
// Make newform 6762.2.a.cj in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6762_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_6762_2_a_cj();".
// 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_6762_2_a_cj();".
// 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 := [2, 0, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [-3, -6, 0, 1], [3, -4, -2, 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_6762_a();"
function MakeCharacter_6762_a()
    N := 6762;
    order := 1;
    char_gens := [2255, 3727, 3823];
    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_6762_a_Hecke(Kf)
    return MakeCharacter_6762_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, 0, 0, 1], [0, 0, 0, 0], [0, 1, 0, 0], [-1, 0, 0, -1], [-2, 0, -1, -1], [-2, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 1, -2], [2, -2, 1, 1], [1, -1, -1, 0], [-1, 0, 1, 2], [1, -1, -1, -2], [-5, 2, 0, -1], [0, 1, 0, 2], [0, 0, 2, -2], [-6, 0, 1, -1], [-2, 1, -2, 0], [0, 0, -2, -2], [-2, 2, 0, -2], [2, -2, 0, 2], [-4, 0, -2, 0], [-6, 0, -1, -1], [-5, 4, 0, -1], [-2, 0, -3, 1], [-5, 0, 1, 2], [-6, -1, 2, 0], [5, -3, 1, 2], [-1, 0, -1, -2], [1, 0, 3, 2], [4, -4, 2, 2], [-1, 2, -1, 2], [7, 0, 1, 2], [-4, -3, 2, 4], [-3, 0, 1, 4], [0, 0, 1, -3], [-8, 4, -2, -2], [2, -2, -1, 3], [-4, -4, 1, -1], [7, -2, 1, -2], [4, 0, -1, -5], [-8, 0, -2, 2], [-5, 2, 1, 4], [3, 0, 3, 2], [-3, 0, -3, -4], [-2, -2, 0, 2], [4, -2, 1, -5], [3, -4, -1, -2], [-14, 0, 1, -1], [-12, -2, 0, -2], [-8, 0, 0, 0], [5, 4, 0, 1], [3, 0, 3, -2], [-22, 0, 0, 2], [5, 0, -1, 2], [-6, 0, 3, 3], [-12, 0, 1, -1], [-14, 2, -4, 0], [-11, 4, 1, -2], [0, -4, 2, 0], [4, -4, 1, 5], [9, 4, -3, 4], [2, 4, -1, -1], [-8, -4, 1, -1], [-9, 0, -1, 2], [8, 0, 0, 4], [-6, 0, -4, 0], [3, 0, -1, -4], [8, 0, -1, 1], [-7, -2, -1, -2], [-13, 0, 3, 4], [-3, 0, -3, 4], [12, 1, 0, 2], [-1, -1, 1, 2], [-14, 0, -2, 4], [4, -3, 4, -2], [-4, -4, -1, 1], [-18, 2, -4, 0], [-2, 4, -2, 0], [4, 4, -2, 0], [-11, 1, -3, 2], [1, 6, -1, -2], [13, 4, -2, -5], [6, 0, 1, 1], [-11, 0, -3, 0], [-2, -4, 0, 0], [24, 2, -2, -8], [14, 4, -1, -5], [-5, 0, 5, 2], [-7, -4, 5, 6], [-2, 4, 0, -2], [3, -8, 5, 2], [-16, -4, 2, 10], [2, 0, -4, 2], [4, 4, -2, 2], [-8, 4, 1, 3], [16, -4, 7, 1], [22, -4, 0, 0], [32, 2, -2, -4], [-2, 4, -2, 4], [-2, 3, 0, 0], [15, -4, -5, -6], [1, 2, 5, -10], [2, 1, -2, 4], [2, -2, 0, 2], [6, 0, -2, -8], [-13, 0, -5, -4], [6, 0, 2, 4], [0, -2, -6, -2], [0, 6, -1, 1], [-17, 1, 3, 2], [-10, -8, 4, 0], [-4, 4, 2, 8], [2, 0, 0, -6], [9, -6, 1, 2], [6, 4, 0, 4], [22, 0, 5, 5], [1, -2, 5, 2], [-8, 1, 4, -4], [-8, 0, 3, 3], [-1, -4, 3, -10], [-4, -4, -1, 3], [-12, 6, 2, -2], [9, 8, -5, 2], [-2, 5, 0, 4], [12, -1, 0, -10], [19, 0, 0, -5], [-8, -4, 2, 6], [8, -8, 3, -1], [0, 0, -2, -2], [-8, -2, 2, 6], [14, 2, 2, 4], [0, 