
// Make newform 722.2.e.m in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_722_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_722_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_722_2_e_m();".
// 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_722_2_e_m();".
// 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, 0, -1, 0, 0, 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_722_e();"
function MakeCharacter_722_e()
    N := 722;
    order := 9;
    char_gens := [363];
    v := [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_722_e_Hecke();"
function MakeCharacter_722_e_Hecke(Kf)
    N := 722;
    order := 9;
    char_gens := [363];
    char_values := [[0, 0, 0, 0, 0, -1]];
    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, 1, 0, 0, -1, 0], [1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 2, 0], [-2, 0, -2, 2, 2, 2], [0, 1, 1, -2, 1, 1], [0, 0, 2, -2, 2, -2], [0, 1, 4, -4, -1, 0], [0, 0, 0, 0, 0, 0], [0, -2, 0, 2, 2, 0], [-2, 4, 4, 4, -2, 0], [2, 2, -4, -2, 2, 2], [-2, 4, 4, 0, -2, -2], [0, 0, 1, 1, 0, -1], [0, 0, 0, -2, 3, -2], [2, -6, -2, -6, 2, 0], [-4, 6, 0, -2, -2, 0], [2, -2, -1, 1, 2, -2], [2, 0, 0, -2, -2, -6], [3, 0, -6, 0, 3, 0], [4, 0, -4, -2, -4, 2], [7, -4, 4, -3, 0, 3], [-2, 6, -2, 0, 0, 0], [-2, 4, -1, 2, -3, -3], [8, 8, -3, -4, -4, 3], [-1, 0, 0, 0, 0, 1], [-4, -4, -2, 2, 2, 2], [0, -4, 6, 2, 6, -4], [7, -2, 4, -7, -2, -2], [-2, 0, 2, 2, 8, 0], [0, -8, -8, 0, 5, 3], [-2, -2, 6, 10, -8, -6], [-1, -8, -1, 1, 8, 1], [5, 0, 0, -5, -5, 4], [-5, -5, 10, 5, 0, -10], [-2, 12, 2, 4, 0, -4], [2, -14, -14, 0, 10, 4], [-4, 0, 4, 2, 4, -2], [0, -4, 5, -8, 5, -4], [6, 0, 0, -6, -6, -6], [4, 12, 2, -2, -12, -4], [0, -7, 12, 2, 12, -7], [8, 0, 0, 0, 8, 0], [-2, -10, -10, 0, 2, 8], [-16, -5, -4, 12, 0, -12], [-2, -14, 10, 2, 4, 4], [6, 0, 8, 0, 6, 0], [-6, -9, 6, -6, 9, 6], [6, 0, -6, 0, 2, 6], [6, 5, 5, 0, -5, 0], [-8, 0, 0, 0, 8, -8], [4, 0, -4, -13, -4, -9], [0, 10, -8, 0, -8, 10], [-1, -1, -4, 8, -7, 4], [-9, 8, 0, 1, 1, 8], [-16, -1, -4, -1, -16, 0], [-4, 4, -16, -12, 0, 12], [-2, -4, 4, 6, 0, -6], [-14, 0, 14, 8, 6, -6], [0, -12, 6, -4, 6, -12], [-1, 4, 0, -3, -3, 0], [-6, 2, 5, -5, -2, 6], [0, 8, -16, 0, -16, 8], [15, 2, 14, -1, 0, 1], [2, -16, 8, -2, 8, 8], [-8, 5, 16, 5, -8, 0], [10, 10, 4, -4, -6, -4], [0, 7, 11, -6, 11, 7], [-8, 0, 8, 7, 16, -1], [3, 0, -3, -11, -6, -8], [0, 8, -6, 0, -2, -2], [0, -5, 16, 0, 16, -5], [14, -16, 12, -12, 16, -14], [-6, -6, -10, -6, 12, 10], [2, 16, -4, -2, -12, -12], [0, 8, 8, 0, -16, 8], [12, -6, 10, -2, 0, 2], [4, -6, -20, -6, 4, 0], [-4, 12, 2, -2, -12, 4], [-12, -8, -9, 9, 8, 12], [17, -9, 12, -9, 17, 0], [-1, -18, -18, 0, 20, -2], [-6, -4, 4, 10, 0, -10], [-4, -4, 20, -2, 6, -20], [4, 4, 0, -8, -8, -14], [14, 16, 0, 