
// Make newform 6566.2.a.i in Magma, downloaded from the LMFDB on 03 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6566_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_6566_2_a_i();".
// 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_6566_2_a_i();".
// 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_6566_a();"
function MakeCharacter_6566_a()
    N := 6566;
    order := 1;
    char_gens := [4021, 1275];
    v := [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_6566_a_Hecke(Kf)
    return MakeCharacter_6566_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]];
    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], [-3], [-3], [0], [-4], [-5], [2], [-6], [-9], [-1], [5], [-9], [-7], [10], [-4], [-12], [-14], [-2], [1], [-1], [-10], [-4], [6], [-10], [-14], [17], [6], [-12], [-10], [-16], [19], [-6], [-12], [17], [-14], [11], [0], [-13], [-8], [6], [2], [16], [12], [5], [-12], [-2], [-7], [-22], [4], [-7], [10], [24], [-8], [-13], [28], [-4], [20], [1], [-10], [18], [-14], [24], [8], [13], [15], [3], [20], [4], [6], [-20], [18], [-15], [-1], [-26], [8], [3], [-6], [-8], [24], [-21], [-30], [-23], [-16], [-2], [14], [-20], [-21], [-7], [-32], [0], [6], [-12], [26], [36], [18], [0], [-30], [15], [-24], [8], [22], [-33], [9], [31], [7], [29], [1], [-30], [-36], [-20], [8], [-25], [-15], [-14], [30], [12], [-4], [-47], [-34], [9], [1], [-38], [-3], [-38], [-36], [2], [-19], [-48], [-31], [-41], [-20], [21], [48], [-32], [6], [34], [-24], [12], [8], [-32], [55], [33], [5], [-48], [4], [-6], [-4], [-42], [50], [-13], [25], [14], [-16], [-28], [-3], [-25], [-2], [-11], [-23], [-18], [59], [41], [-35], [-6], [25], [-48], [24], [44], [-2], [-26], [-53], [-4], [-61], [6], [-4], [-22], [50], [50], [-13], [8], [-53], [42], [31], [-24], [24], [10], [24], [61], [34], [-31], [-62], [-59], [-14], [-57], [-14], [-30], [-30], [-26], [-45], [22], [-54], [56], [5], [19], [-68], [-24], [36], [7], [-42], [20], [62], [-61], [-44], [0], [3], [-26], [67], [46], [38], [-58], [40], [-56], [32], [16], [27], [71], [-24], [-66], [2], [30], [-46], [45], [-69], [44], [-40], [24], [-6], [-63], [39], [-2], [24], [76], [11], [72], [47], [5], [-28], [9], [30], [-36], [-34], [28], [-62], [-40], [9], [-42], [0], [50], [6], [64], [7], [45], [-11], [-19], [-58], [4], [18], [-50], [-26], [40], [78], [-16], [-83], [17], [12], [-23], [56], [-31], [27], [23], [2], [1], [27], [-53], [-28], [9], [53], [-31], [-24], [46], [-83], [-19], [-39], [16], [18], [-46], [-35], [21], [-78], [-43], [-6], [87], [59], [15], [-64], [-32], [-29], [-2], [0], [-46], [65], [14], [-9], [81], [36], [-25], [68], [-12], [58], [-54], [2], [-10], [-70], [-66], [71], [22], [-36], [-14], [-12], [30], [-1], [-78], [-53], [12], [52], [-66], [32], [40], [-42], [-54], [-43], [-50], [59], [5], [2], [22], [-25], [20], [54], [-42], [56], [-36], [33], [-61], [90], [-39], [-8], [-17], [-78], [-22], [-50], [-20], [-8], [7], [80], [-60], [34], [36], [-8], [-40], [50], [31], [0], [-40], [-56], [-70], [-32], [-14], [59], [28], [10], [70], [-30], [-78], [16], [-27], [-64], [-88], [-30], [-39], [-30], [-21], [-65], [-73], [10], [58], [-88], [44], [-50], [-42], [18], [-31], [14], [-22], [65], [-60], [32], [11], [56], [68], [-48], [-43], [36], [0], [-16], [-102], [5], [-38], [-16], [30], [-38], [74], [12], [-31], [-49], [78], [16], [87], [22], [21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6566_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6566_2_a_i();".
// 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_6566_2_a_i(:prec:=1)
    chi := MakeCharacter_6566_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_6566_2_a_i();".
// 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_6566_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6566_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![3, 1]>,<5,R![3, 1]>,<11,R![4, 1]>,<13,R![5, 1]>],Snew);
    return Vf;
end function;