0], [-506, 0], [-556, 0], [0, -42], [802, 0], [698, 0], [0, 408], [156, 0], [0, 338], [0, 218], [0, -182], [682, 0], [894, 0], [0, 246], [0, 198], [758, 0], [0, -350], [-2, 0], [0, -384], [674, 0], [-656, 0], [-598, 0], [0, 350], [338, 0], [656, 0], [0, 132], [-82, 0], [0, 474], [-536, 0], [0, 182], [0, -488], [558, 0], [0, 174], [-214, 0], [0, 14], [318, 0], [0, -28], [684, 0], [0, 670], [842, 0], [0, 454], [156, 0], [0, 56], [-964, 0], [0, 426], [1284, 0], [0, 28], [-418, 0], [0, 76], [-534, 0], [0, 58], [0, 678], [0, -182], [674, 0], [0, -594], [0, 166], [0, 198], [-962, 0], [-24, 0], [-1342, 0], [0, -454], [0, 726], [-278, 0], [0, 624], [-486, 0], [-1018, 0], [0, 0], [-1962, 0], [0, 702], [506, 0], [-842, 0], [-1906, 0], [0, 602], [1244, 0], [0, 302], [0, 702], [0, 422], [1236, 0], [-1124, 0], [-562, 0], [0, -380], [-1426, 0], [0, -406], [1878, 0], [0, -694], [0, -54], [0, 530], [262, 0], [-936, 0], [0, -882], [0, -554], [0, 786], [-702, 0], [-1944, 0], [1858, 0], [0, -42], [0, 306], [982, 0], [0, -862], [0, -672], [-962, 0], [646, 0], [0, -770], [1626, 0], [0, -610], [0, -1164], [782, 0], [-1586, 0], [0, 72], [0, -638], [0, 294], [-1598, 0], [-1598, 0], [0, 132], [-1514, 0], [0, 154], [2438, 0], [0, -678], [0, 992], [-926, 0], [0, -930], [984, 0], [2762, 0], [-1682, 0], [0, -124], [-2224, 0], [0, -894], [-2382, 0], [0, -370], [0, 630], [0, 1008], [0, 786], [-1482, 0], [1186, 0], [2238, 0], [0, -462], [0, 148], [-586, 0], [0, 586], [-1718, 0], [0, -1162], [-604, 0], [0, 440], [0, 450], [-1062, 0], [0, 1286], [0, -174], [682, 0], [-676, 0], [0, -966], [1906, 0], [766, 0], [0, 742], [-802, 0], [1944, 0], [0, 1062], [-838, 0], [0, 602], [0, 364], [1976, 0], [0, 922], [2042, 0], [1984, 0], [634, 0], [-138, 0], [-114, 0], [-814, 0], [0, -834], [318, 0], [1902, 0], [0, -342], [0, -194], [3518, 0], [0, 1498], [282, 0], [-1836, 0], [0, 876], [-2026, 0], [0, 966], [-878, 0], [0, 588], [0, -110], [0, 730], [0, 666], [0, 294], [-1976, 0], [0, 450], [898, 0], [0, -966], [0, -1266], [2984, 0], [2634, 0], [0, 22], [-1438, 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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![6, 1]>,<11,R![20, 0, 1]>,<13,R![16, 1]>],Snew);
    return Vf;
end function;