5, -2, -4], [14, -2, 0, 6], [3, 0, 2, -3], [-25, 0, -2, 3], [-14, 4, 0, 8], [-5, 0, 6, 7], [-10, 2, -2, 6], [3, 4, -5, -4], [-14, 6, 0, 0], [13, -4, -3, -8], [20, 3, 0, -12], [4, 0, -3, 7], [4, 0, -4, -8], [-19, 4, -2, -1], [-21, 6, 1, -2], [15, 0, 3, -4], [0, 4, 4, -12], [-4, 10, -2, -4], [0, 0, -5, 9], [0, 2, -6, -6], [14, -2, 3, -5], [19, -7, 5, 2], [-9, -4, -1, -4], [-20, 2, -2, -2], [11, 4, -3, -14], [-13, 4, 4, -1], [27, -8, -4, 1], [-3, 4, -3, -8], [6, 6, -4, 4], [12, 8, -2, 10], [12, 4, -2, -8], [-5, -6, 3, 10], [0, 0, 2, -6], [12, 0, -2, -6], [-16, 0, 1, 7], [30, 2, -4, -8], [-14, 8, -1, -5], [14, 7, -2, -8], [-5, 0, -4, 3], [-11, -6, 1, 0], [-14, 4, 0, -4], [10, 0, -7, -3], [-28, -4, -3, -5], [0, 2, 4, 0], [-6, 2, -4, -8], [-15, -8, 5, 12], [14, 0, -5, 9], [20, -8, -2, 2], [-15, 4, 5, 2], [0, 6, -6, 4], [9, 2, -5, 2], [-5, 6, 1, 6], [-21, 0, 2, 3], [-20, 0, 6, 8], [12, 0, -6, 8], [-21, 0, -1, -10], [39, -2, 0, -5], [-14, 6, 2, 8], [17, -6, -7, -8], [26, -7, -2, -4], [14, -8, 3, -5], [22, -7, -6, 4], [-14, 4, 3, -5], [28, -8, 2, 4], [0, 7, -6, -8], [-22, 8, -11, -7], [-19, -4, 2, 3], [0, 3, 0, 10], [-30, 4, 3, -5], [-29, -4, 2, -3], [-16, 4, 3, 9], [-23, 0, -3, -10], [2, 4, -7, 5], [17, 4, 3, 2], [-12, 10, 2, -2], [-36, 2, 4, 10], [12, 8, -6, 6], [-17, -2, 5, 12], [-16, 0, -1, 1], [-14, 2, 2, -8], [8, -4, 2, 4], [-3, 2, 2, -1], [8, -8, 5, 3], [-4, 0, -6, 14], [15, -2, -1, 4], [-7, -2, -3, 8], [-23, -2, 5, 2], [10, 4, 0, 0], [24, 0, -2, -2], [-24, 0, 2, -4], [20, 0, 10, 10], [20, 4, 0, -16], [-30, 0, -4, -4], [-24, -4, 5, -9], [-34, -8, 4, 14], [11, 4, -3, -10], [32, -2, 8, 4], [16, -2, -2, 8], [-20, 0, 2, 16], [0, 0, -10, -6], [14, 0, 8, 0], [-6, 4, -8, 4], [28, 4, -4, 4], [5, 12, -4, -3], [-14, -1, -10, 6], [-11, 8, -5, 2], [-30, -2, 5, 13], [2, -1, -2, -4], [12, 0, 0, 0], [12, 2, 7, 13], [16, -8, -2, -4], [-19, -6, 1, 8], [5, -4, 0, -3], [-20, -4, 1, -9], [-9, 12, -9, -10], [13, 3, -9, 2], [5, -6, 1, 0], [19, -1, 11, 2], [2, 8, 3, -5], [-23, 0, -9, -10], [33, -4, 10, -1], [0, -4, 3, 13], [-5, 6, -3, 6], [-3, 5, -11, 6], [14, -4, -8, -12], [0, 0, -5, 9], [2, -8, 10, 8], [-18, -2, -2, 8], [-2, 5, -10, -8], [-9, -4, 6, -11], [-14, -8, 7, -11], [30, -6, 2, 4], [30, -4, -4, 0], [-10, 11, 4, -4], [-14, 8, 3, 11], [17, 3, 3, -14], [-10, -12, 4, -4], [39, 4, -4, 7], [-14, 5, 6, 4], [-21, -4, -6, 1], [18, -2, -2, -4], [34, -8, 4, -2], [6, 4, -5, 7], [7, -12, 5, 2], [30, 8, 0, 8], [20, -4, -6, -16], [39, 8, -5, -2], [15, -4, -10, 1], [26, -2, 10, 8], [10, -12, 8, 6], [-38, 0, 7, 5], [11, 0, 15, 12], [-4, 4, 0, -12], [-18, 2, -8, 4], [-54, -1, 4, 0], [-16, 4, 7, 5], [-5, 12, 1, 6], [18, 13, -4, 0], [25, 8, -3, -2], [-8, 2, 4, 2], [-15, 8, -3, 2], [2, -6, -8, 12], [-40, -4, 7, 7], [-7, 4, 11, -14], [8, 8, 1, 7], [22, 0, -4, -10], [3, 8, -1, -8], [-19, 0, -2, -5], [4, -3, 