
// Make newform 6762.2.a.cl in Magma, downloaded from the LMFDB on 18 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_cl();".
// 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_cl();".
// 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 := [30, 8, -13, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-7, -2, 1, 0], [10, -8, -3, 1]];
    Rf_basisdens := [1, 1, 1, 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_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, -1, 0, 0], [0, 0, 0, 0], [0, 1, 0, 1], [2, 0, 1, 0], [1, 0, 0, 1], [-2, 0, 0, 0], [-1, 0, 0, 0], [3, 0, -1, 0], [-2, 0, 0, 0], [6, -2, 0, 0], [0, 2, 0, 0], [2, 0, 0, 0], [-1, -2, 0, -2], [5, 1, 2, 0], [4, -4, 0, 0], [0, 0, -2, 0], [-1, -1, -2, 0], [1, 0, 2, -2], [1, 0, -2, 0], [8, -1, 0, -1], [-2, 0, 0, -2], [-2, 2, 2, 0], [-8, 0, 0, 0], [7, 0, 1, 0], [-3, 4, 2, -1], [8, 2, 0, 2], [4, -1, -2, 1], [1, -1, 0, -2], [0, -2, -2, 0], [-6, 2, -1, -2], [3, 2, 0, -3], [3, 0, 1, 2], [7, -3, 0, -2], [-2, -2, 2, 0], [10, 1, 2, 1], [-1, 0, -1, 2], [0, 0, -3, 0], [-5, -2, -1, 4], [0, 2, -1, 4], [-6, -3, 2, -3], [-2, 2, 2, 0], [-2, 4, -1, 0], [13, 0, 3, 0], [-4, 3, 0, -1], [5, 2, -1, 2], [-12, 6, 0, 0], [8, 3, 0, -1], [4, 5, 2, 3], [-4, 4, 0, 4], [-9, 0, -1, 2], [10, 4, 0, 2], [4, 1, 4, -3], [14, 4, 0, 2], [6, 2, -2, 2], [-1, -2, -1, -4], [4, 2, 2, 2], [-8, -4, -1, -2], [3, -8, -2, -5], [5, 3, 0, -2], [9, -1, 0, 2], [-15, 2, 1, 2], [13, 2, 0, 2], [-4, 4, 0, 4], [-2, 8, 0, 0], [-6, 6, 4, 0], [10, 0, -2, 0], [-4, -2, -3, -4], [-16, 2, -1, 2], [24, 2, 0, 0], [10, 2, 0, -2], [3, -5, -2, -4], [2, 3, 2, 3], [5, -5, 2, 0], [6, 4, 0, -4], [-4, 0, 4, -2], [-10, 0, -3, -2], [9, -2, 0, -1], [-3, 2, -4, 0], [2, -3, -4, 3], [30, 1, 0, 1], [6, -4, 2, 4], [0, -6, -2, -6], [2, -6, -4, -2], [10, -10, -1, -2], [-8, 4, -2, -2], [-4, -4, 0, 0], [6, 0, 0, 0], [4, 0, 4, 0], [-4, 1, -2, -3], [8, 6, 0, 4], [-6, -4, 2, 2], [-8, 4, -1, 6], [-3, -2, 1, -6], [8, -6, -6, 2], [7, 4, 1, 4], [9, -9, -4, -2], [-7, 5, 0, 6], [4, -4, 1, 4], [7, -4, -3, 2], [-2, -8, -2, -6], [-18, 5, 0, -1], [-1, 1, 0, -6], [-9, -7, -2, 4], [-5, -2, -3, 0], [-4, -8, 3, -4], [-2, -4, -4, -2], [-21, -4, 2, -2], [-10, -4, 0, 0], [4, 6, -2, 0], [12, -1, 0, -3], [-5, 10, 4, 1], [-1, 1, 2, 6], [5, 8, 2, -1], [-15, 8, 2, 1], [-12, 0, 2, 0], [15, -4, -1, -2], [15, 2, -3, 4], [-6, -5, 2, -7], [-16, 9, 2, 3], [3, 10, 1, -4], [1, 5, 4, -2], [-10, 6, 7, -2], [9, 2, 3, -2], [-15, 1, 0, -2], [-8, -1, 2, 1], [-18, -6, -1, -4], [0, 9, 4, 1], [-20, -7, -2, -1], [-11, 0, -3, -6], [-4, 0, -2, 2], [-24, -2, -2, -6], [16, 0, 0, 0], [2, 2, 2, 2], [4, 10, 2, -2], [3, 1, -6, 2], [7, -7, 0, -4], [7, -5, -2, 6], [-12, 2, -6, 0], [9, 2, -1, 6], [13, 6, -1, 