
// Make newform 4598.2.a.n in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4598_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_4598_2_a_n();".
// 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_4598_2_a_n();".
// 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_4598_a();"
function MakeCharacter_4598_a()
    N := 4598;
    order := 1;
    char_gens := [3269, 3631];
    v := [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_4598_a_Hecke(Kf)
    return MakeCharacter_4598_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], [0], [0], [3], [0], [1], [5], [1], [5], [6], [2], [-3], [-2], [-4], [-3], [2], [1], [0], [3], [-4], [15], [10], [0], [6], [2], [10], [14], [-3], [-17], [16], [-10], [-12], [3], [-14], [20], [6], [12], [4], [8], [-15], [-21], [-2], [9], [-20], [-2], [7], [28], [-8], [13], [22], [-19], [-24], [-10], [-18], [2], [8], [9], [29], [22], [-24], [-14], [6], [7], [-17], [1], [-11], [7], [-4], [-16], [-10], [-30], [-1], [19], [-10], [25], [6], [10], [12], [12], [-28], [-4], [35], [0], [2], [26], [-38], [14], [-13], [12], [8], [-14], [5], [-12], [-2], [-4], [-21], [-35], [24], [-27], [2], [-11], [-36], [-41], [8], [-34], [-13], [-26], [-35], [-30], [18], [-34], [-40], [-27], [8], [-25], [-22], [28], [24], [8], [31], [-26], [-6], [7], [-36], [-28], [30], [-32], [-5], [52], [-24], [2], [-22], [50], [-4], [-43], [22], [-51], [8], [-53], [42], [44], [-36], [-28], [-3], [-23], [24], [-36], [24], [-18], [-10], [27], [-19], [-4], [36], [43], [-12], [-9], [14], [-26], [-15], [40], [26], [37], [-45], [24], [-42], [-10], [8], [6], [24], [-17], [30], [24], [10], [52], [59], [-18], [38], [-34], [-14], [55], [61], [-43], [-8], [-24], [-24], [-17], [21], [-60], [0], [14], [50], [36], [1], [60], [-54], [-45], [-6], [-54], [-21], [31], [50], [-45], [34], [-66], [-42], [37], [28], [10], [5], [-2], [28], [-35], [-28], [22], [-40], [-16], [-6], [30], [-20], [-33], [-8], [-60], [59], [-24], [-28], [4], [24], [72], [4], [26], [67], [62], [-26], [56], [-7], [22], [-4], [-64], [68], [-3], [-74], [39], [-14], [-6], [-37], [-21], [-76], [-59], [-24], [-38], [42], [51], [-56], [-56], [8], [-20], [31], [6], [-3], [-50], [11], [12], [61], [14], [55], [34], [-53], [52], [28], [58], [-29], [1], [-21], [26], [72], [-28], [-2], [28], [-30], [0], [56], [-39], [10], [-22], [15], [77], [33], [-40], [-38], [42], [54], [56], [-3], [-11], [-40], [-26], [82], [61], [12], [-23], [-73], [-16], [67], [-8], [78], [-36], [-3], [76], [-74], [-35], [46], [66], [31], [65], [-40], [-60], [32], [-62], [-2], [-11], [54], [-55], [-78], [66], [84], [52], [19], [-18], [-44], [-28], [5], [-20], [-78], [18], [-12], [66], [26], [-25], [-39], [42], [39], [-71], [-38], [74], [21], [-86], [25], [58], [78], [8], [10], [50], [8], [-29], [-48], [-31], [-12], [-39], [-40], [28], [-85], [24], [-80], [-80], [84], [-22], [88], [30], [-2], [-23], [33], [-87], [-23], [-10], [-79], [-56], [74], [-18], [-46], [45], [93], [71], [49], [-44], [-6], [65], [6], [-48], [38], [-96], [-86], [38], [0], [74], [40], [90], [24], [-36], [-4], [14], [-23], [-24], [-34], [-33], [41], [36], [42], [-45], [20], [-64], [-24], [69], [52], [54], [-66], [-94], [-86], [39], [90], [85], [38], [-15], [83], [64], [-27], [69], [73], [84], [-33]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4598_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4598_2_a_n();".
// 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_4598_2_a_n(:prec:=1)
    chi := MakeCharacter_4598_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_4598_2_a_n();".
// 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_4598_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4598_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>,<7,R![-3, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;