
// Make newform 1088.2.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1088_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_1088_2_a_p();".
// 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_1088_2_a_p();".
// 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 := [-3, 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_1088_a();"
function MakeCharacter_1088_a()
    N := 1088;
    order := 1;
    char_gens := [511, 69, 513];
    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_1088_a_Hecke(Kf)
    return MakeCharacter_1088_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, 0], [-1, 1], [0, -2], [-1, 1], [3, 1], [-2, 2], [-1, 0], [-2, -2], [-3, -1], [0, 2], [-1, -3], [-8, -2], [-6, 0], [-2, -6], [0, 4], [-6, 4], [-6, -2], [4, 2], [-8, 4], [-3, 3], [2, 0], [-7, 3], [6, 2], [6, 2], [2, 4], [6, 2], [-4, 4], [-9, 1], [4, -6], [6, -8], [-10, -6], [3, 5], [6, 2], [13, -1], [-6, 0], [-10, 2], [-2, 0], [-5, 9], [-3, -5], [12, -2], [-18, -2], [-8, -2], [12, 0], [14, -4], [0, 2], [-1, -3], [-11, -5], [2, -6], [15, -3], [10, -6], [6, 0], [0, -12], [2, 0], [0, 4], [-18, 6], [-6, -10], [24, -2], [8, 4], [4, 6], [-6, 8], [-5, 9], [-30, 0], [4, -12], [9, 7], [14, -4], [-12, -2], [-2, 10], [14, 12], [15, -11], [10, 12], [-6, 8], [-6, 10], [11, 5], [-2, -6], [7, 1], [-6, -2], [-6, 18], [16, 6], [18, -8], [-10, -16], [9, -9], [-2, -14], [9, -5], [14, 6], [-19, 3], [0, -12], [6, 0], [-10, 6], [-6, 12], [-16, -8], [-18, -2], [-9, 5], [-13, -11], [6, -18], [1, 15], [15, -7], [-6, -16], [-6, 0], [-8, -12], [-8, 6], [-41, 1], [30, 2], [18, -10], [6, 8], [1, 7], [-10, 2], [-18, 6], [-18, -8], [0, 16], [-10, -12], [-37, 1], [34, -8], [6, 24], [-17, -3], [38, -2], [-6, 4], [-17, 9], [24, 4], [0, -6], [-36, 4], [22, 4], [14, 8], [0, -14], [27, -11], [31, -3], [6, -6], [-8, -14], [21, -13], [-16, -12], [-2, -12], [-2, 6], [-9, -19], [17, -1], [-14, 2], [6, -6], [-10, -6], [18, 10], [-11, 7], [-18, -12], [6, 8], [1, -21], [0, 26], [-7, -5], [-3, -13], [-26, 8], [27, 13], [16, 14], [-18, -20], [10, -10], [24, 8], [-20, -6], [-18, 12], [-20, 0], [27, 9], [-5, 17], [-21, -3], [-52, 0], [18, 16], [14, 8], [-24, 10], [-51, -1], [-18, -14], [38, 6], [18, 22], [-42, 0], [-3, 19], [5, 3], [-32, -6], [14, 8], [0, -2], [0, 0], [-2, -6], [21, -9], [-10, -14], [2, 14], [-6, -16], [-11, 15], [24, -10], [-22, -6], [10, -6], [-40, 0], [9, 19], [-20, -6], [-6, -32], [-6, 22], [-42, 10], [-20, 2], [-8, 0], [-10, 16], [27, 1], [14, -4], [-39, -1], [-14, 2], [-6, 2], [-39, -5], [-18, -4], [2, 20], [-32, -6], [30, 20], [12, -20], [24, -10], [29, -9], [-14, 16], [38, 14], [36, -4], [18, 10], [-4, -20], [-6, -22], [18, 8], [4, 20], [26, 4], [18, -14], [5, 15], [-42, 6], [-15, -13], [-34, -20], [-4, -12], [18, -24], [-39, -5], [30, 