
// Make newform 1694.2.a.z in Magma, downloaded from the LMFDB on 03 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1694_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_1694_2_a_z();".
// 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_1694_2_a_z();".
// 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 := [16, 4, -9, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, -5, -1, 1], [-5, -1, 1, 0]];
    Rf_basisdens := [1, 1, 4, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_1694_a();"
function MakeCharacter_1694_a()
    N := 1694;
    order := 1;
    char_gens := [969, 365];
    v := [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_1694_a_Hecke(Kf)
    return MakeCharacter_1694_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]];
    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 := [[1, 0, 0, 0], [-1, 1, 0, 0], [-1, -1, 1, -1], [-1, 0, 0, 0], [0, 0, 0, 0], [-4, 0, -2, 1], [2, -1, 2, 1], [-4, 1, -2, 1], [-2, 0, 0, 1], [-4, 0, -4, -1], [-2, 0, 0, -1], [-1, -1, -1, -3], [1, -1, 0, 1], [2, -2, -1, 0], [-5, 1, -7, -1], [2, 2, 2, 1], [-3, 4, -1, -1], [-7, -1, 1, -2], [2, 3, 0, 1], [-2, 2, 6, 1], [-4, -4, 5, -1], [-6, 0, 2, -3], [-4, 4, -5, 2], [4, 0, 3, 2], [7, -3, 4, 1], [-8, 0, -2, 1], [3, 5, -3, 2], [0, -4, -3, 0], [-1, 3, -1, 2], [6, -3, -2, -2], [3, 1, 5, 2], [0, 3, -2, 2], [1, 3, 4, 1], [-8, 0, 4, -2], [-3, 5, 5, 3], [2, 2, 4, 2], [-8, -2, -4, -5], [9, -2, 3, -4], [15, -1, 9, 4], [-2, 2, -4, -3], [-1, 0, 5, -3], [-2, -4, 2, 2], [9, -3, 11, 0], [-4, 2, -10, 1], [-1, -1, 5, -1], [-12, 0, -6, 2], [-7, -1, -6, -4], [-11, 5, 3, 4], [3, -6, 5, -1], [-7, -3, -3, -2], [-3, -1, -4, -7], [1, 1, -7, 5], [-5, -2, 11, -2], [-4, -4, 4, 2], [3, -3, -6, 2], [-7, 3, -3, -7], [17, -3, 7, -3], [-9, 5, 1, 4], [-12, 0, -6, 2], [4, -8, 7, -5], [-4, 0, 12, 1], [13, 1, -3, 0], [9, -1, 0, -6], [2, -4, 2, 3], [-13, 8, -17, 0], [-4, -4, -8, 1], [-5, -3, 2, 3], [5, -4, 5, -9], [-4, -1, 6, -5], [0, -8, 4, -4], [1, 0, -7, 3], [-7, 1, -11, 2], [0, -8, 20, -5], [14, -2, 26, -1], [7, -7, 6, -6], [-9, -1, 7, -5], [-23, 3, 1, 1], [2, -2, 12, 1], [21, 3, 0, 2], [-10, -4, -12, 3], [6, 6, -7, 5], [0, -2, 2, -3], [-4, -2, -4, -7], [4, -1, -4, -3], [-10, -2, -14, 4], [-18, 6, -15, 1], [7, 10, -17, 5], [-9, 9, -4, 3], [-7, 5, -7, 7], [9, 3, 19, 4], [0, -8, 4, -1], [4, -2, 12, -2], [8, -10, 10, 3], [-8, 4, -11, -7], [5, -1, 6, 4], [12, 6, 20, -3], [-2, 4, -18, 4], [-17, 8, -5, -2], [4, 9, -26, 5], [-10, 12, -6, -2], [-6, -5, 8, -9], [-22, 2, -14, 3], [3, 9, 14, -1], [-13, 4, -9, -3], [0, 8, -4, 7], [-1, -2, -17, 5], [6, -15, 12, -1], [14, 1, 10, -6], [4, 4, 12, 1], [-11, -1, 22, 0], [5, -15, 3, -4], [5, 5, -15, 9], [28, 3, -4, 7], [-9, 4, -3, 0], [7, -7, 1, 4], [5, 7, -4, 16], [-9, 1, 6, 6], [6, 4, 4, -1], [-6, 8, 4, 2], [-5, -12, 9, -3], [-3, 1, -5, -3], [12, -13, 4, 3], [10, 0, 8, -2], [-13, -7, -5, 7], [-19, -2, -9, -8], [-18, -2, -36, 0], [-3, 11, -17, 6], [7, -1, -3, -10], [-30, -2, -14, -5], [15, -3, 9, -3], [-2, -5, 