
// Make newform 6762.2.a.cg in Magma, downloaded from the LMFDB on 20 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_cg();".
// 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_cg();".
// 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, 12, -11, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-6, -1, 1, 0], [2, -10, -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_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], [0, 1, 0, 0], [0, 0, 0, 0], [1, 1, 1, -1], [-1, 1, 0, -1], [0, -1, 1, 0], [-1, -1, 0, -1], [1, 0, 0, 0], [-2, -1, 0, -1], [-4, 1, 2, 0], [1, 1, 0, -1], [3, -1, 0, -1], [2, 0, 0, 0], [-3, 1, 0, -2], [2, -1, 0, 0], [5, 2, -2, 1], [-8, 2, 0, 0], [6, 2, 2, -2], [4, 0, 0, -1], [1, 1, 2, -2], [-2, 2, 1, 2], [4, -1, 0, 2], [-5, 3, 0, 1], [-1, 2, -2, 3], [-1, -1, -2, 0], [-7, 2, 1, 1], [8, -1, 0, 2], [3, -2, 1, 1], [1, 3, 4, -1], [-2, -3, 2, -2], [-1, 2, 4, 1], [3, 2, -3, 1], [1, 1, 0, -2], [2, -2, 4, 2], [-4, 3, -2, 2], [-2, -1, 1, 2], [8, 0, 0, -1], [-2, 4, 4, -2], [-5, 1, 2, -2], [2, 2, -4, 4], [-3, 2, 5, -1], [7, -3, 0, -3], [2, 3, 2, -2], [7, -7, -2, 2], [-7, 2, -5, 1], [7, -1, 0, 4], [-8, 1, -2, 0], [17, 1, 3, -1], [-16, -1, 3, 0], [-5, -1, 4, -3], [-1, 1, 0, 2], [-3, -6, 4, 1], [1, -1, 5, -1], [-3, -3, -2, -1], [0, 8, -2, -2], [4, -7, 0, -1], [-14, -1, -4, 0], [2, 4, 6, -4], [2, 5, 3, -2], [2, -4, 4, -2], [4, -7, 2, -2], [-12, -4, -4, 3], [0, 6, 0, -3], [-1, -4, -2, 3], [23, 2, -2, 1], [-1, 5, 8, -3], [-5, 6, -2, -3], [2, 2, 4, 0], [7, 1, 2, -3], [5, -7, 0, 5], [-4, -4, -6, 2], [-7, 2, -4, 1], [17, -8, -1, 3], [4, -6, -6, 2], [9, 1, 8, -7], [-16, 6, 2, -2], [-13, -1, -2, -3], [-1, 2, 3, -7], [1, -4, 4, -6], [7, -2, 1, -1], [-3, 4, -7, 3], [-6, -6, 4, -4], [18, -6, 2, 6], [-6, -5, 4, 2], [1, 4, -4, 3], [-9, 1, -6, 3], [17, 0, 2, 1], [2, 0, -8, 0], [16, -2, -4, 4], [7, 0, -1, -1], [11, -1, 0, -1], [8, -5, -4, 2], [15, 0, -4, -3], [3, -9, 4, 0], [4, -6, 2, 0], [6, -3, 8, -1], [4, 2, 4, 0], [14, -6, 4, 4], [-21, 3, 4, 1], [12, 4, 4, -5], [10, 7, -4, 0], [17, -1, -1, 3], [-10, 2, -8, 2], [21, -3, -6, 1], [-11, 2, 8, -4], [7, -8, 2, -5], [-6, -4, -10, 6], [4, -12, 0, 1], [2, -3, -2, 2], [-18, 7, 6, -2], [11, -8, -3, -3], [3, -2, 5, 1], [-11, -1, -4, 3], [-4, -2, 15, 0], [-6, 3, 5, 2], [7, 9, -4, 5], [25, 1, -4, 0], [2, 1, -4, 1], [1, 2, -1, -3], [-15, 2, 3, 3], [3, 0, 4, -2], [-18, 9, -2, -2], [-3, 10, 0, -5], [-2, -6, -4, 1], [-12, -7, 2, 0], [10, -3, -3, -2], [-10, 4, -12, -2], [-14, 14, 3, -6], [14, -5, -1, 2], [13, -7, -8, -4], [19, 1, 2, -5], [-7, -2, 8, -3], [-17, -3, 8, -3], [-1, -3, 10, -5], [-7, 2, 8, -3], [-4, 4, 0, 4], [10, -2, 2, 8], [8, 9, -2, 0], [-15, 1, 8, 1], [1, 9, 12, -6], [-4, 9, 0, 1], [22, 0, 10, -2], [15, 