
// Make newform 525.2.i.g in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_525_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_525_i_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_2_i_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_525_2_i_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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_525_i();"
function MakeCharacter_525_i()
    N := 525;
    order := 3;
    char_gens := [176, 127, 451];
    v := [3, 3, 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_i_Hecke();"
function MakeCharacter_525_i_Hecke(Kf)
    N := 525;
    order := 3;
    char_gens := [176, 127, 451];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 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 := 2;
    raw_aps := [[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 0], [1, 1, 1, 2], [0, 1, 2, 1], [3, 0, 0, 1], [0, 3, 2, 3], [1, 4, 1, 0], [-2, -3, -2, 0], [4, 0, 0, 3], [0, 2, -3, 2], [-7, 1, -7, 0], [-2, 0, 0, -3], [9, 0, 0, -1], [-2, 8, -2, 0], [0, 2, -4, 2], [0, 1, 0, 1], [-4, -6, -4, 0], [0, -9, 1, -9], [-2, 0, 0, 1], [0, -5, 5, -5], [-1, 4, -1, 0], [-4, 0, 0, 1], [8, -3, 8, 0], [8, 0, 0, -2], [0, -3, -2, -3], [3, -3, 3, 0], [14, 1, 14, 0], [0, -6, -5, -6], [10, 0, 0, 4], [11, 0, 0, -5], [2, 6, 2, 0], [0, -5, 10, -5], [3, 0, 0, -6], [-12, -4, -12, 0], [0, -6, 2, -6], [0, 10, 2, 10], [-8, -8, -8, 0], [-16, 0, 0, -3], [0, -2, 0, 0], [0, 4, 14, 4], [-5, 0, 0, -4], [-2, -12, -2, 0], [0, -3, 11, -3], [10, 0, 0, 11], [0, -6, 14, -6], [-14, 0, 0, -10], [-20, 0, 0, -2], [0, -5, -16, -5], [3, 8, 3, 0], [-14, -6, -14, 0], [6, 0, 0, -8], [0, 4, -16, 4], [-4, 0, 0, 1], [-20, -7, -20, 0], [0, 2, -4, 2], [0, 10, -6, 10], [-6, -10, -6, 0], [0, 13, 5, 13], [4, 0, 0, -4], [0, 3, 9, 3], [4, 0, 0, -8], [-3, 0, 0, -1], [0, -5, -10, -5], [5, 15, 5, 0], [10, -7, 10, 0], [5, 16, 5, 0], [-9, 0, 0, -7], [0, 12, 8, 12], [6, 0, 0, 16], [0, -7, 4, -7], [28, -1, 28, 0], [0, -1, 21, -1], [15, 3, 15, 0], [-5, 0, 0, -14], [4, 6, 4, 0], [0, 19, -10, 19], [-3, -15, -3, 0], [8, -6, 8, 0], [0, -8, -15, -8], [-6, 0, 0, -2], [19, 0, 0, -6], [0, 2, 34, 2], [9, 0, 0, -11], [-6, 4, -6, 0], [-8, 6, -8, 0], [10, 0, 0, 12], [-11, -7, -11, 0], [-20, 0, 0, 5], [9, 0, 0, -1], [18, -8, 18, 0], [0, -16, -12, -16], [0, -7, 9, -7], [4, 0, 0, 10], [11, -14, 11, 0], [10, 0, 0, -6], [6, -14, 6, 0], [0, -22, -8, -22], [-9, 1, -9, 0], [-21, -8, -21, 0], [12, 0, 0, -14], [0, -4, 14, -4], [0, 20, -14, 20], [10, 19, 10, 0], [0, 12, 11, 12], [0, -11, 9, -11], [-26, 0, 0, 3], [12, 1, 12, 0], [0, -10, -28, -10], [-27, 0, 0, -2], [3, -3, 3, 0], [0, -10, 4, -10], [6, 0, 0, -2], [0, 12, 23, 12], [16, 0, 0, 0], [0, -15, 8, -15], [-9, 0, 0, 11], [0, 13, -28, 13], [12, -3, 12, 0], [-16, 0, 0, 8], [0, 8, 21, 8], [-15, 0, 0, -9], [-6, 7, -6, 0], [0, 1, -22, 1], [-19, -8, -19, 0], [20, 0, 0, -15], [-20, 22, -20, 0], [-26, 6, -26, 0], [-9, 0, 0, -3], [-15, 3, -15, 0], [0, -10, -21, -10], [-8, 0, 0, -9], [3, -16, 3, 0], [-2, 0, 0, 28], [28, -5, 28, 0], [1, 0, 0, 12], [0, 5, -32, 5], [0, 2, -16, 2], [-26, 0, 0, -11], [0, -16, 18, -16], [-6, 0, 0, 12], [2, -11, 2, 0], [0, -20, 6, -20], [-10, 0, 0, 2], [0, -4, -13, -4], [-20, 0, 0, -9], [3, 0, 0, 29], [0, 3, 2, 3], [-10, -4, -10, 0], [10, -14, 10, 0], [-18, 0, -18, 0], [-16, 0, 0, 4], [-1, 0, 0, 5], [-16, 15, -16, 0], [0, -13, 27, -13], [0, 0, 0, -3], [41, 4, 41, 0], [10, 29, 10, 0], [15, 0, 0, 5], [0, 9, -24, 9], [-12, 5, -12, 0], [-40, 0, 0, 6], [-5, 0, 0, 13], [-20, 4, -20, 0], [0, -21, 16, -21], [0, -39, 2, -39], [0, -4, -25, -4], [0, 3, -5, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_525_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_525_2_i_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_525_2_i_g(:prec:=4)
    chi := MakeCharacter_525_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_2_i_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_525_2_i_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_525_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![4, 0, 2, 0, 1]>],Snew);
    return Vf;
end function;