
// Make newform 6762.2.a.ct 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_ct();".
// 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_ct();".
// 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 := [1, 2, -3, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1, -2, -2, 1], [-3, -4, 2, 0], [-2, 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], [2, 1, 0, 0], [0, 0, 0, 0], [1, 0, 1, 0], [3, -1, 0, -1], [-1, 1, 0, -1], [1, -1, 1, 2], [1, 0, 0, 0], [2, -2, -2, -2], [-1, -1, 1, 2], [1, -1, 0, 1], [-2, -1, -2, 0], [1, 1, 0, 3], [3, 1, 1, -2], [1, 1, -2, -1], [4, -2, -2, -2], [4, -1, -2, 0], [5, 5, 0, -1], [-6, -1, 1, -1], [2, -4, -1, -1], [-8, -3, -1, 1], [-1, 1, 1, 0], [5, 3, -2, 3], [1, -7, 0, 1], [3, 1, 0, 3], [0, 0, 2, 2], [5, -4, 1, 4], [-3, 7, 0, -1], [-12, 1, 1, 1], [-8, -3, -3, 1], [-6, 4, -2, -4], [-12, 2, 0, -2], [8, -2, 0, 0], [3, 3, 2, -1], [-2, -3, -1, 5], [4, 4, -3, -3], [-2, -5, -1, -1], [11, 7, -1, 0], [-1, 7, 2, 1], [-2, 4, 2, -4], [8, 3, 2, 0], [-6, -3, 1, -3], [2, 6, -4, -2], [4, 1, -3, 1], [-10, 0, 4, 2], [-4, -5, 1, -1], [3, 9, -1, 0], [-1, -13, -1, 2], [0, -9, 4, 2], [-8, -8, 2, 0], [-16, 1, 1, -3], [-1, 5, 4, 3], [13, 5, 1, 2], [4, -5, -2, -2], [-20, 8, 0, 0], [11, -5, -2, -7], [-3, 0, 2, -1], [-8, 6, 4, 6], [2, -6, 2, -2], [-3, -10, -2, 1], [8, 0, 1, 5], [-2, -8, 0, -6], [13, -6, 2, 7], [-3, -5, 4, 1], [4, -15, 1, 1], [12, -4, 0, 0], [4, -1, 3, 3], [-10, 5, -3, -7], [3, -1, -2, 5], [2, -2, 3, -5], [-2, -5, -1, -1], [-2, 8, 0, -2], [-1, -19, 0, -3], [7, 12, -1, 0], [-4, 6, 2, -6], [-7, 1, -2, 5], [-13, -7, -2, 1], [-8, 2, -4, 2], [12, 9, -4, 0], [5, 5, 1, -6], [15, 3, 2, 5], [-2, 16, -2, 0], [3, -9, 0, -5], [-11, 0, -2, 3], [6, -3, -3, -7], [-24, 6, 2, 6], [-12, 0, 2, 0], [1, 3, 4, 1], [-4, 2, 2, 10], [1, 0, -2, 1], [-6, 2, 2, -8], [0, -10, 2, -2], [2, -11, -3, -7], [14, -5, 3, 3], [-10, 6, -4, -2], [9, -1, 4, 1], [-3, 7, -2, -1], [11, 3, 1, 0], [-4, -4, 2, -8], [8, 4, 0, 8], [-7, 3, 0, 7], [9, 10, -8, -1], [-20, 0, -2, 4], [5, 1, -4, -5], [-6, 14, -1, 7], [-4, 4, -6, 2], [4, 3, -4, 8], [-8, -4, -4, 4], [2, -20, -3, 1], [11, -9, -7, -10], [15, -3, 0, 5], [-18, 4, 0, -4], [-27, 11, -3, -6], [2, 1, 5, -11], [-12, -4, 2, -4], [-7, 3, 1, 2], [5, 12, -4, 5], [-4, -6, 4, 6], [15, 7, 4, 1], [-8, 17, -2, -8], [10, 0, -4, -8], [-14, -1, 0, 4], [8, 15, 5, -1], [20, -12, 2, 2], [13, 17, 2, 3], [11, 9, -4, 7], [-11, 2, 2, -1], [-2, 6, -2, 0], [14, -1, -6, -6], [2, -13, -5, -5], [-12, -3, -1, 5], [6, 2, -4, 2], [13, -9, 8, 7], [20, 4, 1, -1], [-9, 3, -2, 9], [-2, 8, 5, 11], [5, 1, 3, -4], [-10, 0, 7, -3], [-12, -12, 4, 0], [4, -12, 4, 0], [6, -13, -3, -5], [2, -25, 1, -1], [11, -8, -5, -8], [-3, -19, 8, 5], [-18, 6, 0, 