
// Make newform 1950.2.a.q 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_q();".
// 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_q();".
// 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], [-1], [-3], [1], [-1], [-8], [4], [-7], [1], [4], [-6], [-12], [3], [5], [-9], [5], [-11], [8], [0], [-8], [7], [-8], [6], [1], [-14], [-18], [2], [-2], [12], [-10], [12], [-14], [10], [13], [-15], [-4], [20], [-5], [10], [1], [-10], [6], [6], [-26], [12], [24], [-3], [14], [26], [3], [-10], [18], [-7], [30], [1], [1], [-18], [-18], [-14], [6], [-12], [18], [3], [20], [28], [13], [-14], [-28], [14], [-15], [-28], [13], [11], [-32], [34], [26], [-28], [-2], [6], [-12], [12], [-10], [-8], [-12], [36], [-6], [26], [-11], [24], [23], [-25], [18], [25], [-22], [34], [38], [-2], [-8], [-6], [-6], [-36], [3], [-36], [-14], [-39], [34], [-26], [-35], [-22], [-34], [-34], [-28], [-12], [-27], [20], [24], [-21], [0], [-46], [-43], [22], [27], [-43], [15], [38], [-26], [14], [28], [-47], [-29], [28], [43], [-36], [8], [-46], [7], [-11], [-26], [-47], [-52], [-4], [3], [47], [16], [8], [10], [-24], [-9], [0], [-21], [48], [-56], [8], [28], [-14], [-34], [21], [-60], [47], [57], [7], [42], [52], [-11], [-8], [-17], [-20], [-33], [-12], [20], [46], [-20], [46], [-15], [-55], [10], [-6], [-9], [32], [-56], [-30], [12], [-49], [-42], [43], [-13], [36], [-5], [2], [-21], [-40], [-52], [-26], [13], [-28], [-53], [-6], [-22], [8], [-32], [38], [-39], [9], [-58], [63], [-34], [-44], [38], [67], [-58], [4], [-51], [45], [28], [-2], [9], [-15], [16], [-10], [48], [40], [17], [0], [-17], [10], [-46], [10], [59], [-46], [-60], [53], [18], [48], [32], [10], [0], [24], [-12], [12], [0], [50], [76], [12], [62], [68], [-56], [-29], [74], [32], [-56], [-60], [-77], [-30], [-45], [-38], [1], [-7], [-22], [22], [48], [12], [-22], [-52], [-12], [-6], [36], [77], [67], [-38], [-17], [2], [54], [-17], [-5], [4], [84], [80], [70], [-58], [32], [64], [2], [5], [24], [45], [30], [63], [-73], [-21], [-46], [-78], [-75], [9], [39], [-40], [54], [-28], [-23], [19], [66], [-10], [12], [-50], [-12], [-48], [-46], [-24], [11], [-48], [-12], [47], [-4], [61], [-62], [1], [40], [56], [-39], [-70], [-58], [-6], [43], [0], [35], [83], [-1], [38], [51], [-6], [-54], [26], [-19], [55], [51], [-2], [-36], [-2], [2], [74], [82], [-26], [80], [4], [76], [-13], [-11], [-45], [-15], [-17], [-48], [74], [84], [18], [1], [-15], [-19], [13], [-51], [-80], [-38], [84], [0], [98], [86], [-70], [-73], [-29], [-78], [-46], [-56], [-69], [66], [-37], [-33], [-30], [8], [9], [-46], [-38], [12], [-47], [22], [-52], [24], [-95], [-83], [-76], [58], [-84], [60], [59], [62], [69], [-83], [-7], [-51], [-22], [34], [32], [-49], [73], [-42], [-42], [-42], [34], [42], [-71], [-12], [31], [-51], [-30], [30], [-82], [-29], [-92], [-76], [62], [-26], [-63], [-56], [1], [64], [24], [-60], [44], [-62], [-21], [6]];
    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_q();".
// 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_q(: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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1950_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![3, 1]>,<17,R![1, 1]>,<23,R![-4, 1]>,<31,R![-1, 1]>],Snew);
    return Vf;
end function;