
// Make newform 322.2.i.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_322_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_322_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_322_2_i_b();".
// 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_i_b();".
// 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, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_i();"
function MakeCharacter_322_i()
    N := 322;
    order := 11;
    char_gens := [185, 281];
    v := [11, 9];
    // 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_i_Hecke();"
function MakeCharacter_322_i_Hecke(Kf)
    N := 322;
    order := 11;
    char_gens := [185, 281];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 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 := [[0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [1, 0, 1, -1, 0, 0, 1, -1, 0, -1], [2, -2, 2, -2, 2, 0, 1, -1, 1, -1], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 1, -1, 0, 0, 2, -2, 1, -2, 2], [0, 0, 0, -1, 1, -1, 0, -1, 1, -1], [0, 1, 0, 0, 0, 1, 0, 0, 2, -2], [-1, 1, -2, 2, -1, 1, 0, 0, 0, 0], [-3, -1, 1, 1, -3, 1, -3, 1, -1, 1], [-1, 0, -3, 3, -3, 0, -1, 0, -1, 1], [0, 3, 0, 0, -1, 0, 1, -1, 0, 1], [0, -2, 3, 2, -2, -3, 2, 0, 0, 0], [-1, -3, 0, -3, -1, 0, -1, -5, 5, 1], [-3, 2, -5, 3, -3, 3, -3, 5, -2, 3], [2, 0, -1, 4, -1, 3, -3, 1, -4, 1], [0, -5, 5, -7, 4, -7, 4, -7, 5, -5], [4, -4, 0, -1, 6, -5, 3, -5, 6, -1], [5, -3, 5, 0, 5, -1, 7, -7, 1, -5], [-1, 4, -4, 1, 2, 1, -4, 4, -1, 0], [1, -7, 2, 0, 7, 0, 2, -7, 1, 0], [-2, 1, 3, -3, -1, 2, 0, 6, -1, 6], [-4, 4, 0, 6, -7, 8, 0, 8, -7, 6], [-3, 0, 0, 0, 0, 0, 0, 3, 0, 8], [7, -8, 3, -4, 4, -4, 4, -3, 8, -7], [2, 1, -4, 1, 2, 0, -1, -5, 5, 1], [1, 1, -4, 4, -4, 4, -1, -1, 0, 2], [-6, 0, 0, 6, 0, -2, -4, -1, -4, -2], [-3, 5, -3, 0, 5, 3, 2, -2, -3, -5], [9, -7, 5, -4, 5, -7, 9, 0, 10, -10], [-1, -5, -2, 1, -2, 1, -2, -5, -1, 0], [-1, -3, 3, 1, 0, 1, -3, 11, -3, 1], [0, -6, -7, 1, 1, -2, 1, 1, -7, -6], [15, 0, 9, -10, 9, -6, 6, -9, 10, -9], [1, 0, -1, -6, 0, 0, 0, 0, 6, 1], [1, 2, -2, -1, 0, 1, -7, 9, -7, 1], [0, 0, 0, 2, 0, -2, 0, -2, 0, 2], [6, 1, 2, -2, -1, -6, 0, -2, -4, -2], [0, 5, -1, 7, -1, 7, -1, 5, 0, 0], [0, 0, -7, -1, -7, 0, 0, 0, 3, -3], [7, -10, 10, -7, 0, -15, 13, -15, 13, -15], [-9, 4, -9, 4, -9, 0, -9, 5, -5, 9], [9, -3, 3, -9, 11, -11, 9, -3, 3, -9], [0, -4, 2, 0, 4, 14, 4, 0, 2, -4], [4, -9, 15, -5, 5, -15, 9, -4, 0, -3], [2, -2, 0, -2, -12, 4, -1, 4, -12, -2], [-7, -4, -7, 0, -2, -1, -11, 11, 1, 2], [-5, 5, 4, -4, -5, 5, 0, 12, -5, 12], [0, -4, -6, -7, -1, 8, -1, -7, -6, -4], [2, 5, -7, 7, 3, -3, -7, 7, -5, -2], [2, 0, -2, -1, 8, -4, 