6], [-11, -11, 0, 1], [-6, 0, -1, 7], [-6, 20, 0, -2], [-12, -1, 7, -5], [-6, -5, 9, 7], [-27, 9, -2, -13], [8, 9, -5, 1], [-15, -19, 5, 0], [7, 2, 1, 14], [2, -15, -3, 5], [-8, 9, -1, -7], [-6, -7, 0, 2], [-7, -15, 6, -1], [30, -13, 4, 4], [6, 9, 1, 1], [2, 7, 5, -1], [4, 2, 2, -10], [-25, 10, 0, 9], [16, -4, -2, 8], [-20, -18, -6, 6], [-2, -9, 9, 7], [-9, 7, 4, 3], [-36, -14, -4, 2], [11, 5, 10, 9], [9, -24, -7, -8], [19, -5, 2, -7], [0, 17, 7, 9], [-8, 9, 1, -15], [7, -4, 2, 9], [11, 3, -8, -9], [26, -14, -4, -6], [0, 16, -2, -12], [-16, 8, 0, -12], [24, -12, -3, -7], [10, 9, 7, 1], [-33, 5, -3, -4], [-10, 1, -5, -7], [-42, 10, -1, -5], [-18, 2, 0, 6], [6, 12, -3, 3], [-10, -4, 4, -8], [7, 3, 1, 12], [-10, -10, -10, -2], [5, -11, -7, -8], [-16, 24, 3, 9], [-20, -3, 7, 1], [-3, 2, -1, -2], [31, 13, -6, -3], [15, 7, -4, 1], [-5, -5, 0, -1], [12, 7, -9, -1], [-5, -29, 2, 3], [7, -7, 8, 11], [3, 8, 4, 3], [-11, 3, -4, 1], [-15, -2, 4, -7], [23, -1, 0, -1], [1, 3, 8, 13], [-23, -1, -5, -12], [11, -15, 2, -3], [-2, -2, -16, -10], [-12, -3, 7, 1], [-20, 8, -6, 0], [2, 6, 2, -12], [-4, -30, -2, -2], [3, 37, 4, -1], [14, 8, -6, -8], [-13, 7, -5, 10], [33, 11, 3, -6], [-31, -23, -2, 3], [-8, 22, -6, 2], [-12, -14, -7, -9], [-16, 1, -5, -7], [52, -14, 4, 10], [12, 9, 1, -7], [4, 10, -2, -6], [-10, 20, -4, -12], [-10, 9, 1, 13], [18, 24, 2, -4], [12, -5, -1, 7], [-1, 1, 10, 9], [10, 7, 9, -1], [-6, -6, 10, 4], [-14, -2, 8, 14], [10, -17, -3, -5], [-7, -21, -3, -2], [30, -3, 5, 1], [-32, -14, 6, -6], [27, 6, 10, 7], [-28, 14, -8, -8], [15, 5, 0, 15], [17, -21, 0, -7], [-24, -4, 4, -12], [7, -11, 1, 2], [3, 9, 6, -5], [-24, -16, 0, 4], [-3, -9, 9, 0], [-6, -4, 8, 0], [-12, -12, -5, -3], [11, -4, -4, -13], [-13, -18, 0, -1], [-7, 13, -7, 6], [19, 1, 2, -9], [28, -11, 7, 17], [-21, -17, -10, 9], [-45, 1, -2, 1], [-36, 4, -4, 8], [-17, 23, -2, -3], [23, -7, 0, -9], [-30, 20, 6, 8], [21, -13, 12, 19], [17, -12, 8, 11], [-11, -9, -6, -1], [16, -7, 2, -10], [6, -8, 14, 4], [33, 21, 0, 3], [-31, -9, 2, 7], [-6, -6, 7, 5], [14, 6, 5, 5], [-11, -15, 7, 0], [-11, 11, -12, -17], [-9, -9, -12, -9], [-31, 5, -4, -1], [-2, 18, 6, -2], [19, -13, -2, -19], [-1, -4, 7, -8], [-7, 23, -6, -13], [14, 10, 16, 10], [14, -21, -2, -8], [7, -2, -4, -9], [16, 6, 2, 10], [-2, -32, -4, -4], [22, 0, 4, 0], [-23, 21, 9, 14], [49, 4, 4, -7], [6, 12, 2, 8], [-30, -8, -2, 0], [-2, 18, 1, 11], [14, 28, -4, 6], [-4, 17, 9, 13], [24, 24, 6, -8], [19, 5, -4, -7], [13, 26, 4, -11], [-10, -15, -2, -8], [1, -15, -2, 5], [-9, 26, 0, -7], [-14, 32, 2, 0], [-39, 4, 6, 1], [20, 0, -6, 8], [-2, 1, 0, 12], [30, -12, -6, -20], [47, 15, 8, -1], [14, 22, -2, 4], [30, 4, -10, -8], [-11, 15, -2, 7], [5, -11, -2, -15], [-12, 