
// Make newform 5610.2.a.cf in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5610_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_5610_2_a_cf();".
// 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_5610_2_a_cf();".
// 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 := [6, 4, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [-1, -4, 0, 1], [-7, 0, 2, 0]];
    Rf_basisdens := [1, 1, 1, 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_5610_a();"
function MakeCharacter_5610_a()
    N := 5610;
    order := 1;
    char_gens := [1871, 3367, 1531, 3301];
    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_5610_a_Hecke(Kf)
    return MakeCharacter_5610_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 := [[-1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, -1, 0], [-1, 0, 0, 0], [2, -1, 0, 0], [-1, 0, 0, 0], [1, 1, 0, -1], [0, 0, 1, -1], [1, -1, 1, 0], [1, -1, -1, 0], [2, 0, 0, 0], [1, 0, 0, 1], [1, 2, -1, 0], [0, 0, 0, 0], [-1, 1, -2, 1], [0, -1, 0, 0], [2, -2, 0, 0], [5, -2, 2, 1], [-4, 1, -4, 0], [4, 1, 2, 0], [5, -1, 0, 3], [2, 0, 0, 2], [4, 0, -2, 0], [2, 1, 1, -1], [-5, 2, -2, -3], [3, -1, -1, 2], [3, 1, -3, 0], [2, 2, 2, -2], [-3, 1, 0, -3], [0, 0, 6, -2], [2, 0, 0, 2], [3, 1, 5, -2], [3, 1, 3, -2], [-1, -2, 0, -1], [6, -2, 0, -2], [9, 0, -2, 3], [-3, 1, -1, -4], [6, -2, 2, 4], [6, 0, 0, 0], [0, 1, 0, 0], [1, 1, 1, 4], [9, -3, 3, 0], [4, 3, 2, 0], [10, -2, 2, 2], [6, 0, 0, -2], [3, -1, 1, 0], [3, -3, 3, -2], [3, 3, 1, -4], [-6, 0, 2, 2], [5, -1, 1, 4], [14, 0, 2, 0], [0, 6, -1, -1], [-4, 3, -2, -2], [9, -5, -1, -2], [-5, -3, 3, -2], [-10, 0, 0, -4], [5, 1, 1, 2], [8, -2, -4, -2], [1, 1, 3, -2], [12, -2, 2, 2], [9, -5, -1, -2], [10, -1, -2, 4], [-4, -1, -6, 2], [-4, -1, 7, -1], [9, 3, -7, 4], [-1, 1, 5, 4], [0, 1, -2, 0], [-6, 2, -8, 2], [3, -3, -8, 3], [-1, 5, -3, 2], [0, 4, -2, 2], [13, -4, 8, 3], [-6, 1, -2, -6], [-14, 0, 0, 2], [-4, -2, -4, 0], [-6, 3, -2, 2], [16, 4, 4, -2], [-13, 4, -1, 0], [2, 0, 2, -2], [3, 1, 1, -4], [4, -2, 2, 0], [2, 3, -1, 3], [-16, 2, -6, 0], [-1, -3, 6, 3], [4, 2, 3, 1], [-13, 0, -3, 2], [2, 4, 0, 0], [7, -8, 6, 1], [8, 2, -4, 4], [-5, -5, 0, 1], [-8, -5, 1, 3], [15, 0, 4, -3], [-3, -2, 6, -3], [-8, -2, -4, 0], [4, -2, -8, 0], [-6, 3, -2, 2], [6, 3, 2, -2], [-7, 4, -5, -4], [8, 0, -2, -4], [18, 0, -2, 0], [9, 1, -7, 4], [-10, 4, -8, -2], [-10, -2, 4, 4], [0, 2, -8, 0], [2, 4, 4, -4], [-14, 4, -3, 1], [-8, -4, 0, -2], [3, -5, 5, 4], [5, 1, 0, -3], [16, 3, -2, -2], [6, 1, 6, -2], [17, -1, -8, 1], [-6, 0, 10, 0], [18, 4, -6, 4], [1, 2, 1, 0], [-1, -3, 5, 4], [4, -6, 6, -2], [-13, 1, 9, 2], [-4, 5, 5, 3], [8, 0, -2, 8], [14, -3, -4, 4], [12, -4, -4, -2], [20, -8, 8, 0], [16, -8, 0, -4], [27, 0, 2, 1], [-14, 0, 8, 0], [-22, 3, -8, -2], [3, 3, 7, -6], [4, 3, -12, 2], [-15, -1, 10, -3], [-2, 6, 4, -6], [19, -1, 5, 0], [-4, 