
// Make newform 3920.2.a.cb in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3920_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_3920_2_a_cb();".
// 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_3920_2_a_cb();".
// 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, -9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-6, -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_3920_a();"
function MakeCharacter_3920_a()
    N := 3920;
    order := 1;
    char_gens := [1471, 981, 3137, 3041];
    v := [1, 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_3920_a_Hecke(Kf)
    return MakeCharacter_3920_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 := [[0, 0, 0], [0, -1, 0], [-1, 0, 0], [0, 0, 0], [-1, -1, -1], [-1, -1, 1], [-2, 0, 0], [1, -1, 1], [-1, 0, 1], [4, -1, -1], [4, -2, 0], [3, 1, -1], [-3, 0, -2], [-4, 1, 0], [-5, -1, -3], [3, -3, -1], [8, 0, 0], [2, -3, -1], [-2, -3, 0], [0, 0, 0], [0, -2, 2], [6, 2, -2], [-10, 1, 0], [0, -1, 1], [2, 0, 0], [6, -3, 1], [-2, -1, 2], [-6, -3, 0], [2, -3, 3], [0, -4, 0], [3, -5, 1], [-3, 7, 1], [4, 0, 0], [0, 2, 0], [8, -1, -3], [10, 0, 2], [-1, -1, -3], [8, 2, -2], [3, 0, -1], [13, -1, 1], [1, -3, 1], [4, 3, 3], [-8, -2, 0], [4, -6, -2], [1, 1, -1], [8, 4, -4], [7, 5, -1], [-4, 6, 2], [0, 6, -2], [-16, -2, 2], [18, 0, 0], [6, 4, -2], [7, -1, -3], [3, 3, 3], [0, 4, -4], [-18, 1, -2], [-2, 3, 1], [0, 4, -4], [14, 4, 0], [-5, -1, 5], [-10, 4, 0], [-13, 1, -1], [4, 5, 2], [-16, -4, 0], [4, 8, -4], [6, 0, 4], [-3, 1, 1], [-2, -2, -6], [-2, 1, 0], [0, 1, -3], [4, 4, -4], [6, -4, -2], [-7, 6, 3], [-2, -2, -6], [-1, 9, 3], [-3, 4, 3], [0, -6, -2], [2, 0, 8], [-15, 2, -2], [-4, 3, -5], [9, 7, -3], [-12, 5, -1], [-8, -2, 0], [-8, -4, 0], [-2, -6, -6], [-18, -5, -2], [21, 4, -2], [-14, 2, 2], [-18, -4, -4], [21, -2, -3], [-16, -3, 2], [-20, -2, 4], [20, 6, 2], [0, 2, 8], [-8, -6, 0], [18, -1, 0], [-4, 5, 1], [17, -11, -1], [-32, 2, 2], [-2, 11, -1], [-2, -7, -4], [-7, -11, -1], [-20, -1, -6], [11, 3, 9], [-4, 4, -8], [16, -6, 2], [12, -6, 2], [0, -8, 4], [-18, -2, 2], [14, -4, -4], [15, 8, -3], [3, 11, 1], [-6, 6, 2], [5, -11, -3], [32, 2, -4], [27, -4, 2], [-6, 0, 4], [-3, 2, -5], [17, 5, -1], [-24, 4, 4], [6, 1, -9], [18, 6, 6], [-1, 3, 1], [-18, 3, 2], [-4, -2, 4], [20, 1, -1], [-16, -9, 3], [4, 6, 0], [-1, -8, -1], [-3, -11, -5], [15, -9, -1], [9, -2, 9], [-18, -2, 2], [14, -4, 0], [21, 1, 3], [23, -1, 5], [-3, 13, -1], [8, -3, 6], [-18, 2, -2], [-21, 2, 2], [5, -7, -7], [-2, -8, 8], [2, -3, -4], [-12, -1, 0], [2, 4, 4], [26, 4, 2], [1, 7, -7], [-6, -8, -4], [8, 0, 0], [3, 12, 3], [-23, 7, -3], [27, -2, -4], [-20, 10, 2], [22, -7, -4], [-12, 3, -4], [-10, -4, -2], [10, -2, 10], [23, -4, 2], [14, -2, -6], [-4, 14, 2], [-28, -1, 6], [-12, -12, 4], [-6, 7, 0], [5, -9, -3], [14, -4, 0], [-9, 2, 7], [40, -6, 0], [18, 6, 10], [11, 10, 2], [1, 7, 1], [-20, 0, -4], [14, 9, 5], [-20, -6, -4], [2, -2, -10], [-4, -14, 4], [-12, 11, -7], [-1, -3, -1], [14, 8, 0], [-17, -6, -3], [-10, 11, 5], [19, 2, -9], [-15, -7, -11], [23, 3, -3], [-10, -8, -4], [-23, 1, 3], [4, 9, -11], [-19, 7, 1], [2, -4, -4], [-7, -8, -6], [-8, 6, -8], [-8, 12, 4], [-30, 8, 0], [-11, 17, 5], [34, 5, 3], [-10, -8, 0], [-38, 0, 0], [21, -2, 0], [-6, -12, -4], [-32, 4, -8], [-4, -2, -6], [-6, -15, 9], [10, 4, 10], [-14, 14, 2], [15, -4, 0], [12, -2, 12], [-5, 13, -5], [-20, 16, 8], [20, 2, 10], [-51, 0, 0], [35, -7, 3], [-20, 8, 8], [22, 9, -5], [10, -3, 2], [-60, -3, 0], [-14, -4, 2], [3, -13, -7], [29, 4, -7], [31, 2, -2], [-21, -5, 