
// Make newform 4650.2.a.bn in Magma, downloaded from the LMFDB on 19 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_bn();".
// 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_bn();".
// 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], [-2], [2], [4], [3], [0], [-6], [-5], [1], [8], [12], [-1], [3], [14], [0], [-8], [-2], [7], [14], [-5], [4], [-5], [-7], [2], [4], [-17], [15], [-6], [3], [-13], [3], [-15], [20], [17], [3], [24], [-12], [-6], [-10], [12], [17], [14], [13], [0], [-8], [-1], [13], [-20], [-6], [-10], [-8], [2], [-22], [24], [-15], [-3], [8], [-18], [-26], [34], [28], [-3], [4], [-2], [-28], [-2], [-22], [-25], [14], [0], [-27], [-21], [-20], [-6], [30], [18], [2], [0], [-15], [-3], [32], [-26], [-30], [39], [-25], [-32], [-18], [-21], [3], [-35], [13], [12], [-25], [-21], [30], [12], [-21], [-33], [8], [3], [-11], [10], [-28], [-27], [18], [34], [-45], [-38], [28], [-36], [28], [25], [-23], [-18], [-1], [28], [4], [20], [-3], [-26], [-17], [-31], [2], [2], [-10], [-10], [18], [-1], [25], [-26], [-8], [-42], [-33], [-15], [-21], [28], [18], [5], [32], [42], [4], [-2], [-40], [5], [9], [-12], [25], [-6], [-27], [7], [-16], [-27], [8], [32], [30], [-30], [38], [-3], [-32], [14], [-32], [-3], [-42], [24], [-13], [18], [30], [-11], [0], [22], [-8], [-31], [-30], [10], [-28], [12], [-46], [10], [38], [12], [-11], [18], [49], [55], [3], [24], [-30], [-23], [44], [-56], [12], [62], [28], [19], [52], [39], [-12], [4], [30], [32], [-22], [55], [35], [3], [-40], [24], [50], [-28], [-22], [27], [-46], [-52], [30], [7], [-32], [-38], [8], [4], [-58], [-10], [20], [-36], [28], [70], [24], [-20], [13], [17], [54], [60], [62], [-18], [-51], [-12], [15], [-66], [20], [72], [49], [-53], [64], [50], [49], [-40], [-47], [-28], [-15], [-56], [-62], [-8], [38], [55], [14], [-45], [-33], [3], [68], [-17], [-66], [48], [-40], [-26], [78], [-20], [20], [-78], [4], [-36], [-13], [-72], [14], [5], [-7], [4], [-57], [-30], [32], [82], [-51], [2], [48], [37], [68], [-8], [-46], [53], [50], [-75], [-48], [63], [54], [32], [44], [-35], [27], [-6], [-10], [63], [-31], [53], [0], [-36], [-23], [13], [-52], [-25], [-50], [-11], [34], [45], [62], [59], [8], [80], [-40], [-58], [74], [-25], [57], [-42], [62], [-56], [-36], [72], [-70], [14], [28], [-66], [-73], [18], [-20], [-6], [-38], [-52], [60], [24], [-13], [43], [-16], [-87], [-55], [67], [64], [-35], [42], [-27], [42], [78], [52], [68], [-38], [-41], [10], [-51], [0], [12], [13], [-51], [-22], [42], [78], [-45], [78], [-76], [8], [64], [62], [32], [-10], [-51], [-40], [-78], [58], [-60], [72], [-46], [30], [-82], [-13], [99], [-17], [38], [60], [-71], [-78], [58], [-11], [-22], [60], [-81], [-15], [-82], [17], [-86], [80], [30], [-68], [-93], [60], [24], [68], [68], [55], [-38], [-37], [17], [4], [-40], [-6], [8], [-36], [-93], [-97], [12], [80], [68], [28], [-16], [30], [88], [28], [-95], [29], [-42], [-51], [85], [-3], [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_bn();".
// 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_bn(: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_bn();".
// 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_bn( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4650_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![-2, 1]>,<13,R![-4, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;