
// Make newform 912.2.ci.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_912_ci();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_912_ci_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_ci_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_912_2_ci_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, 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_ci();"
function MakeCharacter_912_ci()
    N := 912;
    order := 18;
    char_gens := [799, 229, 305, 97];
    v := [9, 18, 18, 13];
    // 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_ci_Hecke();"
function MakeCharacter_912_ci_Hecke(Kf)
    N := 912;
    order := 18;
    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, 1, 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, 0, 0], [0, 0, -1, 0, 0, 0], [-1, -1, 1, 0, 1, -1], [2, 0, -1, -1, 1, 0], [1, 2, 1, 1, -1, 1], [0, -1, -2, -2, -1, 0], [-3, 1, 0, 2, 2, -1], [4, -1, 2, -2, 2, -2], [2, 0, 1, -3, 0, -3], [2, 6, 2, -4, -3, 2], [0, -3, 5, 2, 5, -3], [4, -1, 1, -8, -1, -2], [-1, 2, 4, -2, 1, -8], [2, -2, 1, -4, -2, 1], [7, -2, 7, -1, 1, -6], [-1, 2, 3, -2, -4, -2], [3, -1, 0, -2, -2, -11], [4, -1, 5, 1, 0, -1], [6, 0, -6, 0, 2, 6], [4, 4, -3, -2, -2, 3], [0, -3, -8, -3, 0, 0], [-4, 2, 7, 2, -4, 0], [-2, -3, -2, 1, 2, 3], [0, 4, 1, 1, 4, 0], [-2, 1, 4, 3, -3, -2], [-6, -3, 6, -6, 3, 6], [5, 2, -1, -5, -1, -1], [0, 0, 0, 6, 0, 0], [-7, 7, -3, 1, -6, -3], [-2, -4, 4, 4, 10, 6], [-9, 5, -6, 6, -5, 9], [-5, -1, 0, 4, -4, 0], [0, -3, 6, 6, 0, -6], [-5, 1, -1, -1, 1, -5], [-4, 1, 4, 1, -4, 0], [-1, 9, 9, 0, -2, -7], [-7, -7, -1, 8, -1, 1], [1, 9, 0, 1, -9, 9], [-2, 2, -2, 0, 0, 0], [-7, 4, 4, 11, -11, -2], [-12, -1, -4, 12, 5, 5], [3, -10, 3, -7, 5, 4], [2, -2, 2, -4, 5, 3], [3, -2, 2, 2, -3, -4], [0, -6, 14, 7, 14, -6], [-7, -12, -7, -1, 6, 8], [10, -8, 0, -2, -2, 4], [-7, -7, -13, 11, -4, 13], [-8, 6, 6, 0, -2, -4], [-1, 0, 0, 0, -12, 12], [-5, -5, 17, 3, 2, -17], [5, 0, -4, 5, -4, 4], [2, -8, -2, -2, -8, 2], [2, 10, -3, 1, -20, 1], [-14, 0, -14, -6, 0, 20], [2, -6, 0, 6, -2, 0], [8, 14, 2, -14, -8, -4], [-10, 10, -8, -1, -11, -8], [-3, -5, -6, 3, 11, 11], [-13, 5, -8, 21, -10, 21], [18, 5, 8, -13, 13, -4], [-1, -5, -11, -1, -6, 6], [-2, -3, 5, -3, -2, 0], [10, 6, 1, -5, -1, -6], [-1, 0, 1, -4, -23, -5], [9, -1, -16, -16, -1, 9], [8, -11, 10, -8, 1, 1], [-3, 3, 0, 13, 10, 0], [16, -16, 4, -1, 15, 4], [0, -16, 8, -8, 8, -16], [-2, 9, -17, 2, 8, 8], [10, -7, -6, -17, 17, 3], [2, 7, -8, -8, 7, 2], [-24, -2, 6, 12, -6, 2], [3, 7, 7, 0, 15, -22], [19, -5, -3, -5, 19, 0], [18, 0, -18, -8, 2, 10], [3, 2, 0, -5, -5, -13], [-4, 3, 10, 7, -7, -5], [-9, 10, -9, 11, -5, -2], [-8, -14, 14, 16, 5, -9], [-3, 19, 4, -19, 3, -8], [4, 0, 20, -20, 0, -4], [1, -10, 