
// Make newform 3822.2.a.by in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3822_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_3822_2_a_by();".
// 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_3822_2_a_by();".
// 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 := [8, 8, -7, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, -1, 1, 0], [2, -6, -1, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_3822_a();"
function MakeCharacter_3822_a()
    N := 3822;
    order := 1;
    char_gens := [2549, 3433, 1471];
    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_3822_a_Hecke(Kf)
    return MakeCharacter_3822_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, 0, 0], [1, 0, 0, 0], [1, 1, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [-1, 0, 0, 0], [0, 2, -1, -1], [1, 1, 2, -2], [1, -1, 2, -2], [2, -2, -2, 1], [-1, 1, -3, 1], [1, -1, -1, 2], [5, -1, 2, -1], [3, 1, -2, 2], [5, -3, 0, 1], [3, -3, 0, 1], [5, -1, -5, -1], [-4, 2, -1, 5], [4, -2, -3, 0], [-1, -1, -1, -1], [3, -1, -6, -2], [-3, -3, 5, -3], [6, 2, 4, 0], [8, 0, 1, -2], [-2, 4, 4, -2], [1, 3, -5, 5], [-1, -1, 5, -4], [0, 2, -1, 6], [7, -3, -5, -2], [-1, 3, -7, 3], [5, -1, 7, 1], [6, -4, -1, -3], [8, 2, 5, -3], [4, -4, 6, -4], [-4, -2, -2, 0], [0, -4, -5, 1], [-8, -2, -5, -1], [1, -3, 0, 5], [-6, -4, -2, 5], [3, -1, -5, -3], [-2, 0, -3, -2], [-6, -4, -2, 2], [5, 3, 8, -6], [13, 1, 9, -3], [17, -3, -1, -3], [-13, -1, -5, 4], [-3, 3, 10, 2], [0, -4, 0, -6], [7, -7, 1, 3], [-5, 5, 1, -1], [16, -2, -2, -4], [-1, 3, -9, 3], [8, -2, -4, -6], [0, 4, -1, -5], [-6, 0, 6, -2], [12, -4, -4, 0], [3, 1, 3, -7], [-7, -5, 1, 1], [1, 5, -9, 5], [6, -2, 8, 0], [-2, -2, 10, -2], [12, 2, -2, -4], [2, 2, 4, 4], [-18, 6, 2, 0], [-22, 6, 1, 0], [-3, -7, 3, 1], [-10, -8, -7, 6], [17, -1, 0, 2], [4, -2, 5, 0], [-17, 9, 5, 3], [-5, 5, -2, 7], [0, 6, 2, 4], [-3, 1, 0, 3], [11, -1, 1, 5], [-6, -4, 3, 4], [2, 0, -8, 9], [1, 3, 12, 3], [-6, 2, 12, 0], [10, 2, -6, 2], [5, 9, 4, -4], [8, -6, -7, -5], [11, 1, 6, 3], [0, 8, 6, -6], [-7, -5, 8, 3], [-17, 3, -3, 6], [7, -9, 6, -1], [4, 6, -9, 7], [25, -3, 5, -7], [-3, -11, 2, 4], [0, 0, 13, -5], [-2, -4, 21, -1], [12, -2, 6, -9], [-18, 2, -8, -4], [-13, 3, 10, 3], [-2, -4, 13, -8], [-9, -3, -21, 5], [25, 1, 10, -8], [-8, -6, 5, -5], [-9, 1, -1, -1], [-11, -5, -3, -2], [2, -6, 2, 0], [16, 2, 14, -8], [-20, 8, 13, 5], [0, -6, -22, 2], [-6, -2, -12, 0], [-22, 0, -10, -2], [14, 6, -14, 0], [0, 0, -9, 6], [-19, -5, -8, 2], [5, -5, 2, -9], [19, -9, -3, -10], [3, 1, -21, 2], [4, 8, -7, 9], [25, -9, -4, -2], [-2, -10, 17, -1], [11, -5, -7, 3], [15, 9, -16, 6], [4, 0, 4, -12], [-10, 14, 12, -1], [-27, 5, -6, 5], [2, -4, -8, 8], [-3, -5, 28, -2], [-8, 8, -6, 10], [-20, -4, -5, 5], [-4, 0, -2, -2], [11, 13, -14, 1], [-21, 5, 6, 3], [-9, -3, -1, -7], [-13, -5, -17, 6], [-17, -5, 3, 1], [-11, 1, -8, 5], [6, 2, -8, -8], [-1, 5, -9, 9], [-17, 7, 9, 7], [23, -3, -2, 