
// Make newform 3192.2.a.v in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3192_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_3192_2_a_v();".
// 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_3192_2_a_v();".
// 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 := [2, -4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-6, 0, 2]];
    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_3192_a();"
function MakeCharacter_3192_a()
    N := 3192;
    order := 1;
    char_gens := [799, 1597, 2129, 913, 1009];
    v := [1, 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_3192_a_Hecke(Kf)
    return MakeCharacter_3192_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], [1, 0, 0], [0, 0, 0], [-1, 0, 0], [-1, 1, 0], [-2, 0, -1], [1, -1, 1], [-1, 0, 0], [-2, -2, 1], [-4, 0, 1], [1, -1, -1], [-3, 1, 1], [0, 2, 0], [-2, -2, 0], [-1, -1, 0], [-8, 0, -1], [2, 2, -2], [-4, 2, -2], [1, 3, 0], [-4, 2, -1], [0, -2, 0], [2, 2, 1], [-4, 2, 1], [-6, 0, -2], [-1, -1, 2], [-8, 0, 0], [-5, -3, 1], [0, 2, -3], [-1, 3, -3], [-2, 2, 1], [2, -2, 1], [-2, -4, 3], [-10, -4, 2], [-6, -2, 0], [-1, 3, -1], [2, -2, 3], [6, 0, 2], [6, -2, 0], [2, 2, 0], [-14, -4, 2], [8, -2, -1], [-6, 0, -1], [2, 2, 1], [4, -2, -2], [-13, -5, 1], [0, 4, -2], [15, 1, 0], [-1, 1, -5], [10, -2, 0], [2, 0, 2], [-4, 2, -2], [8, 0, 1], [13, 1, -2], [-2, -4, 3], [-8, -2, 0], [6, 2, 1], [-8, 2, -2], [-8, 0, 0], [6, -4, 2], [-18, 2, -3], [10, -2, 4], [8, -2, -4], [0, 4, -2], [1, 5, 0], [8, 2, 2], [0, 4, 3], [11, 1, -4], [8, -2, -2], [9, -5, -2], [-2, 0, 2], [15, 1, 3], [-8, 4, 1], [-16, 4, -2], [-5, -5, 1], [23, -3, 0], [8, -4, 2], [3, -1, 5], [-4, -6, 4], [-2, 6, -3], [11, -1, -4], [8, 6, -1], [-17, -5, 3], [12, 2, 3], [-1, -1, -6], [-11, 3, -1], [-1, 5, -6], [-30, 2, -1], [14, 8, -4], [-6, -2, 2], [20, 4, 0], [-4, -2, -1], [11, -1, -4], [-16, -4, 1], [-9, -3, 0], [2, 2, 4], [1, 5, 0], [-12, 6, 2], [-10, -8, 0], [-16, 4, -6], [18, 4, -2], [1, -9, 6], [-13, 7, 3], [10, -2, 4], [-14, -2, 3], [12, 0, -4], [14, 4, 0], [-18, 4, -7], [-7, -5, 1], [20, -2, -3], [1, 5, 4], [-31, -1, 1], [12, -2, 0], [-4, -2, -4], [18, 6, 0], [24, 0, 2], [30, 6, -1], [-8, -8, 6], [-5, 3, 6], [13, -7, -3], [4, 2, -1], [0, 6, -1], [10, 0, 6], [-14, 0, -2], [2, -4, -1], [-16, 4, 0], [-29, 3, -1], [0, -6, 6], [-3, 5, -10], [-6, -2, 6], [-6, 8, -4], [8, -4, 0], [4, 2, 5], [-4, 4, 7], [-8, 6, 2], [7, 5, 3], [28, -2, 6], [0, -2, -4], [14, -2, -4], [-22, -4, 2], [12, -2, 2], [12, 0, -2], [17, -7, 5], [12, 4, -4], [-4, -2, 9], [4, -2, 7], [14, -6, -2], [12, 2, -6], [22, -4, -6], [-10, -10, 6], [24, 10, -5], [-7, 5, -7], [-21, -3, -1], [10, 2, -4], [6, -2, 0], [7, -7, 6], [0, -6, -3], [0, -8, 0], [1, -1, -7], [-2, 4, -2], [-36, -2, -2], [-25, -7, 2], [-18, 6, 3], [0, 8, -6], [-18, -2, 6], [-2, -2, 9], [4, 4, -4], [16, 4, -1], [14, -8, -2], [32, -2, -4], [-18, 2, 2], [6, 4, 5], [-50, 0, 1], [22, 10, 1], [6, 8, 2], [37, 3, 1], [21, 15, -7], [-8, 4, -4], [-11, -7, -3], [-13, 5, 7], [28, -6, -2], [-24, 0, -6], [8, 4, 4], [-21, -1, 1], [10, 0, -4], [18, -6, 11], [-30, 6, -2], [-11, -11, 5], [-12, -8, 4], [6, 0, -4], [-23, 1, -4], [-13, 7, 6], [3, 5, 6], [-27, 7, 4], [6, -4, 2], [29, -5, 8], [20, 2, 2], [-10, 8, 6], [-12, 2, -2], [3, 13, -9], [-9, -13, 8], [-24, -8, -1], [-23, 7, -7], [8, 2, -11], [55, 3, -2], [-18, -12, 1], [4, 12, -8], [-22, -6, 4], [-6, 0, -7], [-14, -4, -2], [-18, -2, -4], [10, -12, 8], [-28, -4, 4], [-20, -4, -4], [-32, 0, -2], [26, -2, -2], [-27, -11, 4], [16, 