
// Make newform 5775.2.a.bf in Magma, downloaded from the LMFDB on 30 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5775_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_5775_2_a_bf();".
// 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_5775_2_a_bf();".
// 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, -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_5775_a();"
function MakeCharacter_5775_a()
    N := 5775;
    order := 1;
    char_gens := [3851, 4852, 4126, 3676];
    v := [1, 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_5775_a_Hecke(Kf)
    return MakeCharacter_5775_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], [0, 0], [-1, 0], [-1, 0], [1, -1], [0, -2], [-4, -1], [3, -2], [0, 4], [5, 2], [4, -5], [9, 0], [-2, 4], [-3, 4], [-9, 2], [0, 1], [-7, 2], [-2, -2], [3, -6], [-11, 4], [11, 0], [6, 1], [-9, 0], [-2, -1], [-3, 8], [-8, 3], [3, 6], [-4, 1], [-6, 9], [-11, -3], [9, -10], [-15, 4], [8, -2], [3, 5], [-16, -3], [-5, -8], [13, 2], [21, 1], [-3, 3], [-6, -3], [-19, 2], [-12, -5], [-5, 11], [3, -3], [2, 6], [-7, -4], [-5, 14], [3, 4], [-1, -1], [-12, 3], [-6, -1], [2, -8], [21, 0], [-18, -3], [-24, -1], [30, -1], [14, -2], [7, 3], [-9, 0], [19, -3], [0, 1], [1, -4], [15, 0], [-8, -3], [-27, 1], [-22, 1], [-29, 4], [0, 2], [-13, 12], [-12, 6], [3, 6], [-5, 1], [22, -12], [11, -9], [-18, 8], [-6, -6], [25, 0], [-15, -3], [-4, -9], [3, 12], [-13, -1], [18, -11], [7, -11], [-31, -1], [3, -3], [15, -6], [16, -7], [6, -14], [-17, 14], [-9, -1], [-12, 17], [-2, -3], [-3, 9], [-4, -6], [0, -4], [0, 4], [21, -2], [10, -12], [17, -20], [25, 2], [-12, 1], [3, -4], [6, -6], [8, 7], [-5, 13], [-15, 3], [12, -15], [21, -19], [-31, -2], [-23, -6], [-17, 20], [-18, 12], [14, -14], [-4, 13], [-24, 12], [7, 1], [3, -18], [12, -23], [9, -3], [8, 0], [7, -6], [21, -16], [24, 6], [11, -2], [-9, -8], [-10, -13], [-42, 2], [-38, 1], [4, -6], [-34, 0], [-21, -3], [26, 12], [-20, 15], [36, -12], [5, 6], [-15, 18], [-14, 4], [-39, 9], [-3, -6], [-4, 16], [3, 6], [-23, 15], [-33, 12], [20, -11], [15, 0], [-2, 5], [30, 3], [2, -9], [18, -12], [-2, 6], [-9, -18], [1, 13], [42, -4], [25, 10], [9, 6], [-40, 7], [6, -8], [-26, 10], [27, -12], [3, -23], [9, 20], [10, -17], [42, -4], [-18, -11], [3, 1], [-46, -2], [28, 0], [-10, 16], [27, 4], [-3, -22], [-1, -27], [12, -11], [28, 4], [-1, -11], [-15, -7], [26, -3], [-12, 15], [-38, 0], [8, 5], [19, -3], [18, 1], [-5, -4], [3, -19], [0, -26], [-33, 1], [25, -24], [-53, 6], [-19, 18], [15, -8], [-11, -12], [15, 16], [-1, -9], [-33, -7], [-36, 0], [-18, 3], [-28, -11], [28, -2], [-9, -20], [27, -17], [-18, 15], [29, 14], [-44, 7], [26, -13], [6, 5], [9, -26], [-34, 7], [24, 1], [-9, -5], [26, -17], [-8, 31], [9, -14], [-32, 1], [0, 17], [0, -9], [17, 12], [-8, -12], [-3, 