
// Make newform 4704.2.a.y in Magma, downloaded from the LMFDB on 09 May 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_y();".
// 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_y();".
// 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], [0], [0], [-2], [-1], [2], [-5], [6], [-8], [3], [-9], [-2], [1], [-8], [6], [-6], [2], [-5], [4], [11], [-5], [0], [-12], [-18], [6], [11], [-18], [-3], [-12], [9], [10], [-12], [15], [-20], [16], [22], [-20], [-12], [-18], [4], [-5], [20], [7], [-10], [12], [12], [-8], [-22], [-1], [4], [-22], [-10], [-6], [32], [-8], [26], [8], [-11], [-14], [31], [-28], [25], [10], [-31], [4], [-17], [-31], [-18], [14], [24], [32], [37], [-1], [-25], [2], [-8], [-15], [-22], [-5], [34], [-11], [12], [25], [0], [-22], [24], [1], [-12], [11], [-20], [-6], [7], [0], [-23], [-42], [20], [0], [13], [25], [28], [26], [-20], [26], [-7], [17], [-24], [-14], [20], [-3], [13], [10], [34], [5], [44], [-38], [-13], [-42], [-38], [-6], [-15], [35], [-36], [-18], [29], [-14], [10], [-30], [17], [39], [33], [54], [-31], [38], [50], [-23], [-42], [8], [-6], [-28], [28], [48], [32], [-24], [-49], [4], [21], [-8], [-44], [-18], [10], [54], [19], [6], [-1], [-22], [35], [16], [-51], [-26], [50], [14], [-13], [-26], [30], [2], [-55], [41], [-7], [-4], [-8], [6], [-12], [29], [7], [36], [15], [-40], [0], [37], [-7], [44], [49], [42], [-14], [16], [-58], [25], [-31], [2], [-5], [-8], [4], [-22], [66], [-48], [7], [-10], [54], [26], [34], [29], [-6], [-34], [-58], [-18], [0], [26], [20], [44], [39], [-32], [16], [30], [-42], [41], [56], [-4], [-30], [52], [25], [43], [-24], [-32], [-14], [-30], [-28], [18], [53], [12], [53], [-52], [-61], [-30], [-33], [-4], [-25], [-6], [-16], [-24], [8], [27], [-49], [10], [16], [-18], [-43], [-52], [-28], [-24], [1], [50], [-70], [31], [36], [-36], [-19], [9], [-12], [2], [-5], [-12], [-29], [45], [30], [-45], [2], [-56], [-8], [70], [-23], [56], [27], [-65], [-26], [-8], [8], [39], [-1], [30], [56], [-57], [54], [65], [-68], [-30], [29], [-80], [-65], [-40], [24], [60], [48], [20], [-61], [58], [-41], [-30], [-12], [-47], [-54], [-54], [-56], [64], [81], [-43], [-24], [39], [60], [-37], [-66], [40], [-74], [-76], [-24], [35], [-68], [-78], [-29], [46], [-21], [-29], [-38], [-61], [-18], [3], [71], [11], [-48], [24], [1], [-44], [-41], [-78], [43], [-56], [-22], [-76], [-5], [-61], [59], [-64], [-62], [-19], [-16], [22], [43], [-52], [-10], [-34], [40], [-30], [-38], [-36], [-41], [24], [68], [-54], [-50], [54], [-46], [20], [38], [-66], [-37], [-2], [-44], [-31], [-54], [18], [15], [-36], [-42], [-43], [-19], [70], [26], [2], [40], [-21], [38], [-72], [20], [78], [-40], [-56], [-1], [58], [19], [72], [5], [26], [16], [-63], [-38], [50], [-80], [-10], [-21], [-68], [-50], [-18], [8], [6], [-6], [23], [-62], [16], [-61], [92], [-62], [-12], [-96], [56], [45], [-50], [-66], [-101], [28], [46], [-50], [-11], [40], [72], [50], [48], [52], [-12], [-52], [44]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4704_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![2, 1]>,<13,R![1, 1]>,<19,R![5, 1]>,<31,R![-3, 1]>],Snew);
    return Vf;
end function;