
// Make newform 4730.2.a.b in Magma, downloaded from the LMFDB on 25 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_b();".
// 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_b();".
// 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], [-4], [8], [0], [-6], [0], [8], [6], [-6], [1], [2], [6], [0], [8], [2], [-6], [-14], [-8], [16], [-6], [-2], [-10], [-10], [8], [-10], [-2], [-20], [-20], [-6], [-12], [20], [12], [2], [-8], [8], [-12], [-6], [10], [-18], [-4], [-16], [2], [-12], [8], [20], [-14], [6], [16], [0], [-16], [2], [16], [-18], [-8], [2], [-18], [12], [-12], [-32], [4], [-22], [34], [-22], [-8], [-12], [-16], [-14], [8], [38], [26], [-32], [32], [-30], [-22], [30], [4], [-30], [-30], [-40], [-38], [16], [-6], [10], [10], [10], [-32], [-28], [8], [18], [36], [-18], [40], [-6], [-42], [-4], [-38], [-8], [-28], [24], [-14], [-28], [-22], [12], [14], [12], [32], [0], [28], [30], [-4], [-30], [26], [14], [-28], [26], [12], [-22], [30], [34], [-6], [-34], [-6], [22], [-20], [8], [-22], [-32], [-16], [2], [22], [36], [-26], [-6], [-52], [-6], [42], [-32], [30], [-42], [-36], [50], [-42], [-8], [36], [-26], [4], [28], [-14], [-50], [-40], [-22], [30], [24], [14], [58], [10], [-30], [46], [8], [0], [54], [24], [18], [-18], [-52], [6], [36], [50], [24], [4], [-10], [6], [40], [38], [-24], [16], [8], [-12], [46], [10], [36], [30], [18], [-44], [18], [48], [6], [-36], [16], [-54], [36], [-34], [50], [-18], [60], [-24], [-54], [44], [34], [0], [-6], [30], [-22], [-16], [16], [20], [66], [-42], [-22], [54], [-48], [6], [0], [-60], [52], [14], [-20], [32], [54], [-18], [-28], [34], [-74], [-38], [-56], [4], [66], [12], [16], [44], [66], [-4], [-10], [-64], [26], [20], [-24], [0], [-22], [-62], [74], [-64], [24], [76], [-60], [-64], [16], [46], [-74], [50], [54], [56], [-22], [-28], [30], [-12], [16], [12], [-30], [8], [-18], [-38], [-26], [-58], [-4], [40], [22], [-4], [-10], [28], [-28], [24], [-50], [4], [-30], [-8], [12], [24], [-64], [-16], [-20], [-42], [6], [-22], [38], [82], [18], [-42], [-76], [-8], [-36], [-30], [-64], [-58], [-44], [64], [60], [18], [-24], [-42], [-38], [-42], [-58], [26], [-30], [-62], [24], [-14], [82], [-36], [60], [-6], [-8], [-48], [-90], [-32], [68], [-58], [-52], [14], [38], [38], [-30], [-50], [24], [-70], [-64], [-38], [-56], [28], [-20], [42], [34], [34], [50], [88], [2], [-58], [28], [82], [46], [-20], [70], [-86], [28], [-42], [68], [-86], [-82], [-24], [90], [18], [14], [76], [-24], [-18], [-18], [-32], [32], [4], [74], [-26], [-66], [56], [-92], [-46], [-10], [56], [84], [-80], [2], [-22], [-48], [36], [48], [-78], [-34], [42], [36], [-66], [-76], [-18], [-70], [26], [-76], [6], [10], [-14], [-14], [-98], [-66], [28], [56], [-30], [-82], [100], [2], [70], [-8], [-38], [48], [-8], [14], [74], [24], [-20], [-72], [-66], [12], [68], [54], [18], [24], [30], [-38], [24], [-2], [-84], [-48], [-76], [70], [18], [-36], [76], [8], [40]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : 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![4, 1]>],Snew);
    return Vf;
end function;