
// Make newform 6422.2.a.ba in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6422_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_6422_2_a_ba();".
// 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_6422_2_a_ba();".
// 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 := [9, -1, -7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-4, 0, 1, 0], [3, -4, -1, 1]];
    Rf_basisdens := [1, 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_6422_a();"
function MakeCharacter_6422_a()
    N := 6422;
    order := 1;
    char_gens := [4903, 4733];
    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_6422_a_Hecke(Kf)
    return MakeCharacter_6422_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], [0, 0, -1, 0], [-1, 1, -1, 0], [-2, 0, 0, 1], [0, -1, 0, -1], [0, 0, 0, 0], [-2, 0, 1, 2], [-1, 0, 0, 0], [2, -1, 2, 1], [-2, 3, -2, -1], [-3, -3, -1, 1], [0, 2, 2, -1], [4, -4, 1, 1], [0, 0, 4, -1], [0, 4, 0, -1], [-3, -3, 3, -1], [2, -2, 2, 0], [2, 2, 0, 2], [4, 4, 0, 0], [3, 3, 3, 0], [-2, -2, 0, 0], [-4, 0, 0, 4], [-2, 0, -5, -1], [4, -3, 4, 1], [8, 3, 4, -3], [4, -4, -2, 0], [8, 2, 0, -4], [-3, -1, -3, -3], [0, -6, 2, -1], [-2, -2, -2, 0], [-2, 2, -4, 2], [6, 4, 0, -3], [-4, 6, -4, -2], [4, -6, 2, 3], [-2, -4, 1, 3], [-10, 3, -2, -2], [0, 0, 4, -2], [-6, 4, -1, -3], [-4, 0, 5, 3], [-2, 2, 1, -3], [7, 5, 1, 2], [2, 6, -3, -1], [3, 5, -3, -1], [-8, 0, -9, 1], [8, -6, 8, 5], [2, -1, -6, 1], [-4, 3, 0, 0], [6, 2, 5, 2], [-6, -2, 6, 0], [-10, 5, -2, -4], [-17, -1, 1, 4], [14, 2, 8, -5], [-3, -1, 5, -1], [14, 2, 2, -2], [-10, -2, -4, 3], [0, -2, 3, 5], [-2, -8, 4, 0], [-4, 8, -8, -1], [4, 2, -4, -4], [4, -3, 4, 3], [-2, -6, -4, 0], [-8, 0, -2, 1], [-14, -4, 0, -2], [4, -2, 1, -1], [1, 1, 11, -4], [-6, 8, 0, 4], [-12, 8, -4, -4], [-16, -2, -6, 5], [0, -2, 6, -5], [-12, -2, -1, 0], [20, 0, -4, -4], [0, 0, -6, -2], [-3, -1, -5, 3], [6, 2, -5, -1], [6, -10, -4, -2], [9, 5, -3, 6], [-14, -2, 4, -4], [-5, -3, -3, -9], [11, 5, -1, 5], [-14, -4, 0, 6], [8, -4, -4, -3], [-2, 2, 6, -3], [10, 6, -5, 2], [-20, 0, -10, 3], [-14, -10, 8, -6], [-2, -4, -8, 3], [8, 8, -7, 9], [-12, -2, -10, 6], [-6, -5, 6, 0], [-6, 2, 2, 6], [0, 2, -6, -8], [2, -2, -4, -13], [-12, -9, 2, -3], [0, -2, -6, -9], [18, 0, -1, -3], [-28, 0, -4, 4], [14, -8, -10, -2], [10, 0, -2, -3], [-9, -3, -5, -9], [-19, 3, -11, 0], [-7, 3, -9, -10], [-10, -3, -10, -4], [12, -1, 12, -4], [0, 0, 0, 1], [0, -4, 4, 13], [0, 4, 8, -2], [-12, 4, 12, 4], [2, 0, -4, 6], [2, 8, -4, 2], [-6, -10, 4, 7], [10, 2, 14, 2], [-3, -5, -1, 5], [16, -12, 10, 0], [-3, -3, -7, -5], [0, -16, 2, -1], [24, -4, -6, 0], [-6, 4, 5, 1], [9, -1, 15, 5], [-10, -20, 2, 8], [-6, 2, 9, 3], [-18, 0, 2, -2], [-10, -6, 0, 3], [18, 3, 18, -5], [10, -10, -2, 0], [-4, 3, 4, -5], [-4, -8, -4, 2], [6, 8, 8, 0], [0, 0, -12, 2], [16, -10, 11, 3], [-6, -13, 2, 6], [-24, -5, -4, 11], [11, -5, -1, 2], [-10, 10, 0, -4], [-2, 10, 2, 4], [14, 