
// Make newform 912.2.bo.d in Magma, downloaded from the LMFDB on 19 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_d();".
// 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_d();".
// 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, -1, -2, 0], [1, -1, 3, -1, -2, -2], [0, -1, -2, 0, -2, -1], [1, 1, -2, 1, -2, 2], [0, -2, 1, -1, 2, 0], [0, 5, 0, 0, -3, 0], [-6, 3, 0, 3, 3, 2], [0, -1, 2, -1, 0, 0], [5, -1, 0, -5, 1, 1], [-1, 4, 4, 0, -4, 0], [-3, -3, -2, 1, 0, -1], [-3, 0, 3, 5, -3, 2], [-4, 3, 3, 3, -4, 0], [0, -2, 0, 2, 2, -9], [4, -6, -1, 1, 6, -4], [-3, 2, 0, 1, 1, -1], [6, -11, -4, -11, 6, 0], [-3, 0, 3, 5, -7, 2], [3, -5, 1, -2, 0, 2], [-5, 5, -4, 1, 0, -1], [-1, 4, -3, 1, -1, -1], [7, 7, -1, 0, -7, 1], [7, 4, 8, -8, -4, -7], [-10, -10, 1, 8, 2, -1], [0, 5, 0, -1, 0, 5], [3, -3, 7, -3, -4, -4], [-4, 0, 4, 3, 11, -1], [2, -1, -1, 0, 2, -1], [2, 2, 8, -2, 0, -8], [9, 2, -4, 4, -2, -9], [-10, 10, 0, 0, 0, -3], [-1, -1, -9, 2, -1, 9], [-11, 2, -2, 9, 0, -9], [-8, 0, 0, 0, -3, 3], [2, 0, -2, 3, -6, 5], [0, -11, 10, 2, 10, -11], [-5, -3, 0, 8, 8, -8], [9, -5, 10, -10, 5, -9], [0, 2, 0, 5, 0, 2], [-7, 9, -6, 9, -7, 0], [0, -12, -12, 0, 13, -1], [8, 7, 5, -3, 0, 3], [-1, 11, -6, 1, -5, -5], [8, 5, -8, 5, 8, 0], [6, 5, 5, -5, -5, -6], [-5, 0, 5, -1, -15, -6], [9, -3, -3, 0, 7, -4], [-2, -4, -4, 0, 4, 0], [-4, 0, 4, -3, -10, -7], [0, 8, -6, -16, -6, 8], [2, 2, 17, 2, -4, -17], [8, 1, 0, -9, -9, -9], [-7, 4, 6, 4, -7, 0], [-6, 5, -1, 5, 0, -5], [11, 1, 8, -3, 0, 3], [-1, 0, 1, 8, 11, 7], [0, 7, -17, -1, -17, 7], [5, -11, 0, 6, 6, -9], [-3, -5, -2, 2, 5, 3], [0, -1, 8, 15, 8, -1], [-7, -17, -4, 3, 0, -3], [3, 8, -5, -3, -3, -3], [-19, 11, -7, 11, -19, 0], [-5, -5, -19, 0, 5, 19], [0, -4, -10, -1, -10, -4], [3, 0, -3, -12, -2, -9], [-4, 0, 4, 10, -15, 6], [9, -10, 3, -9, 7, 7], [0, 0, -6, 3, -6, 0], [4, -8, 12, -12, 8, -4], [4, 4, 22, 1, -5, -22], [3, 0, 11, -3, -11, -11], [-16, -8, -8, 0, 15, -7], [16, -13, 18, 2, 0, -2], [-2, 5, 5, 5, -2, 0], [9, 0, -2, 2, 0, -9], [9, 5, 7, -7, -5, -9], [-10, -2, -10, -2, -10, 0], [-13, -15, -15, 0, 2, 13], [-18, -2, -15, 3, 0, -3], [11, 11, 5, 1, -12, -5], [0, -2, 0, 2, 2, 24], [-1, 1, 10, -10, -1, 1], [21, 21, -7, -14, -7, 7], [8, -10, -4, -8, 14, 14], [22, -7, -7, 0, 1, 6], [-6, 0, 6, 3, -19, -3], [7, 0, 15, -7, -15, -15], [0, -2, 