
// Make newform 912.2.bo.e in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_912_bo();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_912_bo_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_912_2_bo_e();".
// 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_912_2_bo_e();".
// 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_912_bo();"
function MakeCharacter_912_bo()
    N := 912;
    order := 9;
    char_gens := [799, 229, 305, 97];
    v := [9, 9, 9, 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_912_bo_Hecke();"
function MakeCharacter_912_bo_Hecke(Kf)
    N := 912;
    order := 9;
    char_gens := [799, 229, 305, 97];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [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, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, -1, 0, 1, 1], [-1, 1, 1, 1, -2, -2], [0, -2, -1, -1, -1, -2], [0, 0, 3, 2, -2, -3], [1, -1, -2, 2, 1, -1], [-2, 4, 2, 2, -2, -1], [3, -2, 0, -1, -1, -4], [-4, 4, 1, 4, -4, 0], [2, 0, -3, -2, 3, 3], [-2, -3, -3, 0, 2, 1], [-3, 6, -3, 0, 0, 0], [2, 0, -2, 2, 7, 4], [0, 1, 5, 1, 0, 0], [-3, 5, 0, -2, -2, -2], [-5, -3, -4, 4, 3, 5], [-1, 2, 0, -1, -1, -5], [0, 2, 2, 2, 0, 0], [6, 0, -6, -6, -3, 0], [3, 2, 4, 1, 0, -1], [-2, -9, 0, 2, 0, -2], [-5, 7, -4, 5, -3, -3], [-1, -1, -4, 7, -6, 4], [5, 2, 7, -7, -2, -5], [-2, -2, 7, 2, 0, -7], [0, -6, 1, -5, 1, -6], [-8, 2, -6, 8, 4, 4], [4, 0, -4, 1, -1, 5], [4, -10, -10, 0, 6, 4], [-1, -1, 11, -4, 5, -11], [-9, 4, -8, 8, -4, 9], [4, 6, 0, -10, -10, 1], [6, 6, -7, -3, -3, 7], [1, -14, 6, 5, 0, -5], [1, -4, -4, 0, -7, 11], [13, 0, -13, -8, -1, 5], [0, 3, 1, -17, 1, 3], [4, -8, 0, 4, 4, 6], [-6, 2, -1, 1, -2, 6], [0, 7, 3, 10, 3, 7], [-2, 7, 11, 7, -2, 0], [10, -5, -5, 0, -1, 6], [-7, 8, -13, -6, 0, 6], [5, -12, -4, -5, 16, 16], [6, -5, 12, -5, 6, 0], [-12, -6, -10, 10, 6, 12], [-2, 0, 2, 13, 1, 11], [6, 10, 10, 0, -8, -2], [1, -4, -4, 0, -8, 12], [12, 0, -12, -7, 1, 5], [0, -14, 0, -5, 0, -14], [4, 4, -2, -8, 4, 2], [7, -12, 0, 5, 5, 10], [0, 8, -2, 8, 0, 0], [-14, -4, -2, 12, 0, -12], [-10, -4, -12, -2, 0, 2], [3, 0, -3, -9, 4, -6], [0, -1, -1, -11, -1, -1], [-8, -3, 0, 11, 11, -17], [-5, 4, -7, 7, -4, 5], [0, 5, 16, -5, 16, 5], [-15, -1, 0, 15, 0, -15], [-1, 3, -1, 1, -2, -2], [-3, -6, -1, -6, -3, 0], [1, 1, -17, 2, -3, 17], [0, 13, -9, -12, -9, 13], [6, 0, -6, 3, -10, 9], [-1, 0, 1, -7, 6, -8], [14, 8, 0, -14, -8, -8], [0, 5, 2, 12, 2, 5], [15, 11, 5, -5, -11, -15], [-10, -10, -7, 14, -4, 7], [10, 6, -12, -10, 6, 6], [1, 13, 13, 0, -2, -11], [4, 15, -1, -5, 0, 5], [0, 2, 28, 2, 0, 0], [-8, -1, -13, 13, 1, 8], [-17, -1, -23, 23, 1, 17], [12, -9, -13, -9, 12, 0], [20, 11, 11, 0, -17, 6], [-6, 4, 9, 15, 0, -15], [0, 0, -10, -10, 