
// Make newform 3762.2.a.bg in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3762_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_3762_2_a_bg();".
// 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_3762_2_a_bg();".
// 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 := [-3, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-4, -1, 1]];
    Rf_basisdens := [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_3762_a();"
function MakeCharacter_3762_a()
    N := 3762;
    order := 1;
    char_gens := [2927, 343, 2377];
    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_3762_a_Hecke(Kf)
    return MakeCharacter_3762_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, 0, 0], [1, 1, 1], [-2, -1, 1], [1, 0, 0], [0, -1, 1], [3, 0, 1], [-1, 0, 0], [5, 0, 1], [-2, -1, -2], [-1, 3, 0], [-2, 0, -2], [3, 3, 2], [-7, 3, 1], [4, -4, -2], [3, 4, -1], [1, -2, -1], [8, 0, 2], [0, 1, -3], [-7, 5, 1], [-1, 0, -1], [-2, -2, -4], [11, 1, -1], [-2, -2, -2], [4, 0, -4], [0, 2, 2], [1, -1, 5], [1, -2, 1], [-1, -2, 3], [8, 0, 0], [-6, 2, -4], [-3, -3, 0], [-2, -1, -5], [5, -3, -3], [-2, -4, 2], [-8, -4, -2], [5, 5, 5], [0, -6, -4], [-6, 0, 4], [-9, -5, -3], [1, 3, 4], [0, -6, -4], [7, -2, -5], [5, 3, -3], [12, 0, 2], [-17, 0, -3], [-9, 4, 1], [6, 0, 2], [5, 0, -5], [-1, 1, 4], [18, 0, -2], [-6, 1, -6], [-1, 1, 7], [-10, 4, 2], [-16, -2, 4], [-7, 7, 3], [4, -10, -2], [0, -7, 5], [0, -4, -2], [-3, -1, -3], [11, -1, 0], [-8, -1, 1], [-10, 2, 6], [5, 8, 3], [0, 5, -4], [13, 8, -1], [14, -11, -2], [3, -3, -4], [6, 0, -2], [6, 8, 8], [5, 6, -7], [-8, -7, 5], [-8, -8, 6], [-2, -5, 6], [-10, -5, 1], [1, 1, -8], [9, -7, 1], [-13, 11, 2], [0, 6, -6], [-3, 1, 5], [-10, 4, 4], [17, 0, -3], [22, -8, -2], [6, -14, -2], [-18, 0, 2], [6, -8, -2], [22, 2, 0], [3, 4, -7], [-8, 0, 4], [-20, 4, -4], [6, 10, 4], [-7, -13, 0], [-3, 3, 9], [5, 9, 4], [12, 4, -2], [-6, -11, -6], [18, -12, -2], [8, -4, -10], [9, 0, -9], [6, 10, -4], [8, 0, -12], [2, -8, -10], [-8, 8, -6], [9, -13, -8], [3, 7, -4], [-8, 3, -7], [-4, 0, 0], [-2, 0, 12], [-15, -9, -3], [21, -1, 0], [8, 2, -2], [-8, 4, -6], [13, 1, 4], [-4, 12, -6], [18, -2, -8], [6, -14, 2], [-2, -12, -6], [-3, 16, 3], [1, -9, -12], [-21, 12, -1], [1, 2, 3], [5, -13, -4], [-20, 3, 0], [-16, -8, -4], [-26, -8, -4], [-4, -12, -4], [1, 9, 6], [5, 4, 3], [5, 2, 7], [10, -12, -8], [3, 5, 11], [28, -8, 2], [-42, -4, -2], [-9, -1, -5], [-1, 16, 9], [-11, -10, -7], [11, -18, -1], [-13, 8, 1], [15, 4, 1], [-5, 6, -11], [3, -14, -13], [-18, -10, -12], [-29, -6, 5], [-21, -10, -3], [21, 10, 5], [23, 9, 11], [26, -4, 2], [-15, 1, 6], [6, 4, 6], [1, -3, 12], [-14, -1, 10], [-35, -7, 0], [30, 2, -2], [8, 8, -4], [25, -2, -9], [-18, 8, -6], [-40, -1, -4], [-10, -15, 13], [5, 10, -15], [-17, -6, 7], [-26, 6, 2], [-4, 0, 6], [-4, 4, 0], [-11, 3, 7], [-4, -4, -4], [27, 3, 0], [3, -3, -14], [24, -6, 2], [-7, -5, -16], [-11, -17, 4], [5, 6, -5], [-15, -8, 3], [13, 15, 7], [21, -4, -5], [26, -20, -2], [15, 7, 5], [7, -3, 10], [8, -12, -14], [-6, 8, -4], [10, -10, -10], [3, -2, 5], [17, -6, 7], [14, -4, 12], [27, 13, -13], [3, 18, -3], [4, -4, -2], [-17, -6, -1], [31, 9, 1], [7, 19, 7], [24, 7, 0], [2, -2, 0], [13, 11, 2], [-17, 9, 13], [21, 8, 7], [47, 5, 2], [-14, 2, -8], [-15, -10, 15], [25, -1, 4], [17, -2, 17], [-17, 10, 1], [11, 7, 16], [-32, -4, -12], [3, 6, -3], [15, -1, -5], [4, -18, 4], [-6, 0, 12], [-23, 2, 9], [-35, -4, -3], [19, -6, 9], [6, 3, -2], [-33, 4, -5], [-7, -5, -15], [14, -6, 10], [2, -21, -11], [10, 0, -6], [2, -10, -18], [-33, 18, 5], [8, -12, -10], [15, 