8, -4], [-2, -4, -8, -5], [8, -2, -10, 0], [13, 7, 19, 4], [-8, -7, 8, -6], [1, 1, 15, -2], [-4, 4, -20, 6], [17, 2, -1, 12], [-24, -4, -20, 4], [10, 1, 14, 1], [16, 0, 11, 0], [20, 4, -2, 6], [22, 0, 30, -1], [-8, 17, -2, 10], [4, -6, 2, 1], [-1, 1, 17, 0], [-12, -2, -16, -11], [-9, -7, -14, -6], [-4, 7, -2, 17], [-13, -13, 17, -9], [-6, 6, -4, -9], [9, -5, 16, -10], [-13, 9, -28, -1], [-23, -3, -29, 2], [3, 4, -7, 2], [13, 3, -9, 8], [2, -8, 6, 1], [10, -3, 2, -8], [-12, 11, 12, 11], [11, 3, 21, 7], [-7, -4, -5, -4], [24, 4, 17, 5], [-11, -11, -25, -1], [7, -3, 7, 5], [-32, 14, -26, -4], [4, -16, 2, 0], [-28, 10, -20, -4], [17, 1, 9, -1], [-7, -5, 6, 7], [-6, 18, 2, 2], [43, -3, 18, 11], [16, -2, 34, 0], [-11, 13, 3, -1], [24, 5, 28, 2], [12, 0, 0, 12], [17, 1, 7, -10], [-19, -5, -4, 4], [-3, -9, 13, -1], [23, -7, 19, -6], [22, 2, 20, -14], [20, 6, 2, 15], [-33, 1, -37, -6], [-28, -6, -18, -7], [3, -9, -7, -1], [23, -9, 3, 0], [1, 1, -9, 2], [-27, -3, -15, 4], [19, -23, 10, -4], [8, -5, -16, 7], [28, -4, 20, 0], [-25, 10, -1, 2], [9, 0, -13, 1], [-26, -3, -20, -6], [-5, -9, -15, -12], [-25, 1, -13, 12], [-12, 10, -14, 3], [-22, -2, -19, -3], [0, -2, 12, 10], [-8, 10, 14, 8], [19, -13, 5, 1], [-22, -8, -22, 9], [-3, -11, 9, 1], [24, -14, 28, -2], [27, -7, 28, -3], [33, 11, 6, -3], [-22, 0, -28, 1], [-24, 0, -18, -5], [-6, 3, 0, 2], [-30, 0, -16, -14], [15, 9, -6, 14], [-20, -1, -8, 7], [-7, -13, -1, -21], [-18, -16, -6, 1], [-27, 11, -4, -6], [11, -9, -15, -3], [-25, 9, -4, -3], [8, 12, -8, 19], [-2, -2, 5, 4], [-33, 1, -17, 2], [-11, 9, -13, 18], [-50, -2, -10, 5], [16, 18, -28, 9], [4, -6, -10, -1], [46, 2, -8, -1], [-36, 0, 8, 5], [-20, 12, -6, -6], [16, 17, 4, -5], [14, 20, -16, 6], [-4, -2, 2, -14], [23, -11, 20, -3], [27, -7, -1, 4], [5, -1, -34, 2], [8, 4, 26, 1], [-8, -6, 10, 4], [-15, -1, -18, -3], [27, 11, -19, 1], [-21, 21, 8, 4], [22, -6, -6, -3], [27, -11, 0, 5], [0, -4, -4, 12], [-18, 1, -24, -5], [-19, -1, -10, 13], [-40, 10, -10, 6], [-68, 6, -12, -3], [22, 9, 18, -3], [54, -8, 28, 7], [40, 0, 2, -10], [-13, 5, -20, 12], [-7, -1, 9, -21], [34, 0, 12, 1], [-45, 13, -35, -7], [2, 24, -36, 9], [-34, 8, -30, -5], [-11, 16, -11, 0], [-18, 6, -4, -13], [34, -17, -8, -15], [-46, -2, -12, -4], [1, 8, 7, 4], [54, -4, 0, 1], [16, -11, 4, -3], [19, -13, -9, -1], [8, 12, -12, 10], [7, 19, -7, 17], [-3, 11, -23, 12], [17, 9, -9, -6], [13, 11, 18, 16], [21, 9, 7, 4], [26, -5, -14, -2], [-17, 0, -11, 4], [-1, -5, 9, 12], [-19, 15, 15, 9], [23, -3, -2, 2], [-25, 6, -17, -13], [-29, 15, -11, -3], [26, -15, 30, -8], [-35, -1, 3, -9], [4, -15, 30, -1], [25, -9, -7, 5], [-4, -23, 4, -4], [3, 13, -1, -5], [-37, 7, -7, 1], [49, -9, 17, 9], [-11, 11, -15, 10], [24, -4, 4, -7], [0, 11, 14, -1], [-25, 19, -1, 7], [4, 1, -4, -4], [-36, -6, 6, 0], [2, 14, -24, 2], [-28, -2, -6, -25], [-1, 1, 5, 2], [32, -8, 5, 11], [-15, 19, -32, 2], [-11, -12, 23, -6], [-2, 10, 8, -8], [38, 0, 24, 15], [-38, 0, 14, -7], [5, 3, 15, -11], [10, 2, 17, -6], [-10, 30, -11, 2], [15, 13, -22, -1], [60, 