
// Make newform 603.2.a.h in Magma, downloaded from the LMFDB on 23 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_603_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_603_2_a_h();".
// 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_603_2_a_h();".
// 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_603_a();"
function MakeCharacter_603_a()
    N := 603;
    order := 1;
    char_gens := [470, 136];
    v := [1, 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;

function MakeCharacter_603_a_Hecke(Kf)
    return MakeCharacter_603_a();
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 := [[1, 1], [0, 0], [3, 0], [1, -3], [1, -2], [-5, 3], [4, -2], [2, -3], [-1, -4], [1, 4], [-1, 0], [1, -3], [2, -1], [0, 3], [7, 1], [9, 0], [-6, 0], [1, -9], [-1, 0], [-7, 2], [-4, 0], [-8, 9], [4, 7], [-1, 2], [-5, 12], [2, 5], [-7, 9], [-15, 6], [0, -3], [-17, 1], [-1, -3], [3, 0], [9, -6], [-3, 0], [2, -13], [-1, 0], [11, -9], [-2, -3], [-14, 16], [22, -5], [10, -2], [-8, 9], [5, -4], [12, 3], [11, -4], [-14, 15], [-7, 3], [6, 6], [-7, 17], [-14, -3], [-8, 4], [11, 5], [21, 3], [-8, -2], [1, 16], [6, -9], [1, -11], [15, 6], [-6, 24], [-7, -1], [-3, -12], [15, -12], [-2, 12], [-12, 24], [-9, -3], [17, -22], [16, -12], [16, -15], [-5, 10], [-6, 0], [-10, 17], [18, 0], [10, -18], [-4, -9], [4, 0], [25, -11], [-4, 14], [-21, -9], [-3, -3], [-16, 18], [10, -20], [-26, -3], [-31, 2], [13, -33], [5, -27], [17, -4], [-20, 7], [-29, 12], [9, 3], [-13, -9], [-7, -10], [-8, -14], [1, 24], [-9, -12], [-25, 0], [-29, -8], [12, -12], [18, 0], [13, -9], [12, 12], [20, -9], [32, -1], [0, 6], [-4, -10], [-8, 21], [19, -6], [-31, 5], [28, 4], [-4, -4], [22, -18], [38, -9], [5, -12], [33, -21], [-44, 6], [-19, 9], [-25, 26], [-6, -9], [28, -2], [2, -4], [17, 8], [21, 18], [8, -27], [5, -10], [-11, 1], [8, 0], [-8, -20], [14, -12], [-47, 4], [28, 0], [4, 0], [13, -15], [11, -22], [15, 18], [19, 12], [12, 0], [-1, -6], [-7, 20], [-13, 12], [-22, 35], [12, -9], [-19, -18], [19, -2], [10, -27], [-16, 2], [25, -18], [-26, 28], [-4, 27], [3, 12], [32, -21], [-21, -9], [17, -18], [23, 2], [11, 6], [19, -26], [-26, 3], [48, -3], [-31, -12], [-5, -8], [21, -6], [39, -24], [27, -15], [1, -20], [-43, 24], [1, -32], [19, -14], [-9, 24], [36, 9], [-21, 30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_603_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_603_2_a_h();".
// 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_603_2_a_h(:prec:=2)
    chi := MakeCharacter_603_a();
    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_603_2_a_h();".
// 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_603_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_603_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, -3, 1]>,<5,R![-3, 1]>],Snew);
    return Vf;
end function;