
// Make newform 6762.2.a.ce in Magma, downloaded from the LMFDB on 03 May 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_ce();".
// 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_ce();".
// 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 := [-6, -15, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-10, -1, 1]];
    Rf_basisdens := [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], [-1, 0, 0], [1, 0, 0], [0, 0, 0], [0, 1, 0], [-1, -1, -1], [1, 1, -1], [2, 0, 1], [-1, 0, 0], [4, -1, -1], [0, 1, 2], [-4, -2, 1], [-6, 0, 1], [0, -2, 0], [-1, 1, 2], [-1, -2, 0], [-4, -1, 1], [10, 0, 0], [-7, -1, -2], [-1, 3, 1], [5, 1, 0], [-4, 3, 1], [-4, 1, 0], [4, 2, -1], [0, -3, -3], [-6, -2, 0], [7, -1, -2], [-10, -1, 0], [-4, -2, 0], [-5, -1, 3], [-12, -1, 0], [1, -4, -3], [1, 3, 2], [2, 0, 4], [-9, 3, 0], [6, 1, 0], [0, 2, -5], [2, 2, -5], [1, 3, 4], [-4, -2, -2], [-3, -3, 2], [-2, -2, -2], [2, 2, -5], [-3, 6, 2], [-2, 0, 2], [0, -2, 2], [-2, 0, -2], [2, 3, -2], [4, 3, -2], [18, 0, 3], [-14, -4, 3], [-16, 0, 0], [10, 3, 5], [12, -5, 0], [4, 4, -1], [-8, 2, 0], [-8, 3, -1], [-4, -1, -4], [-1, 3, 0], [3, -7, -5], [7, -3, -4], [-1, 0, 4], [-14, 6, 3], [-13, -3, -1], [12, -1, 1], [2, 3, 5], [-8, -4, 3], [-8, -3, -5], [11, 3, 2], [15, -3, -1], [-14, -4, 3], [-14, 0, -4], [-11, 2, 3], [6, 2, 4], [-11, 5, 2], [-12, -2, -5], [-10, -2, -8], [1, 1, -1], [-5, 3, 0], [-1, -2, -6], [-4, -6, 2], [4, -2, 6], [4, -4, -4], [-2, -8, 0], [-12, 3, 2], [-11, 4, -1], [-8, -2, -5], [4, 5, 7], [6, -4, 0], [8, -2, 0], [0, -2, 0], [-22, 4, 1], [-10, -1, -6], [9, -2, 1], [-8, 2, -6], [-10, 6, -2], [2, -1, 3], [-3, 3, -2], [-15, 5, 2], [1, -1, 1], [-4, -8, -1], [-4, 7, -2], [-18, -5, 2], [-17, -1, 0], [-11, -3, -9], [12, 5, 2], [-25, -2, -7], [-22, -2, 4], [9, 1, 7], [-8, 3, 0], [12, -7, -4], [-22, 6, 0], [5, -5, 2], [3, -1, -9], [-17, 6, 3], [-7, 7, 5], [6, -2, -1], [-16, 10, 6], [-6, 3, -7], [-22, 4, -2], [-12, -6, 8], [-4, -11, 0], [-15, 4, 0], [5, -2, 7], [12, 0, -7], [-27, -2, 2], [-2, 10, 3], [3, 5, 0], [-24, -3, 3], [6, -2, 9], [-8, -6, 6], [0, 6, -5], [4, -7, -3], [10, 2, 3], [10, -12, -1], [6, -9, 1], [7, -1, 2], [19, 1, 4], [-5, 4, 2], [4, -6, 3], [-6, -4, 4], [-34, 1, 1], [2, -6, -8], [10, 7, -10], [15, -9, 0], [18, -2, -3], [-14, 4, 2], [-12, -8, -1], [18, 8, -7], [-23, -7, 1], [-17, 3, -3], [-11, 3, 0], [2, 6, 1], [32, 4, 0], [-19, -1, -1], [8, -6, 8], [27, -5, -2], [2, 2, -11], [-20, 11, 1], [24, -5, 0], [-33, -3, -4], [22, 0, 2], [10, -7, 0], [-22, 1, 0], [9, 3, 6], [-8, 4, -3], [-24, -5, 2], [24, 2, 1], [30, 7, -1], [-12, 11, 1], [-12, 5, -6], [-11, 3, -6], [14, 6, 11], [-36, 1, -1], [8, -3, -4], [12, 8, -3], [-12, 14, 8], [-14, -3, 5], [-21, 3, 2], [-14, -6, -9], [14, 3, 4], [-14, -9, 2], [5, -1, -5], [52, 0, 1], [22, -8, -5], [31, 4, 0], [-9, 1, 5], [-16, 6, 8], [19, -4, 4], [-5, 5, -7], [-4, -7, 0], [16, -7, 5], [-7, 7, -5], [-16, -10, 0], [-10, 1, -8], [47, -3, 0], [-20, 4, 0], [24, 0, -1], [5, 7, 5], [-25, -5, -5], [15, 0, -2], [0, 2, -2], [3, 5, 2], [28, 2, 6], [-2, -2, -4], [-2, 7, 7], [-5, 7, -2], [35, 4, 11], [8, -2, 9], [24, -6, -7], [21, -7, -2], [20, -7, 1], [-4, -5, -13], [8, -5, 10], [-4, -4, -2], [-20, 10, 4], [2, 7, 0], [-8, -14, -5], [-28, -2, 0], [32, 0, 4], [37, 1, 2], [11, 3, 2], [-6, 4, 