3, -9, 3], [13, -1, 13, 0], [-2, -4, -4, 0], [-11, 7, 2, -1], [2, -2, 4, 2], [-11, 5, 2, -1], [-7, 0, -6, -1], [26, -4, 4, 2], [-17, 5, -1, -4], [-13, 2, -4, 1], [23, 3, -1, -6], [-32, 8, -8, -2], [-10, 1, 8, -6], [11, -7, -1, 4], [9, -1, 3, 6], [-7, 3, -13, 4], [-22, 2, -8, -2], [-1, 5, -1, -8], [17, 2, -9, 2], [17, -4, 14, 1], [6, 0, 0, 6], [-1, -3, 5, 4], [-2, -5, 2, 8], [5, 5, -1, 4], [9, 4, 9, 0], [14, -4, -8, -4], [0, -6, 4, 2], [30, 0, 6, 0], [20, 0, 4, -2], [-14, 0, 2, 0], [8, -3, 2, 6], [-18, 4, -4, -8], [-14, -4, 5, -7], [-27, -1, -3, -2], [-3, 9, -3, 4], [-16, -6, 5, 1], [21, -1, 9, 0], [-8, -1, 11, -7], [2, -4, 10, 2], [19, 0, -6, -5], [-9, -7, 1, 6], [-44, 6, -4, 0], [-20, -1, 10, 0], [36, 0, 4, 0], [-26, -2, 4, 0], [4, -6, 8, 0], [-22, 2, -1, 1], [-9, 3, 15, -2], [-2, 6, 0, -2], [-28, 0, 4, 2], [12, -4, 0, -6], [-28, 0, -4, 0], [-26, -5, 2, 2], [11, -1, 3, 0], [7, 6, 2, -9], [-28, 0, -1, -5], [-7, -8, 2, -3], [-14, 9, 5, 3], [-6, -4, -4, -2], [-17, 1, -9, 2], [-23, 0, -13, 2], [-11, 5, 5, -10], [-21, 3, -10, 1], [-11, -7, 0, 5], [-4, 4, 0, 6], [1, -11, 5, -4], [-15, 5, 1, 2], [-24, 6, 4, 2], [-11, 1, -10, -3], [-20, -11, -1, 3], [-28, 2, 10, -4], [30, -7, -4, -2], [13, 1, 6, -5], [-4, 6, 6, 6], [26, -6, -4, 6], [-7, -2, 8, 9], [13, -5, -7, -6], [-24, 1, 4, -6], [-19, -4, -11, 4], [3, -2, -2, -9], [-10, 4, 7, -5], [1, 6, 14, -7], [10, -6, 6, 2], [-18, -10, -10, 8], [17, 3, -11, 4], [-33, -1, -4, 1], [-21, 1, 8, -5], [4, 8, -1, -9], [-31, 1, -8, -1], [8, -6, 6, -4], [9, -2, 0, 7], [8, 2, -8, 4], [-14, 6, -3, -13], [-27, 1, 8, -11], [12, 5, -12, 0], [10, -2, 2, 0], [6, -2, -2, 8], [-4, -5, -1, -5], [-29, 3, -7, 0], [17, 11, 1, -10], [-17, 5, 15, 4], [-13, 4, -6, 3], [-1, -7, -7, 12], [-38, 6, -10, -6], [16, -6, 10, 12], [21, -5, 7, -4], [-30, -6, 0, 6], [26, -4, -4, 6], [-15, 7, 2, 5], [-19, 5, -5, -4], [-30, -2, 0, 4], [-17, -3, 3, -8], [12, 9, 5, -5], [-3, 10, 0, -5], [-5, -5, 15, -2], [-3, 3, 2, -7], [-27, 9, -9, 0], [14, -5, -4, 16], [-19, 4, 4, 1], [-22, 4, 12, 2], [-10, 2, 12, 0], [5, 5, -11, 4], [-4, -9, 11, -3], [-18, 4, 8, -4], [6, 8, -14, 0], [-3, -5, 5, 4], [-18, -4, 8, -4], [-34, 9, 2, 4], [18, 10, -2, -4], [-26, -6, -8, 4], [1, -5, -3, 8], [1, -3, -2, 3], [23, -1, -15, 0], [-25, 4, -1, 0], [16, 6, -6, -8], [26, -2, 10, -4], [-21, 11, 5, -2], [-45, 7, -1, -4], [21, -12, 12, 7], [-20, 0, -4, -6], [-5, -5, 1, 4], [7, -3, 17, 6], [10, -4, 8, 6], [-39, 7, 4, -1], [-18, -12, -7, 11], [16, -2, 0, 2], [-13, 4, 2, 3], [-1, -3, -5, 10], [16, -4, 20, 0], [28, -6, 12, 4], [-38, -2, 5, 3], [0, 4, 2, -10], [-16, 0, 8, 0], [-9, 2, -22, 5], [-30, 4, -8, -4], [-14, -10, -14, 4], [-3, 1, 7, -10], [10, 16, -4, -4], [14, -12, 16, -2], [0, -18, 6, 0], [12, 8, -2, 2], [15, 3, -9, -8], [-8, -10, -4, 2], [0, 4, 14, 2], [1, 1, -14, 3], [22, 12, 6, -8], [-5, 2, 1, 4], [-5, -5, -7, -6], [23, -13, 