10, 10], [-2, 1, 0, 1, 1, -14], [-11, 6, -14, 14, -6, 11], [-4, -4, -5, -9, 13, 5], [-5, 5, 9, 5, -14, -14], [1, 11, 11, 0, -14, 3], [-6, 0, 6, -12, 12, -18], [12, -3, -8, -12, 11, 11], [0, -4, 2, -3, 2, -4], [18, -15, 0, -3, -3, -9], [5, 1, 4, -5, -5, -5], [3, -5, 22, 19, 0, -19], [-1, 0, 1, -3, -9, -4], [-7, 13, 4, 13, -7, 0], [-2, 15, 0, -13, -13, -8], [0, -2, 3, -23, 3, -2], [10, 10, 7, 10, 10, 0], [1, 5, -17, 5, 1, 0], [1, -2, 0, 1, 1, -25], [2, 2, 14, 16, -18, -14], [1, 13, -7, -1, -6, -6], [31, 2, 2, 0, -6, 4], [-7, -13, -13, 0, 3, 10], [-23, -5, 1, 23, 4, 4], [-17, 1, -9, 9, -1, 17], [17, -24, 0, 7, 7, 2], [13, 10, -3, 10, 13, 0], [13, 4, -17, -13, 13, 13], [-14, -2, -2, 0, -5, 7], [-13, 0, 13, 8, 29, -5], [22, -17, -3, -17, 22, 0], [0, 3, 0, -13, 0, 3], [-18, 19, 0, -1, -1, -14], [-10, 0, 10, 2, -9, -8], [9, -6, -8, -17, 0, 17], [-1, 2, 2, 0, -29, 27], [11, -16, 1, -11, 15, 15], [4, 4, -14, -4, 0, 14], [13, 1, 0, -14, -14, 25], [0, 11, 0, 9, 0, 11], [-14, -24, 19, 14, 5, 5], [-14, -15, -15, 0, 25, -10], [5, -16, 8, -5, 8, 8], [-3, 11, -7, 7, -11, 3], [0, 0, -33, 2, -2, 33], [5, 42, 1, -4, 0, 4], [11, 0, -11, -3, 24, 8], [0, 4, -17, -20, -17, 4], [12, -16, 4, -4, 16, -12], [6, 6, 21, -21, -6, -6], [2, 13, 14, 13, 2, 0], [-8, 13, 5, 13, 0, -13], [-3, 25, 25, 0, -17, -8], [-6, -13, 21, -13, -6, 0], [-11, -11, -10, -9, 20, 10], [0, -11, -2, -13, -2, -11], [-32, -6, -6, 0, -8, 14], [0, -11, -4, -16, -4, -11], [-23, -23, -16, 22, 1, 16], [18, 0, 0, -18, -18, -6], [-10, -10, -4, 11, -1, 4], [19, -15, 4, -15, 19, 0], [-3, 3, 20, 3, -23, -23], [17, 8, 3, -14, 0, 14], [6, 4, -13, 13, -4, -6], [-23, 11, -8, 8, -11, 23], [22, 11, 0, -33, -33, 7], [0, -9, 17, 1, 17, -9], [-6, -16, -23, -17, 0, 17], [-20, 11, 0, 20, -11, -11], [13, -19, 6, -19, 13, 0], [12, 12, -31, -1, -11, 31], [16, 0, -16, -2, 4, 14], [0, 9, 9, 0, 0, -9], [3, -31, 12, -3, 19, 19], [-1, 34, -7, 7, -34, 1], [-10, -10, 23, 0, 10, -23], [-14, 3, -20, 3, -14, 0], [15, -8, -5, -20, 0, 20], [-15, -21, 3, 18, 0, -18], [-12, 11, -5, 5, -11, 12], [26, 1, 23, -23, -1, -26], [0, -5, 8, -10, 8, -5], [13, 0, -13, 2, -19, 15], [0, 29, 18, 18, 0, -18], [20, -20, 18, -20, 20, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_912_bo_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_912_2_bo_e();".
// 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_912_2_bo_e(:prec:=6)
    chi := MakeCharacter_912_bo();
    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_912_2_bo_e();".
// 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_912_2_bo_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_912_bo();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -27, 36, -30, 18, -6, 1]>],Snew);
    return Vf;
end function;