
// Make newform 1127.2.a.d in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1127_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_1127_2_a_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_1127_2_a_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, -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_1127_a();"
function MakeCharacter_1127_a()
    N := 1127;
    order := 1;
    char_gens := [346, 442];
    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_1127_a_Hecke(Kf)
    return MakeCharacter_1127_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], [1, 0], [2, -2], [0, 0], [2, -4], [1, 2], [0, 0], [6, -2], [-1, 0], [1, 4], [9, 0], [-2, 6], [1, -2], [0, -4], [1, -4], [10, -2], [4, 4], [-6, 12], [-6, 10], [-9, 2], [3, -6], [-6, 2], [-4, 4], [-4, 8], [0, 6], [6, -12], [12, -6], [10, -2], [-8, 12], [0, -4], [1, 6], [1, 8], [12, -6], [-5, -4], [14, -2], [-9, 10], [-8, 12], [13, -10], [-12, 16], [6, 4], [-15, 6], [10, 8], [-18, 10], [-3, -12], [-19, 4], [-6, 16], [-12, 0], [4, 12], [-4, -16], [-16, -2], [-9, 4], [-17, -2], [14, -12], [-12, -6], [-3, 2], [-22, 8], [3, -10], [12, -8], [13, -12], [0, -6], [-14, 8], [12, -6], [4, -12], [-7, -16], [-4, -2], [14, -28], [-3, -6], [-4, 22], [4, 16], [-5, -6], [3, 2], [6, 2], [12, 12], [26, 0], [12, 6], [12, -6], [0, 12], [17, -14], [-10, 16], [7, 18], [-24, 4], [-16, -12], [12, -10], [-14, 0], [-13, 28], [-1, 10], [18, -24], [12, -4], [27, 2], [0, 0], [14, 4], [12, -18], [9, -18], [13, 2], [-33, 6], [2, -22], [19, 10], [-4, 28], [4, -6], [21, 0], [37, -10], [-28, -2], [-4, -22], [-2, 10], [20, 6], [19, 6], [-11, 16], [-22, 20], [0, 4], [-21, 2], [16, -28], [-16, 10], [34, -2], [24, -38], [-4, -12], [12, 8], [14, -6], [21, 4], [27, 4], [8, 2], [38, -6], [-1, 0], [-22, 2], [1, 10], [-28, 24], [-24, -2], [12, -36], [-36, 0], [-2, 2], [-8, 8], [5, -6], [-28, -4], [-16, -12], [-14, 24], [23, -14], [42, -2], [-32, 14], [-12, -20], [38, 2], [-18, 24], [33, -20], [30, 12], [7, -30], [0, 28], [14, 0], [-4, 26], [-30, 24], [9, 30], [-19, -12], [1, -2], [-42, 0], [-12, 22], [0, -16], [-7, 16], [-6, -30], [-2, 32], [0, -6], [-43, 14], [46, -18], [-32, 34], [-9, 22], [22, 16], [-15, 26], [-30, 6], [-10, 14], [-2, -10], [-44, 8], [10, -28], [42, -30], [29, -2], [-10, -10], [-9, 18], [12, -12], [48, -2], [23, -8], [0, -18], [-15, -2], [-9, -12], [-38, 6], [-36, 24], [-7, 6], [44, -8], [21, -20], [14, -20], [-34, 40], [-32, 8], [-6, 20], [42, -32], [29, 4], [-4, -16], [17, -30], [-16, -16], [-54, 2], [30, -32], [10, -22], [-12, -12], [22, -22], [-8, 30], [20, -22], [-33, 44], [-12, -14], [24, -24], [7, -10], [-30, 4], [-4, 14], [43, -14], [-6, -18], [-21, 18], [27, -24], [-29, 32], [-29, 24], [-59, -2], [-12, 30], [50, -10], [-12, 24], [-36, 2], [3, -18], [11, 14], [2, 26], [-21, 0], [3, 32], [-6, -18], [9, -48], [-20, 6], [-3, 12], [15, 24], [6, -24], [5, 26], [-30, 6], [-24, -24], [62, -20], [64, 2], [10, -40], [22, 8], [12, -22], [-44, 26], [-24, 0], [-6, -16], [-33, 38], [3, -12], [-12, -14], [21, -20], [21, 0], [-33, 8], [46, 4], [-11, 32], [-59, 8], [-22, 40], [-22, -18], [-30, 4], [0, 38], [46, -28], [22, 2], [0, -30], [6, 0], [21, -30], [14, 22], [14, 0], [-37, 38], [31, 6], [40, 6], [56, -10], [50, -20], [48, 0], [18, -36], [46, -12], [-42, 24], [-40, -10], [14, -50], [-3, -36], [19, -30], [10, 20], [4, 2], [-28, 10], [-5, 38], [-3, 4], [-16, -32], [-6, -24], [-18, -36], [-8, -16], [-57, 16], [-50, -6], [-14, -14], [2, -12], [52, -12], [-24, 8], [10, 30], [54, -28], [32, -24], [-2, 4], [66, -16], [-32, 26], [-18, 0], [22, -20], [13, 12], [30, 28], [24, -16], [-15, 58], [-1, 32], [15, -48], [28, -64], [-30, 24], [12, -6], [-43, -2], [-16, 10], [15, 8], [-12, 48], [24, 30], [58, -4], [-18, -24], [-31, 44], [-42, -18], [-4, 4], [49, -46], [26, 4], [0, 36], [7, -4], [-37, 22], [24, 16], [7, -56], [-32, -18], [-28, 30], [17, 2], [-72, 12], [34, 4], [48, 24], [-30, 0], [35, -60], [54, -24], [12, -44], [49, -60], [-33, 68], [-17, 24], [19, -32], [-6, 24], [15, -28], [-58, -18], [56, -4], [29, -38], [-20, 24], [27, 12], [-32, 4], [30, -44], [-54, -12], [74, -10], [-36, 20], [-55, -6], [20, -24], [-42, 0], [-54, 36], [-27, 0], [5, 24], [-70, 16], [-6, 48], [36, -4], [14, -16], [48, -12], [-6, 26], [9, 8], [-12, 12], [2, -50], [-25, -26], [53, -62], [-18, 64], [-69, 12], [-59, -16], [61, -38], [26, -32], [-30, -30], [73, -38], [12, -24], [11, 34], [-8, 32], [0, -34], [-29, 32], [-1, 32], [38, -2], [-6, 56], [-40, -14], [25, -54], [30, -46], [6, 28], [20, 36], [-9, -36], [-40, 30], [3, -48], [1, -58], [-6, 48], [54, -34], [-4, 14], [24, 12], [-42, 12], [47, -62], [-24, -4], [36, -36], [34, -72], [44, 12], [44, -14], [4, -18], [38, -64], [-7, -2], [-35, 16], [12, -8], [-12, 18], [-83, -10], [-7, 24], [46, -4], [13, -16], [40, 26], [-64, -16], [81, -30], [-1, -28], [-22, -22], [26, -36], [-54, -18], [-53, 10], [-16, 28], [-12, -24], [12, 30], [-60, 6], [12, -30], [-12, -42], [48, 8], [24, 0], [-21, 34], [-19, -22], [6, -60], [26, -32], [-3, -12], [-47, 76]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1127_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1127_2_a_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_1127_2_a_d(:prec:=2)
    chi := MakeCharacter_1127_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_1127_2_a_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_1127_2_a_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1127_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1, 1]>,<3,R![-1, 1]>],Snew);
    return Vf;
end function;