2, 18, -3], [10, 14, 12, -7], [11, -11, 8, 2], [22, -1, 24, 7], [38, 6, 5, 19], [-19, -9, 10, -16], [5, -9, 43, 1], [-46, 2, -34, 3], [-34, 6, -18, -21], [18, 6, 10, -3], [8, -8, 34, -7], [-8, 3, -28, 10], [16, 9, -30, 5], [4, -20, -4, -8], [-13, 9, 16, 5], [-4, 0, 7, -10], [-48, -6, -18, -5], [16, 8, 35, 4], [-2, -14, 1, -22], [8, 0, 23, 3], [5, -33, 16, -5], [-23, -11, -15, -8], [9, -13, 45, -8], [4, -5, 36, -10], [-28, 12, 13, 15], [26, -22, -6, -7], [-19, -1, -11, 2], [-36, 16, 0, 5], [10, 16, 8, 2], [-2, 0, -18, -1], [-2, 18, -10, 21], [14, 4, 14, 5], [-43, 7, -8, -4], [56, 1, 2, -3], [3, -15, 11, -6], [-14, -6, 26, 4], [30, -3, 46, 6], [-1, -2, -29, -10], [32, 6, -8, 5], [-5, 19, 23, 13], [16, -24, -17, -19], [-56, -4, -6, -2], [-26, 12, -34, 11], [-20, 10, -16, 21], [20, -8, -4, 11], [-12, 24, -20, 13], [-40, -8, -24, 10], [-31, -9, 19, -11], [-29, 7, -41, 12], [3, 18, 25, 18], [41, -9, 20, 5], [13, -11, 7, 3], [-11, -5, -43, -5], [15, 1, 45, 1], [32, -4, 37, 1], [11, 1, -21, 18], [-20, 1, -42, -1], [35, 0, 19, 17], [-7, -11, 39, -7], [39, 7, 11, 12], [14, -12, 40, -13], [-18, 12, 0, -14], [-37, 6, -3, 2], [-1, 1, -17, -11], [9, -5, -10, 13], [-19, 7, -1, 12], [-8, 32, 6, 7], [48, -12, 9, -3], [27, -7, 54, -3], [-19, -25, 1, -1], [-5, 23, 17, 22], [8, 10, 12, -1], [35, 1, 19, 4], [-12, -22, -2, 11], [-51, -2, -49, -8], [22, -14, 50, 1], [14, 17, -26, 15], [-25, 21, -10, 4], [-73, 9, -38, -15], [42, 14, -16, 0], [2, 2, 13, -10], [14, -18, 36, -20], [-3, -11, 31, -14], [20, -12, 3, -11], [-12, -2, 48, -1], [-18, 0, 4, 4], [-17, 7, -41, 9], [-22, -9, 44, -5], [-22, 14, -24, 11], [-36, 3, 24, 4], [20, 14, 40, 18], [-46, 6, -29, -5], [1, -29, 5, -15], [18, -28, 12, -18], [-22, 8, -32, -6], [-7, 5, -19, 6], [-8, 11, -20, 6], [13, -17, 24, 9], [-13, 11, 23, -9], [-31, -5, -53, 11], [23, -19, 18, -8], [-27, -5, 1, -11], [30, -11, 2, -9], [-12, -8, 20, -13], [-24, -10, -10, 1], [1, -11, -15, -20], [29, -38, 1, -12], [35, -15, 11, -3], [-17, 2, -23, 2], [53, 6, 29, -13], [20, -2, -2, 0], [7, -15, 3, 16], [-50, -19, -8, 9], [70, -18, 29, 13], [1, 21, 5, 16], [-11, -1, -45, -8], [14, 10, 14, -9], [40, 15, -8, 10], [-9, -23, -5, -12], [-38, 2, -18, 2], [25, -21, 3, 1], [22, 18, 44, 6], [54, 10, 41, -11], [20, -20, 29, -5], [3, -9, 31, 5], [-10, 6, 7, 3], [-18, -9, -22, 1], [43, -27, 14, 0], [16, -6, 32, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1694_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1694_2_a_z();".
// 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_1694_2_a_z(:prec:=4)
    chi := MakeCharacter_1694_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(2999) - 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_1694_2_a_z();".
// 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_1694_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1694_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![11, -13, -6, 3, 1]>,<5,R![-20, -40, -5, 5, 1]>,<13,R![-284, -26, 51, 14, 1]>],Snew);
    return Vf;
end function;