2, 2, 6], [-4, 27, -11, -5], [15, 33, -10, -1], [-4, 6, 2, -2], [38, -4, 10, 8], [-12, -15, 2, -8], [-4, 8, -4, -12], [-10, 22, 2, 6], [19, 8, -8, -1], [-15, -23, -6, -3], [-4, 10, 8, 2], [4, 7, -7, -1], [-18, -28, -2, 0], [45, -11, 4, 13], [-10, -8, 12, 6], [-10, -11, -5, 5], [11, -5, -16, -17], [-58, 8, 2, 8], [-27, 10, 2, -11], [2, -29, 9, 3], [15, -7, -12, -5], [-8, -14, -10, 6], [26, -1, -11, -13], [-25, -9, -2, 9], [37, -5, 10, 5], [50, 10, 6, -4], [-16, -5, -3, -9], [-39, -27, 2, -1], [4, -12, -6, -8], [28, 38, 4, 2], [-19, -12, -2, -5], [-36, 18, -6, -6], [-21, 17, 2, 15], [-14, 34, 2, 4], [-15, 15, 0, 1], [33, 3, -6, -13], [28, 23, 6, -10], [-50, 2, 0, -6], [6, 30, 2, 10], [-4, 20, 10, 16], [29, 7, -2, 7], [4, 14, -6, 6], [17, -31, 6, 5], [-14, 34, 0, -6], [29, 27, 10, -9], [-10, 22, 0, -2], [-26, -2, -4, 6], [36, -10, 7, 11], [0, 2, 2, 10], [20, 8, 2, -8], [-56, -19, 9, -7], [30, 12, 6, 0], [-31, -17, -8, -1], [34, -24, -5, -5], [-2, -4, -6, -20], [-3, -18, 4, 7], [-8, -32, 4, -4], [8, -20, -4, 8], [21, 1, -6, -1], [2, -37, 0, 2], [16, -6, -10, -18], [-3, -17, 2, -3], [16, -16, 8, 8], [18, -16, 4, 16], [3, 1, -14, -7], [44, -22, -2, 6], [44, 4, 2, -4], [46, -3, -7, 1], [56, -4, -2, -2], [12, 22, 6, 2], [45, 23, 0, -5], [26, 10, -6, -4], [-10, 24, -4, 4], [14, 18, 6, 8], [6, 22, -12, 2], [-19, -13, 8, -15], [-15, -1, 14, 3], [26, 25, 12, 6], [10, 6, 13, 19], [-30, -23, 11, -3], [12, 3, 3, -1], [-36, 16, -2, 8], [-1, 18, 2, -5], [3, 2, 6, 17], [-24, -19, -7, -7], [7, 19, 4, 11], [-29, 9, 2, -9], [34, -12, -6, 4], [-12, 45, -3, -7], [19, 5, -6, -19], [8, -30, 10, 14], [-20, -24, -6, -6], [-18, 29, 9, 5], [14, -39, 3, 5], [-54, -6, -8, -2], [3, -27, 2, 1], [23, -8, 11, 12], [-28, -12, 10, 8], [19, 17, 12, 3], [12, -28, -12, -4], [-8, 21, 11, 5], [-23, -19, 0, -7], [7, 31, -12, -5], [-13, -3, 11, 20], [-5, -7, 4, -15], [-8, 24, 12, 8], [21, 0, 8, 7], [-34, 6, -14, -16], [40, 24, -3, -1], [32, 12, 6, -4], [17, -5, 10, -7], [19, 6, 1, 2], [32, 15, -13, 3], [27, 5, -16, -9], [4, 4, -24, -16], [-5, 1, 11, 4], [23, -5, 8, 7], [-52, 10, -2, 2], [21, -3, -10, -1], [34, 5, -4, 4], [0, -24, 0, 0], [-40, -20, 16, 0], [92, -3, 0, -4], [17, -3, -6, -9], [-9, 34, -3, -14], [2, 32, -10, 4], [-8, -28, 14, 2], [45, 13, 1, -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_ct();".
// 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_ct(: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_ct();".
// 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_ct( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, -4, 1]>,<11,R![16, 48, -20, -4, 1]>,<13,R![-112, 16, 36, -12, 1]>,<17,R![-16, -48, -12, 4, 1]>],Snew);
    return Vf;
end function;