3, -4], [1, -1, 22, 1], [18, -10, -4, -2], [-20, -8, 8, 0], [31, -8, -2, 1], [-18, 12, -5, -1], [-20, 0, 8, 6], [3, 5, -13, -2], [-6, -5, 2, -10], [24, -8, 6, 6], [46, -8, 0, 4], [-39, 8, -12, -3], [-5, 10, 2, 3], [45, -3, 15, 0], [-30, 4, -6, -2], [38, -3, 6, -4], [-19, 4, -6, -7], [-17, 14, -6, 5], [1, -1, -3, 4], [39, -7, 7, 6], [20, 5, 3, 3], [-17, -2, -4, 11], [-7, 5, 13, -10], [8, 6, -16, 6], [-19, 1, 9, 0], [2, 8, 0, -6], [33, -3, -9, -8], [-24, 10, -12, 0], [34, -6, 26, 2], [-16, -4, -4, -2], [-35, -1, 1, 0], [21, 1, 2, -7], [0, -4, 8, 4], [22, 0, -8, 10], [34, 2, 4, 6], [6, 4, -10, 2], [39, 5, -1, 4], [2, -8, -2, 2], [-38, 0, 6, -4], [-37, 0, 10, -19], [38, 4, -12, 8], [-21, 17, -9, 0], [0, 8, 18, -2], [23, 5, -6, 5], [58, -6, 6, 4], [-10, -4, 4, 8], [15, 10, 2, -1], [22, 7, -3, 1], [-3, -13, 13, -4], [7, 17, -4, -3], [25, -8, -2, 7], [35, -4, 15, -4], [42, 3, 6, 4], [-29, -3, 16, -5], [22, -4, 8, 8], [3, 11, -1, 10], [-24, -6, -8, -4], [-15, 9, 9, 0], [14, -4, -6, 8], [27, -1, -17, 12], [-13, -18, -6, 5], [-26, 2, -8, -6], [22, 0, -2, 0], [47, -8, 2, 7], [24, -10, -2, 12], [0, 0, -8, -10], [14, -6, -2, 8], [-7, 13, -8, -7], [-42, -1, 4, -10], [0, -16, -4, 8], [8, -10, 2, -6], [51, -10, 2, 1], [-17, -1, -13, 6], [-8, 6, 10, -8], [30, -10, 0, 6], [2, 7, -3, 5], [-5, 19, -7, -6], [-50, 6, -4, -4], [-22, 4, 8, -8], [20, -2, -6, -4], [-27, 3, 15, -6], [40, 1, -6, 2], [39, -7, 5, -2], [3, 11, -7, 8], [10, 8, -7, 5], [-20, 14, 2, -6], [0, -9, 19, 3], [-54, 12, -18, -8], [20, -18, 8, 12], [-14, 2, -14, -2], [35, -3, 11, 0], [36, 1, 12, 0], [37, -8, 8, 3], [60, -2, -12, 0], [-32, 13, -4, -6], [-44, -2, 12, -4], [34, -4, 12, 4], [28, 4, 2, 6], [25, 12, -8, -3], [-16, 8, -10, 4], [-34, 8, -4, 0], [14, -6, -12, 18], [11, 7, -10, 9], [75, 0, -2, 5], [31, -3, -18, 11], [-41, 10, -6, -7], [58, -2, 0, 4], [-29, 0, -4, 9], [21, -1, 5, -2], [-18, 9, 14, 2], [-46, 4, 8, 0], [-7, 15, -4, -9], [-21, 1, -13, -10], [15, 7, 9, -2], [6, -6, -4, -8], [55, -12, 22, 3], [-11, 2, 2, 9], [-37, 7, -10, -15], [-50, 12, 4, 6], [-4, -17, 10, 4], [8, 10, -16, 10], [1, 1, 20, -1], [-13, -2, 6, -1], [23, 2, -10, 1], [5, -3, -20, 13], [-16, -4, 0, -4], [43, 6, -2, 9], [21, 7, -11, 6], [20, -1, 3, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5610_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5610_2_a_cf();".
// 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_5610_2_a_cf(:prec:=4)
    chi := MakeCharacter_5610_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_5610_2_a_cf();".
// 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_5610_2_a_cf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5610_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![16, 24, -4, -5, 1]>,<13,R![64, 56, -12, -6, 1]>,<19,R![-256, 208, -16, -8, 1]>,<23,R![-48, -80, -36, -1, 1]>],Snew);
    return Vf;
end function;