
// Make newform 3969.2.a.q 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_q();".
// 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_q();".
// 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
        poly := [-1, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, -1, 0], [0, 0, 0], [-1, -1, 1], [0, 0, 0], [2, -1, 1], [1, -2, 4], [-2, 0, 1], [1, 2, -1], [4, 1, 2], [3, 5, -4], [1, -3, -3], [-1, 3, 3], [0, -1, -1], [-1, -1, -1], [-1, -2, -3], [2, -3, 2], [1, 5, 0], [-2, -3, 0], [-4, 0, 3], [3, 3, -6], [7, -4, -1], [-7, 0, 3], [6, 4, 1], [-4, 3, -7], [1, 7, -8], [7, -2, -1], [-2, 3, 0], [1, 0, 4], [5, 0, -3], [12, -2, 4], [-4, -13, 5], [5, -7, 10], [5, -7, 7], [-5, 10, -5], [-2, 7, -7], [-1, 9, -12], [1, -10, 5], [2, 0, -3], [3, 14, -4], [-13, 7, 3], [-7, 0, -1], [7, 3, 3], [1, -1, 9], [5, -3, 0], [15, 3, -6], [-2, 3, 0], [-1, 2, 2], [-5, -6, 6], [7, 1, 2], [-5, 12, 0], [7, 9, -5], [3, -8, -2], [10, 3, 0], [6, -6, -9], [-16, -4, -2], [-1, 14, -17], [7, 9, -2], [-8, 3, -3], [2, 6, 3], [18, 8, -7], [-5, 15, -3], [-3, -7, 2], [-2, -8, 7], [16, -1, -3], [1, 17, -10], [-8, 11, -3], [14, 4, 1], [5, 3, 12], [5, -8, 15], [-8, 10, -8], [-8, -5, 2], [-2, -6, -5], [-5, 5, 5], [5, 1, -5], [-1, 5, -10], [11, -1, -11], [-7, -4, 13], [19, 3, 3], [4, -11, -7], [4, -12, 18], [12, -13, 8], [-13, 6, -15], [5, -15, -4], [-17, 10, -14], [-11, 11, 8], [-2, -4, -6], [0, 6, -3], [-19, 12, -15], [-6, -2, 7], [-4, -3, 6], [23, -3, 11], [25, 4, -7], [-7, 1, 16], [-9, 13, 7], [2, 6, -18], [12, -9, -9], [8, 11, -2], [13, 6, -2], [-11, -15, 12], [11, 10, -2], [8, -1, 20], [-1, 21, -4], [10, -13, 21], [-17, -10, 6], [-13, 13, -14], [16, 9, 0], [-18, 7, 13], [-19, -6, -7], [14, -14, 12], [22, -5, 1], [-17, -22, 8], [-1, 6, -24], [24, -5, -11], [22, -15, 0], [2, -1, 17], [14, -13, 10], [19, 13, -2], [7, -18, 19], [-8, -2, 23], [-3, -28, 5], [-5, 5, -22], [14, 9, -3], [-1, 1, -24], [32, 0, 2], [4, 9, -15], [6, -12, 6], [2, -6, 12], [11, 21, -16], [13, -17, 19], [-11, -6, -12], [-31, 3, 0], [-6, 10, -23], [11, 16, -11], [14, 8, -13], [-10, 2, 16], [-20, -9, 12], [1, -12, 16], [19, 15, -9], [-24, 2, -19], [8, 12, -7], [4, -9, 24], [-5, -22, -3], [32, 0, -12], [3, 3, 18], [-5, -10, -1], [-9, -8, 10], [-11, 0, 15], [4, -17, 14], [31, 12, 0], [-2, 12, -11], [2, 0, -9], [-33, -21, 15], [5, 3, -21], [-4, 17, -26], [17, -18, -6], [9, 11, -22], [8, 3, 9], [-25, 19, -12], [-2, 16, -17], [31, -1, -15], [-8, 17, -12], [18, -24, 12], [-22, -28, 11], [26, 18, -19], [17, 16, -24], [-32, 13, -4], [35, 0, 15], [-2, 12, -24], [14, 14, -25], [-13, -4, 19], [41, -1, -2], [28, -9, -3], [7, -8, 8], [-19, -8, 4], [-5, -6, -6], [-18, 11, -31], [-7, -7, -7], [-19, 21, -1], [16, 31, -20], [37, -3, 15], [2, 7, -26], [9, 4, -29], [8, 14, -16], [-19, 18, 11], [-10, 1, 6], [19, -2, 23], [-7, -27, 9], [-14, -22, 5], [-16, 21, 21], [1, 12, 19], [25, -1, -16], [-15, -25, 23], [-19, -27, 12], [20, -5, 3], [17, -27, -1], [-32, -1, -12], [-13, -20, -11], [5, -36, 21], [-36, -8, 1], [11, -6, 20], [-25, 16, -30], [-5, 10, -32], [43, -3, 0], [16, 21, -15], [-6, -27, 33], [40, -3, -5], [-26, -30, 24], [16, -17, -4], [12, -10, 8], [-14, -9, 27], [14, -21, -15], [21, 8, 5], [20, 14, 2], [-10, -11, 27], [-2, -34, 3], [-17, -27, 6], [-28, 19, -32], [25, 7, 17], [40, -12, -5], [21, 7, 4], [11, -27, 0], [22, 4, -8], [16, -20, -4], [-1, 7, -17], [-6, -10, 8], [20, 2, 29], [-28, -11, 27], [20, -3, 2], [34, -1, 23], [-5, 