4, -8, 1, 2], [-6, -7, 7, -3, 9, -9, 3, -7, 7, 6], [3, -11, 2, 7, -7, -2, 11, -3, 0, -9], [-2, 7, -7, 2, 0, -1, 2, 2, 2, -1], [-10, -1, 5, 1, -4, 1, 5, -1, -10, 0], [-4, -6, -3, 3, 6, 4, 0, 10, -2, 10], [5, -5, 0, 7, 8, -11, 0, -11, 8, 7], [0, -7, 0, 3, 10, 0, 10, 3, 0, -7], [3, 0, -2, 0, 3, 0, -1, -11, 11, 1], [1, 0, 3, -5, 4, -11, 11, -4, 5, -3], [1, -2, 5, -2, 1, 0, -8, -3, 3, 8], [0, 2, 0, -4, 8, -5, 8, -4, 0, 2], [1, 3, 2, -1, 2, 3, 1, 0, -9, 9], [13, 3, 13, 0, -5, 0, 5, -5, 0, 5], [4, -2, 2, -4, 0, 8, -5, -5, -5, 8], [19, -8, 8, -6, 6, -8, 8, -19, 0, -1], [-2, 2, -4, 2, -2, 0, -12, -1, 1, 12], [2, -4, 15, 3, -3, -15, 4, -2, 0, -1], [-5, 1, -5, 0, 3, -7, -3, 3, 7, -3], [-3, 5, 10, 7, -6, 7, 10, 5, -3, 0], [2, -2, -9, 9, 2, -2, 0, 2, -14, 2], [3, 9, 3, 0, 2, 12, -11, 11, -12, -2], [-1, 7, -4, 8, -8, 4, -7, 1, 0, -8], [0, 0, 5, 5, 5, -9, 9, -5, -5, -5], [-6, 0, 0, 0, -6, 0, -13, 5, -5, 13], [6, -11, 11, -6, 0, -12, 18, -4, 18, -12], [-8, 5, -8, 0, -7, 16, -2, 2, -16, 7], [-12, 17, -15, 25, 0, 25, -15, 17, -12, 0], [2, 3, 16, -5, 16, 3, 2, 0, 11, -11], [-3, 3, 0, -5, -4, -14, 5, -14, -4, -5], [-3, -4, -15, -4, -3, 0, 2, 15, -15, -2], [-9, 4, 1, 4, -9, 0, -12, 17, -17, 12], [0, 12, 4, -2, -4, 5, -4, -2, 4, 12], [2, 6, 4, -3, 4, 6, 2, 0, 13, -13], [3, -22, 7, -7, 22, -3, 0, -10, 6, -10], [-6, 9, -6, -5, -27, -5, -6, 9, -6, 0], [6, 9, 14, -16, 14, 9, 6, 0, 5, -5], [12, 5, -5, -12, 0, 4, -2, 8, -2, 4], [7, -14, 10, -8, -4, 4, 8, -10, 14, -7], [-21, 0, -6, 0, -15, 15, -15, 15, 0, 6], [16, -10, 24, -5, 4, -4, 5, -24, 10, -16], [0, -1, 8, -4, -6, 5, -6, -4, 8, -1], [13, 4, 1, -1, -4, -13, 0, -6, 19, -6], [-2, 10, -1, 1, -10, 2, 0, -3, -13, -3], [-24, 37, -24, 0, -16, 8, -10, 10, -8, 16], [0, -7, 15, -10, 5, -8, 5, -10, 15, -7], [-6, 20, -11, 11, -19, 19, -11, 11, -20, 6], [10, -26, 16, -26, 13, -13, 26, -16, 26, -10], [6, -10, 6, 0, 5, -16, 14, -14, 16, -5], [8, -9, 21, 2, 21, -9, 8, 0, 9, -9], [10, -4, 4, -10, 0, 1, 4, 14, 4, 1], [6, -4, -2, -4, 6, 0, 0, 14, -14, 0], [4, 13, 3, 13, 4, 0, -3, -4, 4, 3], [-2, 18, -18, 2, 0, 10, -4, 20, -4, 10], [-7, 8, 0, 5, 0, 8, -7, 0, 0, 0], [-15, 17, -11, 11, -17, 15, 0, 24, -10, 24], [-2, -7, 1, -7, -3, -7, 1, -7, -2, 0], [-1, -2, 2, 1, 0, 6, -3, -4, -3, 6], [-12, 6, -22, 6, -12, 0, -11, 23, -23, 11], [19, 0, -11, 4, -6, 1, -1, 6, -4, 11], [6, 12, 3, -4, 3, -3, 4, -3, -12, -6], [-15, 5, -11, 7, -7, 11, -5, 15, 0, 5], [-1, -28, -1, 0, 3, -6, 16, -16, 6, -3], [4, 2, 2, -2, -2, -4, 0, -3, -13, -3], [-5, 5, 3, 5, -27, 5, 3, 5, -5, 0], [-6, 6, 0, 10, 11, 2, 6, 2, 11, 10], [-23, 14, -23, 25, -9, 9, -25, 23, -14, 23], [-2, 0, -6, 23, -4, 11, -11, 4, -23, 6], [11, 