
// Make newform 6762.2.a.cp 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_cp();".
// 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_cp();".
// 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 := [34, 8, -13, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [10, -10, -3, 1], [10, -6, -3, 1], [-38, 2, 7, -1]];
    Rf_basisdens := [1, 4, 2, 4];
    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, 1, 0], [0, 0, 0, 0], [1, -1, 1, -1], [1, -1, 0, 0], [-3, -1, 1, 0], [-1, 0, -1, 1], [1, 0, 0, 0], [2, 1, 0, 0], [1, 1, 1, -1], [-1, -2, 1, -1], [1, 0, 1, -1], [6, 2, 0, 2], [4, 3, -1, 1], [1, 2, -1, 0], [2, 1, -2, 0], [-2, -4, 0, 0], [6, 3, -2, -1], [11, 1, 0, 0], [2, 1, 1, -1], [-2, -1, 0, 0], [-3, -1, 3, -1], [1, 2, 1, 1], [-2, 1, -2, 0], [2, -2, 2, -4], [0, 3, -2, -1], [-1, 1, 1, 3], [6, -2, -4, 2], [5, -3, 1, -2], [9, -1, -1, -1], [2, -4, 1, -1], [2, 1, 0, 1], [-2, 0, -4, 0], [0, 1, 2, -3], [3, 5, -1, 3], [-3, -6, 1, 1], [-7, -6, -1, -1], [-2, 1, -7, 1], [4, -2, -4, 0], [-6, 3, -1, 1], [-2, 2, -4, 4], [-9, 2, -1, 1], [11, -2, 0, -2], [4, -4, 4, -2], [2, 2, -4, 2], [-4, 0, 4, 2], [-13, -5, 3, -1], [-7, -3, -3, -1], [-7, 0, 1, -1], [-1, 0, 1, 1], [12, 4, 0, 4], [10, 3, 2, 2], [1, 1, -3, -1], [-7, 4, -1, 1], [-2, -6, 6, -2], [-2, 1, 2, 4], [-1, -1, -1, 1], [4, -5, -3, 3], [11, -5, 1, 0], [8, 1, 2, -1], [-23, 0, -3, 0], [13, -2, -3, -1], [7, -1, -2, -4], [-4, 3, 0, -2], [-4, 1, 2, 4], [1, -4, -5, 1], [-2, -3, 2, -4], [6, 5, 1, -1], [7, -3, 2, -6], [7, 4, -1, 1], [6, 4, -8, 0], [0, 2, -4, -1], [12, -2, 2, -2], [0, 1, -2, 1], [1, 6, -3, 5], [2, -6, 0, 0], [5, 1, 0, 6], [-2, 5, 0, -1], [-7, -2, 4, 2], [-6, 2, 0, 2], [4, 2, -4, 0], [0, -4, -8, 0], [6, 4, 0, -4], [-11, -9, 1, -5], [8, 4, -1, 1], [3, -2, 1, 3], [-8, 1, 0, 6], [-14, 0, -4, 4], [16, 2, -4, -4], [-4, 2, -4, 4], [-9, 4, -1, 5], [-1, -1, -5, -1], [-6, -2, 11, -1], [-12, 2, -2, 0], [-6, 6, 0, -4], [-18, 1, -6, 2], [2, -7, 4, -3], [-4, 5, 0, 1], [-11, -5, 10, 2], [-27, 4, -3, 1], [-9, 5, 7, -3], [-13, 1, 7, -1], [-22, -3, 4, -1], [-11, -7, 5, 2], [9, -3, 1, -5], [8, 6, -7, 3], [-20, -6, 0, -2], [13, 3, 8, 0], [15, 3, 3, 5], [1, 1, -1, -1], [-8, 6, -4, -2], [-6, 1, 0, -3], [9, 7, -5, 4], [-8, 2, -6, 1], [5, 5, -9, 4], [3, -2, 3, 5], [16, 6, -4, -4], [-2, 1, 4, -2], [-14, 0, -2, 4], [18, 6, 4, 6], [-3, -3, 1, -5], [7, 4, -7, 0], [2, 2, -7, 7], [7, -4, -7, -1], [3, 2, 5, -4], [7, 2, 11, 1], [2, -5, 5, -5], [0, -3, 6, -2], [5, -2, -7, 3], [-12, 6, -4, 0], [-3, 2, -5, 9], [-12, -3, 6, -6], [23, 2, -3, -7], [-13, -4, 1, -11], [-4, 3, -2, 4], [18, -3, 2, -1], [0, -3, 4, 3], [1, 4, 1, 0], [15, 10, -3, 3], [2, -4, 10, -4], [-16, -9, 2, -4], [8, -2, 4, 