6], [45, 0, 5], [4, 11, -3], [13, 5, 11], [-20, 10, 3], [43, -3, -1], [41, 4, 5], [9, 3, 0], [-17, 13, 0], [11, -9, -2], [0, -1, -1], [30, -4, -6], [43, 3, 7], [-16, 12, 2], [-12, -9, -13], [-8, 7, -8], [25, 1, 2], [-26, -8, 3], [-38, -8, 8], [18, 2, 1], [-8, 11, -4], [11, -11, -2], [-4, -2, -5], [-30, 4, 1], [-8, 5, -8], [24, 4, 4], [-14, 0, -1], [10, -10, -1], [0, 4, 9], [-7, -1, 7], [16, -6, 0], [0, 2, -8], [14, 3, -7], [-24, -11, -3], [4, -4, 9], [-14, 4, -7], [-24, -9, 11], [-16, 3, 4], [17, 1, 2], [0, -5, -10], [21, 11, -9], [-16, 6, -3], [20, 0, -1], [16, -2, 0], [39, 4, 0], [32, -12, -3], [-11, -17, 3], [10, -1, -9], [-35, 5, -4], [10, 2, 1], [32, -5, 3], [39, 8, 7], [-33, -8, -8], [4, 10, 0], [-35, -8, 5], [33, -11, 3], [40, -6, -6], [2, 16, -2], [-37, 7, 1], [-23, 6, 3], [-18, -14, -4], [-6, 12, 9], [14, 12, 14], [-20, 0, -10], [-4, -7, -15], [-21, 10, 0], [-16, -5, 10], [-4, 8, -8], [-18, -12, -8], [6, -11, 4], [26, 0, -5], [62, 4, 4], [49, 1, -6], [-19, -12, -4], [-57, -1, -4], [-8, 2, 8], [-39, -1, 0], [38, 0, -8], [-16, 10, 4], [31, -2, -10], [43, -10, -1], [-40, 1, 7], [-43, 3, 6], [-38, 6, 6], [-19, 2, -3], [-32, 14, 2], [24, -6, -3], [8, 14, 15], [28, -2, 2], [53, 1, 2], [39, 3, 15], [2, -6, -3], [-20, 12, 5], [-46, -8, -4], [16, 5, -7], [30, 12, -1], [-16, -7, -1], [28, -6, -5], [19, 1, -11], [-44, 2, 6], [-2, -3, -19], [12, 10, -8], [3, 7, 18], [-18, -7, -13], [-6, 6, -11], [-16, 10, -7], [-28, 6, 10], [-28, -9, 2], [37, 3, -11], [-4, 11, 15], [-10, 0, 8], [1, 13, 8], [-53, 7, -3], [1, 11, -4], [12, -16, 0], [-18, -16, 0], [49, 4, 10], [-16, -9, 17], [-47, -5, -11], [-9, 19, 6], [34, 3, 7], [-32, 5, -11], [24, 13, 12], [-13, 4, -5], [21, -25, -8], [-24, 10, 7], [24, 10, 2], [-16, -3, 2], [12, 4, 13], [39, -10, 2], [-28, -2, -4], [-11, 2, -9], [12, -4, 9], [10, -4, -10], [48, 10, 3], [0, 1, 0], [-10, -12, -7], [-1, -1, 9], [14, 0, -10], [-6, 18, 6], [-14, 10, 11], [10, -20, -12], [-12, 4, -10], [-19, -10, -4], [0, -12, 4], [-8, 20, 8], [-2, -5, -5], [-4, -5, 1], [-16, 8, 2], [61, -1, 2], [28, 11, -8], [-17, -13, -12], [-3, 5, 4], [4, 10, 10], [-12, -8, -4], [38, 3, -3], [9, -5, -4], [54, -6, 4], [-14, -6, -16], [-23, -13, -16], [-6, -8, 2], [-54, 2, 0], [-49, -1, 3], [-1, -3, 12], [26, 4, 12], [0, 0, -9], [45, 11, -5], [24, -12, -3], [-66, 5, 1], [-30, -4, -6], [17, -8, -3], [46, -8, 4], [-4, 2, 15], [36, -1, -3], [17, 12, 13], [37, -9, 0], [-35, 7, 7], [62, 7, -5], [-13, 19, 8], [-22, 18, 4], [36, 4, -10], [29, -21, -3], [24, -5, 11], [-14, -1, 6], [10, 8, -4], [24, 8, 13], [13, 11, -8], [-8, -5, 9], [-57, 6, 6], [16, 11, 17], [-20, -1, -4], [15, -15, 2], [26, 4, 14], [-36, -11, 13], [17, -5, -15], [-2, -10, -8], [5, 1, 6], [-12, 6, 12], [-12, 3, -16], [-48, 16, 1], [-6, -6, 9], [49, -6, -9], [43, -11, -6], [44, 10, -2], [36, -13, 4], [-36, -14, -9], [16, -2, 5], [-52, -4, -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_ce();".
// 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_ce(:prec:=3)
    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_ce();".
// 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_ce( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-1, 1]>,<11,R![-6, -15, 0, 1]>,<13,R![-22, -24, 3, 1]>,<17,R![126, -36, -3, 1]>],Snew);
    return Vf;
end function;