
// Make newform 945.2.a.i in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_945_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_945_2_a_i();".
// 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_945_2_a_i();".
// 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];
        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_945_a();"
function MakeCharacter_945_a()
    N := 945;
    order := 1;
    char_gens := [596, 757, 136];
    v := [1, 1, 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;

function MakeCharacter_945_a_Hecke(Kf)
    return MakeCharacter_945_a();
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, 4], [-3, 5], [6, -1], [1, 1], [5, 1], [1, -3], [3, 0], [-3, 0], [0, 1], [-9, -2], [7, -6], [6, -1], [7, -6], [3, 3], [6, -11], [7, -3], [-5, -4], [4, -11], [-7, 12], [11, -4], [8, -13], [0, 4], [3, 7], [5, 6], [-2, -3], [-13, 9], [-2, 11], [6, 7], [-12, -4], [-7, 4], [-8, 5], [11, -16], [-2, -2], [-14, 4], [-5, -2], [1, -10], [-5, -6], [17, -10], [17, 1], [-2, -9], [-11, 16], [6, 13], [0, -6], [-17, 4], [-4, -15], [-11, 10], [-21, -5], [11, -20], [-12, 11], [2, -14], [8, -4], [-12, -5], [17, -18], [20, 3], [-2, 11], [-18, 7], [14, -21], [16, -18], [-24, 12], [27, -13], [-13, -10], [19, -8], [9, -3], [17, -19], [4, 10], [29, -10], [-11, 25], [-5, -6], [2, -21], [11, 3], [-2, 16], [31, -4], [-14, -5], [-2, -20], [13, -1], [-2, -4], [17, -4], [-37, 3], [16, -7], [4, -5], [15, 12], [21, -21], [-26, 8], [-24, 13], [11, -19], [-5, 11], [20, 4], [2, 11], [2, 4], [-2, 11], [-29, 20], [-15, 11], [19, -4], [11, -12], [11, -19], [2, 19], [23, -10], [-25, 11], [16, -3], [17, -8], [32, -7], [-1, 18], [13, -17], [7, -26], [14, 20], [-17, -7], [-13, 12], [19, 0], [20, -13], [31, 0], [-32, 28], [-17, 33], [17, 1], [14, -8], [-9, -25], [16, -2], [-27, 7], [8, 8], [-23, -4], [-15, -3], [5, -14], [-5, -10], [14, 2], [2, 19], [-13, -19], [32, -9], [26, 10], [-8, 31], [2, 8], [12, -27], [13, 5], [5, 16], [-31, 11], [-12, 22], [-10, 9], [-9, 10], [-15, -11], [-11, 18], [11, -2], [23, -42], [13, -2], [-2, -17], [1, 0], [-19, -16], [-2, -26], [-12, 8], [-45, 14], [-30, -4], [3, -8], [30, 10], [19, -27], [32, -4], [4, -5], [-30, 21], [-46, -4], [-5, -14], [-4, 27], [-23, 38], [21, 22], [36, -12], [18, -32], [-25, 21], [-17, 8], [-1, 18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_945_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_945_2_a_i();".
// 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_945_2_a_i(:prec:=2)
    chi := MakeCharacter_945_a();
    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_945_2_a_i();".
// 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_945_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_945_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, -1, 1]>,<11,R![-19, -2, 1]>],Snew);
    return Vf;
end function;