0, -16, -14], [-7, -7, -12, 12, -5, 12], [4, 3, 8, -4, -11, -11], [-4, -5, -5, 0, 1, 4], [12, 0, -12, -2, 4, 10], [-14, -10, 4, 14, 6, 6], [0, -3, -4, -18, -4, -3], [12, -12, 0, 0, 0, -4], [-10, 14, -28, 10, 14, 14], [-18, -5, -18, 0, 0, 0], [1, 0, -1, 14, -6, 15], [12, -14, 8, -14, 12, 0], [6, -8, 0, 2, 2, -16], [0, -8, -9, -16, -9, -8], [-12, 12, -4, 12, -12, 0], [-8, 6, 0, 6, -8, 0], [6, 4, 0, -10, -10, -11], [-24, -24, 6, 10, 14, -6], [-14, -11, 17, 14, -6, -6], [-3, 12, 12, 0, 8, -20], [4, -5, -5, 0, -12, 17], [-16, -1, -3, 16, 4, 4], [2, -1, 6, -6, 1, -2], [17, -1, 0, -16, -16, 20], [14, -24, -16, -24, 14, 0], [-16, -5, 1, 16, 4, 4], [-16, -10, -10, 0, 14, -4], [18, 0, -18, -18, -8, 0], [-13, 17, 0, 17, -13, 0], [0, -4, -2, 27, -2, -4], [6, 2, 0, -8, -8, -4], [5, 0, -5, -9, -16, -4], [17, 0, 17, 0, 0, 0], [-14, 12, 12, 0, -16, 4], [-4, 0, 2, 4, -2, -2], [10, 10, -1, -16, 6, 1], [-6, 10, 0, -4, -4, 16], [0, -28, 8, -17, 8, -28], [6, -20, 20, -6, 0, 0], [19, -14, -14, 0, 4, 10], [-19, 10, 0, 19, -10, -10], [2, 2, -2, 2, -2, -2], [-14, -14, 12, 16, -2, -12], [-8, 12, 12, 20, 0, -20], [6, 0, -6, 12, 4, 18], [0, 12, -22, -14, -22, 12], [25, -2, 28, -28, 2, -25], [24, -18, 22, -22, 18, -24], [2, 2, -34, 2, 2, 0], [-6, -2, -10, -4, 0, 4], [20, 13, 13, 0, -12, -1], [4, -16, -15, -16, 4, 0], [-10, -10, 20, 16, -6, -20], [0, 19, -9, -18, -9, 19], [-4, -18, -18, 0, -4, 22], [0, -13, 9, -4, 9, -13], [-4, -4, -32, -12, 16, 32], [0, 12, 0, -12, -12, 6], [12, 12, 0, -28, 16, 0], [-11, 17, 10, 17, -11, 0], [14, -16, -8, -14, 24, 24], [-14, 26, 6, 20, 0, -20], [-24, -2, -14, 14, 2, 24], [1, -32, -3, 3, 32, -1], [-27, 17, 0, 10, 10, -6], [0, -12, 2, 4, 2, -12], [0, -2, -28, -28, 0, 28], [-20, 8, 15, 20, -23, -23], [-17, -12, 2, -12, -17, 0], [-12, -12, 12, 10, 2, -12], [-9, 0, 9, -2, 22, -11], [-34, 12, 12, 0, -4, -8], [36, 0, -10, -36, 10, 10], [-16, 8, -5, 5, -8, 16], [9, 9, -8, -12, 3, 8], [6, 18, 4, 18, 6, 0], [4, -20, -12, -16, 0, 16], [25, -20, 32, 7, 0, -7], [16, 16, 0, 0, -16, -16], [-21, 24, -17, 17, -24, 21], [0, 36, -17, 12, -17, 36], [-4, 0, 4, 18, -10, 14], [-12, -26, 2, 14, 0, -14], [22, -2, -2, -2, 22, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_722_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_722_2_e_m();".
// 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_722_2_e_m(:prec:=6)
    chi := MakeCharacter_722_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_722_2_e_m();".
// 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_722_2_e_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_722_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, -3, 12, -19, 15, -6, 1]>,<5,R![64, 0, 0, 8, 0, 0, 1]>,<7,R![576, 0, 144, 48, 36, 6, 1]>,<13,R![64, -96, 96, -64, 12, 6, 1]>],Snew);
    return Vf;
end function;