
// Make newform 4704.2.a.l in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4704_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_4704_2_a_l();".
// 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_4704_2_a_l();".
// 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_4704_a();"
function MakeCharacter_4704_a()
    N := 4704;
    order := 1;
    char_gens := [1471, 1765, 3137, 4609];
    v := [1, 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_4704_a_Hecke(Kf)
    return MakeCharacter_4704_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 := [[0], [-1], [1], [0], [1], [0], [-8], [-4], [4], [-5], [7], [8], [4], [10], [6], [-1], [9], [-2], [2], [6], [2], [-9], [-3], [-6], [-1], [2], [-16], [15], [10], [4], [13], [-19], [18], [10], [6], [7], [20], [20], [-18], [-18], [20], [20], [0], [-11], [18], [20], [22], [-19], [7], [24], [-8], [12], [15], [1], [18], [-30], [-17], [11], [-24], [6], [-6], [3], [-4], [28], [1], [-5], [20], [1], [28], [6], [-24], [6], [-3], [-4], [-28], [-34], [-34], [4], [-32], [23], [24], [2], [0], [30], [19], [29], [-24], [15], [-14], [-40], [-12], [-18], [-31], [-27], [34], [-12], [-9], [14], [8], [22], [8], [9], [29], [-12], [14], [23], [15], [-36], [30], [19], [25], [44], [6], [-10], [33], [38], [2], [10], [-31], [16], [18], [-19], [-27], [-37], [-12], [27], [14], [10], [19], [-14], [34], [-38], [-7], [-26], [0], [-19], [18], [-38], [-51], [-32], [-2], [-1], [16], [-43], [-20], [-8], [42], [-6], [-18], [-46], [32], [18], [-28], [4], [-12], [-6], [-32], [34], [-29], [-51], [40], [-6], [7], [59], [50], [-24], [-5], [10], [-1], [-3], [-57], [42], [-4], [-35], [-23], [52], [-10], [-51], [40], [42], [-41], [15], [50], [22], [14], [9], [-46], [46], [51], [-4], [41], [-57], [-12], [-18], [-19], [-10], [46], [-26], [-26], [-34], [-15], [0], [-38], [10], [-24], [-69], [-16], [-9], [-48], [-18], [-38], [61], [-1], [39], [-20], [-22], [13], [28], [58], [2], [0], [-56], [-42], [19], [-61], [-8], [18], [-36], [-13], [-36], [2], [44], [-33], [42], [14], [-24], [-29], [13], [12], [44], [4], [60], [69], [-50], [24], [0], [63], [20], [-32], [-12], [-26], [-10], [14], [-38], [39], [-39], [-26], [-64], [-57], [13], [-8], [69], [-38], [-64], [30], [46], [53], [72], [-40], [5], [64], [-18], [-83], [49], [53], [-8], [-33], [-44], [-66], [40], [-18], [59], [-56], [-16], [-2], [32], [-65], [-27], [17], [-20], [6], [87], [52], [-12], [16], [37], [-48], [-10], [-76], [-42], [-10], [45], [25], [-51], [-22], [34], [69], [30], [58], [10], [-42], [-78], [-24], [-56], [32], [-42], [-15], [40], [-3], [16], [24], [-22], [-77], [0], [-12], [43], [50], [-14], [18], [-79], [-10], [-19], [-40], [48], [22], [4], [-28], [42], [-15], [5], [30], [-72], [75], [37], [-45], [75], [-74], [-2], [-38], [-69], [36], [19], [-62], [81], [-14], [-62], [66], [1], [-60], [86], [62], [-32], [74], [-84], [84], [-21], [-14], [-88], [-31], [-33], [8], [66], [-5], [16], [20], [32], [24], [43], [14], [-90], [-50], [-46], [72], [-48], [-2], [90], [80], [-76], [32], [36], [9], [38], [21], [-58], [30], [23], [33], [54], [20], [45], [-76], [84], [-4], [42], [79], [-27], [2], [-86], [-14], [-19], [55], [65], [11], [-70], [-14], [-33], [48], [-80], [-40], [18], [49], [-18], [-54], [93], [34], [10], [47], [24], [30], [28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4704_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4704_2_a_l();".
// 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_4704_2_a_l(:prec:=1)
    chi := MakeCharacter_4704_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_4704_2_a_l();".
// 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_4704_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4704_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<11,R![-1, 1]>,<13,R![0, 1]>,<19,R![4, 1]>,<31,R![-7, 1]>],Snew);
    return Vf;
end function;