15], [-39, -7], [-39, -5], [-43, -1], [-43, -6], [36, 3], [19, -17], [15, 18], [-28, 6], [15, -16], [21, 4], [1, -12], [-18, -18], [-35, 12], [-16, 0], [-31, 23], [-3, -11], [49, -7], [-6, -23], [23, -2], [39, -14], [15, -9], [42, 3], [18, -5], [5, -14], [-8, 16], [-10, -10], [15, -11], [-3, 32], [-20, 31], [0, 18], [-31, 25], [15, -38], [-50, 21], [15, -6], [3, 15], [14, 20], [-3, 6], [12, 16], [26, -26], [-5, 4], [21, 1], [-8, 28], [-26, 19], [-33, 10], [-31, 8], [10, 21], [57, -4], [-13, 8], [-15, 0], [-12, -1], [43, -11], [-24, -5], [2, 22], [-53, 15], [13, -26], [-13, 3], [10, 2], [-53, 4], [3, -13], [-31, 15], [26, 8], [-9, 20], [-30, -6], [11, 20], [21, -21], [-46, 15], [-38, -17], [30, 14], [-32, -16], [21, -18], [-37, 35], [-21, -16], [12, -12], [-18, 24], [-18, 13], [-63, 6], [13, -18], [15, -9], [68, 8], [-24, -13], [-9, 0], [7, 24], [16, 27], [-36, 6], [29, 14], [-15, 0], [-28, 7], [-5, 20], [45, -28], [2, -30], [-12, 5], [25, 26], [0, 30], [36, -21], [-12, -6], [43, -13], [33, 7], [-19, -13], [15, 11], [-18, 42], [37, -17], [18, -9], [29, 3], [-65, 19], [-15, -21], [28, -23], [27, -35], [50, 6], [56, 1], [1, -35], [-3, 36], [-30, -12], [-13, -31], [57, -11], [23, -28], [48, 12], [-1, 30], [-42, 9], [29, -24], [-42, 12], [-31, 12], [10, 1], [-29, 15], [-57, 21], [36, -14], [-13, -28], [-12, 30], [-42, 25], [26, -4], [4, -35], [-33, 40], [-57, -12], [-31, 32], [-32, -5], [27, -12], [31, -9], [11, -16], [48, -15], [15, -36], [30, 21], [-6, -8], [-9, -16], [10, -43], [48, 6], [-48, -6], [48, -24], [-8, 27], [-8, -20], [-9, 30], [1, 28], [-28, -19], [-39, 6], [5, -7], [39, -27], [-6, -7], [8, -47], [-20, 0], [-9, 9], [24, -18], [-35, -8], [12, -25], [-59, -12], [-3, 12], [48, -40], [52, 13], [-24, 18], [26, -13], [-12, -27], [-40, 18], [43, -8], [4, -37], [6, 6], [26, 18], [0, 32], [-24, 12], [-5, -26], [-42, 23], [-41, -2], [-13, -38], [72, 5], [-55, -20], [66, 5], [50, -27], [-84, 0], [55, -8], [-33, 16], [-18, 27], [-19, -16], [-35, 26], [-27, 8], [-68, 6], [33, -30], [34, -29], [30, 28], [3, 9], [23, -16], [-17, 6], [-3, 17], [-3, 37], [-11, 40], [57, -28], [3, 12], [54, 9], [-20, 15], [-18, -11], [-6, 1], [-44, 31], [-42, 12], [21, 28], [-3, -27], [35, -4], [-33, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5775_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5775_2_a_bf();".
// 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_5775_2_a_bf(:prec:=2)
    chi := MakeCharacter_5775_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_5775_2_a_bf();".
// 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_5775_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5775_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 1, 1]>,<13,R![-3, -1, 1]>,<17,R![-12, 2, 1]>],Snew);
    return Vf;
end function;