5], [21, -11, -10, 2], [-5, -9, 0, -4], [25, -1, -2, 4], [-5, 7, 1, 11], [-7, 5, 13, -7], [-11, 9, -8, 0], [-16, 10, -10, 12], [-23, 13, -1, -1], [-6, 6, 15, 1], [12, 6, -1, 9], [11, -5, 0, 7], [-8, -2, 0, -4], [0, -6, -18, 6], [16, -8, 8, 0], [-20, 2, 24, 2], [-3, 15, -2, -1], [-22, -4, -1, 5], [-19, 7, -14, -2], [12, 8, 10, 0], [0, -8, -22, 6], [-17, 9, -2, 6], [-22, -4, 2, 0], [9, -13, -2, -5], [8, -8, -13, -8], [-44, 2, 0, 2], [0, 8, -21, 1], [-18, 0, -12, 6], [2, -2, -13, 1], [8, 2, -6, -8], [10, -22, -3, 9], [12, 2, 20, -9], [-20, -6, -4, 2], [-12, 14, -12, -2], [13, 1, 23, -5], [-5, -3, 13, -13], [-34, -6, 1, 5], [27, -1, 13, -9], [6, -10, 10, 4], [4, -6, 16, -2], [-33, -1, -7, 12], [-24, -6, 14, -6], [-16, -2, 11, 8], [1, 1, 3, 5], [5, 5, -18, -5], [10, 4, -7, 3], [-28, 4, -17, -3], [-32, 2, -2, -4], [4, 2, -22, -2], [-21, -7, -20, 7], [-7, 13, 9, 1], [9, -5, 3, -11], [10, -8, 0, -6], [-3, -5, 8, -14], [-14, 8, -20, 8], [6, 0, 2, 17], [-11, 7, -26, 3], [-12, 4, 19, -5], [16, 4, 2, 6], [21, -7, -8, -10], [13, 9, -8, -7], [22, -12, 5, 11], [5, 9, 1, 5], [-21, -5, 5, 3], [8, -8, -3, 6], [-15, 7, -5, -15], [10, 8, 2, 6], [-5, -1, 7, 4], [-5, -9, -3, 11], [-4, -20, 11, 4], [-20, 16, -4, 12], [-20, 6, -10, 6], [-43, 19, 9, 3], [-13, 3, -4, -3], [20, -12, 17, 9], [1, 15, -7, -7], [43, -9, 4, -6], [-2, 4, 14, 6], [8, -16, -14, -2], [16, -4, -12, 6], [-12, -2, -16, -3], [-9, -13, 4, 4], [26, -4, 14, -8], [2, 0, 15, 3], [15, -11, -7, 7], [-23, -7, -13, 13], [14, 4, 10, -6], [27, -11, 3, -7], [-10, 14, 13, -7], [-34, 14, 2, -2], [8, -28, -1, 3], [-7, -7, -21, 5], [8, -10, -15, -5], [-10, 2, 8, 10], [29, 1, 5, -4], [-2, 14, -21, 9], [35, -9, -3, -9], [-1, -7, 5, -7], [-32, 16, -3, -1], [16, -2, 28, -8], [-2, 6, -22, 4], [-10, 8, 0, -5], [17, -7, -18, 10], [26, -4, 12, 2], [27, -1, 8, -1], [-17, 9, -11, 3], [20, -8, -12, -6], [-4, 4, -2, -16], [1, -11, 21, 0], [-29, -7, -26, 19], [-2, -6, 11, 0], [-21, 9, 11, 11], [37, 7, 7, -7], [-7, -9, 23, -1], [-31, 13, 16, 9], [3, 1, 34, -8], [31, -3, -7, 4], [-12, 0, -1, 12], [-3, 7, -17, -7], [4, 0, 19, 0], [-12, 12, 4, -4], [-46, 18, 5, 1], [54, 4, 12, -6], [26, -12, 6, 0], [-43, 9, 17, 3], [33, -3, 0, -2], [7, -5, 33, -15], [14, -20, 7, -2], [-48, 12, -6, 14], [18, -28, 1, 7], [28, 0, 27, -12], [-9, -7, -27, 13], [-11, 5, -15, 3], [-18, -18, 7, 0], [-28, 6, -13, 22], [31, -7, -14, -7], [26, 8, -8, -4], [11, 7, 4, -9], [-19, -3, 2, -2], [-42, -12, 0, 6], [7, -3, 12, 21], [-24, -8, -24, 8], [6, -18, -7, 5], [5, -9, 5, -26], [16, 10, 4, -12], [-34, 14, 4, 8], [-18, -10, 4, -10], [2, -18, 17, -7], [39, 1, 27, -21], [-26, 10, -6, 22], [39, 1, 20, -6], [18, -14, 24, -4], [-9, 13, 0, 10], [-19, 17, -21, 19], [-2, 2, -10, -10], [20, 6, 35, -5], [-10, 2, -22, 3], [-38, -4, 15, -7], [0, -2, 1, -3], [0, -12, 0, -4], [-4, 2, -36, 6], [7, -1, -35, -9], [23, -5, -20, 1], [30, 0, 28, -12], [-2, -8, 13, 7], [-9, 9, 18, 16], [14, 10, -7, 4], [2, 8, -12, 