8, -6], [6, -4, 2, -12], [1, 10, -3, -4], [13, 1, -1, 2], [17, 12, -3, 10], [-13, 6, 5, -2], [-8, 0, -5, 11], [-29, 0, -7, -2], [-16, 4, -6, 0], [37, 4, -6, -5], [-10, -1, -2, -10], [-50, 10, 4, 0], [-22, 6, -4, 2], [28, 6, 0, -12], [-16, 4, -3, -9], [26, 8, 0, -10], [-12, 0, 2, -10], [7, -4, -10, 1], [-14, 6, -2, 2], [10, -8, 4, 8], [2, 6, 8, 4], [-38, -4, 1, 9], [-19, -4, 5, 8], [-17, 2, -1, -2], [4, -8, 1, 11], [-18, -1, 6, -8], [-45, 0, -7, -2], [-1, -2, 7, 4], [18, 1, -12, -4], [52, -6, -2, 0], [1, 6, 5, 10], [-14, -8, -1, -1], [-8, -4, 10, 14], [25, 3, -1, -2], [14, 0, 4, -10], [-11, 7, 3, -8], [-36, -4, 1, -9], [-2, 8, 6, -16], [-13, 0, -13, -8], [14, -4, 12, 4], [-11, 2, -7, 14], [40, 0, 9, 7], [-24, -4, -4, -4], [-10, 1, 10, 10], [-5, -2, 5, 6], [23, -4, -12, -13], [22, -2, -2, -4], [0, -4, 2, -14], [21, -8, 8, 11], [0, 0, -4, 0], [21, -6, -5, -8], [-29, 0, -5, 2], [-32, 0, 0, 0], [25, -5, -1, 4], [33, -10, -5, -6], [-16, -4, -8, 16], [44, 4, -2, -12], [-10, 18, -10, -10], [10, -16, 8, 14], [0, 3, -2, -16], [19, 4, -3, -14], [-12, -4, -6, -10], [8, -9, 12, -4], [19, -4, 11, 4], [51, -4, -1, -10], [-30, 8, 3, 7], [-10, -16, 2, 4], [14, 2, 8, -4], [-12, 16, 2, 6], [41, 0, -7, -16], [-4, 8, -14, 6], [-28, -9, 2, 4], [8, -8, -8, 12], [10, 4, 8, 12], [-40, -4, -4, -12], [19, -6, -7, 14], [-49, 8, 0, -1], [12, -8, 7, -3], [42, 8, -8, 2], [-24, 4, -3, -7], [26, -6, 16, -4], [-24, 4, 4, 12], [46, -6, 10, 6], [14, 4, 4, 0], [37, -8, 0, -3], [51, -4, -7, -2], [31, 8, -2, -3], [-25, 1, 1, -6], [20, 8, -10, 18], [26, -12, -4, -8], [-48, 4, 3, 9], [-29, -12, 5, -18], [55, 4, 1, 2], [-4, 8, -2, -2], [6, -14, 4, -4], [-73, 0, 3, 4], [-20, -12, 3, -15], [-15, 5, 11, -14], [-7, 6, -3, 10], [39, 0, 6, 9], [12, 0, 2, 20], [-26, 6, -8, -2], [34, 0, 1, -7], [-72, 4, -3, -5], [-16, -10, 1, -1], [23, -9, 1, 0], [-10, -8, 0, -8], [-50, 4, 8, 0], [36, 0, -10, -6], [-46, 6, 2, 8], [-11, -6, 9, -14], [2, -7, -2, 12], [14, -1, 2, 4], [12, 4, -6, 12], [20, -16, 13, 11], [26, -12, -8, -10], [-2, 6, -13, -13], [22, 8, -7, -11], [-22, -8, 4, -14], [-24, -3, -2, -8], [-12, 0, -5, -5], [-2, 4, -2, 16], [-2, 8, -8, -2], [20, -12, -8, 4], [-12, -4, 7, 13], [-24, 5, -4, 4], [28, 0, 2, 12], [1, -4, 15, 6], [39, -10, -4, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6762_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6762_2_a_cj();".
// 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_6762_2_a_cj(:prec:=4)
    chi := MakeCharacter_6762_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_6762_2_a_cj();".
// 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_6762_2_a_cj( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-16, -32, -12, 3, 1]>,<11,R![32, 0, -24, -2, 1]>,<13,R![-8, -20, -6, 5, 1]>,<17,R![-64, -184, -4, 10, 1]>],Snew);
    return Vf;
end function;