
// Make newform 5184.2.a.cc in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5184_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_5184_2_a_cc();".
// 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_5184_2_a_cc();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1, -7, -1, 1], [-4, -6, 0, 1], [1, -9, -3, 2]];
    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_5184_a();"
function MakeCharacter_5184_a()
    N := 5184;
    order := 1;
    char_gens := [2431, 325, 1217];
    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_5184_a_Hecke(Kf)
    return MakeCharacter_5184_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, 0, 0], [0, 0, 0, 0], [-1, -1, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0], [-1, 1, 0, 0], [0, 1, 0, 0], [0, 0, 1, -1], [0, 0, 0, -1], [-3, -1, 0, 0], [0, 0, 2, -1], [-4, 0, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, -1], [-4, 0, 0, 0], [0, 0, 1, 2], [-5, 3, 0, 0], [0, 0, 1, 2], [0, 0, 0, -2], [8, 1, 0, 0], [0, 0, -2, 1], [0, 0, -2, -1], [-8, -2, 0, 0], [9, 0, 0, 0], [-5, 3, 0, 0], [0, 0, 0, -1], [0, 0, -1, 1], [-4, 4, 0, 0], [-15, -1, 0, 0], [0, 0, 2, -2], [0, 0, -2, 1], [-1, -4, 0, 0], [0, 0, 1, -4], [-15, -1, 0, 0], [0, 0, 0, -1], [-7, -3, 0, 0], [0, 0, 0, -4], [0, 0, 2, -1], [3, 1, 0, 0], [0, 0, 0, 4], [-12, -6, 0, 0], [0, 0, -4, 3], [1, 0, 0, 0], [-6, -2, 0, 0], [0, 0, 2, 4], [0, 0, 2, -3], [0, 0, 0, 1], [0, 0, 3, -4], [1, -1, 0, 0], [14, 5, 0, 0], [0, 0, -4, 3], [7, 2, 0, 0], [0, 0, -3, 3], [-7, -4, 0, 0], [0, 0, 4, -3], [-4, -4, 0, 0], [0, 0, 4, -4], [-3, 3, 0, 0], [-9, 5, 0, 0], [0, 0, -2, 3], [-17, 1, 0, 0], [0, 0, 3, 5], [0, 0, 4, -5], [-1, -6, 0, 0], [5, -1, 0, 0], [0, 0, 6, -1], [-9, 0, 0, 0], [0, 0, -1, -2], [13, -3, 0, 0], [-1, -6, 0, 0], [0, 0, -6, 6], [0, 0, -2, -7], [7, -3, 0, 0], [0, 0, 1, 1], [0, 0, -2, -5], [11, 9, 0, 0], [4, 0, 0, 0], [-1, 8, 0, 0], [-25, -2, 0, 0], [0, 0, -2, 3], [19, -1, 0, 0], [0, 0, -2, 2], [-8, 3, 0, 0], [0, 0, 2, 5], [0, 0, -3, 4], [8, -3, 0, 0], [1, 8, 0, 0], [-19, 3, 0, 0], [0, 0, -6, -1], [0, 0, 5, 3], [0, 0, 0, 1], [0, 0, 2, -2], [0, 0, 3, -8], [0, 0, -5, 2], [0, 0, -4, 6], [17, 9, 0, 0], [-14, -9, 0, 0], [0, 0, 4, -2], [-2, 2, 0, 0], [0, 0, -1, 2], [-12, 2, 0, 0], [0, 0, 3, 2], [33, -2, 0, 0], [0, 0, 3, -8], [8, -3, 0, 0], [0, 0, 1, -8], [-16, -10, 0, 0], [0, 0, 2, 7], [-23, -6, 0, 0], [0, 0, 6, 1], [4, 12, 0, 0], [31, -2, 0, 0], [0, 0, -1, 2], [0, 0, 0, 0], [-7, 2, 0, 0], [0, 0, 3, 0], [0, 0, 6, 2], [31, -3, 0, 0], [0, 0, 2, 9], [-7, -3, 0, 0], [-7, -9, 0, 0], [-29, -9, 0, 0], [0, 0, -5, -3], [0, 0, 6, 5], [28, -2, 0, 0], [-11, -7, 0, 0], [0, 0, 4, 4], [0, 0, -4, -5], [-33, -3, 0, 0], [0, 