6], [38, 2, 0, -4], [-14, -16, -13, 6], [21, 3, 8, -22], [1, -11, 23, -7], [-7, 17, -12, 7], [18, 4, 17, -17], [3, 9, 7, 1], [-20, -10, 18, 2], [-23, -5, 7, 5], [-22, 12, 0, -18], [19, -1, 19, -7], [-5, -7, 16, 8], [26, 0, 10, -24], [-5, 9, -14, -7], [0, 0, 24, -8], [33, 3, -1, -3], [15, -21, -18, 4], [-2, 4, 22, -8], [9, -9, 15, -21], [15, 19, 17, -9], [-1, -1, 5, 3], [-48, 4, 1, -7], [0, -2, -13, -5], [-5, -7, -30, 7], [-25, 3, -34, 9], [11, 23, 0, -8], [-23, 19, -14, -16], [-6, 10, -40, 13], [-33, 13, -15, -2], [18, -6, 24, 5], [11, 7, 20, -23], [53, -5, 11, 5], [37, -15, 6, -21], [-44, 8, -12, 10], [17, -17, 27, -22], [7, 17, -30, 5], [20, 6, -4, 8], [-27, -3, -9, 0], [5, -5, 42, -13], [33, -3, 17, 7], [13, 9, 6, 4], [8, -2, 36, -2], [35, -1, -15, 5], [-4, 6, -33, -6], [-44, 22, 9, 13], [22, 4, -5, 18], [29, 5, 20, -7], [20, 6, 14, 14], [14, -18, 20, 0], [-27, 21, 1, -3], [42, -12, 0, 10], [17, 17, -22, 3], [-13, -5, -29, -1], [-32, 2, -3, 23], [10, -8, -16, 1], [-11, 19, 1, -23], [15, -21, -5, 7], [9, 3, 27, -11], [41, -17, 5, -18], [0, 10, -9, -11], [-7, -5, -20, -16], [-28, 4, 0, 0], [20, -16, 2, 6], [62, 8, -2, -2], [69, -3, 12, -6], [16, -8, 11, -9], [-9, 17, 22, 4], [6, -8, -5, 0], [13, 7, 45, -15], [37, 15, 23, -19], [-54, 10, 4, 9], [9, -7, -11, -10], [-33, 3, 9, 9], [-32, -8, -14, 4], [49, -5, -24, 2], [7, 23, 14, -8], [-6, -8, 17, -7], [18, -10, -1, -2], [44, -14, 4, 16], [-14, 20, -3, 5], [6, 10, 13, 15], [-8, 24, -2, -6], [42, 2, 8, 4], [-40, -6, 26, -5], [0, -28, 13, 1], [18, -12, -11, -25], [21, 1, -34, -5], [-39, 11, -13, 19], [-36, 22, 12, 6], [-12, -6, 34, -12], [6, 14, -18, -10], [1, 7, -3, 25], [-23, 3, 19, -15], [22, -16, 26, 2], [-27, 11, 13, -7], [34, -12, -17, 7], [15, 21, -4, -2], [15, -29, 9, 17], [6, 4, 46, 2], [1, 17, -3, 9], [-34, 10, 5, 21], [25, -9, 26, -29], [-2, 8, -50, 2], [-19, 13, 11, 0], [3, 5, -10, 3], [10, -16, -5, 23], [35, -11, -16, 14], [-28, 8, -24, -4], [-22, 2, -21, 6], [14, 18, 10, -5], [-43, -15, -14, 15], [-29, 5, -36, -4], [13, -11, -4, -6], [20, -34, -4, 0], [-31, -21, 21, -5], [1, -3, 37, 4], [-13, 21, 2, -9], [-4, -20, 22, 4], [-32, -4, -10, 9], [-4, 12, 10, -14], [56, 2, 34, -16], [24, 14, 20, -10], [3, 15, -16, -14], [24, 6, -16, 14], [2, 4, 19, -8], [-6, 4, -38, -10], [43, -11, 14, -8], [-32, 8, -14, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3822_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3822_2_a_by();".
// 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_3822_2_a_by(:prec:=4)
    chi := MakeCharacter_3822_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_3822_2_a_by();".
// 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_3822_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3822_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 12, 5, -6, 1]>,<11,R![-32, 48, -17, -2, 1]>,<17,R![316, 12, -53, -2, 1]>,<29,R![254, 108, -41, -6, 1]>],Snew);
    return Vf;
end function;