
// Make newform 525.4.d.e in Magma, downloaded from the LMFDB on 24 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_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_525_4_d_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, 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, 2], [0, 3], [0, 0], [0, 7], [-21, 0], [0, 24], [0, 22], [-16, 0], [0, -25], [-167, 0], [10, 0], [0, 133], [-168, 0], [0, -97], [0, 400], [0, -182], [-488, 0], [28, 0], [0, 967], [-285, 0], [0, -838], [469, 0], [0, -406], [-324, 0], [0, 114], [34, 0], [0, -1976], [0, -828], [1467, 0], [0, 783], [0, 515], [-1526, 0], [0, 1822], [3042, 0], [-349, 0], [-2167, 0], [0, 1204], [0, 3252], [0, 1276], [0, 1140], [-2724, 0], [4474, 0], [3932, 0], [0, 3441], [0, 201], [1384, 0], [3964, 0], [0, -762], [0, 5764], [-1510, 0], [0, 5937], [-1456, 0], [-588, 0], [6510, 0], [0, 2898], [0, 2953], [-3394, 0], [-1284, 0], [0, 3506], [5823, 0], [0, -3830], [0, 3572], [0, -7278], [3786, 0], [0, 448], [0, 4081], [1071, 0], [0, -2102], [0, 11471], [6448, 0], [0, -10050], [6597, 0], [0, -6500], [0, 8665], [217, 0], [0, -4640], [3021, 0], [0, 7826], [-3405, 0], [-4052, 0], [-11322, 0], [-1757, 0], [-9304, 0], [0, -5752], [-2344, 0], [0, -15132], [5055, 0], [0, 3795], [8300, 0], [0, -5052], [0, -2482], [-14176, 0], [0, -1039], [9255, 0], [-14396, 0], [0, -6998], [6400, 0], [15158, 0], [0, 4120], [3291, 0], [0, 20547], [0, 3169], [0, -24878], [1425, 0], [-14649, 0], [0, -2108], [0, -24366], [0, 17892], [-8077, 0], [-7836, 0], [0, 24092], [0, 13647], [0, -12813], [300, 0], [24615, 0], [-24117, 0], [0, -14020], [0, -2430], [0, -27514], [20772, 0], [32132, 0], [0, 23542], [0, -30814], [0, 9009], [-25860, 0], [24170, 0], [8426, 0], [30752, 0], [0, -11570], [0, 32836], [22351, 0], [0, -21720], [-6352, 0], [0, 7685], [-12, 0], [24832, 0], [0, 28242], [0, -12542], [0, -11058], [17307, 0], [-2706, 0], [-38862, 0], [0, 27217], [0, -11301], [-6786, 0], [-8358, 0], [0, -16500], [0, -3372], [-39026, 0], [0, -41487], [0, 158], [36594, 0], [0, -17839], [0, -13996], [0, 31168], [1881, 0], [-42447, 0], [-7514, 0], [0, -33284], [-10962, 0], [0, 6172], [0, 26263], [0, 5920], [13372, 0], [0, 35155], [0, -1858], [-16401, 0], [0, 22952]];
    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_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_525_4_d_e(: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_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_525_4_d_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![4, 0, 1]>,<11,R![21, 1]>],Snew);
    return Vf;
end function;