
// Make newform 4650.2.a.f in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4650_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_4650_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_4650_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_4650_a();"
function MakeCharacter_4650_a()
    N := 4650;
    order := 1;
    char_gens := [3101, 2977, 1801];
    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_4650_a_Hecke(Kf)
    return MakeCharacter_4650_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], [-5], [4], [0], [-5], [9], [-2], [1], [-8], [6], [-1], [12], [13], [10], [-14], [-14], [-9], [9], [5], [-6], [3], [18], [-7], [-8], [-11], [-2], [-13], [-20], [-6], [-2], [16], [7], [0], [-13], [-4], [9], [6], [-16], [17], [8], [22], [10], [-11], [-13], [-16], [-15], [-19], [1], [-6], [24], [-28], [7], [8], [-22], [-25], [2], [12], [-26], [14], [8], [-8], [-6], [-34], [-24], [-2], [18], [4], [4], [-9], [-18], [-1], [-11], [-12], [-16], [3], [-25], [-28], [30], [-26], [-32], [11], [20], [-13], [6], [22], [36], [-30], [-20], [-21], [-24], [15], [6], [-22], [-20], [-36], [-17], [0], [-16], [-7], [44], [-21], [-10], [2], [6], [10], [-39], [-12], [-13], [-4], [-9], [16], [31], [-2], [5], [-5], [-18], [30], [28], [-38], [-51], [-3], [27], [29], [-17], [-30], [-1], [14], [10], [43], [18], [20], [15], [-35], [27], [23], [2], [-5], [55], [-2], [2], [46], [29], [-21], [-21], [-10], [0], [31], [6], [6], [-23], [-8], [-4], [18], [-8], [-11], [54], [-48], [10], [52], [20], [2], [-22], [52], [-3], [18], [8], [-3], [12], [-29], [-8], [-58], [-4], [13], [44], [21], [-7], [-21], [24], [-23], [-6], [-8], [-18], [-24], [27], [8], [10], [9], [-19], [-45], [48], [-36], [-12], [10], [26], [39], [45], [30], [41], [-25], [43], [-35], [-6], [-2], [-66], [16], [-18], [4], [-57], [48], [12], [27], [24], [3], [-16], [30], [24], [54], [-29], [-70], [20], [23], [1], [9], [42], [-12], [-64], [42], [-30], [20], [54], [-45], [-44], [-33], [-3], [34], [21], [-58], [-28], [-2], [-58], [50], [-12], [61], [12], [-67], [6], [44], [-1], [-40], [3], [47], [-12], [24], [-38], [7], [12], [-62], [-35], [-10], [33], [35], [16], [-3], [5], [2], [18], [-18], [48], [-68], [-51], [-47], [28], [32], [-57], [46], [20], [-13], [-20], [-68], [84], [12], [52], [14], [-31], [-54], [-24], [18], [27], [45], [28], [60], [64], [-18], [56], [-24], [21], [-87], [22], [3], [-40], [-3], [-26], [-48], [-60], [-12], [-88], [-54], [-16], [-40], [-30], [19], [40], [-2], [-24], [72], [-16], [-55], [-58], [43], [-49], [79], [75], [-70], [-79], [-4], [52], [33], [-10], [82], [80], [0], [-81], [7], [-62], [41], [-19], [10], [20], [-64], [-86], [-7], [-28], [-42], [-34], [49], [80], [76], [-4], [-37], [89], [82], [-86], [-16], [0], [87], [-24], [46], [-47], [-69], [21], [-16], [90], [29], [-26], [-64], [-77], [-74], [-61], [-78], [-34], [4], [10], [37], [15], [16], [-8], [33], [-84], [-60], [-22], [-42], [-22], [10], [22], [50], [55], [-63], [8], [-55], [22], [-36], [85], [22], [-51], [-70], [-100], [98], [7], [-75], [-59], [-70], [-96], [26], [-10], [-38], [41], [45], [6], [0], [-78], [66], [-10], [-53], [56], [1], [-53], [16], [-76], [102], [35], [-16], [26], [-53], [69], [55], [-26], [30]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4650_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4650_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_4650_2_a_f(:prec:=1)
    chi := MakeCharacter_4650_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_4650_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_4650_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4650_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1, 1]>,<11,R![5, 1]>,<13,R![-4, 1]>,<19,R![5, 1]>],Snew);
    return Vf;
end function;