8], [18, 4, 4, 6], [-12, -7, 2, 1], [14, -8, 3, 0], [-8, 8, 0, -4], [0, -2, -2, 6], [16, -2, 2, 2], [-7, 0, 1, 2], [-5, -2, 0, 2], [-2, 2, -5, 0], [-1, 0, -2, -5], [8, -12, 4, 0], [-1, 12, -1, 6], [-23, 3, 0, 2], [-18, 2, 6, 0], [-3, -2, 2, 1], [4, 10, 0, 10], [20, -10, 0, -4], [-4, -6, -4, 4], [-6, 12, 3, 2], [8, -3, -2, -5], [-28, 18, 4, 0], [-4, -9, -2, -1], [-11, -1, 2, -2], [-36, -2, 2, 0], [22, -6, 0, -6], [4, -4, -2, -8], [20, 4, 6, -2], [-6, -4, -2, -4], [-8, -2, -6, 2], [-12, 12, 1, 6], [-18, -6, -6, 0], [12, 4, 4, 4], [-10, -4, 0, 0], [24, 5, -2, 3], [-43, -2, 1, -2], [7, 4, 1, 8], [-19, 5, 0, -4], [-28, 5, -2, -1], [8, -2, 0, -12], [4, 1, 2, -7], [6, 6, 1, 4], [-8, 14, 2, 0], [8, 10, 0, -6], [-7, -3, -4, -4], [-4, 8, -7, 2], [20, 2, -2, 6], [3, 8, 0, -4], [-26, 18, 5, 0], [5, 2, 7, 0], [16, -14, 0, -2], [-25, 9, -4, -4], [22, 0, -2, 0], [-6, 5, 8, -5], [25, -12, -4, 1], [4, -2, -4, -4], [2, 3, 6, -1], [11, 1, -6, 2], [8, 4, -3, -4], [-21, 15, -2, -2], [-20, -2, 0, 8], [2, -2, -7, 0], [-10, 10, 2, 8], [34, -1, 2, 1], [-28, 6, 2, 0], [5, 5, 6, -4], [-32, -6, 5, -4], [-16, -6, -2, -8], [-28, -4, 2, -4], [-16, -10, 5, -8], [-17, 12, -1, -4], [-12, 11, -4, 3], [-10, 4, -4, 10], [29, 4, 5, 6], [-36, 0, -8, -2], [10, 0, -6, 6], [0, 4, 2, -6], [16, -10, 4, 8], [-9, 8, 7, 0], [-18, -2, -3, -6], [17, 6, 4, -11], [-16, -2, -6, -2], [3, -6, -4, 3], [-26, 2, -2, 2], [-2, 8, -1, 4], [2, -6, -2, -8], [27, -2, -2, 2], [31, -8, -2, -7], [-2, -10, -1, 2], [10, -12, 3, 8], [-5, 5, 6, 4], [-8, 3, -4, 11], [-22, 8, 4, -2], [-45, 1, -2, -2], [-16, 14, 0, 0], [-38, -4, 2, 0], [-12, -8, 4, -10], [-14, 0, -1, -4], [20, -4, -4, -4], [16, -1, 4, 11], [-17, 6, -5, -2], [-46, 2, -4, -2], [4, 0, -11, 4], [-2, 2, -4, -8], [21, 0, 5, -2], [12, -18, -6, -8], [0, -6, -8, -2], [-12, 18, 2, -2], [-24, 6, -8, -6], [10, 14, -4, 12], [5, -2, -2, 3], [-8, 20, -4, 0], [-10, 14, 2, 10], [36, 2, -2, 0], [-12, -12, 0, -8], [0, 5, 4, -5], [8, -10, -6, -6], [-28, 2, 0, 8], [-17, 14, -3, 4], [-1, -10, -8, -9], [-22, -8, 0, 2], [-10, -4, -3, 2], [28, 1, 0, 3], [36, -4, 10, 6], [-34, -14, 4, -10], [35, -1, 0, -2], [12, -1, -6, 5], [-9, -3, 2, 0], [17, -18, -1, 0], [-34, 18, 7, 0], [10, -14, 4, -2], [-4, 0, 8, -8], [17, -11, 2, 4], [12, 10, 1, 4], [26, -8, 2, -2], [6, 2, 1, 8], [-20, -10, -1, -12], [16, -4, 0, 8], [18, -13, -2, -3], [10, 2, -3, -8], [19, -17, -6, -8], [-16, -10, 2, 4], [-54, 2, 2, -4], [37, -6, -1, -2], [39, -8, -1, -2], [42, 4, -2, 0], [7, -1, -12, 6], [-2, 12, 0, 6], [30, -4, -2, 2], [18, 4, 2, 2], [-22, -3, 0, 3], [22, 8, 6, 6], [24, -6, -2, -2], [0, -2, -5, 6], [-17, -7, 4, -6], [33, 0, 0, -5], [1, 10, 9, 8], [8, -6, 1, -2], [-5, -10, 1, -6], [10, 10, -6, 