6], [-5, 5, -7, 3], [4, -1, 3, -5], [5, -6, 7, 3], [28, 0, -2, 2], [1, 4, -7, 9], [21, 4, -5, -5], [15, 15, -8, 2], [19, 3, 4, 2], [4, 1, 0, -1], [3, -6, 9, 1], [20, 8, 8, 4], [-17, -5, 7, 0], [-2, 2, -8, -6], [-16, 7, -6, 3], [-29, -2, 1, 1], [-14, -1, -4, -4], [1, -7, -3, 3], [-12, -1, -13, 1], [-8, -4, -10, -2], [3, -7, 3, -5], [23, -1, 5, -1], [10, -3, 10, -7], [-11, 4, -5, 1], [-3, 3, -3, -5], [1, -2, 5, 5], [0, 11, -4, 8], [-18, 1, 6, 0], [1, 3, -15, -1], [40, 7, -1, 1], [11, 10, -5, 5], [-10, 1, 14, 0], [5, 1, 5, 3], [-7, 8, 1, -3], [-18, -10, 4, 6], [2, -1, 6, -2], [-20, -1, -6, -1], [-1, 2, 3, 5], [15, 3, 3, 3], [-11, -3, 11, -7], [3, -1, 10, -4], [-5, -8, 9, -5], [17, 8, 3, 3], [3, 12, -9, -2], [-23, -19, 2, -4], [6, 6, -8, 2], [-1, 4, 0, 4], [-23, 3, -4, 6], [5, -7, 1, -5], [8, -1, 0, -10], [-33, -9, 5, -4], [42, -2, -6, 2], [-35, 3, -7, 1], [-20, 7, -10, 5], [6, 4, -8, 10], [3, 4, 3, -1], [-15, -3, -5, 4], [25, 5, -14, -2], [-19, 0, -3, 0], [2, -2, -12, 6], [8, 17, -7, -1], [0, -6, -2, 0], [8, 2, -8, -6], [-10, -3, -2, -6], [16, -5, -8, 3], [2, 4, -3, 1], [9, 6, -1, 13], [9, -2, -3, 9], [16, -15, 3, -3], [-2, -5, -8, -4], [-16, 7, -2, 2], [13, -15, 7, -5], [-2, 0, 2, -14], [-2, -2, 8, 2], [1, 3, 5, 9], [-9, 2, -7, -1], [-30, 10, -4, 2], [22, 8, 8, 6], [-10, -3, -9, 3], [40, -1, -8, 3], [16, 0, -10, 2], [-24, 0, 4, -5], [4, 1, -4, -2], [1, 9, -6, 6], [-7, -10, -7, 1], [-33, -13, 6, 4], [32, 4, -6, -9], [-32, 1, -1, -5], [-22, 1, 5, -3], [0, 3, -2, -1], [-20, -21, 6, -6], [-4, -4, 12, 6], [-3, -1, 7, 0], [2, 0, 0, -6], [32, -1, 0, -10], [13, -3, 9, -1], [-18, -1, 1, 1], [-33, 0, 5, 1], [-12, -8, 2, 2], [23, -2, -1, 5], [9, 15, -7, -1], [12, 1, -1, -5], [-21, -2, -15, -1], [-3, 16, -13, 3], [-29, 1, 5, -11], [-52, 4, -4, 4], [-7, 4, -3, 3], [-9, -6, -11, -3], [-23, -4, 3, -7], [31, 5, 5, -2], [10, -6, -10, -2], [42, 6, 6, 2], [20, 1, -4, 8], [14, -1, 18, 0], [13, -12, -5, 5], [19, 0, 7, 5], [14, 13, 8, 4], [-7, 3, 9, 3], [36, 5, 0, 5], [3, 13, 9, 1], [-1, 11, -2, 0], [3, 2, 17, -1], [-11, 16, -9, 5], [-2, 6, 0, 2], [-1, 4, -7, -6], [1, 12, 5, 1], [7, -5, -1, -8], [20, 5, 2, -4], [0, -11, 15, -3], [1, 10, 5, -1], [-6, -13, 2, 4], [-12, -16, 4, -3], [-7, 0, 6, -10], [8, 14, -2, 8], [-8, 12, -3, -3], [45, 13, 4, 6], [24, 10, -8, -4], [18, 0, -10, 8], [-29, -15, 6, 4], [-6, -6, 14, 1], [4, 14, 4, 6], [-9, -12, -5, 1], [-8, -20, 12, -6], [6, 8, -6, 18], [-22, -15, -6, -4], [-9, -10, 9, -10], [-1, 3, 5, 9], [24, 0, -6, 4], [-2, 20, -16, 4], [47, 7, -1, 7], [-5, -8, 11, -11], [-24, -12, 10, -6], [26, 5, -9, 3], [-53, -12, -1, -8], [54, 11, -2, 5], [6, -10, 14, -6], [-16, 13, 11, -5], [-2, -16, 2, 4], [-2, 2, -10, 