
// Make newform 4598.2.a.q in Magma, downloaded from the LMFDB on 24 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_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_4598_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_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], [1], [0], [1], [0], [-2], [-6], [-1], [-9], [-3], [8], [-7], [0], [10], [9], [-9], [-9], [-8], [-4], [12], [-2], [-14], [12], [6], [-16], [0], [14], [15], [19], [12], [-8], [18], [-6], [-8], [12], [10], [-22], [-10], [-6], [-3], [15], [-22], [-9], [22], [12], [-16], [-5], [-16], [-12], [-4], [-27], [-9], [-2], [18], [-12], [24], [-3], [16], [4], [-6], [16], [-33], [-17], [-12], [17], [-18], [5], [-32], [12], [10], [-15], [12], [23], [19], [-13], [-12], [6], [-34], [-12], [4], [24], [-1], [18], [-34], [16], [-6], [6], [-17], [-24], [-16], [-6], [21], [2], [-24], [-10], [9], [-27], [-6], [7], [10], [-23], [-30], [-9], [0], [-38], [2], [24], [-15], [-12], [10], [10], [-14], [45], [14], [35], [24], [26], [45], [-6], [12], [-19], [46], [3], [-36], [-40], [-24], [-16], [-24], [-19], [4], [-14], [-30], [20], [-46], [21], [7], [21], [7], [30], [-45], [-35], [48], [23], [-36], [38], [24], [46], [42], [-40], [36], [-14], [45], [-10], [0], [-19], [18], [16], [45], [19], [-18], [12], [30], [-59], [39], [-30], [0], [-28], [52], [-38], [-6], [36], [-13], [-48], [-23], [-10], [-45], [10], [42], [-26], [-20], [-19], [-33], [-31], [-48], [57], [24], [7], [17], [-50], [-45], [-28], [36], [20], [-18], [18], [60], [-50], [-46], [-3], [12], [51], [-8], [50], [40], [36], [-18], [47], [-45], [-9], [47], [43], [54], [-19], [-33], [0], [20], [7], [-36], [12], [-24], [-5], [4], [-54], [32], [9], [-20], [-72], [-48], [-44], [6], [-64], [49], [70], [-12], [38], [51], [14], [-42], [-24], [-24], [-39], [-2], [5], [47], [0], [-9], [47], [-12], [-59], [-57], [-44], [-51], [-12], [-52], [54], [-54], [-22], [55], [-6], [-47], [70], [-57], [40], [70], [42], [20], [-27], [-42], [1], [66], [-73], [65], [-31], [-32], [-44], [-25], [66], [-35], [-14], [-60], [-12], [-43], [-72], [-47], [-44], [15], [-34], [45], [-13], [30], [60], [-12], [24], [33], [37], [-30], [-16], [6], [39], [10], [49], [-30], [-29], [-48], [-10], [-16], [-87], [5], [-24], [-2], [21], [12], [-30], [-73], [-63], [-62], [54], [36], [-61], [30], [-5], [-49], [66], [2], [-63], [20], [65], [-19], [24], [0], [94], [9], [-29], [-78], [-20], [-54], [-7], [36], [-10], [7], [17], [51], [33], [2], [3], [81], [44], [-4], [-24], [-24], [-20], [-34], [-6], [-26], [-17], [12], [-33], [18], [57], [54], [28], [87], [6], [-21], [-34], [-46], [-18], [40], [34], [-12], [20], [-48], [33], [61], [-46], [-60], [0], [-14], [3], [-26], [-3], [-87], [-32], [-87], [-5], [6], [8], [32], [-56], [48], [44], [9], [30], [-28], [-6], [94], [52], [-30], [-4], [-24], [94], [-3], [40], [-48], [51], [-56], [62], [-6], [-31], [36], [-80], [-48], [27], [-32], [-5], [-36], [-24], [26], [-18], [36], [21], [-5], [-48], [48], [86], [33], [51], [-51], [92], [-87]];
    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_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_4598_2_a_q(: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_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_4598_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4598_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<5,R![0, 1]>,<7,R![-1, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;