4, 8, -8], [-18, 4, -10, -6], [-26, 10, -2, -3], [-4, 6, -2, 6], [18, -3, -6, 3], [8, 3, 8, 2], [16, -2, 0, 0], [8, 6, 8, -5], [2, -1, 18, 7], [-24, 2, -12, 4], [25, -7, 11, 1], [16, 1, -2, -9], [0, 6, -13, -2], [-10, 20, -10, -6], [-6, -8, -8, 2], [18, -7, 6, 2], [6, -4, 2, 5], [9, -15, 3, -2], [-22, 0, -18, 5], [-18, 8, 6, 4], [-33, -7, 1, -6], [8, 4, 8, -6], [-4, 14, -9, -1], [0, -6, 6, -8], [-27, 1, 1, -7], [-8, 6, -14, -3], [-23, 1, 1, -7], [10, -4, 28, -5], [36, -10, 8, 3], [-3, 15, -9, -2], [4, -13, 4, 5], [-26, -14, 1, 0], [-6, -6, 4, 2], [-2, 8, -16, 0], [24, -9, -4, -5], [-6, -14, 0, 4], [24, 8, -6, -8], [-12, 4, 2, 11], [2, 15, 2, -5], [10, -2, 0, 0], [8, -4, 12, -6], [-10, 8, -19, -8], [-9, 3, -13, 7], [12, -13, -6, 3], [-26, 10, 14, 4], [-2, 4, 2, 6], [8, -16, -2, -2], [48, -4, 0, 0], [-26, 7, 2, 7], [-9, 9, -9, -9], [4, 4, 10, -2], [-6, -4, 6, 14], [11, -1, 9, -3], [20, -6, 10, 4], [10, 4, 24, 0], [10, -12, 22, -4], [22, -6, -4, 3], [-25, -5, -1, 3], [-8, 1, -16, 0], [6, -16, -6, -2], [20, 4, -10, 7], [22, 10, 4, -18], [-4, 6, -14, 2], [-38, -8, 2, -6], [2, 10, 8, 6], [8, -8, -4, -14], [24, -16, 6, 1], [12, 6, -2, 14], [14, -2, 16, -3], [-20, 16, -4, -5], [-47, 5, -11, 1], [2, -4, -4, 12], [12, -2, 14, 5], [-6, 5, 0, -5], [-6, 10, -18, -2], [5, -9, -21, 2], [-22, -3, 0, 5], [-12, -8, 6, 14], [-8, 2, -7, -13], [-26, 8, -2, 2], [22, -3, 22, 8], [-8, 12, -27, 3], [-15, 11, 7, -5], [-20, -14, -8, 9], [-31, -3, -13, 5], [4, 10, 7, -6], [-22, 0, 12, 10], [18, -18, 20, 7], [15, 1, -9, -1], [-26, 10, 0, 5], [8, 9, -16, 8], [-22, -6, -18, 4], [-6, -12, -12, -1], [14, 2, -4, 8], [8, 8, 12, -2], [11, -9, -13, -13], [0, 2, 10, -12], [18, -10, 13, 15], [-8, -8, -9, 13], [-42, 12, -6, -7], [-2, 6, 2, 3], [-12, -14, 0, -3], [-18, -16, 8, 0], [-4, -14, -1, 6], [6, 2, -6, -2], [10, -2, 4, 8], [-16, 4, -4, 8], [12, -26, 10, 7], [12, 6, -11, 13], [-42, 6, -4, -7], [-4, -21, 8, 13], [-25, 17, -19, -1], [-18, 2, -16, -7], [48, -2, 6, 8], [-12, 10, -4, 0], [24, -20, -12, 0], [-17, -7, 15, 0], [12, -4, 0, 2], [-12, 8, -18, 11], [34, 0, 2, -3], [-10, -6, -10, -18], [-20, -7, 4, 15], [10, 8, -4, -2], [6, 12, 8, 6], [57, 5, 3, -3], [4, -2, 6, 0], [48, -6, 16, 8], [-4, 0, -1, 10], [21, 11, -7, -12], [34, 8, -4, -10], [34, -6, -2, 2], [7, -7, 5, 10], [12, 30, -6, -9], [22, 2, 4, -6], [32, 0, 10, -2], [-26, 0, 4, 10], [-10, 23, -18, -7], [38, -2, -2, -10], [-12, -8, -4, 2], [17, -5, -9, 5], [20, -14, 24, 12], [-36, -2, 2, 8], [59, -9, 11, 5], [18, 20, -16, 0], [-8, -5, -12, -11], [28, 16, 4, -15], [40, 4, 4, 0], [-14, 22, -12, -7], [17, 3, 11, 13], [2, 0, -20, 0], [20, 4, 4, 12], [-1, -3, 5, -17], [-10, 8, -21, 0], [20, 0, 14, -5], [-6, 4, 8, 25], [-18, -4, -18, 6], [2, -4, 20, -6], [-22, -10, -10, -3], [44, 15, 8, -15], [22, -18, 4, -4], [-37, -9, -15, 1], [-40, -2, 2, 6], [26, -6, 12, 4], [-28, 30, -16, -8], [-38, -2, -20, 7], [-16, 7, -20, -3], [20, -7, 24, 