
// Make newform 1950.2.a.g in Magma, downloaded from the LMFDB on 26 September 2025.
// 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_g();".
// 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_g();".
// 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 := 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]];
    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], [2], [-1], [-4], [2], [6], [-2], [-4], [6], [-6], [-8], [-8], [-10], [-14], [10], [-4], [8], [-10], [-8], [-12], [-18], [-6], [-14], [-6], [8], [12], [12], [18], [0], [-2], [16], [0], [8], [10], [4], [0], [-10], [0], [6], [0], [-6], [-26], [24], [-28], [16], [-12], [-16], [4], [-12], [-30], [-12], [12], [10], [-14], [16], [-2], [-18], [4], [14], [-28], [8], [-8], [30], [18], [32], [24], [-28], [26], [20], [-22], [2], [26], [12], [-26], [-6], [-18], [-2], [16], [28], [-8], [0], [32], [-8], [6], [26], [-4], [36], [16], [-12], [40], [8], [30], [-18], [-16], [38], [-8], [-8], [-44], [-14], [16], [-42], [-36], [-26], [-36], [-14], [-16], [-10], [-18], [-26], [34], [42], [-32], [-2], [20], [6], [26], [40], [24], [-12], [-22], [-12], [2], [-30], [-32], [24], [-14], [2], [-2], [4], [48], [22], [14], [-22], [-14], [28], [-34], [14], [-2], [-12], [-26], [12], [42], [16], [42], [40], [-4], [24], [30], [-46], [-48], [6], [32], [-32], [-24], [46], [-16], [40], [-12], [48], [28], [-28], [-42], [-4], [32], [-38], [-30], [-2], [-42], [-40], [16], [10], [-64], [50], [-10], [0], [26], [-14], [8], [-56], [-50], [-22], [-36], [18], [-38], [28], [6], [-20], [28], [36], [-20], [48], [36], [-68], [62], [-22], [-54], [-18], [-12], [8], [22], [-14], [-6], [-22], [-12], [-16], [6], [48], [8], [42], [-34], [36], [60], [-62], [22], [14], [-60], [-46], [10], [-52], [-30], [-32], [20], [18], [0], [-16], [10], [-6], [-14], [60], [-32], [18], [-28], [-32], [-50], [10], [24], [48], [28], [40], [10], [-24], [-42], [-48], [32], [54], [66], [-54], [58], [-6], [-20], [-2], [30], [-30], [-38], [4], [-18], [62], [-2], [12], [32], [62], [-38], [-12], [44], [-54], [28], [-62], [2], [-68], [58], [-56], [-28], [-20], [-44], [-16], [30], [-60], [18], [-68], [-74], [-48], [20], [24], [-20], [10], [28], [2], [34], [16], [-2], [30], [-34], [54], [40], [26], [12], [28], [56], [-46], [40], [-12], [40], [22], [-32], [34], [36], [34], [-42], [-72], [42], [-36], [24], [-42], [-54], [-40], [54], [6], [-60], [38], [-46], [-28], [30], [50], [-22], [36], [-18], [-66], [4], [-46], [16], [44], [-10], [84], [-52], [-54], [38], [-62], [-34], [48], [-36], [-40], [-34], [-44], [22], [-44], [40], [50], [-22], [-82], [44], [-34], [38], [-36], [40], [-14], [-48], [-4], [-50], [42], [6], [30], [12], [4], [30], [-72], [26], [12], [64], [36], [86], [56], [22], [-18], [-40], [-18], [14], [-84], [72], [38], [72], [-82], [-62], [-4], [0], [-82], [-4], [-78], [26], [-70], [-6], [48], [44], [68], [48], [-86], [88], [-36], [12], [4], [-48], [42], [88], [48], [46], [22], [-84], [58], [-36], [46], [-20], [-20], [92], [-74], [-72], [-94], [-102], [66], [42], [-74], [24], [84], [6], [50], [-64], [58], [-26], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1950_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1950_2_a_g();".
// 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_g(: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_g();".
// 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_g( : 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![-2, 1]>,<17,R![4, 1]>,<23,R![-6, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;