8, 0], [-8, 6, 2], [-28, -2, -7], [-25, -5, 5], [-25, -5, 1], [-25, -7, 3], [18, -10, -5], [-12, -12, 0], [20, 0, -8], [22, 0, 8], [-24, 6, -4], [8, 2, -9], [-25, 1, -3], [-27, -9, 6], [38, -8, -2], [-34, -2, 0], [32, 4, -3], [-38, -2, -1], [-30, 2, -2], [29, -3, 8], [-24, -2, -4], [-27, 1, -5], [-27, -1, -8], [-32, 4, -6], [22, -8, 11], [-12, -8, 8], [-10, 6, 0], [1, 5, -13], [28, -6, -4], [-15, 1, -2], [-50, -2, -4], [-40, 0, -2], [25, -1, -8], [-24, 12, 9], [-6, -4, -2], [-11, 7, 5], [-16, 0, -3], [-5, 7, -6], [12, -8, 12], [27, -3, 4], [4, -6, 0], [-46, 2, 2], [2, -8, 2], [-26, 4, -2], [-34, -18, 7], [-6, -8, -3], [-18, 4, -4], [12, 2, 9], [31, 5, -5], [-36, -8, 4], [-28, -4, 3], [5, -1, 9], [-33, 1, 0], [3, -1, -3], [-66, -4, 3], [37, -5, 4], [20, 10, -4], [0, 0, 10], [-47, -3, 4], [-38, -6, 4], [-11, -17, 8], [21, 9, 3], [-46, -8, 0], [10, -8, -11], [-20, -8, 2], [-24, 0, 8], [17, 5, -8], [-20, -2, -1], [-38, 10, -2], [12, 4, -5], [14, -4, -10], [2, -14, 15], [-24, 12, 2], [26, -4, 2], [3, -1, -9], [-16, 14, -7], [22, 2, 1], [-12, 8, 0], [17, 5, -9], [6, -10, 2], [15, 9, 1], [-14, -2, -6], [-22, -2, -8], [48, 0, 2], [20, -2, 12], [8, -4, 1], [44, 4, -6], [54, -4, -3], [10, -6, 4], [-46, 4, 4], [-18, -4, 5], [-40, 6, -7], [-20, 12, -7], [-12, -2, 10], [25, 5, 2], [-27, 5, 3], [-10, 6, -13], [-49, -3, -2], [-4, -16, 3], [1, -11, 0], [-20, -10, 3], [40, 14, -5], [-12, -14, -2], [52, 6, 0], [6, -2, -2], [2, -4, 4], [28, -10, 8], [4, 4, -11], [12, -14, 4], [49, -3, 6], [39, 9, -2], [-42, 6, 0], [-20, 10, 3], [9, 5, -3], [30, 8, 2], [30, 6, -7], [22, -10, 8], [-4, 10, -13], [30, 6, -8], [-32, 2, 11], [-31, -3, -3], [-24, -14, 4], [2, -12, 0], [6, 2, 2], [15, 7, -3], [26, 0, 6], [22, 8, 8], [-2, 6, -1], [6, 6, -3], [-32, -10, 3], [-4, 6, -10], [21, -13, 10], [18, 22, -6], [28, -4, 4], [-42, -2, 2], [28, -6, 8], [-9, 15, -15], [15, -7, -5], [-5, -13, 5], [-18, 16, 0], [55, -1, 8], [-26, 8, 5], [36, 10, -4], [-12, 2, -9], [42, 20, -10], [21, 7, -3], [-4, 18, -9], [-38, 8, 1], [44, -4, -2], [-52, -2, 0], [-18, 6, -14], [-38, -10, 5], [52, -2, -2], [37, 11, -2], [-20, 0, -8], [-24, 8, -13], [2, 6, 7], [-44, -20, 4], [-4, 2, 16], [-42, -10, -2], [14, 6, 5], [4, 6, 10], [19, 17, -9], [-11, -15, 10], [-18, -4, 2], [-4, -2, -6], [-12, 4, 14], [-50, -12, 8], [48, 20, -8], [-46, -6, -4], [-8, 0, 2], [24, 10, 0], [30, 6, -4], [70, -2, 8], [0, -18, 6], [42, -4, -8], [-5, -3, 2], [4, -8, 8], [4, -14, 13], [-2, 8, 4], [25, 15, -9], [38, -4, 4], [-13, -7, 12], [-5, -9, -11], [0, -2, 13], [4, -2, -2], [-48, -12, 3], [0, 14, 4], [12, -2, -6], [42, 2, -4], [-6, -20, 16], [-10, -10, 1], [22, -10, 8], [-40, -10, 5], [17, -3, -8], [31, -5, 1], [-60, 6, -11], [26, 26, -4], [-26, 12, -10], [-46, -6, 6], [24, -2, 1], [-31, 7, 9], [-63, -1, -7], [6, -14, -2], [38, 6, -3], [-32, -14, 9], [-2, -14, -3], [4, 0, 4], [-9, 3, -12], [14, -8, -10], [0, 6, 3], [16, -10, -10], [12, -4, -14], [17, -3, 14]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3192_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3192_2_a_v();".
// 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_3192_2_a_v(:prec:=3)
    chi := MakeCharacter_3192_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_3192_2_a_v();".
// 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_3192_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3192_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![-16, -12, 4, 1]>,<17,R![64, -24, -4, 1]>],Snew);
    return Vf;
end function;