3, -3, 9], [-20, 0, -12, 2], [21, -1, 4, -1], [0, -4, -6, 4], [-11, -3, 2, -5], [12, 2, 12, -7], [24, 0, 8, 3], [-13, -7, 0, 5], [15, 0, -1, -3], [-3, -5, -4, -1], [3, -3, -12, 6], [1, 1, 8, -9], [-20, -6, 0, 4], [11, -6, 7, -9], [-17, -1, -6, 9], [1, -10, 6, -5], [16, -9, -2, 4], [24, 4, -2, -4], [-11, 0, 5, -9], [14, -7, -6, 6], [-9, -5, 3, -7], [-8, 10, -4, 0], [11, -11, 0, -5], [10, -5, -16, -2], [-7, -3, 0, -9], [9, -2, 12, -1], [7, -4, 14, 1], [-10, 9, 0, 8], [28, -4, 2, -6], [3, 5, 4, -7], [5, -4, -2, 1], [14, -5, -6, 2], [-16, 3, 9, -4], [1, -7, -4, -6], [-12, 1, -8, 1], [14, -12, -2, -6], [8, -3, 3, 0], [-2, -5, -14, 2], [-17, 3, -3, 1], [-43, 1, 0, 3], [-9, 1, -12, -1], [21, -5, 2, 3], [-22, -3, -12, 8], [11, 9, -6, 3], [-46, -2, 2, -2], [-9, 8, 8, -4], [19, 7, 12, -3], [29, 6, -8, 2], [-5, -16, -4, 7], [-11, -5, -10, 5], [14, -12, 8, 4], [-9, 7, 11, -3], [21, 4, 5, -1], [-26, -6, 8, 0], [-40, 5, 3, -4], [15, -5, 0, 9], [-7, -13, -8, 7], [-20, 13, -2, 4], [16, -6, 4, 2], [4, -6, -16, 0], [6, 4, 8, -6], [-9, -10, 7, 3], [-19, 0, -14, 3], [10, 12, 6, -2], [23, -14, -2, -3], [7, -5, 8, -5], [-11, -3, 4, 3], [-6, -14, 0, 4], [-12, 7, 8, -7], [-8, 0, -7, 0], [-2, -1, 0, -4], [7, -11, -4, 4], [4, 4, -14, 2], [-14, 1, 8, 4], [-13, -3, 6, -1], [22, 0, -8, 8], [-41, 7, 2, 4], [0, -2, -2, -2], [-29, -2, 3, -9], [-8, -16, -2, 4], [-26, 8, 7, -2], [21, 12, 8, -5], [-5, 1, -4, -5], [-5, 11, 16, -3], [-30, -4, 0, 1], [-12, 0, -15, -4], [-10, 2, 22, 0], [12, 6, 8, -2], [16, -2, -10, -2], [-24, 8, 1, -12], [-4, -4, 10, -4], [9, 7, -12, 3], [4, -4, 12, -2], [-13, -12, -6, 5], [-34, 3, 0, -2], [10, -10, -8, 0], [11, 7, -12, -3], [-15, 4, 9, -3], [16, -4, 4, 1], [4, 13, 4, -8], [-32, -2, 0, -4], [-29, 7, -4, -7], [-26, 16, 8, 1], [-6, 11, -10, 10], [30, 0, -10, -2], [-23, 11, 2, -1], [13, 9, -6, 3], [-23, -1, 8, 3], [18, 5, 3, -2], [-4, 2, 8, 4], [-4, 12, -2, -6], [-17, 18, 2, -3], [13, 4, -6, 1], [-6, 15, -9, 6], [19, -15, 6, 9], [13, -10, 6, -13], [39, 2, -12, 2], [-11, 10, 5, -3], [-38, 3, 8, 0], [23, 5, -2, -3], [-14, 17, -1, -2], [-29, 3, -10, 3], [-8, -12, 6, -2], [10, 3, 14, -10], [18, -1, -13, -2], [-27, 7, -2, -3], [10, -3, -4, 11], [26, -6, -4, 6], [41, 1, 10, -3], [29, -10, 6, -3], [-15, 10, 8, -3], [12, -7, 10, 0], [6, -10, 6, -2], [-25, 12, -6, -9], [-17, -11, 8, -9], [-6, 12, 0, -10], [33, -2, -1, -3], [-25, 5, 0, 9], [-27, 18, 0, 5], [6, -6, 4, 0], [5, -15, 4, -1], [-17, -5, 0, 4], [21, -1, 16, -4], [5, -16, -6, 7], [-14, -3, 10, -10], [4, 5, -12, 6], [-30, 0, 2, 2], [-10, 6, -2, -6], [31, 3, 11, -3], [-11, 23, -2, -9], [-10, 2, -14, 2], [-10, -12, 14, 2], [-32, 15, 2, -6], [15, -4, 9, 3], [15, -15, 6, 4], [4, -8, 6, -12], [-9, 7, 0, -2], [20, -4, -2, 