14], [-35, 19, 10, 2], [-15, -4, -8, 3], [10, 10, -2, -2], [43, 5, 0, 2], [7, -6, 5, 8], [6, 8, 7, 4], [16, -20, -4, 0], [28, -4, 2, 8], [24, -12, -2, -8], [15, 16, 3, 6], [-21, 1, -4, 0], [-9, -16, 4, -5], [13, 14, -3, 2], [-12, 4, -2, -4], [22, 10, -8, 2], [6, -2, -10, -2], [15, -4, 7, 2], [-6, -10, -10, 2], [50, -6, 4, 2], [1, -19, 2, -2], [-10, 4, 10, -10], [0, -10, -2, -12], [2, 6, -2, 4], [41, -10, -2, -9], [28, -8, -2, -4], [44, -8, 10, 8], [3, -8, 5, 10], [-22, 4, -10, -2], [28, 8, 4, 10], [2, 8, -3, -2], [-23, -8, 1, -4], [16, -24, -2, -6], [20, -6, 9, 6], [1, -1, 4, -8], [-22, 2, 5, -4], [14, -11, -4, -7], [4, 11, 6, 1], [-46, 0, 3, 0], [4, -3, 4, 1], [16, 4, 5, -12], [-5, 13, -6, 6], [7, -2, -11, 0], [-20, -6, -2, -6], [38, 0, -10, 6], [-18, 14, 5, -2], [-26, -12, -7, -16], [-43, -8, -1, -10], [-20, -4, -4, 6], [30, 2, 2, -4], [-41, -8, -7, -2], [39, 10, -2, 12], [13, 4, 1, -12], [-21, 4, -8, 1], [36, 3, 8, -11], [-8, -8, 6, -12], [2, 6, 12, -4], [4, 1, -8, 13], [-6, -12, -6, 2], [39, 6, 6, 6], [16, 17, -4, -1], [4, -20, -2, -10], [-9, -16, -1, -6], [14, 1, -6, 3], [34, 10, 6, 8], [-4, -10, -9, -4], [36, 6, 12, 12], [32, 10, 2, 6], [-14, -4, 4, -6], [22, -20, 2, 4], [8, -12, -8, -12], [15, 12, 12, 6], [-6, 8, 2, 10], [-35, -12, 6, -9], [30, -18, -1, -8], [24, -6, 0, 6], [28, 12, 4, 6], [24, 4, 6, -8], [-61, 18, 4, 1], [-38, -12, 4, 0], [-14, -6, 2, -4], [-21, -4, -10, -1], [-10, -12, -8, -10], [6, -16, -16, 2], [75, -7, 0, 2], [-12, -12, -1, 8], [-40, 0, -2, 10], [10, -10, 4, -2], [23, -9, -10, -2], [-54, 6, 0, -6], [-20, 2, -10, 6], [-35, 2, -9, 4], [-22, -12, -3, 14], [-39, -8, 3, -2], [46, 6, 8, -2], [36, -10, 0, 4], [15, -20, -2, -3], [9, -1, -10, -2], [-49, 3, 4, -2], [29, -2, 7, -8], [4, -14, 3, -4], [50, 8, 0, 2], [-16, -5, -4, 11], [-33, -7, -10, -14], [-3, -4, -11, 0], [-32, 4, 4, 6], [16, -3, 4, 7], [38, -10, -4, 6], [1, 13, 4, 14], [-42, 6, -6, 2], [-15, 22, 8, 0], [49, -2, 1, 0], [22, -3, 2, 11], [-5, -9, -6, -14], [24, 12, 6, 2], [-32, 10, -4, 8], [-49, -2, 0, -6], [-6, 24, 8, 16], [-57, 12, 6, 3], [-12, -2, -4, 6], [-34, 2, 6, -4], [6, -5, -10, 7], [-17, -20, 3, -2], [-18, -8, -15, 10], [32, 2, -9, 4], [7, 12, 11, 0], [-32, 2, -2, 10], [16, 16, -6, 0], [23, -4, -7, 6], [39, 0, -1, -2]];
    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_cl();".
// 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_cl(: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_cl();".
// 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_cl( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![24, 20, -13, -2, 1]>,<11,R![180, 12, -31, 0, 1]>,<13,R![-60, 164, -27, -6, 1]>,<17,R![3, -34, -28, -2, 1]>],Snew);
    return Vf;
end function;