
// Make newform 4650.2.a.z in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4650_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_4650_2_a_z();".
// 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_4650_2_a_z();".
// 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_4650_a();"
function MakeCharacter_4650_a()
    N := 4650;
    order := 1;
    char_gens := [3101, 2977, 1801];
    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_4650_a_Hecke(Kf)
    return MakeCharacter_4650_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], [-2], [0], [2], [0], [-4], [2], [-10], [-1], [-2], [2], [4], [4], [2], [-6], [-8], [8], [-8], [-10], [12], [4], [-14], [-4], [0], [-18], [4], [2], [6], [8], [-2], [-16], [10], [-12], [-16], [-18], [-4], [-18], [-2], [-12], [-20], [8], [-16], [6], [24], [4], [4], [12], [0], [-6], [-8], [18], [-12], [14], [-6], [14], [-12], [-22], [-2], [-4], [6], [-4], [-20], [6], [30], [14], [2], [32], [-10], [12], [24], [32], [-26], [24], [-38], [14], [26], [2], [-2], [-18], [-14], [24], [-2], [-16], [20], [2], [-18], [10], [-8], [4], [-4], [-4], [0], [-14], [-16], [26], [30], [-20], [-30], [-8], [30], [-12], [6], [-22], [32], [12], [18], [-32], [-2], [6], [-6], [38], [-30], [-8], [-10], [44], [34], [42], [6], [42], [-6], [-26], [-28], [-8], [-20], [20], [-8], [-26], [-22], [54], [30], [-32], [-22], [-6], [6], [-14], [12], [2], [42], [-40], [30], [36], [32], [-4], [24], [26], [-22], [22], [46], [54], [-18], [-44], [-48], [-12], [32], [8], [-34], [-20], [-50], [52], [8], [-32], [18], [-42], [6], [40], [-42], [50], [-54], [20], [32], [36], [-40], [-32], [58], [-28], [40], [46], [-8], [-42], [56], [-14], [-64], [16], [-10], [46], [4], [46], [52], [-42], [-4], [6], [42], [36], [-36], [30], [-34], [42], [36], [60], [16], [-42], [-50], [-66], [54], [-40], [52], [-34], [0], [30], [-26], [10], [-28], [-40], [38], [-58], [38], [-14], [-66], [52], [-40], [74], [-70], [-44], [-28], [26], [-56], [0], [30], [-66], [-44], [-40], [18], [20], [-42], [-62], [-6], [36], [0], [4], [-38], [0], [-32], [-16], [-16], [-28], [-24], [26], [52], [-70], [-38], [-2], [-54], [-46], [74], [-70], [-4], [-6], [24], [50], [12], [16], [2], [-20], [-32], [36], [-54], [44], [-26], [-54], [44], [70], [60], [24], [-30], [-28], [40], [-58], [-48], [0], [-24], [-72], [-50], [-28], [-72], [-70], [-42], [-80], [2], [80], [44], [12], [-18], [82], [-18], [-40], [-58], [-28], [4], [36], [-66], [60], [76], [-34], [44], [24], [-10], [-24], [46], [90], [-30], [-42], [68], [-12], [-50], [36], [24], [56], [50], [18], [-42], [0], [54], [-46], [2], [48], [-8], [-22], [42], [-82], [46], [-40], [0], [-84], [84], [92], [-6], [6], [-8], [66], [12], [70], [-8], [-94], [-62], [14], [-28], [-56], [-54], [50], [-18], [60], [60], [58], [-80], [40], [48], [-76], [-60], [-18], [70], [78], [2], [76], [22], [38], [84], [22], [-22], [-56], [-12], [-24], [16], [-34], [-36], [-4], [40], [-18], [2], [-46], [-8], [-20], [20], [38], [-72], [-64], [-26], [76], [30], [-14], [-10], [-6], [48], [-56], [-72], [-60], [50], [90], [-2], [56], [-58], [-42], [-78], [-14], [-16], [-46], [-18], [20], [24], [50], [-74], [48], [-98], [-20], [-60], [-48], [-18], [66], [-76], [-2], [22], [-30], [-66], [76], [-58], [60], [-26], [-2], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4650_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4650_2_a_z();".
// 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_4650_2_a_z(:prec:=1)
    chi := MakeCharacter_4650_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_4650_2_a_z();".
// 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_4650_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4650_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;