
// Make newform 322.2.e.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_322_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_322_e_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_322_2_e_c();".
// 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_322_2_e_c();".
// 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 := [7, -38, 64, -40, 38, -10, 1, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [-6620, 11263, -5301, 2308, -909, 338, -55, 60], [-21668, 13756, -23147, 8788, -716, 957, -743, 74], [14204, -38221, 14321, -22495, 6885, -314, 938, -655], [6734, -18187, 12372, -13037, 3332, -312, 674, -367], [18369, -49196, 25443, -33224, 9308, -288, 1557, -962], [88979, -241837, 128118, -149399, 42410, -2449, 7290, -4270]];
    Rf_basisdens := [1, 1, 8102, 8102, 8102, 4051, 4051, 8102];
    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_322_e();"
function MakeCharacter_322_e()
    N := 322;
    order := 3;
    char_gens := [185, 281];
    v := [1, 3];
    // 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_322_e_Hecke();"
function MakeCharacter_322_e_Hecke(Kf)
    N := 322;
    order := 3;
    char_gens := [185, 281];
    char_values := [[-1, 0, 0, 0, 0, 0, 1, 0], [1, 0, 0, 0, 0, 0, 0, 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+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, 0], [0, 0, 0, 0, 1, 1, -1, -1], [-1, 0, 0, 1, 0, 0, 1, 1], [0, -1, 0, 1, 0, 0, 0, 1], [0, 1, 1, 0, 1, 0, 2, 0], [-2, 1, -2, 1, -1, 1, 0, 0], [0, 0, 0, 0, 0, 0, -4, -1], [1, 0, -2, 1, 0, -2, -1, 1], [-1, 0, 0, 0, 0, 0, 1, 0], [0, 0, -1, 2, -1, 1, 0, 0], [0, 1, 1, 0, -1, -2, 0, 2], [-6, 3, 1, 2, -3, 1, 6, 2], [0, 1, -2, 3, -1, 1, 0, 0], [5, -3, 2, -4, -1, 1, 0, 0], [-5, 2, 1, 1, -2, 1, 5, 1], [0, 2, 2, 0, 1, -1, 1, -5], [0, 1, 1, 0, 1, 0, -2, 2], [-8, 6, -2, 1, -6, -2, 8, 1], [0, 0, 0, 0, 3, 3, -2, -3], [2, -3, 4, -1, 1, -1, 0, 0], [0, -3, -3, 0, -1, 2, -2, -2], [6, -6, 0, -5, 6, 0, -6, -5], [2, 2, -1, 4, 1, -1, 0, 0], [-3, -4, 2, -3, 4, 2, 3, -3], [1, -4, 1, -6, -3, 3, 0, 0], [0, 2, 2, 0, -1, -3, -3, 5], [5, 2, -1, -4, -2, -1, -5, -4], [-2, 1, -2, 2, -1, -2, 2, 2], [0, 4, 4, 0, -2, -6, 13, 4], [0, -1, 3, 4, 2, -2, 0, 0], [-7, 6, -7, 8, -1, 1, 0, 0], [1, -5, -3, 1, 5, -3, -1, 1], [0, -1, -1, 0, -4, -3, -1, 1], [-4, 0, -3, 0, -3, 3, 0, 0], [-4, 0, 2, 6, 0, 2, 4, 