2], [17, -14, -5, 4], [0, -6, -8, -3], [-16, -21, 8, 2], [-22, 11, -4, -1], [32, 6, -2, 6], [10, 2, 1, 3], [4, 6, -12, 8], [39, -6, -1, 7], [19, -14, 7, -1], [36, 10, -8, -12], [10, -5, 10, 1], [45, 17, -3, 2], [3, 18, -5, -3], [5, 0, -3, 13], [6, 8, -8, 12], [6, -9, -6, 8], [-1, 8, -3, 9], [14, 21, -10, 0], [-1, -10, 19, -5], [-27, 5, 9, 4], [12, 2, 2, -4], [-28, -9, 0, -12], [4, -2, 2, 4], [-20, 1, 4, 5], [52, 5, -10, 0], [-27, 6, -7, 3], [-19, -6, -7, -7], [-12, 18, -6, 4], [-5, 13, 7, 1], [-35, -5, -4, -2], [-4, -7, 10, 6], [20, -4, 0, -10], [12, 3, 11, 7], [7, -13, 25, -6], [-32, -7, 9, -5], [24, -4, 12, -4], [-14, -4, -4, 0], [19, 4, 4, 10], [0, 7, -2, 2], [13, -1, 10, 2], [20, -3, -20, 1], [-18, 5, 8, 10], [16, -7, -6, 2], [-5, -1, -9, 1], [2, 4, -21, 1], [14, 15, -9, -1], [-9, -2, 3, 7], [-4, 2, -10, -8], [21, -1, 1, 3], [-29, -12, 3, 3], [-43, 14, -8, 6], [14, -10, 12, -14], [0, 2, 8, -1], [-7, -16, 7, 5], [-4, -16, 14, 2], [-51, 6, -5, 1], [5, 11, 1, 15], [25, 4, -7, 7], [-9, 1, 6, 12], [16, 0, -12, 2], [22, -6, 2, -12], [17, 2, -1, 11], [0, -4, 16, -18], [-18, 8, 0, -10], [11, -2, -2, 10], [-34, -8, 4, 2], [52, 4, 2, -4], [-18, -9, 14, -10], [-4, -15, 20, -10], [-42, 0, -6, 2], [-12, 5, -9, 11], [27, -7, 3, -3], [-6, 15, 0, 1], [8, -3, 3, 5], [14, 2, 2, -6], [-4, 8, -18, -4], [46, -9, -8, 6], [-28, 5, -18, 3], [-6, -14, 4, 4], [18, -2, -4, -4], [56, 5, -8, 7], [10, 12, -6, 0], [12, 6, 0, -18], [29, 7, 1, 12], [30, 3, -17, 3], [16, -12, -12, 10], [9, -16, 13, -7], [-5, 3, -17, 12], [65, 12, 5, 9], [-8, -15, 14, 4], [-36, 0, -2, -6], [-50, 8, -1, 3], [18, 12, 4, -4], [-7, 2, -11, -7], [2, 7, 12, 8], [-22, -16, 0, -11], [36, 5, 0, 7], [3, -13, 13, -4], [-8, -19, 10, -4], [0, 11, -11, 11], [12, -10, -8, 6], [-26, 12, 4, 2], [-51, -7, 15, 0], [22, 13, 12, 0], [11, 7, -9, 7], [6, -8, 6, -12], [39, -8, 3, 3], [-40, 7, 8, -3], [-46, -11, 14, 4], [-11, 14, -4, 14], [-42, 17, 6, -4], [-49, -19, 5, 1], [-22, -31, 8, -7], [32, 8, -4, -6], [-4, 7, 6, 2], [61, 1, -2, -4], [-12, -2, 2, -14], [-12, -9, 4, -13], [-30, -10, 6, 2], [29, -15, 5, -5], [-11, -20, 17, -3], [27, -10, -7, 13], [56, 2, -11, -3], [0, -19, 13, -5], [24, 2, -2, -4], [83, 9, 3, 5], [23, 6, -9, -9], [-43, -10, 5, -11], [46, 0, -14, 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_cp();".
// 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_cp(: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_cp();".
// 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_cp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-7, 2, 1]>,<11,R![-188, 156, -19, -6, 1]>,<13,R![-2, 36, -1, -6, 1]>,<17,R![-146, -68, 15, 10, 1]>],Snew);
    return Vf;
end function;