14], [34, 13, -14, 4], [-48, 2, 3, 0], [-5, -4, 0, 9], [-24, -2, 4, -8], [25, -11, 6, 2], [1, -8, -6, 11], [-30, 16, 0, 2], [-27, 1, -4, 7], [-5, 7, 16, 1], [73, 1, 4, -4], [24, 6, -10, 10], [22, 3, 11, 6], [38, 14, -12, 1], [-15, 7, 20, -5], [8, -4, -18, 2], [-11, -2, 8, -1], [42, 8, 16, -9], [-7, 10, 0, -1], [-5, -19, 10, 3], [-15, 1, -12, 13], [-24, -4, -2, 0], [19, -4, -6, -7], [-4, -14, 4, 4], [-23, 4, 3, -9], [9, 10, 6, -3], [-27, 15, -4, 7], [30, -4, 12, -7], [36, -12, -2, -8], [-28, 9, -8, 12], [9, -1, 10, 11], [28, -7, 4, -1], [-12, -12, 6, -2], [4, -2, 10, 0], [11, -1, -18, 3], [-22, 8, 4, -2], [-1, -8, 9, -3], [-7, 6, -5, -1], [-2, -7, -2, 14], [-16, 2, 7, 0], [-5, -9, 12, -3], [12, 4, 4, 0], [24, -5, 4, 7], [5, 2, 8, 9], [-56, 7, 4, 2], [-27, 2, 24, -1], [4, -6, 16, 2], [42, -16, -4, -1], [-30, -10, -10, 4], [-44, -3, 2, 6], [30, -22, -8, 1], [35, -18, -8, 6], [41, -13, -6, 8], [30, 8, 5, -2], [-4, 11, 1, 0], [38, 8, 0, -2], [-5, 3, -12, -7], [21, -15, -19, 3], [-29, -3, -14, 5], [48, -8, -8, 7], [19, 8, -9, 5], [18, -4, 10, -4], [-30, -2, 8, 3], [21, -20, -7, 1], [26, -22, -4, -2], [20, 9, -6, 4], [-26, -14, 20, -4], [14, -28, -6, 6], [19, 8, -12, 11], [31, 4, 6, 9], [14, 6, -4, 18], [-17, -5, -4, 2], [-12, 9, -4, -6], [7, 16, 17, -13], [38, -10, -12, 6], [-20, 8, 14, -4], [24, 2, 2, -6], [3, 22, 6, -5], [13, 12, -15, 11], [-6, -2, -12, 0], [20, 4, 8, 0], [-1, -12, -1, -3], [0, -12, 6, 8], [18, 4, 10, -10], [-31, -1, -4, 7], [-14, -12, -4, -6], [8, -8, 14, -14], [-29, -1, -10, 11], [-21, -5, 16, -1], [29, -23, -6, 13], [5, 8, 0, 5], [25, 11, 2, -6], [29, -24, -12, 9], [-29, -9, 4, -10], [43, -11, 6, 3], [33, -6, 6, -5], [52, -8, 9, 16], [-26, 12, 12, -2], [-25, 15, -4, -9], [10, 5, 16, -5], [16, 4, 12, 10], [-42, 0, 2, -6], [53, -12, -5, 1], [11, 3, 8, -7], [-4, -3, 12, -1], [2, 5, -20, 6], [-19, 6, 3, -5], [13, -7, -14, 1], [8, -14, -16, 6], [1, -20, 0, 3], [29, -12, 0, -6], [-24, 9, 12, -3], [31, -5, 15, 5], [46, -4, -4, 12], [14, -2, -6, 12], [-1, 22, -6, 1], [4, -14, -8, 11], [-14, -6, 12, 4], [29, -4, 15, -5], [-2, -14, -10, 6], [-22, -3, -14, 4], [-10, -5, 15, -18], [44, -4, -8, -3], [15, -4, -4, 15], [46, -8, 0, 4], [-43, 3, -6, 2], [-20, 1, 20, -6], [-17, 11, 0, -7], [10, 4, -4, 5], [13, -1, -4, 8]];
    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_cg();".
// 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_cg(: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_cg();".
// 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_cg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![8, 12, -11, -2, 1]>,<11,R![-199, 170, -22, -6, 1]>,<13,R![-56, -112, -34, 2, 1]>,<17,R![-62, -82, -25, 2, 1]>],Snew);
    return Vf;
end function;