
// Make newform 6699.2.a.d in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6699_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_6699_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_6699_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_6699_a();"
function MakeCharacter_6699_a()
    N := 6699;
    order := 1;
    char_gens := [2234, 5743, 1828, 6469];
    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_6699_a_Hecke(Kf)
    return MakeCharacter_6699_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], [2], [-1], [1], [-2], [-2], [0], [-8], [-1], [8], [-10], [6], [4], [8], [14], [0], [6], [-4], [-8], [2], [8], [-4], [6], [-2], [-6], [-16], [-8], [14], [14], [-16], [12], [-2], [-4], [-6], [16], [10], [16], [16], [6], [-4], [14], [0], [14], [18], [16], [28], [16], [-12], [10], [14], [-4], [2], [-4], [-2], [-24], [-14], [20], [-26], [-2], [-4], [18], [8], [0], [34], [26], [32], [14], [-8], [14], [6], [24], [-16], [-10], [32], [12], [-14], [-2], [18], [-30], [24], [-18], [-12], [-26], [-8], [4], [-2], [2], [26], [32], [12], [-24], [-32], [-36], [-4], [-40], [2], [6], [-44], [-6], [-20], [18], [-12], [22], [20], [-2], [-24], [18], [24], [-14], [-4], [14], [38], [-28], [24], [22], [4], [-12], [42], [-20], [-2], [10], [-22], [-36], [44], [-6], [22], [-36], [-8], [-34], [28], [0], [12], [6], [34], [-46], [42], [-44], [2], [22], [-20], [2], [4], [-44], [10], [24], [54], [-30], [-28], [-16], [-42], [30], [4], [-40], [-16], [48], [-40], [22], [26], [-42], [-36], [-2], [8], [12], [42], [-48], [48], [22], [34], [26], [-32], [-58], [12], [14], [-32], [-26], [-20], [10], [28], [46], [-36], [36], [6], [42], [56], [50], [-30], [20], [30], [-24], [-14], [12], [-8], [-30], [4], [-66], [-2], [46], [-6], [24], [-46], [-40], [-62], [-38], [-28], [18], [-40], [0], [-18], [12], [-42], [42], [4], [-12], [0], [-62], [-24], [6], [48], [-22], [-22], [-64], [46], [-44], [-4], [10], [38], [8], [-28], [-16], [14], [-52], [32], [-58], [-28], [-32], [-18], [74], [20], [16], [-60], [-60], [-24], [-2], [42], [56], [48], [-56], [44], [40], [-14], [18], [48], [-26], [-14], [8], [-26], [32], [10], [74], [32], [28], [-26], [62], [-2], [12], [-14], [54], [-52], [-70], [-10], [76], [-38], [-40], [10], [-24], [84], [62], [-50], [20], [64], [72], [64], [6], [-32], [-8], [2], [42], [56], [82], [6], [24], [-34], [60], [34], [46], [-72], [42], [-12], [-8], [-70], [58], [-72], [-28], [-16], [34], [-36], [70], [16], [-74], [-24], [2], [6], [52], [-12], [-54], [-84], [84], [-62], [46], [-32], [-38], [30], [-68], [6], [30], [60], [76], [-48], [42], [2], [46], [8], [-12], [12], [-80], [-2], [6], [70], [24], [86], [42], [74], [0], [6], [12], [66], [56], [-72], [-38], [-28], [10], [2], [-8], [-46], [30], [48], [-60], [-18], [0], [-10], [-54], [-8], [-80], [-44], [-6], [58], [-88], [70], [4], [-20], [84], [-6], [-32], [62], [-12], [72], [-54], [42], [58], [-30], [-62], [-32], [-66], [76], [60], [-12], [-66], [-4], [-48], [-54], [74], [28], [-20], [8], [58], [-32], [70], [-28], [10], [-18], [-10], [48], [6], [-82], [-8], [14], [78], [92], [-48], [66], [-14], [12], [44], [30], [-78], [0], [-24], [78], [96], [-66], [46], [8], [60], [74], [42], [-84], [-74], [-28], [-56]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6699_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6699_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_6699_2_a_d(:prec:=1)
    chi := MakeCharacter_6699_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_6699_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_6699_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6699_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![-2, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;