
// Make newform 504.4.s.e in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_504_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_504_s_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_504_4_s_e();".
// 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_504_4_s_e();".
// 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_504_s();"
function MakeCharacter_504_s()
    N := 504;
    order := 3;
    char_gens := [127, 253, 281, 73];
    v := [3, 3, 3, 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_504_s_Hecke();"
function MakeCharacter_504_s_Hecke(Kf)
    N := 504;
    order := 3;
    char_gens := [127, 253, 281, 73];
    char_values := [[1, 0], [1, 0], [1, 0], [-1, 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 := 4;
    raw_aps := [[0, 0], [0, 0], [7, -7], [-14, -7], [0, 7], [-52, 0], [0, 72], [-20, 20], [-48, 48], [243, 0], [0, -95], [-352, 352], [296, 0], [158, 0], [-142, 142], [0, -375], [0, 279], [-246, 246], [0, 730], [-338, 0], [0, 542], [305, -305], [-1123, 0], [-426, 426], [-369, 0], [0, 1270], [-1832, 1832], [-999, 999], [0, 1666], [1832, 0], [-1931, 0], [979, -979], [0, -486], [-2630, 0], [-1878, 1878], [0, -2103], [0, -1404], [1536, -1536], [3634, 0], [858, -858], [0, -2420], [-2672, 0], [-1712, 1712], [0, 3395], [-510, 0], [0, -276], [3198, 0], [-5091, 0], [0, -3895], [6428, -6428], [-3824, 3824], [-568, 0], [0, -3535], [-4335, 0], [3574, -3574], [0, 5466], [0, -5815], [3237, -3237], [0, 3176], [3282, 0], [0, -2182], [-3021, 0], [-1304, 0], [0, 6208], [-3337, 3337], [8637, -8637], [-1940, 1940], [-5527, 0], [0, 9564], [-918, 0], [0, 11480], [3654, -3654], [0, 10067], [-8800, 8800], [1136, 0], [4290, -4290], [0, -862], [9140, -9140], [-14232, 14232], [0, 10001], [10256, 0], [-4502, 0], [0, 12356], [862, 0], [1365, -1365], [8931, -8931], [-11228, 0], [-12151, 12151], [-18534, 0], [17096, 0], [-2140, 2140], [0, -13266], [0, 19183], [21693, 0], [-19658, 19658], [-19436, 0], [3089, -3089], [0, 1950], [-13132, 13132], [-2930, 2930], [19824, 0], [0, -19169], [0, 7267], [-4892, 4892], [0, -22126], [0, 1097], [-9969, 0], [-6424, 6424], [0, 17046], [-18877, 0], [13783, -13783], [0, -18192], [13054, 0], [0, -7246], [2817, 0], [0, 15786], [-17426, 0], [0, -25834], [27983, -27983], [28296, 0], [0, -21254], [-28259, 0], [31779, -31779], [0, -9747], [24512, -24512], [-14325, 0], [-12078, 12078], [-10658, 10658], [19099, 0], [-1406, 1406], [0, -6730], [1166, 0], [16967, -16967], [11878, 0], [27324, -27324], [-11971, 0], [0, -338], [0, -14114], [-1563, 0], [0, -31568], [2626, 0], [361, -361], [0, 29544], [-24163, 0], [0, 39196], [-6528, 0], [1046, 0], [0, 32910], [-22682, 22682], [-7450, 7450], [-10080, 10080], [32958, 0], [-19784, 0], [-11480, 11480], [0, 45352], [39906, 0], [-17528, 17528], [17066, -17066], [-30821, 0], [0, 4179], [9776, -9776], [-14778, 0], [8759, 0], [-33579, 33579], [0, 42634], [0, 29148], [0, 44797], [0, -10654]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_504_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_504_4_s_e();".
// 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_504_4_s_e(:prec:=2)
    chi := MakeCharacter_504_s();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_504_4_s_e();".
// 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_504_4_s_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_504_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![49, -7, 1]>],Snew);
    return Vf;
end function;