23, -9], [11, -17, -11], [22, -10, 8], [20, 2, -10], [-7, 4, 3], [-2, -20, -2], [19, 43, -31], [-50, 2, -10], [-2, -15, 4], [33, -17, 13], [3, -6, 9], [14, -7, 13], [25, 5, -10], [16, 23, -19], [-7, -9, 36], [6, 35, -22], [-10, -2, -15], [7, 1, 10], [-2, 29, 3], [17, -14, 43], [-3, 18, -6], [-16, 17, -1], [-10, -12, -22], [38, 1, -12], [-2, 4, 7], [19, -38, 5], [4, 30, -11], [5, 18, -15], [-41, -18, 21], [3, 21, -9], [-17, -28, -4], [26, 19, -14], [18, -10, 5], [16, 21, -15], [7, -14, -20], [16, -16, 24], [-2, -24, 30], [-3, -12, 3], [15, 31, 1], [-16, 15, 21], [-46, 1, 6], [28, -16, -13], [11, -6, -9], [-5, 36, -6], [-28, 12, -33], [25, 27, -24], [31, -18, 6], [10, 19, -10], [-1, 24, -3], [2, -12, -6], [35, 7, -15], [-17, -4, 24], [26, 27, -9], [12, 13, -20], [1, -11, -32], [10, -12, 15], [7, 6, 19], [-7, -12, 21], [-3, 31, -32], [7, -9, -18], [48, -12, 21], [-52, 23, 10], [-26, 12, 4], [13, 34, 5], [-3, 8, -13], [-22, -37, 26], [37, 17, 3], [-23, -9, -6], [30, 10, 16], [-31, 9, 17], [1, -8, 28], [43, 0, 12], [-26, -51, 34], [5, -5, 4], [-63, -8, 13], [-43, -12, -6], [-13, -1, -10], [-10, 11, 4], [-17, -15, 36], [55, -8, 26], [32, 25, -11], [41, 7, 9], [-28, 24, 5], [-47, -7, 21], [-28, -6, -15], [24, -18, 45], [-5, -39, 24], [-18, 22, -14], [-31, 1, 9], [31, -32, 16], [6, -8, -20], [10, -1, 50], [47, -20, 25], [27, 15, 6], [23, 14, 8], [-28, 29, -44], [16, -30, 39], [8, -33, 36], [46, 6, -15], [46, 17, -27], [-54, -9, -12], [8, -36, 30], [-43, -32, 18], [-5, 24, 0], [-11, -38, 23], [-49, 3, -3], [-12, 21, -24], [-7, -43, 41], [-25, -4, -29], [-8, -2, -41], [1, -6, -21], [23, -18, -18], [-15, 23, -7], [9, -31, 29], [23, -15, 0], [-17, 23, -39], [-12, 6, -15], [46, 15, 9], [-22, -38, 40], [57, -2, 10], [-31, 12, -43], [-35, 6, -9], [-4, 6, -24], [24, -17, 22], [16, 11, -4], [-34, 3, 6], [15, 15, -36], [26, -25, 34], [34, -3, 37], [-14, -26, -16], [-3, -46, 35], [14, 38, -22], [-25, -23, 18], [11, 27, -25], [61, 8, -9], [-32, 18, -36], [8, 12, -42], [-18, 7, -20], [2, 39, -18], [38, -37, 14], [-10, -10, -35], [13, 9, 12], [49, 31, -28], [-24, 38, -4], [43, -10, 29], [-37, 21, -3], [-41, -13, 27], [-12, -9, -33], [-17, -12, -21], [5, -3, -13], [-50, 6, 0], [-11, 19, -52], [3, -41, 13], [-4, -13, -10], [-19, -19, 4], [7, -2, 22], [-56, 0, -8], [-13, -33, 24], [22, 30, -24], [-4, 4, -23], [42, -31, 29], [-13, 14, -16], [-37, -5, -12], [44, 30, -19], [-17, 21, -30], [-5, -19, 39], [-7, 0, -54], [-17, 9, -57], [-27, 16, -59], [-7, 47, -4], [14, -38, 36], [-29, 29, -7], [31, 51, -35], [-1, -29, 40], [-55, -31, 16], [-17, -21, 7], [-23, -9, 0], [8, -15, 39], [18, -19, 32], [-32, -6, -9], [41, -41, 0], [-2, 8, -4], [-2, -7, -15], [-27, 15, -24], [-7, 18, -39], [-4, -10, 23], [-10, -42, 38], [55, 3, 7], [-14, 32, -31], [-24, -39, 27], [-7, -61, 34], [14, 41, -5], [7, 15, -18], [12, 23, 14], [-57, 11, -1], [-11, 45, 9], [-44, 14, 6], [-26, -22, 30], [18, -21, 9], [40, -16, -7], [-74, -2, 2]];
    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_q();".
// 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_q(:prec:=3)
    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_q();".
// 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_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3969_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 0, -3, 1]>,<5,R![-3, 0, 3, 1]>,<11,R![-3, 9, -6, 1]>,<13,R![107, -33, -3, 1]>],Snew);
    return Vf;
end function;