
// Make newform 400.4.c.j in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_400_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_400_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_400_4_c_j();".
// 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_400_4_c_j();".
// 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;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_400_c();"
function MakeCharacter_400_c()
    N := 400;
    order := 2;
    char_gens := [351, 101, 177];
    v := [2, 2, 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_400_c_Hecke();"
function MakeCharacter_400_c_Hecke(Kf)
    N := 400;
    order := 2;
    char_gens := [351, 101, 177];
    char_values := [[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 := 4;
    raw_aps := [[0, 0], [0, 2], [0, 0], [0, 8], [60, 0], [0, -43], [0, 9], [44, 0], [0, 24], [186, 0], [-176, 0], [0, 127], [186, 0], [0, -50], [0, -84], [0, 249], [-252, 0], [-58, 0], [0, 518], [-168, 0], [0, -253], [272, 0], [0, 474], [1014, 0], [0, -383], [-1314, 0], [0, -224], [0, -774], [-278, 0], [0, 279], [0, -172], [-780, 0], [0, 333], [884, 0], [114, 0], [40, 0], [0, -77], [0, 1090], [0, -1848], [0, -651], [-4308, 0], [1550, 0], [-48, 0], [0, -529], [0, -1857], [-1768, 0], [4036, 0], [0, 340], [0, -1194], [3874, 0], [0, -1581], [5424, 0], [-3886, 0], [5100, 0], [0, 1089], [0, -3072], [-822, 0], [-8480, 0], [0, -569], [5706, 0], [0, -1514], [0, -1695], [0, 2078], [-6552, 0], [0, 683], [0, 1299], [3292, 0], [0, 3097], [0, 5010], [3130, 0], [0, -2097], [-4104, 0], [0, -3748], [0, 2921], [-412, 0], [0, 1284], [-13086, 0], [0, 5227], [-10830, 0], [8566, 0], [13884, 0], [4286, 0], [-6336, 0], [0, 4487], [-2968, 0], [0, -6186], [-11394, 0], [0, -179], [-7530, 0], [0, -6884], [0, -6690], [-6336, 0], [0, 2504], [-12900, 0], [-8116, 0], [0, -2472], [5466, 0], [10074, 0], [0, -6914], [-15226, 0], [0, 6614], [0, -4245], [0, -5142], [-1770, 0], [-6068, 0], [0, 10753], [0, -6054], [0, -7737], [-2520, 0], [-12790, 0], [0, -5788], [0, -10063], [0, -13971], [-22540, 0], [5128, 0], [-12798, 0], [0, -10574], [0, -8232], [0, 12117], [-22836, 0], [26318, 0], [0, -14401], [0, 1263], [0, -11538], [-7868, 0], [21510, 0], [-30014, 0], [-816, 0], [0, 4976], [0, 16973], [23420, 0], [0, -7296], [-9056, 0], [0, -8777], [-36438, 0], [9022, 0], [0, -735], [0, -2626], [0, 6147], [-15546, 0], [-19364, 0], [-7314, 0], [0, 5992], [0, -6750], [44602, 0], [35448, 0], [0, -6295], [0, 12453], [23204, 0], [0, 9924], [0, 13771], [-20718, 0], [0, 12586], [0, 6432], [0, 11546], [14208, 0], [-26584, 0], [-162, 0], [0, -14867], [17142, 0], [0, -13218], [0, -13941], [0, -6328], [-2916, 0], [0, -3447], [0, -22632], [-52016, 0], [0, -6929]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_400_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_400_4_c_j();".
// 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_400_4_c_j(:prec:=2)
    chi := MakeCharacter_400_c();
    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_400_4_c_j();".
// 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_400_4_c_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_400_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![16, 0, 1]>,<7,R![256, 0, 1]>,<11,R![-60, 1]>],Snew);
    return Vf;
end function;