
// Make newform 4704.2.a.d 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_d();".
// 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_d();".
// 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], [-2], [0], [2], [0], [-2], [0], [-2], [6], [-4], [6], [2], [0], [0], [-6], [12], [-12], [12], [10], [-12], [-12], [12], [-14], [12], [-14], [4], [10], [2], [-6], [-12], [-12], [6], [4], [-6], [0], [-12], [-12], [0], [-14], [-2], [0], [2], [6], [-6], [-24], [-24], [-24], [12], [-24], [6], [-2], [-12], [12], [-22], [-10], [-10], [20], [-6], [-18], [24], [10], [24], [-24], [0], [-6], [-24], [14], [-26], [-12], [-10], [34], [0], [14], [-24], [24], [-18], [-12], [30], [-12], [12], [-6], [2], [24], [-24], [14], [18], [10], [2], [-36], [-12], [-24], [24], [38], [-24], [-24], [14], [14], [20], [18], [-36], [-30], [-12], [30], [-12], [24], [-12], [22], [-14], [-24], [24], [2], [-18], [44], [-12], [6], [24], [-48], [-42], [10], [36], [-6], [2], [38], [-44], [6], [10], [24], [-28], [48], [36], [26], [-48], [-38], [2], [-24], [2], [-4], [-22], [18], [20], [-54], [12], [-22], [0], [24], [36], [50], [24], [22], [-22], [-34], [-24], [-48], [48], [34], [-24], [50], [-48], [46], [26], [-54], [-12], [12], [30], [0], [-24], [36], [-18], [58], [-14], [-12], [-46], [-54], [24], [14], [-36], [-18], [52], [36], [12], [12], [6], [22], [-26], [10], [-2], [-28], [-6], [48], [36], [2], [0], [-38], [34], [-10], [-6], [30], [2], [-24], [-54], [-48], [24], [60], [-12], [10], [48], [-14], [-30], [-24], [-34], [-58], [-24], [36], [0], [0], [-24], [22], [62], [54], [58], [-4], [66], [48], [12], [54], [58], [22], [-44], [-58], [-14], [-48], [24], [42], [-72], [48], [-36], [-54], [-34], [72], [-34], [52], [-24], [18], [34], [-72], [52], [-36], [36], [14], [-50], [-34], [2], [-12], [-38], [50], [-42], [24], [2], [-48], [-12], [14], [-24], [36], [-34], [-28], [66], [-58], [-36], [-18], [-36], [-48], [-60], [72], [12], [-52], [-62], [66], [-18], [-36], [24], [-12], [0], [-12], [68], [82], [34], [-30], [28], [2], [-54], [-60], [-66], [-36], [-10], [10], [48], [70], [36], [-44], [24], [-18], [60], [26], [36], [-2], [62], [24], [-62], [-6], [-24], [54], [6], [-24], [-34], [-12], [-12], [-50], [36], [6], [-48], [18], [-14], [72], [42], [-84], [12], [0], [-22], [-42], [-82], [-42], [28], [-84], [12], [-60], [-70], [-62], [-36], [-20], [46], [-18], [-26], [72], [78], [-62], [24], [48], [-48], [-86], [-24], [54], [-6], [48], [14], [94], [24], [-36], [42], [-14], [62], [-70], [82], [-22], [20], [18], [-26], [-60], [78], [-2], [0], [-94], [42], [68], [66], [-12], [-58], [-72], [-10], [36], [-22], [-30], [72], [18], [-4], [-72], [-48], [0], [60], [24], [-30], [38], [82], [48], [-38], [-58], [24], [-50], [12], [78], [-84], [90], [36], [-46], [-2], [72], [18], [90], [4], [12], [-24], [-66], [-34], [0], [90], [-34], [38], [-28], [-82], [-96], [30], [-36], [10], [-36], [72], [34], [-34], [6], [-20], [-48]];
    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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4704_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![-2, 1]>,<13,R![0, 1]>,<19,R![0, 1]>,<31,R![4, 1]>],Snew);
    return Vf;
end function;