0, -3, 1], [0, 0, -8, 9], [0, 0, 2, 5], [-6, -12, 0, 0], [-17, 1, 0, 0], [-25, -3, 0, 0], [-12, 2, 0, 0], [0, 0, 6, 3], [9, -11, 0, 0], [16, 1, 0, 0], [0, 0, -9, 9], [5, -1, 0, 0], [0, 0, -2, -5], [0, 0, 0, 4], [-12, -6, 0, 0], [0, 0, -2, 1], [11, -3, 0, 0], [7, -11, 0, 0], [0, 0, 3, 0], [0, 0, 0, -8], [-17, -9, 0, 0], [16, 6, 0, 0], [0, 0, -1, -7], [0, 0, 2, 5], [0, 0, -7, 6], [0, 0, -10, 3], [0, 0, -2, -6], [-33, 1, 0, 0], [-8, -6, 0, 0], [-9, 11, 0, 0], [0, 0, 5, 2], [8, -7, 0, 0], [0, 0, -8, 9], [0, 0, 12, -10], [-31, -6, 0, 0], [0, 0, -2, -13], [0, 0, -8, 0], [-11, 9, 0, 0], [-8, -10, 0, 0], [1, 9, 0, 0], [0, 0, -5, -2], [-1, -9, 0, 0], [0, 0, 6, -7], [17, 10, 0, 0], [0, 0, -8, 7], [17, -9, 0, 0], [0, 0, -1, 8], [28, -2, 0, 0], [0, 0, -4, 4], [5, -3, 0, 0], [0, 0, 4, 5], [0, 0, -3, -4], [-31, -5, 0, 0], [32, -9, 0, 0], [0, 0, -8, -7], [-19, 5, 0, 0], [34, 2, 0, 0], [0, 0, -10, 5], [15, -6, 0, 0], [0, 0, -4, -12], [8, 1, 0, 0], [0, 0, -5, 4], [0, 0, -11, 11], [51, -1, 0, 0], [0, 0, -3, 3], [1, 0, 0, 0], [17, 18, 0, 0], [-27, 9, 0, 0], [33, 8, 0, 0], [0, 0, -6, 6], [-25, 5, 0, 0], [0, 0, 4, 5], [9, 9, 0, 0], [-41, -4, 0, 0], [0, 0, 5, 3], [52, 6, 0, 0], [0, 0, -8, -8], [0, 0, 11, -8], [25, 0, 0, 0], [0, 0, 5, 8], [32, 15, 0, 0], [-15, -5, 0, 0], [0, 0, 12, -11], [0, 0, 2, -7], [0, 0, -10, 11], [31, 6, 0, 0], [0, 0, -6, 7], [-33, -16, 0, 0], [0, 0, -10, 2], [41, -7, 0, 0], [-41, 5, 0, 0], [0, 0, 10, -3], [-9, 16, 0, 0], [0, 0, -2, 2], [0, 0, -3, 6], [29, -9, 0, 0], [49, 3, 0, 0], [0, 0, -4, 12], [0, 0, -8, -1], [0, 0, 6, 13], [17, 19, 0, 0], [0, 0, 0, 12], [0, 0, 0, -15], [-7, -4, 0, 0], [0, 0, 2, 3], [0, 0, 8, -1], [-7, 7, 0, 0], [-4, -18, 0, 0], [0, 0, -3, -2], [0, 0, -6, -2], [0, 0, 9, 0], [0, 0, 0, 4], [0, 0, -10, 9], [44, -6, 0, 0], [31, 6, 0, 0], [0, 0, 2, -3], [0, 0, -4, 12], [0, 0, 7, -10], [0, 0, -6, 5], [0, 0, 6, 10], [-17, 17, 0, 0], [0, 19, 0, 0], [0, 0, -10, 9], [-25, -11, 0, 0], [-21, -11, 0, 0], [0, 0, -7, 5], [-4, 4, 0, 0], [0, 0, -2, -9], [-38, 10, 0, 0], [40, 3, 0, 0], [0, 0, 2, 5], [0, 0, 11, 6], [1, -19, 0, 0], [-28, -6, 0, 0], [-25, -3, 0, 0], [0, 0, 1, -8], [12, 16, 0, 0], [-31, 8, 0, 0], [0, 0, 9, 2], [49, 7, 0, 0], [7, 3, 0, 0], [0, 0, -8, -12], [63, 0, 0, 0], [0, 0, -12, -5], [17, 10, 0, 0], [0, 0, 12, -4], [0, 0, -6, -3], [43, 9, 0, 0], [8, -3, 0, 0], [0, 0, -5, -4], [0, 0, 0, 17], [0, 0, -8, 7], [0, 0, -4, 13], [-11, -9, 0, 0], [0, 0, 3, -6], [0, 0, -2, 0], [32, 22, 0, 0], [-25, 15, 0, 0], [0, 0, 2, -1], [-42, 5, 0, 0], [-35, -3, 0, 0], [0, 0, -5, 11], [9, -16, 0, 0], [0, 0, 5, 8], [5, -3, 0, 0], [15, 5, 0, 0], [0, 0, -12, 3], [-3, 7, 0, 0], [0, 0, 8, -12], [0, 0, 10, -7], [31, 