
// Make newform 4730.2.a.h in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4730_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_4730_2_a_h();".
// 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_4730_2_a_h();".
// 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_4730_a();"
function MakeCharacter_4730_a()
    N := 4730;
    order := 1;
    char_gens := [947, 431, 1981];
    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_4730_a_Hecke(Kf)
    return MakeCharacter_4730_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], [-1], [2], [-1], [-4], [0], [2], [-6], [-9], [-4], [-4], [0], [1], [-6], [3], [0], [-1], [2], [12], [11], [-7], [15], [0], [-13], [6], [-10], [3], [-16], [0], [-7], [0], [12], [-13], [15], [-22], [-22], [-13], [3], [-24], [-15], [23], [9], [20], [12], [-4], [20], [-19], [6], [23], [27], [21], [2], [-6], [6], [-24], [-15], [-4], [-10], [24], [-13], [0], [11], [-18], [-4], [15], [20], [-22], [-36], [26], [-9], [3], [26], [-13], [2], [-33], [-6], [29], [3], [-13], [-15], [8], [-36], [-16], [8], [18], [-30], [-10], [-12], [8], [-12], [21], [14], [-30], [5], [12], [30], [18], [8], [14], [17], [-6], [12], [-6], [14], [-4], [36], [21], [-36], [-10], [14], [-34], [-18], [-28], [23], [42], [-22], [33], [42], [-33], [-25], [41], [45], [48], [-37], [-18], [-7], [42], [-1], [-13], [-16], [30], [23], [-46], [-21], [2], [0], [32], [-30], [-36], [38], [-24], [2], [45], [20], [-21], [-46], [0], [-40], [-39], [20], [3], [20], [0], [-10], [-45], [-25], [36], [-55], [48], [30], [-57], [47], [12], [3], [-48], [-1], [-10], [-43], [3], [54], [-22], [-36], [-16], [-7], [-30], [-28], [54], [32], [-19], [-25], [24], [-22], [-51], [36], [6], [35], [-28], [20], [42], [-31], [36], [14], [-12], [-48], [60], [-10], [41], [-24], [0], [-66], [8], [56], [14], [12], [42], [-37], [-3], [15], [2], [-19], [-27], [8], [72], [0], [35], [-58], [-45], [57], [-18], [5], [23], [-42], [38], [12], [8], [-15], [-33], [26], [-48], [2], [-7], [-40], [-63], [-22], [-24], [-58], [-42], [-72], [30], [-21], [14], [-46], [41], [-51], [0], [17], [12], [-4], [-3], [-58], [30], [-48], [56], [-3], [-78], [-34], [44], [24], [56], [-34], [18], [-16], [14], [-24], [-1], [78], [45], [-4], [-12], [17], [-28], [44], [-76], [14], [-40], [-18], [-55], [38], [-30], [36], [-22], [39], [-58], [-28], [-57], [17], [57], [-64], [-72], [18], [-6], [18], [-30], [-28], [-84], [14], [-6], [-45], [23], [-10], [-45], [-25], [66], [20], [38], [54], [-34], [87], [-73], [0], [-54], [-60], [11], [-24], [2], [6], [33], [-73], [-18], [-73], [-82], [-66], [38], [-42], [65], [20], [44], [-87], [42], [-34], [78], [68], [54], [-79], [42], [20], [-6], [14], [38], [-19], [30], [-78], [-31], [-18], [-81], [20], [80], [90], [-51], [-37], [-64], [45], [-46], [-58], [54], [-51], [30], [78], [42], [71], [-90], [-96], [-36], [-22], [-64], [-27], [56], [-19], [63], [-19], [-24], [-6], [-43], [-64], [75], [-15], [92], [78], [-46], [-18], [39], [80], [21], [68], [90], [8], [11], [23], [-78], [71], [54], [-3], [38], [45], [-64], [-19], [-42], [-31], [-54], [50], [51], [-67], [-42], [21], [-34], [32], [-72], [71], [54], [80], [-93], [51], [20], [-43], [21], [-72], [-10], [75], [-54], [-30], [2], [15], [75], [-34], [-6], [60], [6], [-34], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4730_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4730_2_a_h();".
// 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_4730_2_a_h(:prec:=1)
    chi := MakeCharacter_4730_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_4730_2_a_h();".
// 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_4730_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4730_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 1]>,<7,R![-2, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;