
// Make newform 1950.2.a.x in Magma, downloaded from the LMFDB on 24 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_x();".
// 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_x();".
// 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], [4], [1], [-4], [7], [4], [5], [4], [9], [-5], [-10], [3], [9], [-6], [4], [-7], [-15], [12], [7], [6], [14], [-16], [-10], [8], [5], [-11], [-14], [-5], [17], [-19], [-4], [-4], [-4], [-14], [-24], [9], [13], [-4], [-4], [10], [-12], [-16], [-3], [-16], [-2], [6], [-17], [8], [16], [8], [-5], [8], [30], [5], [32], [26], [17], [8], [-12], [21], [2], [-23], [-28], [-28], [-34], [11], [-34], [-21], [-15], [7], [22], [-20], [5], [5], [-15], [-30], [-24], [23], [14], [19], [-19], [-7], [25], [-27], [4], [-24], [-26], [3], [15], [22], [-40], [-23], [36], [-4], [36], [-14], [18], [6], [-38], [39], [8], [-28], [-8], [12], [29], [-12], [-3], [-3], [2], [7], [8], [40], [-18], [-5], [-36], [-10], [33], [-7], [19], [-14], [0], [3], [34], [-38], [-12], [28], [-3], [-37], [-21], [-23], [8], [35], [40], [-8], [-32], [-30], [18], [20], [42], [-19], [-4], [-34], [0], [19], [-50], [-2], [39], [43], [0], [40], [-36], [50], [-18], [29], [11], [-22], [52], [-8], [36], [-44], [-21], [18], [-48], [13], [-10], [6], [14], [-50], [17], [-18], [-24], [-31], [10], [4], [2], [-43], [-50], [50], [60], [6], [53], [-36], [25], [-16], [41], [20], [48], [41], [-44], [-20], [-30], [61], [60], [-60], [26], [-66], [6], [-12], [27], [54], [-17], [16], [-11], [-26], [11], [-50], [40], [29], [61], [16], [-60], [45], [-14], [-16], [-66], [56], [42], [26], [40], [9], [52], [-27], [-30], [-6], [-4], [-15], [-62], [-32], [-62], [-48], [-54], [-16], [48], [-18], [38], [35], [-30], [-50], [-54], [25], [29], [-22], [-54], [28], [48], [27], [6], [-53], [-6], [51], [65], [-29], [-24], [-52], [13], [57], [16], [9], [-25], [15], [64], [-66], [52], [-30], [33], [76], [-10], [-70], [-7], [-38], [-5], [41], [-76], [54], [-55], [48], [45], [-24], [62], [82], [-26], [-51], [-26], [26], [-20], [-84], [60], [-42], [53], [1], [-42], [42], [-66], [-48], [-26], [-60], [-76], [9], [18], [0], [-1], [40], [-52], [-15], [50], [-65], [-88], [-56], [-18], [-78], [-62], [27], [-13], [-54], [48], [-42], [-60], [-70], [28], [63], [-80], [47], [-13], [5], [-37], [52], [81], [-15], [62], [-60], [-34], [55], [22], [58], [27], [22], [29], [-55], [-20], [16], [48], [-36], [24], [76], [60], [-45], [-31], [-81], [-46], [-94], [39], [96], [-46], [-11], [42], [54], [-9], [-51], [0], [-40], [70], [46], [-2], [20], [44], [17], [5], [56], [-60], [3], [11], [-60], [6], [0], [-2], [38], [57], [34], [-71], [-48], [-34], [-15], [59], [44], [-8], [-47], [2], [62], [42], [30], [16], [48], [-23], [-56], [30], [-22], [-8], [7], [36], [-102], [7], [-26], [32], [-98], [-22], [-11], [62], [6], [-74], [-36], [-78], [33], [3], [-15], [-1], [-34], [12], [31], [-41], [11], [7], [-4], [-54], [60], [99], [84], [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_x();".
// 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_x(: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_x();".
// 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_x( : 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![-4, 1]>,<17,R![4, 1]>,<23,R![-4, 1]>,<31,R![-4, 1]>],Snew);
    return Vf;
end function;