6], [0, -2, -2, 0, -4, -2, -7, 4], [0, -4, -4, 0, -1, 3, 2, 3], [3, -1, -2, 3, 1, -2, -3, 3], [8, 0, 2, 6, 2, -2, 0, 0], [-8, -1, 4, -5, 1, 4, 8, -5], [0, -1, -1, 0, 2, 3, 0, 5], [5, 6, 1, 1, 7, -7, 0, 0], [0, -4, 9, -8, 4, 9, 0, -8], [0, 0, 0, 0, -3, -3, 4, 6], [-7, 6, -5, 5, 1, -1, 0, 0], [0, -1, -1, 0, -6, -5, 2, 11], [-17, 1, -2, 1, -1, 1, 0, 0], [1, 0, -5, -7, -5, 5, 0, 0], [0, 2, 2, 0, -1, -3, 5, 7], [-9, 3, -5, 5, -3, -5, 9, 5], [-13, -3, 1, -1, 3, 1, 13, -1], [0, 3, 3, 0, 6, -6, 0, 0], [0, 5, 5, 0, 1, -4, 5, -2], [6, 3, -4, 5, -1, 1, 0, 0], [0, 8, -2, -1, -8, -2, 0, -1], [0, -2, -2, 0, -1, 1, 7, -5], [0, 1, 1, 0, -1, -2, -19, -4], [-3, -3, -2, -4, 3, -2, 3, -4], [0, -9, -9, 0, -8, 1, -5, 2], [4, -2, 0, -5, -2, 2, 0, 0], [0, 1, 1, 0, -2, -3, 18, 0], [-3, -7, 3, -8, -4, 4, 0, 0], [7, -3, 4, 2, 1, -1, 0, 0], [0, 1, 1, 0, -5, -6, -4, -3], [-4, 5, 2, -4, -5, 2, 4, -4], [13, 3, -6, -1, -3, -6, -13, -1], [-7, -4, 1, 1, 4, 1, 7, 1], [-16, 5, -5, 5, 0, 0, 0, 0], [0, 11, 11, 0, 8, -3, -6, -7], [-3, -3, 5, 1, 2, -2, 0, 0], [0, -2, -2, 0, -1, 1, -20, -5], [-20, 0, -1, -2, 0, -1, 20, -2], [0, 1, 1, 0, -10, -11, 24, 8], [-6, 6, -5, -7, -6, -5, 6, -7], [8, -6, 11, -1, 5, -5, 0, 0], [-4, 4, -10, 5, -4, -10, 4, 5], [0, -3, -3, 0, 1, 4, -10, -2], [-4, -6, 2, 7, 6, 2, 4, 7], [-3, 1, 7, 0, -1, 7, 3, 0], [0, -8, -8, 0, 1, 9, -12, -3], [4, 0, 5, -5, 5, -5, 0, 0], [-15, 10, -7, 12, 3, -3, 0, 0], [0, -4, -4, 0, 2, 6, -1, -5], [-9, 2, -3, 3, -1, 1, 0, 0], [1, 3, 3, 9, -3, 3, -1, 9], [-2, -3, 0, -4, 3, 0, 2, -4], [2, 8, -1, 4, 7, -7, 0, 0], [1, 8, -4, 18, -8, -4, -1, 18], [-11, 8, -5, 4, 3, -3, 0, 0], [-9, -7, 6, -12, -1, 1, 0, 0], [10, 4, -4, 18, -4, -4, -10, 18], [0, -3, -3, 0, -6, -3, -4, 8], [0, 3, 3, 0, 6, 3, -25, -5], [-11, 8, -5, 7, 3, -3, 0, 0], [-6, 7, -3, 3, -7, -3, 6, 3], [-29, 3, -13, 4, -10, 10, 0, 0], [13, -4, 4, -2, 4, 4, -13, -2], [0, -3, -3, 0, 1, 4, -9, 11], [10, 6, 3, 3, -6, 3, -10, 3], [21, -18, 9, -14, 18, 9, -21, -14], [-2, -5, -9, 0, -14, 14, 0, 0], [0, -11, -11, 0, -2, 9, 6, 1], [0, 5, 5, 0, 4, -1, 20, -4], [-7, 0, -13, 12, 0, -13, 7, 12], [0, -9, -9, 0, -6, 3, -11, 0], [0, 13, 13, 0, 7, -6, 7, 1], [6, -12, 5, -13, -7, 7, 0, 0], [-15, -4, 11, -6, 4, 11, 15, -6], [0, 2, 2, 0, -4, -6, 16, 3], [13, 4, 1, -8, 5, -5, 0, 0], [25, -9, 3, -6, 9, 3, -25, -6], [0, -2, -2, 0, -2, 0, -7, -4], [-10, 