
// Make newform 8022.2.a.i in Magma, downloaded from the LMFDB on 19 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8022_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_8022_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_8022_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_8022_a();"
function MakeCharacter_8022_a()
    N := 8022;
    order := 1;
    char_gens := [2675, 6877, 2311];
    v := [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_8022_a_Hecke(Kf)
    return MakeCharacter_8022_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], [3], [-1], [0], [4], [-7], [0], [-3], [5], [6], [8], [0], [4], [3], [-5], [-6], [-5], [12], [8], [7], [6], [-3], [-6], [16], [0], [19], [-4], [7], [-3], [8], [15], [-22], [-16], [4], [15], [-13], [-10], [-8], [12], [-2], [-7], [1], [-3], [-24], [0], [-14], [-23], [10], [-9], [-9], [0], [-28], [-2], [-8], [-21], [-18], [31], [3], [24], [4], [21], [-8], [4], [-25], [5], [-28], [-28], [-17], [22], [-7], [34], [4], [-4], [-19], [-8], [35], [2], [-3], [-24], [-20], [-13], [31], [0], [-20], [0], [-24], [11], [-2], [12], [28], [-6], [12], [-11], [-40], [-16], [8], [-18], [20], [34], [32], [21], [-3], [-6], [21], [-26], [23], [-42], [-39], [-14], [13], [-22], [33], [-29], [4], [-10], [28], [-12], [-26], [-36], [-25], [-1], [15], [-8], [20], [-6], [35], [-47], [40], [18], [-13], [-24], [17], [23], [-8], [-14], [-6], [47], [52], [10], [50], [-53], [-14], [52], [16], [14], [47], [14], [-42], [16], [38], [13], [29], [-15], [-9], [23], [49], [-6], [-22], [-15], [24], [9], [-14], [-24], [-18], [-21], [36], [-31], [-37], [-48], [25], [3], [-4], [-19], [10], [-14], [-18], [37], [37], [27], [-17], [-36], [41], [28], [60], [-54], [-30], [-54], [40], [30], [10], [-43], [-12], [26], [10], [-44], [56], [47], [-12], [20], [-26], [3], [5], [-60], [27], [-48], [-16], [14], [10], [38], [46], [26], [-65], [-60], [-27], [-22], [-19], [21], [33], [42], [-20], [14], [-24], [-46], [-54], [-15], [-1], [-16], [-6], [60], [-2], [6], [-26], [6], [-56], [18], [37], [10], [-59], [-52], [38], [-23], [-69], [-42], [-63], [-19], [-56], [-16], [4], [-32], [-57], [36], [36], [-58], [-14], [30], [-52], [-12], [-53], [46], [74], [-36], [-26], [-26], [17], [-40], [54], [-5], [-4], [39], [-26], [-47], [26], [70], [-34], [19], [-27], [68], [-2], [-60], [-28], [3], [18], [-34], [14], [56], [46], [-4], [-63], [-15], [50], [-39], [-40], [53], [-41], [-18], [-2], [-46], [30], [4], [26], [39], [21], [-4], [20], [59], [0], [-14], [11], [28], [24], [81], [86], [-76], [-9], [-7], [5], [35], [60], [19], [-60], [67], [65], [57], [60], [-56], [22], [85], [-86], [30], [-87], [-34], [-51], [-48], [70], [80], [-94], [-27], [58], [36], [54], [22], [33], [15], [-5], [-24], [76], [3], [10], [-22], [4], [-38], [57], [-56], [10], [-70], [-52], [8], [-48], [-22], [24], [-78], [-18], [56], [-18], [60], [-48], [0], [56], [32], [10], [39], [95], [-24], [16], [60], [-77], [-53], [66], [-28], [15], [54], [-55], [-30], [64], [4], [13], [16], [-44], [-12], [-49], [-10], [4], [12], [-60], [2], [21], [60], [15], [-64], [70], [66], [61], [57], [-1], [-11], [-52], [-43], [97], [48], [-70], [60], [71], [46], [-2], [12], [12], [38], [-5], [29], [-59], [26], [66], [12], [29], [-63], [-8], [64], [76], [-40], [-44], [30], [-18], [-86], [-4], [-102], [-11], [43], [67]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8022_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8022_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_8022_2_a_i(:prec:=1)
    chi := MakeCharacter_8022_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(3067) - 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_8022_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_8022_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8022_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-3, 1]>,<11,R![0, 1]>,<13,R![-4, 1]>],Snew);
    return Vf;
end function;