17, 0, 17, -2], [0, -5, 0, 5, 5, 6], [9, -4, -5, -9, 9, 9], [12, 3, -2, -14, 0, 14], [-11, 0, 11, -2, -19, -13], [9, 7, -1, 7, 9, 0], [-10, 3, 0, 7, 7, 11], [0, 14, -11, 2, -11, 14], [-10, -9, 21, -9, -10, 0], [5, -11, -9, -11, 5, 0], [-7, 0, 0, 7, 7, 21], [16, 16, 13, -15, -1, -13], [1, 0, 7, -1, -7, -7], [-5, 8, 8, 0, 14, -22], [-11, 1, 1, 0, -11, 10], [-26, 5, -4, 26, -1, -1], [4, 2, 7, -7, -2, -4], [4, 7, 0, -11, -11, 29], [-7, 7, 22, 7, -7, 0], [23, 1, 1, -23, -2, -2], [22, -3, -3, 0, 0, 3], [11, 0, -11, -1, 4, 10], [-4, 0, -28, 0, -4, 0], [0, 18, -6, 16, -6, 18], [-13, -3, 0, 16, 16, -11], [1, 0, -1, -2, 6, -1], [0, 7, 6, 6, 0, -6], [-13, 5, 5, 0, 9, -14], [5, 1, -15, -5, 14, 14], [-6, -6, -5, 5, 1, 5], [11, -23, 0, 12, 12, 0], [0, 1, -5, -17, -5, 1], [-20, -10, -5, 20, 15, 15], [26, 0, 0, 0, 0, 0], [-7, 18, -28, 7, 10, 10], [1, -17, 16, -16, 17, -1], [16, 16, 4, -19, 3, -4], [-5, 14, 0, 5, 0, -5], [21, 0, -21, -12, -13, 9], [0, 11, -6, 17, -6, 11], [-6, -18, 2, -2, 18, 6], [10, 5, 21, -21, -5, -10], [6, 4, -18, 4, 6, 0], [2, 16, 6, 4, 0, -4], [19, -5, -5, 0, 5, 0], [-6, 3, 25, 3, -6, 0], [-12, -12, 19, -2, 14, -19], [0, 11, -11, -7, -11, 11], [9, 11, 11, 0, -22, 11], [0, 6, -1, -14, -1, 6], [1, 1, 25, -4, 3, -25], [11, -30, 0, 19, 19, 17], [7, 7, -13, -2, -5, 13], [12, 3, -8, 3, 12, 0], [19, 12, -14, -19, 2, 2], [8, 19, 6, -2, 0, 2], [0, -22, 2, -2, 22, 0], [34, -8, 33, -33, 8, -34], [-4, 23, 0, -19, -19, 4], [0, -12, 21, 0, 21, -12], [7, 12, 24, 17, 0, -17], [16, 17, -9, -16, -8, -8], [6, -14, -11, -14, 6, 0], [-21, -21, -10, 40, -19, 10], [8, 0, -8, 4, -1, 12], [-9, 0, 0, 0, 15, -15], [-2, -7, 7, 2, 0, 0], [8, 13, 15, -15, -13, -8], [0, 0, -1, 7, -7, 1], [14, -9, 0, -9, 14, 0], [20, -5, -1, -21, 0, 21], [4, -34, 12, 8, 0, -8], [26, 17, 0, 0, -17, -26], [2, 6, 0, 0, -6, -2], [0, 1, -1, 3, -1, 1], [-20, 0, 20, 27, 10, 7], [24, 33, 8, -16, 0, 16], [15, 7, -18, 7, 15, 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_d();".
// 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_d(: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_d();".
// 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_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_912_bo();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 12, 60, -46, 12, -3, 1]>],Snew);
    return Vf;
end function;