6, 0, 0], [-28, 6, 0, 0], [0, 0, 8, 8], [0, 0, 10, -3], [0, 0, 2, -9], [-24, 6, 0, 0], [0, 0, 10, -15], [9, 9, 0, 0], [0, 0, 4, 5], [-20, -12, 0, 0], [0, 0, 2, -11], [-20, -12, 0, 0], [-15, 11, 0, 0], [0, 0, 6, 17], [0, 0, 12, 4], [24, -15, 0, 0], [0, 0, 15, 0], [0, 0, 6, 1], [-49, 4, 0, 0], [7, 21, 0, 0], [0, 0, -13, 14], [-17, -6, 0, 0], [-20, -8, 0, 0], [0, 0, -6, 6], [-41, -15, 0, 0], [10, -19, 0, 0], [0, 0, -3, 13], [0, 0, 11, -10], [0, 0, 6, 1], [52, 6, 0, 0], [-3, 21, 0, 0], [-33, -5, 0, 0], [0, 0, -2, 3], [0, 0, -9, 10], [0, 0, -3, -5], [0, 0, 0, -12], [14, 0, 0, 0], [39, 22, 0, 0], [9, -18, 0, 0], [0, 0, 8, -2], [-13, -21, 0, 0], [-31, 15, 0, 0], [15, 19, 0, 0], [0, 0, 10, -1], [13, -17, 0, 0], [0, 0, -5, -5], [-20, -16, 0, 0], [0, 0, 1, -10], [0, 0, -2, 11], [-26, -20, 0, 0], [0, 0, 2, 13], [-80, -1, 0, 0], [-55, -1, 0, 0], [0, 0, -2, 9], [23, 3, 0, 0], [-6, 16, 0, 0], [0, 0, -10, 1], [0, 0, 8, 4], [-47, 3, 0, 0], [0, 0, 8, 9], [-13, -3, 0, 0], [1, 4, 0, 0], [0, 0, 12, -2], [0, 0, -9, -8], [0, 0, 15, -15], [-7, -18, 0, 0], [-67, -3, 0, 0], [0, 0, -2, -4], [8, 3, 0, 0], [0, 0, -14, 17], [0, 0, 9, -9], [0, 0, 18, -9], [13, 1, 0, 0], [0, 0, 0, -9], [4, 14, 0, 0], [0, 0, 5, 16], [0, 0, 10, 8], [64, -6, 0, 0], [16, 33, 0, 0], [1, -1, 0, 0], [11, -7, 0, 0], [9, 2, 0, 0], [0, 0, -8, 16], [-33, 10, 0, 0], [0, 0, -7, 12], [0, 0, 4, 20], [0, 0, -4, 13], [-55, 1, 0, 0], [0, 0, -12, 0], [0, 0, -6, -17], [17, 10, 0, 0], [29, -7, 0, 0], [0, 0, 13, -11], [0, 0, -21, 12], [0, 0, 0, 7], [17, 19, 0, 0], [0, 0, 4, -10], [-81, -4, 0, 0], [0, 0, -11, -16], [-15, -23, 0, 0], [49, 15, 0, 0], [80, -3, 0, 0], [0, 0, 2, 7], [-9, -7, 0, 0], [-42, -4, 0, 0], [0, 0, -16, 18], [13, 23, 0, 0], [-63, -14, 0, 0], [0, 0, 7, -12], [0, 0, 6, -9], [-7, 18, 0, 0], [-21, 17, 0, 0], [0, 0, -17, 11], [0, 0, 1, -16], [-23, -13, 0, 0], [-36, -10, 0, 0], [0, 0, -10, 2], [0, 0, -8, -9], [16, 1, 0, 0], [0, 0, 6, 3], [-61, -9, 0, 0], [-4, 24, 0, 0], [0, 0, -2, -15], [0, 0, 1, -6], [40, -18, 0, 0], [-39, 5, 0, 0], [0, 0, -5, -18], [0, 10, 0, 0], [0, 0, -8, -12], [0, 0, -2, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5184_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_a_cc();".
// 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_5184_2_a_cc(:prec:=4)
    chi := MakeCharacter_5184_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_5184_2_a_cc();".
// 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_5184_2_a_cc( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5184_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 1, 1]>,<7,R![108, 0, -27, 0, 1]>,<11,R![27, 0, -36, 0, 1]>],Snew);
    return Vf;
end function;