7, -12], [57, -3, -2], [-19, -11, -4], [-20, -8, 2], [-40, -2, -4], [13, -18, -7], [-11, 6, 7], [34, -4, 0], [-10, -6, -6], [25, 3, 13], [20, 10, 8], [15, 9, -13], [19, -13, -1], [-9, 12, -5], [-6, 22, 2], [11, -11, -8], [2, -2, -12], [-34, -9, -3], [-1, -25, 4], [38, 12, 12], [-4, 4, -14], [-5, 11, 0], [11, -3, 5], [-11, -1, 14], [-31, -8, -1], [39, -2, 5], [4, -20, -14], [-15, -9, 4], [18, 6, -6], [3, 13, 5], [-2, 6, 20], [1, 18, 19], [8, 8, -18], [27, 4, -3], [9, -16, -3], [32, -6, -2], [38, 4, -2], [-3, 17, 1], [19, -7, -9], [-2, 14, -4], [-24, -2, 12], [26, 16, -16], [8, 22, -12], [26, -6, 12], [12, -4, -6], [-38, -3, 2], [57, -7, -1], [7, -10, 9], [31, 10, -7], [52, -3, 8], [-23, 8, 11], [-10, 22, -2], [-9, -14, -25], [24, 3, -4], [52, 9, 3], [-40, 7, -4], [-1, -15, 7], [-43, -8, -1], [-30, -14, 8], [-4, -21, -12], [-3, -3, 13], [41, -13, 5], [-42, -8, 12], [-11, 18, -7], [20, -5, -4], [-18, 5, 10], [5, -17, 1], [6, 2, 0], [-22, -9, 3], [35, -1, -4], [-21, 14, -5], [-3, 11, -7], [-49, 5, -4], [-30, -24, 18], [-5, 9, 8], [18, -6, -18], [-1, -29, -3], [-14, 24, 8], [19, 9, 18], [16, 20, -10], [17, 18, 13], [19, 17, 7], [-28, 6, -6], [14, -27, -3], [-21, 14, -7], [0, 23, 4], [-4, -4, 4], [14, 12, 2], [-34, 17, 11], [-15, 10, -9], [-12, 4, -8], [-16, -12, 12], [-11, 31, 12], [3, 5, -17], [5, -9, 20], [35, -11, -11], [-27, 11, 4], [-26, -8, 0], [-4, -4, -10], [36, -2, -2], [-28, -1, 4], [30, 4, 8], [25, 0, 11], [-35, 10, 1], [39, 6, 19], [-26, 16, 10], [-1, -20, -13], [18, 2, 6], [6, -5, -7], [-24, -13, 8], [-30, -6, 6], [24, 14, -16], [-20, 8, -12], [-41, 6, 9], [8, -15, 12], [-14, -12, -2], [39, 11, 15], [-8, 10, -8], [15, 12, -9], [17, 15, 0], [-27, 14, 1], [40, -1, -8], [29, -22, -1], [16, 5, -8], [12, -23, -13], [-27, -5, 17], [31, 12, 21], [-78, 2, 0], [5, -1, -8], [-10, -19, 7], [-54, -6, -4], [16, -24, -14], [12, 8, -22], [36, -22, -4], [37, 5, 6], [10, 0, 24], [-37, 3, -12], [1, -17, 13], [21, -8, 5], [45, 19, -3], [-27, -2, -15], [61, 1, 0], [-2, -24, -4], [16, -18, 2], [-39, -3, 9], [25, 16, 9], [24, -10, 2], [66, -4, 6], [-14, 12, 2], [4, -30, -16], [-37, 11, -1], [48, 4, -14], [-18, 13, 11], [1, 26, -5], [26, -17, 2], [4, -36, -16], [50, 16, 12], [-6, -25, -19], [1, 15, 0], [14, -20, -2], [-27, -8, -27], [4, 12, 6], [-30, 12, 6], [-15, 1, 12], [-54, -3, 11], [36, 6, 2], [-55, -6, -9], [-10, -16, 2], [21, -28, -5], [-17, -27, -17], [27, 19, 36], [-49, -5, -5], [2, 24, -4], [-11, -20, 5], [46, 0, 18], [-2, 2, -20], [15, -21, -16], [-9, 19, 7], [36, 8, 12], [-48, 4, -6], [41, 11, -16], [-18, -8, -2], [25, 13, 0], [23, -5, 21], [-16, 20, 2], [-6, 8, 14], [-11, 17, 7], [-17, -2, -3], [-37, 3, 14], [4, -20, -2], [29, -26, -13], [-16, 4, -26], [-18, 24, 8], [18, 10, -18], [17, -2, 11], [11, -19, -9], [-51, 2, 9], [0, 0, -6], [28, -2, 8], [27, -21, -4], [-32, -7, -5], [-4, 2, -28], [-26, -8, -16], [-9, 10, -1], [19, 0, 7], [-27, -18, 13], [-2, 0, 6], [-21, 34, 9], [22, -15, 7], [15, -4, 5], [-8, 10, 16], [10, -19, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3762_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3762_2_a_bg();".
// 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_3762_2_a_bg(:prec:=3)
    chi := MakeCharacter_3762_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_3762_2_a_bg();".
// 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_3762_2_a_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3762_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![18, -9, -3, 1]>,<7,R![-57, -6, 6, 1]>,<13,R![-29, -18, 0, 1]>,<17,R![4, 18, -9, 1]>],Snew);
    return Vf;
end function;