2, -6, -1, -4, 4, 0, 0], [0, 4, 4, 0, 2, -2, 20, -11], [-3, 5, 1, 4, 6, -6, 0, 0], [0, 4, 4, 0, -4, -8, -5, -5], [-8, -11, 7, -11, -4, 4, 0, 0], [0, -2, -2, 0, 7, 9, -28, -9], [-18, 5, -1, 9, -5, -1, 18, 9], [10, 0, 2, -17, 2, -2, 0, 0], [0, -12, -12, 0, -2, 10, -10, -12], [9, 5, 3, 3, 8, -8, 0, 0], [-16, -1, 0, 5, 1, 0, 16, 5], [0, 4, 4, 0, -8, -12, 13, 14], [-5, -5, 10, 6, 5, 10, 5, 6], [-5, -2, -3, 4, -5, 5, 0, 0], [-5, -9, 12, -18, 9, 12, 5, -18], [-10, -1, 0, -4, 1, 0, 10, -4], [-2, -1, -8, 9, -9, 9, 0, 0], [9, -14, 9, -3, 14, 9, -9, -3], [0, 8, 8, 0, 9, 1, -6, -9], [-20, 12, -16, 16, -4, 4, 0, 0], [6, -5, 7, 4, 5, 7, -6, 4], [2, 3, -5, 7, -2, 2, 0, 0], [-4, 0, 7, -10, 0, 7, 4, -10], [0, 1, -10, 18, -9, 9, 0, 0], [0, -5, -5, 0, 1, 6, -25, 6], [0, -1, -1, 0, -3, -2, 7, -5], [26, 2, -1, -5, 1, -1, 0, 0], [0, -5, -5, 0, 3, 8, -1, -2], [10, -3, 4, 17, 1, -1, 0, 0], [8, -3, 4, -1, 3, 4, -8, -1], [0, -7, -7, 0, -9, -2, -2, 7], [3, 11, 2, 3, 13, -13, 0, 0], [0, 2, 2, 0, 15, 13, -2, 1], [12, 10, 1, 3, 11, -11, 0, 0], [-6, -1, 9, -9, 8, -8, 0, 0], [0, -4, -4, 0, 4, 8, -24, 3], [-11, 16, -4, -2, -16, -4, 11, -2], [-10, 0, 6, 1, 0, 6, 10, 1], [-9, 5, -13, 4, -5, -13, 9, 4], [25, 1, 9, -8, 10, -10, 0, 0], [-7, -8, 8, 13, 0, 0, 0, 0], [23, -5, 2, -1, 5, 2, -23, -1], [0, -6, -6, 0, 5, 11, -3, 6], [-3, -9, 2, -22, -7, 7, 0, 0], [31, -12, 9, -3, 12, 9, -31, -3], [-33, 2, -4, 6, -2, -4, 33, 6], [16, -7, 5, -20, -2, 2, 0, 0], [0, 1, 1, 0, 10, 9, 1, -19], [8, 2, -6, -10, -2, -6, -8, -10], [40, -2, 7, -10, 5, -5, 0, 0], [12, 5, 1, 13, 6, -6, 0, 0], [-8, 5, 3, -3, -5, 3, 8, -3], [0, 2, 2, 0, 0, -2, -5, 14], [0, -4, -4, 0, -3, 1, 10, 5], [0, -9, -9, 0, 4, 13, 1, 17], [0, -14, -14, 0, -6, 8, -20, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_322_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_322_2_e_c();".
// 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_322_2_e_c(:prec:=8)
    chi := MakeCharacter_322_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    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_322_2_e_c();".
// 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_322_2_e_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_322_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![81, -81, 162, 63, 81, 9, 10, 1, 1]>],Snew);
    return Vf;
end function;