2], [-2, 16], [-31, -9], [-42, 8], [-19, 27], [-12, 16], [-2, -22], [-6, -24], [-33, -15], [-34, 10], [-3, -29], [-50, 4], [31, -15], [38, -18], [54, 6], [16, 4], [6, -34], [-10, -12], [0, -18], [27, 21], [-18, -2], [21, 3], [-32, 8], [20, 0], [-14, 18], [30, -8], [-15, -1], [23, 5], [-3, 3], [34, 6], [18, 2], [-2, 0], [-6, 20], [-18, -6], [2, -40], [66, 2], [12, 0], [40, 2], [-17, 9], [24, -14], [-34, 6], [-55, 15], [-12, -20], [-8, 34], [16, -14], [-6, -4], [-20, 24], [54, 12], [-18, -16], [-5, 33], [-42, 16], [-32, -6], [40, -16], [-34, 0], [26, -22], [-10, -24], [14, 22], [6, -6], [-26, -6], [-10, -6], [18, -10], [-60, 0], [41, -9], [-3, -9], [-38, 4], [-5, -27], [-12, -28], [14, -24], [36, 2], [-10, 2], [6, 14], [0, 34], [-9, -23], [54, 14], [63, 5], [-56, 6], [60, -14], [20, 12], [30, 26], [-57, -11], [10, -2], [14, -6], [-30, -6], [5, 39], [-15, -5], [-17, 25], [38, 0], [24, -8], [16, -34], [-24, 16], [-2, -16], [-63, -13], [-48, -10], [-18, 32], [10, -6], [-24, 16], [38, 0], [6, 6], [33, 31], [-22, -12], [-42, 0], [-17, 13], [-22, 24], [-30, 24], [32, -24], [-30, 24], [-34, 24], [25, 3], [-17, -27], [33, 15], [0, 2], [16, 26], [24, 6], [29, 27], [-48, 8], [49, 15], [-51, -5], [10, 28], [-18, 16], [-58, 0], [50, -6], [-2, 12], [66, -8], [0, -26], [-4, 12], [-18, -14], [-3, 19], [52, 6], [4, 0], [3, 17], [-24, 22], [58, 6], [26, 28], [-42, -24], [-31, -33], [-14, 2], [30, 2], [18, -38], [-3, -1], [30, -12], [-18, -30], [28, -6], [-66, -8], [-12, 16], [-27, 27], [-38, -30], [2, -48], [24, 18], [-16, 12], [-10, -40], [66, -18], [19, 1], [45, -1], [6, -46], [-52, 0], [28, -26], [-3, -41], [27, -35], [38, -26], [-6, 0], [26, -32], [0, -34], [-18, -40], [-49, 13], [-6, -12], [13, -29], [21, 7], [-58, -6], [-14, -22], [13, 39], [-12, 20], [14, 24], [0, -14], [-84, -4], [64, 0], [-42, 18], [-22, -16], [80, -4], [6, 16], [-59, -21], [-42, -6], [28, 18], [54, -14], [8, 0], [-6, 16], [42, 16], [-37, -15], [-38, -14], [-18, 2], [-26, 18], [9, 27], [-34, 36], [-6, -46], [72, -8], [-5, -43], [14, 16], [36, 2], [-33, -31], [-61, -7], [30, 40], [-24, 28], [42, 12], [28, 30], [9, -41], [30, 14], [-10, 36], [6, -48], [-51, -33], [-6, 8], [-44, 24], [-33, 25]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1088_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1088_2_a_p();".
// 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_1088_2_a_p(:prec:=2)
    chi := MakeCharacter_1088_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_1088_2_a_p();".
// 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_1088_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1088_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 2, 1]>,<5,R![-12, 0, 1]>,<7,R![-2, 2, 1]>],Snew);
    return Vf;
end function;