
// Make newform 525.4.d.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_d_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_525_4_d_b();".
// 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_525_4_d_b();".
// 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_525_d();"
function MakeCharacter_525_d()
    N := 525;
    order := 2;
    char_gens := [176, 127, 451];
    v := [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_525_d_Hecke();"
function MakeCharacter_525_d_Hecke(Kf)
    N := 525;
    order := 2;
    char_gens := [176, 127, 451];
    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, 4], [0, 3], [0, 0], [0, -7], [62, 0], [0, 62], [0, 84], [-100, 0], [0, 42], [10, 0], [-48, 0], [0, -246], [-248, 0], [0, -68], [0, 324], [0, -258], [-120, 0], [622, 0], [0, 904], [-678, 0], [0, 642], [-740, 0], [0, -468], [-200, 0], [0, -1266], [232, 0], [0, 1792], [0, -1906], [90, 0], [0, -458], [0, 804], [812, 0], [0, 414], [1620, 0], [-2370, 0], [-568, 0], [0, -266], [0, 272], [0, -1876], [0, 152], [-610, 0], [1042, 0], [-2038, 0], [0, 2602], [0, 2354], [-1680, 0], [-668, 0], [0, 1832], [0, 4944], [5470, 0], [0, 2802], [1170, 0], [-2338, 0], [2792, 0], [0, 7024], [0, -2438], [6780, 0], [-1928, 0], [0, 5554], [1942, 0], [0, -4828], [0, 6152], [0, 5884], [9132, 0], [0, 9382], [0, 3114], [1532, 0], [0, -4166], [0, -11366], [-9310, 0], [0, 8572], [4790, 0], [0, 5424], [0, -1838], [4260, 0], [0, -9048], [11490, 0], [0, -1866], [13662, 0], [13210, 0], [-6960, 0], [8162, 0], [16602, 0], [0, -7738], [840, 0], [0, -6618], [-3090, 0], [0, 5914], [-15968, 0], [0, 1172], [0, 5304], [-5740, 0], [0, 8944], [-5558, 0], [19820, 0], [0, -1848], [-340, 0], [10212, 0], [0, 9332], [-8998, 0], [0, -3416], [0, -526], [0, 6712], [-4190, 0], [3032, 0], [0, 5434], [0, 464], [0, -11748], [-7650, 0], [-22878, 0], [0, 704], [0, -24958], [0, -8826], [-21220, 0], [-3268, 0], [13062, 0], [0, 28012], [0, 3844], [0, 28482], [9330, 0], [8782, 0], [0, 10562], [0, -26016], [0, -8898], [30572, 0], [-30618, 0], [8130, 0], [27840, 0], [0, 14624], [0, 20862], [13920, 0], [0, -25578], [33472, 0], [0, 25934], [26952, 0], [-23450, 0], [0, -39568], [0, -12356], [0, -21736], [38310, 0], [2132, 0], [5002, 0], [0, 3612], [0, -27666], [-12890, 0], [9340, 0], [0, 33082], [0, 7544], [-8180, 0], [0, -10518], [0, 14134], [6492, 0], [0, -38228], [0, -43076], [0, -32236], [-46518, 0], [-17840, 0], [-7000, 0], [0, 36114], [-4748, 0], [0, 42694], [0, 16742], [0, -9956], [-26388, 0], [0, -786], [0, -51888], [-51928, 0], [0, -386]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_4_d_b();".
// 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_525_4_d_b(:prec:=2)
    chi := MakeCharacter_525_d();
    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_525_4_d_b();".
// 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_525_4_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![16, 0, 1]>,<11,R![-62, 1]>],Snew);
    return Vf;
end function;