1], [1, 3, 1], [14, -1, 1], [-32, -2, 4], [19, -18, -6], [27, 11, -3], [-30, 11, -2], [-2, -11, -5], [-2, 2, -2], [-38, -12, 6], [17, -10, 5], [16, -10, 4], [13, 9, 7], [-27, -5, 9], [34, 0, 6], [-11, 14, -8], [36, -1, 0], [42, 5, -4], [-12, -17, 7], [3, -5, 13], [-7, -1, -15], [18, -20, -6], [14, 8, -4], [-17, -7, -5], [4, -2, 2], [-8, -17, -1], [10, -10, -10], [2, 4, 6], [-17, -4, -7], [-7, -3, 13], [-4, 0, 4], [1, 9, -13], [6, 18, 2], [5, 0, 4], [15, -14, -7], [-11, 6, 12], [34, 8, 12], [33, -5, 5], [8, 5, -3], [-42, 7, -6], [-39, 1, 7], [30, 14, 2], [17, 14, 1], [22, 1, -12], [30, -8, -8], [10, -8, -8], [10, -10, -6], [-41, -1, -13], [-32, -2, 2], [3, 3, 9], [0, 9, 8], [13, 3, -7], [-50, -3, -3], [2, -8, 4], [6, -18, -6], [-14, -2, -10], [22, 0, -4], [11, -21, -7], [-4, 6, -10], [32, 7, -5], [17, 2, 8], [15, 5, -1], [15, 2, -7], [24, -12, 4], [-13, -9, 5], [-10, 9, -7], [4, -3, 0], [26, -10, -6], [-4, 4, -4], [-15, 3, 1], [-18, -12, -6], [15, -13, 1], [-20, 9, -13], [12, 13, 10], [-6, 20, 4], [-27, -7, -3], [-5, -11, -1], [18, -5, -7], [14, -6, 6], [-6, -10, 6], [7, -17, -1], [12, -6, 2], [-36, 6, 2], [9, -9, 5], [40, 10, 4], [-14, -5, 6], [-24, -2, -8], [16, -10, 10], [6, -4, 0], [-16, 17, 5], [-12, -14, 0], [5, 13, -5], [13, -4, -7], [-12, 2, -2], [0, 11, 5], [-40, 1, -8], [-23, -12, 13], [11, -4, 0], [13, 15, -15], [18, 18, 2], [8, -10, 14], [28, -3, 13], [32, 18, -12], [-10, 0, -8], [-6, -3, -1], [11, 12, 1], [-20, -14, 6], [-39, 0, -6], [11, 15, -5], [26, 13, 16], [4, 6, 10], [42, 6, 6], [44, 2, -2], [22, -6, 14], [-50, 6, 6], [10, 5, -2], [41, 1, 1], [-42, 3, 0], [-4, 10, -2], [12, 10, -2], [-1, -14, -4], [-3, -15, 7], [50, 6, 14], [-42, -6, -2], [44, -3, 3], [-26, -2, -6], [-38, -12, -8], [20, 8, 8], [-10, 5, -1], [14, 13, 0], [-54, 8, -2], [-33, 1, -9], [-5, 3, -5], [0, 10, 14], [-14, 19, 13], [5, 8, 1], [2, -19, 1], [-10, -14, 2], [-48, 2, 4], [-8, -10, -12], [-20, 10, 2], [-26, -11, 12], [29, -5, -3], [-22, 0, 0], [10, 19, -2], [15, -27, 3], [-24, 13, -4], [10, 24, -12], [-39, -7, -17], [49, -11, -5], [-13, -12, -6], [3, 13, -1], [-13, -1, -5], [-2, 9, 14], [-6, 0, 0], [42, 18, -10], [-10, -28, 8], [3, 9, 15], [-14, 16, -2], [28, 14, -2], [54, 4, -4], [34, -14, -2], [22, -11, 5], [10, 16, -4], [-9, 6, 11], [0, 0, -16], [39, 15, -5], [55, -6, -5], [-16, 10, 16], [57, 3, 1], [-36, -15, -4], [22, -23, -10], [-2, -8, -8], [18, 6, -10], [-24, -10, 4], [-44, -6, -14], [-14, 28, 6], [26, -6, 2], [26, 6, 22], [-15, -6, -6], [-29, 5, -5], [0, 13, 15], [18, -13, 1], [24, -2, 2], [17, 8, -9], [6, 22, 2], [-18, 19, -7], [-48, -4, 4], [-17, 1, -13], [0, -17, -3], [16, 11, -12], [69, -3, 5], [-46, -14, 14], [-42, -12, 4], [22, 24, -12], [-55, 7, 1], [88, -2, 2], [-24, -6, -18], [14, 4, 18], [1, 9, -13], [6, -12, 16], [3, 6, 11], [-24, -5, 13], [-1, -23, -13], [-46, 15, 4], [-32, 6, 0], [-12, 18, 6], [-57, 11, -3], [12, -6, -6], [27, -14, 8], [-1, 5, -1], [20, -14, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3920_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3920_2_a_cb();".
// 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_3920_2_a_cb(:prec:=3)
    chi := MakeCharacter_3920_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_3920_2_a_cb();".
// 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_3920_2_a_cb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3920_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![6, -9, 0, 1]>,<11,R![-44, -24, 3, 1]>,<13,R![-68, -24, 3, 1]>,<17,R![2, 1]>],Snew);
    return Vf;
end function;