
// Make newform 1950.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1950_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_1950_2_a_f();".
// 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_1950_2_a_f();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1950_a();"
function MakeCharacter_1950_a()
    N := 1950;
    order := 1;
    char_gens := [1301, 1327, 301];
    v := [1, 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_1950_a_Hecke(Kf)
    return MakeCharacter_1950_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], [-4], [0], [1], [0], [5], [0], [3], [-4], [-7], [3], [2], [9], [9], [6], [8], [5], [-3], [-4], [11], [-6], [6], [8], [18], [-16], [9], [11], [18], [-7], [3], [3], [8], [-12], [-16], [-10], [-16], [-21], [21], [12], [-16], [18], [20], [24], [-7], [-4], [14], [6], [-7], [24], [-24], [8], [9], [-12], [-6], [-21], [-28], [14], [-15], [32], [24], [-7], [-18], [-1], [0], [20], [-22], [-9], [2], [-3], [-3], [-19], [2], [20], [15], [27], [-31], [-30], [-4], [-27], [-22], [-33], [11], [29], [-3], [21], [8], [24], [26], [-33], [-21], [-34], [0], [11], [12], [-24], [-36], [-22], [38], [2], [-42], [-21], [-24], [8], [32], [0], [3], [-24], [5], [23], [-22], [33], [32], [-16], [18], [23], [-12], [-18], [-21], [-25], [29], [42], [36], [17], [30], [14], [12], [-52], [5], [-7], [-39], [5], [32], [27], [32], [36], [-40], [18], [6], [-4], [-6], [23], [36], [-22], [24], [-37], [-54], [-34], [-27], [35], [12], [-52], [36], [-10], [-30], [-55], [3], [14], [48], [36], [24], [32], [-39], [-18], [-24], [41], [26], [26], [-42], [30], [47], [42], [44], [5], [30], [-28], [6], [47], [2], [-10], [36], [14], [27], [-36], [-57], [20], [-19], [-40], [-48], [-1], [-24], [8], [18], [33], [-36], [-52], [-22], [-30], [-42], [-12], [23], [-34], [-25], [48], [-3], [2], [-9], [30], [-4], [11], [51], [8], [-60], [57], [62], [-40], [54], [-48], [-66], [-34], [-4], [33], [-16], [-63], [-10], [30], [24], [35], [-66], [-4], [2], [20], [-30], [68], [-48], [38], [42], [9], [18], [54], [38], [-37], [-61], [30], [42], [8], [24], [-55], [54], [-13], [-6], [33], [-55], [27], [0], [-4], [-55], [33], [20], [-61], [51], [17], [80], [-30], [-28], [-18], [-63], [68], [-42], [-70], [5], [26], [-49], [23], [-4], [66], [23], [-76], [39], [0], [50], [-30], [26], [41], [-6], [38], [-36], [56], [0], [66], [-63], [-57], [-6], [26], [-30], [-16], [-42], [-36], [-52], [-25], [42], [8], [27], [-52], [-40], [-75], [-10], [27], [-52], [36], [42], [66], [50], [81], [11], [-18], [48], [74], [12], [14], [32], [9], [-16], [-39], [-13], [23], [-49], [72], [-39], [71], [-18], [44], [-66], [77], [18], [-34], [69], [2], [47], [-79], [60], [36], [80], [36], [-48], [-76], [-76], [39], [-15], [77], [-70], [30], [53], [-40], [66], [-63], [-42], [-18], [-75], [29], [-12], [96], [-6], [-22], [50], [72], [-4], [-79], [-81], [20], [-60], [-75], [-49], [-76], [-30], [60], [2], [66], [23], [-42], [-33], [-52], [-78], [83], [-87], [-4], [-28], [-19], [42], [50], [-78], [54], [8], [48], [-49], [20], [42], [-70], [12], [-31], [-60], [-10], [9], [-6], [-88], [-58], [-30], [41], [-18], [-70], [-42], [-12], [-82], [-49], [-75], [45], [-67], [-102], [-48], [81], [95], [-15], [-75], [56], [-54], [-36], [3], [20], [-60]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_a_f();".
// 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_1950_2_a_f(:prec:=1)
    chi := MakeCharacter_1950_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(2999) - 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_1950_2_a_f();".
// 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_1950_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![0, 1]>,<17,R![0, 1]>,<23,R![0, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;