0], [22, -4, -2, -13], [8, -2, 3, 5], [16, 0, 16, 0], [12, 4, 28, 5], [-39, -17, 5, -5], [40, 15, 4, -4], [-14, 15, 2, 9], [-2, -14, 4, 16], [44, 8, 12, -16], [-12, 2, 9, 5], [4, 2, 19, -6], [12, -18, -6, -11], [9, -1, -23, 1], [-46, -6, 8, 5], [14, 18, -6, 9], [-2, -4, 13, 11], [-20, 13, -14, -17], [4, 12, -12, -8], [-26, -9, -8, 3], [30, 25, -8, 3], [8, 0, 28, 6], [-30, 14, -9, -11], [-42, -11, 14, -4], [0, 12, -18, -4], [30, -18, 22, 8], [10, -28, 0, 8], [-18, 0, -22, -8], [10, 6, -2, -14], [20, 6, -4, 18], [8, -5, -2, -17], [22, 1, -2, 1], [2, -9, 6, -17], [-16, 11, -16, -5], [-2, -4, -15, -17], [3, 3, -9, 5], [-10, -2, -2, -12], [54, 8, -12, 6], [-6, 18, 14, -8], [14, -11, 18, 11], [-16, -18, 10, -5], [3, 23, -3, 2], [-60, -6, 0, 9], [42, -12, 10, 2], [-26, -18, -14, 7], [-24, 24, 3, -6], [-8, 4, -16, -24], [-14, -24, 12, -6], [26, 6, -4, 3], [-58, 0, -10, -6], [-30, 4, -10, -2], [-28, -12, -5, 17], [0, 6, 33, -6], [-16, 2, -18, 2], [-38, -11, -10, 13], [36, -8, 24, 2], [-2, 14, 4, -11], [-6, -8, -21, 1], [-28, 26, 8, 1], [-30, -6, 12, 7], [32, -3, 22, 1], [-6, -9, 24, 11], [28, 6, 16, -6], [6, 2, -18, 6], [22, -18, -4, 0], [36, -16, 22, 5], [-38, 9, -2, 0], [16, -24, 21, 19], [-12, -24, -2, -7], [44, 0, 14, -1], [-10, 10, 24, 15], [-20, -2, 4, -11], [-10, 4, 8, 18], [-41, 9, -7, -9], [16, 28, 2, 4], [12, 4, -6, 16], [-48, 6, -15, -13], [-44, 10, -12, -2], [-4, -6, -1, -3], [-40, 32, -24, -10], [6, -12, -6, 10], [24, 34, -12, 4], [-8, 16, 0, -4], [-52, -6, -4, 2], [42, 8, -4, 6], [-22, 2, 5, 8], [-40, -14, -2, -13], [-12, 12, 10, -14], [-8, 19, -24, -9], [-12, -4, -6, -8], [-36, -9, -16, 3], [-26, 12, -2, 5], [2, 16, 11, -1], [32, -12, 0, -5], [14, 38, -13, -2], [-20, -2, -22, 15], [-38, 7, -6, -2], [-70, 0, -10, -1], [-42, 2, 1, -1], [24, 30, -3, -14], [-54, 20, -10, -9], [25, -7, 1, -7], [17, 5, 1, -6], [9, -9, -3, 9], [-18, 4, -42, 1], [-19, 19, 3, -1], [34, -15, 6, 0], [-38, 12, -8, -14], [22, 4, 12, -26], [-32, 23, -20, -11], [-12, -12, -8, -1], [-42, -2, -36, 12], [-24, 24, -15, -6], [-36, 4, -26, 1], [-24, 7, 16, -8], [-14, 4, -14, 4], [2, 26, -1, 16], [34, 18, 2, -18], [-2, 22, -2, 10], [8, -34, 8, 10], [10, 29, -14, -3], [-20, 4, 6, 8], [2, -38, 14, 2], [-11, 7, 19, -2], [17, 3, 1, 13], [-38, 0, 4, 4], [-22, -22, 8, -23], [-34, 0, 14, 0], [6, 4, 17, -13], [-43, 3, -13, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6422_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6422_2_a_ba();".
// 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_6422_2_a_ba(:prec:=4)
    chi := MakeCharacter_6422_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_6422_2_a_ba();".
// 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_6422_2_a_ba( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6422_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, 7, -5, -2, 1]>,<5,R![-3, -11, -9, 2, 1]>,<7,R![-8, -24, 6, 7, 1]>],Snew);
    return Vf;
end function;