
// Make newform 3969.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3969_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_3969_2_a_a();".
// 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_3969_2_a_a();".
// 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_3969_a();"
function MakeCharacter_3969_a()
    N := 3969;
    order := 1;
    char_gens := [2108, 3727];
    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_3969_a_Hecke(Kf)
    return MakeCharacter_3969_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], [0], [-1], [0], [-5], [5], [3], [-1], [-3], [1], [0], [3], [-5], [-1], [0], [9], [0], [14], [4], [12], [-3], [8], [-9], [-13], [9], [-17], [1], [-17], [-9], [1], [-12], [-1], [9], [-9], [-3], [5], [14], [-11], [-19], [-14], [-19], [14], [-8], [-10], [-2], [-3], [13], [-19], [-3], [1], [-3], [15], [-11], [-28], [-29], [-5], [3], [-1], [19], [29], [-28], [-5], [-28], [0], [-14], [6], [8], [-29], [-4], [-19], [11], [11], [3], [-25], [-12], [27], [9], [-15], [-3], [-14], [9], [-1], [9], [14], [0], [-36], [-30], [22], [19], [13], [-27], [25], [19], [-13], [31], [0], [-29], [3], [-1], [-25], [-29], [37], [-28], [34], [32], [-31], [-37], [15], [24], [9], [1], [19], [-27], [-25], [-40], [9], [19], [31], [-3], [-27], [14], [-29], [42], [9], [28], [-30], [-6], [-27], [-47], [-27], [-9], [15], [31], [2], [27], [-23], [31], [-28], [23], [9], [28], [-22], [-24], [12], [25], [-37], [37], [11], [1], [39], [-53], [-42], [44], [-29], [5], [-27], [17], [-14], [-42], [-14], [20], [26], [13], [57], [-18], [-3], [-37], [17], [-26], [55], [51], [33], [-3], [47], [-56], [-5], [55], [37], [0], [14], [8], [-9], [-1], [43], [24], [-17], [19], [59], [-53], [-25], [41], [-27], [17], [42], [37], [42], [-10], [-6], [-9], [57], [38], [37], [-27], [-67], [28], [-53], [-57], [-3], [57], [56], [3], [-33], [-29], [-28], [-56], [-70], [-52], [11], [-17], [-27], [59], [47], [-31], [-67], [37], [-1], [14], [-19], [-56], [-36], [46], [-31], [41], [10], [9], [-3], [-55], [-31], [15], [-56], [-33], [-57], [0], [31], [-33], [-28], [-28], [28], [4], [12], [-1], [-69], [-32], [-19], [11], [11], [47], [3], [42], [13], [5], [-55], [70], [70], [70], [28], [54], [-37], [13], [-74], [-55], [47], [17], [-25], [5], [55], [25], [-9], [-53], [-56], [-56], [60], [19], [-19], [1], [39], [19], [-27], [31], [-42], [-47], [-25], [-3], [-75], [-29], [-70], [-28], [19], [1], [-47], [57], [-3], [19], [-41], [3], [30], [-33], [33], [25], [31], [0], [65], [-78], [60], [-16], [-59], [-9], [24], [-19], [57], [-53], [31], [42], [-40], [-75], [-43], [73], [28], [20], [54], [-62], [66], [-47], [-3], [-65], [56], [2], [-57], [-23], [-85], [-65], [29], [51], [-29], [62], [-44], [-31], [-20], [-9], [-41], [29], [-9], [85], [87], [31], [70], [-57], [3], [53], [57], [27], [-70], [-75], [20], [-59], [22], [-65], [-65], [30], [51], [27], [81], [-27], [28], [-53], [-28], [40], [25], [99], [93], [67], [57], [100], [65], [-9], [-53], [89], [14], [-67], [9], [55], [-42], [-75], [28], [20], [-38], [-31], [75], [-15], [-18], [85], [-31], [31], [27], [87], [83], [-93], [29], [28], [84], [-98], [-22], [-100], [-20], [-57], [15], [-45], [-53], [-42], [27], [9], [-43], [-55], [-47], [-14], [42], [76], [-30], [-87], [-87]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3969_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3969_2_a_a();".
// 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_3969_2_a_a(:prec:=1)
    chi := MakeCharacter_3969_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_3969_2_a_a();".
// 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_3969_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3969_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![1, 1]>,<11,R![5, 1]>,<13,R![-5, 1]>],Snew);
    return Vf;
end function;