10, -11, 20, -11], [1, -9, 0, 8, 8, -6], [19, 1, -5, -5, 1, 19], [18, 4, -9, -9, 9, -4], [-11, 15, 15, 0, -15, 0], [-22, -22, 8, 10, 12, -8], [4, -3, 20, -2, -20, 3], [13, -4, 2, 13, 6, -6], [15, -13, 4, -19, 26, -19], [0, 15, -18, 1, -18, 15], [-12, 3, -15, -3, 12, 30], [0, 0, -5, 13, 13, -5], [12, -4, 12, -11, 2, -1], [9, -4, -10, 1, 8, 1], [3, -8, -23, 3, -15, 15], [12, 0, -12, -10, -21, 2], [-4, 0, 4, -11, 11, -15], [10, -9, 21, 11, 0, -11], [-20, 8, -20, 20, -8, 20], [0, 6, 5, -5, 5, 6], [3, -6, 6, -6, -10, -16], [-9, 8, -8, 18, -19, -11], [0, 4, 11, -29, 11, 4], [4, 17, 2, 13, -13, -1], [18, -14, 1, -17, 0, 17], [-5, 0, 5, -8, 9, -13], [10, -5, -1, -5, 1, 5], [-10, 7, 7, 0, -2, -5], [-17, -17, 19, 0, 17, -19], [9, 0, -9, -19, -11, -10], [-5, -19, -3, -5, 16, -16], [-5, -2, -8, 13, 4, 13], [24, -24, -5, -18, 6, -5], [-10, -1, -6, 1, 10, 12], [7, -21, 21, -14, 12, -9], [0, -9, 9, 3, 9, -9], [-12, -28, 2, -2, 28, 12], [-11, -3, -5, 16, 6, 16], [9, 9, 3, 9, -6, 6], [-8, -5, -11, 4, 11, 5], [-2, 8, 8, 0, -27, 19], [22, 18, 4, -11, -4, -18], [6, -11, 0, 5, 5, -13], [-14, -19, -30, 30, 19, 14], [-31, -18, 8, 18, 31, -16], [-12, 12, 2, 23, 11, 2], [-24, 12, -5, 24, -7, -7], [0, -8, -28, -8, 8, 14], [3, 0, 0, -3, -3, -6], [-13, 0, 13, 7, -4, -6], [3, -27, -7, -27, 3, 0], [1, 12, 12, 0, -11, -1], [-17, 0, 17, 25, -17, 8], [10, -10, -19, -19, -10, 10], [7, -5, 13, -7, -8, -8], [-4, 8, -8, 8, 14, 22], [-8, -3, -17, 8, 20, 20], [-19, -12, 0, 0, 12, 19], [4, 12, 8, 4, 0, -4], [-5, -4, 9, 9, -4, -5], [8, 0, -8, 15, -10, 23], [22, 13, -4, -11, 4, -13], [-7, 4, -32, 4, -7, 0], [9, -26, 0, 17, 17, 24], [-3, 17, -6, 20, -20, 3], [-21, -1, 6, 15, 2, 15], [13, 11, 0, -13, -11, -11], [-5, -19, 6, 19, 5, -12], [0, -5, 10, -24, 10, -5], [-22, 32, -22, 1, -16, 21], [-9, 33, -3, 3, -33, 9], [-2, -2, -8, -14, 16, 8], [-2, -3, -3, 0, -5, 8], [-10, -9, -28, 5, 28, 9], [4, -35, 0, 31, 31, 6], [20, -37, 10, -10, 37, -20], [-21, 8, -21, 5, -4, 16], [-31, -28, 4, 28, 31, -8], [19, 10, -13, -10, -19, 26], [-23, -20, -6, 3, -3, 3], [-15, -2, 0, 17, 17, -1], [8, 9, -9, 8, -18, 18], [21, 21, -25, -4, -17, 25], [-30, 14, 3, 14, -30, 0], [2, 0, -2, -10, 16, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_912_ci_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_912_2_ci_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_912_2_ci_b(:prec:=6)
    chi := MakeCharacter_912_ci();
    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_ci_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_912_2_ci_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_912_ci();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, 27, 36, 30, 18, 6, 1]>,<7,R![3, -18, 45, -54, 33, -9, 1]>],Snew);
    return Vf;
end function;