
// Make newform 1950.2.a.z in Magma, downloaded from the LMFDB on 23 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_z();".
// 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_z();".
// 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], [2], [2], [-1], [2], [-4], [0], [4], [8], [6], [-6], [4], [-8], [2], [10], [-14], [-16], [-4], [8], [-8], [12], [6], [12], [4], [6], [-4], [-18], [6], [6], [-6], [-14], [4], [12], [-16], [10], [-8], [12], [14], [-18], [-6], [-12], [0], [10], [-24], [-4], [-14], [12], [-22], [-14], [-12], [-18], [18], [-18], [-8], [-20], [16], [-2], [-18], [16], [26], [-28], [20], [-20], [-18], [-12], [20], [12], [14], [-34], [32], [-10], [-22], [8], [24], [16], [-18], [-18], [-26], [-14], [34], [16], [12], [32], [4], [30], [-28], [20], [6], [28], [36], [-26], [-22], [36], [-36], [-40], [-10], [16], [22], [-20], [-14], [-44], [6], [0], [28], [36], [34], [32], [-46], [18], [-2], [46], [-24], [-8], [-38], [32], [36], [-34], [10], [30], [-36], [-10], [12], [-28], [12], [34], [-24], [34], [14], [28], [16], [0], [-26], [38], [2], [22], [-8], [14], [-22], [16], [24], [34], [-12], [18], [40], [-18], [34], [20], [36], [42], [-10], [24], [0], [-28], [16], [-48], [-50], [56], [-44], [-36], [6], [58], [50], [-18], [56], [-16], [-14], [30], [-26], [42], [50], [-32], [-8], [32], [-58], [32], [12], [-46], [-50], [-10], [34], [-26], [-22], [-24], [-24], [34], [-8], [-6], [52], [28], [36], [4], [60], [36], [-26], [-34], [-22], [42], [36], [36], [32], [-14], [46], [-30], [2], [48], [-4], [-42], [24], [20], [0], [-34], [-36], [-36], [-2], [22], [-58], [0], [2], [22], [8], [-6], [22], [68], [54], [6], [8], [10], [18], [34], [12], [-8], [-42], [56], [-44], [-2], [-62], [6], [-12], [52], [16], [-14], [54], [-18], [-12], [2], [-30], [-12], [-24], [70], [18], [52], [22], [-54], [54], [-38], [-32], [-78], [-40], [22], [12], [26], [-10], [10], [0], [44], [-78], [-56], [-74], [-10], [52], [-32], [-56], [44], [-14], [-68], [-10], [6], [-78], [0], [-32], [-20], [-6], [-52], [60], [-80], [-50], [-56], [2], [-8], [28], [-50], [18], [62], [24], [-8], [14], [-18], [-44], [44], [38], [-80], [-36], [16], [76], [-68], [46], [60], [34], [-72], [-24], [54], [12], [48], [54], [-54], [44], [24], [-54], [0], [-88], [20], [-10], [-18], [-10], [-16], [-36], [36], [54], [-14], [2], [-8], [-4], [-40], [84], [26], [30], [-46], [70], [74], [-30], [-24], [-40], [-46], [46], [-50], [-68], [28], [-82], [-40], [-52], [-16], [38], [74], [-6], [-32], [58], [-66], [-28], [34], [66], [36], [-6], [24], [88], [66], [-54], [-22], [-18], [-20], [-24], [20], [-16], [22], [-54], [8], [84], [14], [-48], [60], [26], [-30], [38], [-8], [-52], [0], [14], [68], [-72], [-46], [26], [66], [-24], [-40], [-40], [0], [10], [40], [-12], [-6], [-98], [18], [42], [40], [-48], [94], [22], [24], [-2], [-48], [-38], [28], [64], [-76], [100], [72], [-58], [18], [48], [-48], [106], [-24], [-30], [48], [2], [-4], [10], [4], [72]];
    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_z();".
// 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_z(: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_z();".
// 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_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-2, 1]>,<11,R![-2, 1]>,<17,R![-2, 1]>,<23,R![0, 1]>,<31,R![-8, 1]>],Snew);
    return Vf;
end function;