
// Make newform 4730.2.a.g 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_g();".
// 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_g();".
// 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], [0], [1], [0], [-1], [-2], [2], [8], [4], [6], [0], [-10], [-6], [1], [4], [10], [12], [-2], [12], [0], [2], [4], [-4], [-14], [10], [18], [4], [12], [2], [14], [-16], [0], [22], [-4], [-18], [16], [6], [-16], [-8], [6], [-12], [22], [8], [-14], [-18], [-8], [0], [0], [-12], [30], [-22], [12], [22], [12], [-26], [-16], [-2], [-28], [22], [-6], [4], [6], [28], [-24], [-18], [2], [4], [2], [28], [6], [10], [12], [-20], [-10], [-28], [8], [-10], [-22], [18], [-10], [-28], [-26], [28], [38], [20], [-4], [2], [10], [-30], [-8], [8], [-28], [-20], [-16], [-36], [-8], [6], [-30], [4], [-6], [-12], [-10], [-4], [-30], [-32], [-18], [8], [42], [-24], [6], [0], [-2], [10], [-28], [8], [-6], [20], [8], [-10], [-20], [22], [10], [6], [-36], [12], [-22], [-18], [-24], [16], [46], [-24], [24], [0], [-10], [38], [-6], [14], [-4], [34], [-6], [40], [18], [4], [44], [-18], [-40], [-34], [-38], [36], [24], [38], [-46], [-44], [-48], [36], [8], [12], [-54], [-22], [-22], [12], [26], [32], [-52], [26], [-56], [-8], [22], [-2], [42], [24], [-2], [40], [42], [-48], [26], [-32], [38], [-16], [30], [-16], [4], [-2], [42], [16], [-58], [-2], [8], [50], [16], [42], [-44], [60], [-66], [60], [-26], [10], [-14], [-30], [-24], [18], [-24], [-2], [-2], [52], [-42], [-40], [-36], [-42], [4], [66], [62], [44], [28], [40], [-54], [64], [14], [24], [-54], [22], [28], [-30], [-52], [-60], [-22], [-30], [-16], [-32], [-16], [38], [-4], [-68], [14], [52], [-40], [-6], [6], [44], [32], [-56], [32], [12], [14], [-46], [36], [-8], [-68], [-4], [8], [78], [58], [12], [26], [46], [-64], [38], [28], [-18], [-70], [-64], [4], [-14], [-34], [-18], [-60], [30], [-6], [-4], [-10], [6], [80], [30], [-56], [-62], [24], [-28], [-42], [66], [-48], [40], [-40], [-24], [-42], [20], [48], [2], [86], [64], [-30], [-66], [-52], [42], [-64], [14], [82], [8], [-78], [-36], [4], [10], [-18], [-28], [52], [20], [70], [-60], [14], [-88], [-58], [48], [54], [74], [-16], [0], [-30], [60], [-52], [34], [-66], [-12], [38], [22], [-28], [42], [34], [-44], [-28], [-16], [86], [-74], [18], [-72], [-4], [28], [-60], [-2], [66], [-62], [-8], [10], [-58], [-42], [32], [74], [-84], [38], [4], [-48], [58], [-36], [-10], [6], [16], [50], [-14], [-56], [48], [-78], [36], [-58], [54], [-32], [-48], [20], [-26], [-82], [72], [70], [-20], [28], [8], [-34], [68], [-58], [-36], [-28], [34], [-90], [10], [-58], [-54], [16], [74], [4], [84], [80], [-38], [92], [28], [50], [-82], [-4], [52], [96], [-54], [52], [-62], [4], [78], [-70], [-102], [48], [-86], [-34], [16], [-2], [58], [16], [36], [98], [-42], [64], [-44], [98], [38], [84], [28], [50], [48], [-66], [38], [16], [-52], [-74], [-98], [-24], [94], [4], [0]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4730_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<7,R![0, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;