0, 4, 9, 0, 0, -9, -4, 0, -11], [-9, 12, 14, 2, -2, -14, -12, 9, 0, -3], [-21, 15, -21, 0, -5, 9, -18, 18, -9, 5], [-17, 7, -20, 21, -20, 7, -17, 0, -4, 4], [-9, 9, -7, 27, -7, 9, -9, 0, -7, 7], [-9, 9, 0, 16, -6, -5, 3, -5, -6, 16], [0, -17, 16, -16, 7, -27, 7, -16, 16, -17], [-9, 0, -10, 11, -18, 7, -7, 18, -11, 10], [6, -12, 12, -6, 0, 5, 12, 19, 12, 5], [-6, 16, -11, 11, -16, 6, 0, 11, -19, 11], [27, -25, 13, -16, 13, -25, 27, 0, 7, -7], [-20, 11, -7, 18, -18, 7, -11, 20, 0, 21], [-7, 16, -7, 9, -10, 10, -9, 7, -16, 7], [27, -34, 24, -33, 28, -28, 33, -24, 34, -27], [0, -15, 23, -5, 16, -1, 16, -5, 23, -15], [2, -11, 2, 0, 21, -2, 0, 0, 2, -21], [22, -29, 16, -18, 20, -18, 16, -29, 22, 0], [-6, 17, -8, 6, -35, 6, -8, 17, -6, 0], [-3, 3, 0, -25, 15, -8, 24, -8, 15, -25], [-4, 15, 4, -2, 2, -4, -15, 4, 0, 11], [-25, 4, -6, 4, -25, 0, -1, 4, -4, 1], [9, -14, 9, 0, 7, -21, 10, -10, 21, -7], [-2, 7, 0, 0, -7, 2, 0, -8, -22, -8], [15, -2, 5, -7, 5, -2, 15, 0, 6, -6], [0, -2, 10, -2, -3, 22, -3, -2, 10, -2], [-8, -19, -1, -19, -8, 0, 4, -15, 15, -4], [0, 0, 2, 3, -10, -8, 8, 10, -3, -2], [-3, 0, 1, 3, 7, 5, -5, -7, -3, -1], [9, -2, 2, -9, 0, 12, -12, -14, -12, 12], [10, -8, 12, -16, -8, -16, 12, -8, 10, 0], [6, -29, 10, -12, 10, -29, 6, 0, 1, -1], [-9, 22, -9, 0, -7, -16, -1, 1, 16, 7], [17, 6, -6, -17, 0, 8, -17, 4, -17, 8], [-19, 18, -14, 10, -15, 15, -10, 14, -18, 19], [0, 6, 9, -5, 13, 2, 13, -5, 9, 6], [10, -6, -7, -18, 18, 7, 6, -10, 0, -13], [-6, 6, 0, -19, -10, 0, 5, 0, -10, -19], [9, -9, 0, -6, 8, -3, 33, -3, 8, -6], [-17, 17, -15, 26, -26, 15, -17, 17, 0, 21], [-23, 0, -8, 19, -14, 16, -16, 14, -19, 8], [-20, 3, -9, -3, 3, 9, -3, 20, 0, -2], [-6, 15, -12, -12, -12, 15, -6, 0, -16, 16], [4, -1, -6, 22, 8, 22, -6, -1, 4, 0], [-6, 9, -15, 15, -9, 6, 0, -2, -3, -2], [-5, 5, 0, 12, -21, 10, -13, 10, -21, 12], [-10, 0, 8, 21, 9, -7, 7, -9, -21, -8], [22, -12, -4, -12, 22, 0, 18, -12, 12, -18], [-3, 4, -4, 3, 0, -16, 3, -20, 3, -16], [-10, 2, 8, -8, 8, 2, -10, 0, -2, 2], [-2, 6, 18, 5, -10, 5, 18, 6, -2, 0], [4, 29, 4, 0, 12, 3, 8, -8, -3, -12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_322_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_322_2_i_b();".
// 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_i_b(:prec:=10)
    chi := MakeCharacter_322_i();
    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_i_b();".
// 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_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_322_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -20, 147, -135, 126, -100, 75, -37, 14, -5, 1]>],Snew);
    return Vf;
end function;