
// Make newform 416.4.i.a in Magma, downloaded from the LMFDB on 28 January 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_416_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_416_i_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_416_4_i_a();".
// 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_416_4_i_a();".
// 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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, 0, 1, 0], [0, 0, 0, 2]];
    Rf_basisdens := [1, 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_416_i();"
function MakeCharacter_416_i()
    N := 416;
    order := 3;
    char_gens := [287, 261, 353];
    v := [3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_416_i_Hecke();"
function MakeCharacter_416_i_Hecke(Kf)
    N := 416;
    order := 3;
    char_gens := [287, 261, 353];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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]];
    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 := 4;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [-11, -2, 0, 1], [0, 0, 0, 0], [0, 0, 0, 0], [0, -23, 9, 23], [-47, 26, 47, -52], [0, 0, 0, 0], [0, 0, 0, 0], [0, 71, -65, 71], [0, 0, 0, 0], [0, 99, 107, 99], [0, 118, -115, 118], [0, 0, 0, 0], [0, 0, 0, 0], [-259, 286, 0, -143], [0, 0, 0, 0], [415, -117, -415, 234], [0, 0, 0, 0], [0, 0, 0, 0], [-549, 296, 0, -148], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1670, 0], [-594, 0, 594, 0], [0, 485, 299, 485], [0, 0, 0, 0], [0, 0, 0, 0], [-1746, 0, 0, 0], [1001, 332, -1001, -664], [0, 0, 0, 0], [0, 0, 0, 0], [-803, 694, 803, -1388], [0, 0, 0, 0], [-1757, -235, 1757, 470], [0, 0, 0, 0], [-143, -1962, 0, 981], [0, 0, 0, 0], [0, 0, 0, 0], [4082, 0, -4082, 0], [0, 0, 0, 0], [-1971, -1430, 0, 715], [0, 0, 0, 0], [0, 18, 2681, 18], [0, 0, -1174, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [6390, 0, 0, 0], [-598, 0, 0, 0], [0, 0, 0, 0], [-2655, -1318, 2655, 2636], [0, 0, 0, 0], [0, 2024, -767, 2024], [0, 0, 0, 0], [-3406, 0, 3406, 0], [0, 0, 0, 0], [4563, 329, -4563, -658], [3715, -2896, 0, 1448], [0, 0, 0, 0], [-4709, 863, 4709, -1726], [0, 0, 0, 0], [0, 0, 0, 0], [-6838, 0, 0, 0], [5137, -2338, 0, 1169], [0, 0, 0, 0], [6183, 208, 0, -104], [0, 0, 0, 0], [0, 0, -9470, 0], [0, 3212, 1649, 3212], [0, 0, 0, 0], [0, 0, 0, 0], [-6461, -1593, 6461, 3186], [0, 0, 0, 0], [0, 0, 0, 0], [-187, 7670, 0, -3835], [-9614, 0, 9614, 0], [0, 3970, -1199, 3970], [3573, -3730, -3573, 7460], [0, 0, 0, 0], [5445, 6706, 0, -3353], [0, 0, 0, 0], [-2431, -4338, 2431, 8676], [0, 0, 0, 0], [0, 0, 0, 0], [-16114, 0, 16114, 0], [0, -2614, 8253, -2614], [1159, 4915, -1159, -9830], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -4499, 7135, -4499], [-11869, 740, 0, -370], [0, 0, 0, 0], [2961, -12230, 0, 6115], [0, 0, 0, 0], [0, -6209, -4313, -6209], [0, 0, 0, 0], [0, 0, 26806, 0], [0, 0, 0, 0], [1727, -13752, 0, 6876], [0, 0, 0, 0], [7751, 12184, 0, -6092], [0, 0, 0, 0], [0, -6012, -8515, -6012], [0, 0, 0, 0], [0, 4887, 11611, 4887], [7733, 6616, -7733, -13232], [0, 0, 0, 0], [0, 0, 0, 0], [14425, 3718, -14425, -7436], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -33358, 0], [0, 0, 0, 0], [0, 5517, -12925, 5517], [0, -8658, -2231, -8658], [4054, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [20030, 0, 0, 0], [-18405, -2101, 18405, 4202], [0, 0, 0, 0], [0, 0, 0, 0], [-19359, -4366, 0, 2183], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -35046, 0], [-31882, 0, 31882, 0], [0, 0, -9650, 0], [-39542, 0, 39542, 0], [0, 0, 0, 0], [41954, 0, -41954, 0], [0, 9926, -11635, 9926], [0, 0, 0, 0], [0, 0, 1850, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 10435, 11583, 10435], [0, 0, 0, 0], [10189, 22734, 0, -11367], [-22997, -10028, 0, 5014], [0, 0, 0, 0], [0, 0, 0, 0], [-21257, -7461, 21257, 14922], [0, 12952, -3575, 12952], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [15433, -11870, -15433, 23740], [-25973, -12168, 0, 6084], [-31378, 0, 0, 0], [0, 0, 0, 0], [56758, 0, -56758, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -5734, -28303, -5734], [0, 0, 0, 0], [0, 0, 0, 0], [26443, 8541, -26443, -17082]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_416_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_416_4_i_a();".
// 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_416_4_i_a(:prec:=4)
    chi := MakeCharacter_416_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_416_4_i_a();".
// 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_416_4_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_416_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![0, 1]>],Snew);
    return Vf;
end function;