
// Make newform 5290.2.a.p in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5290_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_5290_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_5290_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 := [4, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-5, 0, 1]];
    Rf_basisdens := [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_5290_a();"
function MakeCharacter_5290_a()
    N := 5290;
    order := 1;
    char_gens := [2117, 2121];
    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_5290_a_Hecke(Kf)
    return MakeCharacter_5290_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], [-1, 0, 0], [-1, 0, -1], [-2, 1, 0], [-1, 0, 1], [-4, -1, 0], [1, 2, -1], [0, 0, 0], [0, 2, -2], [1, 2, 1], [2, -4, 0], [2, -1, 0], [0, -2, 2], [-2, 2, 2], [-4, 0, 2], [-6, 2, -2], [-4, 3, 2], [2, 2, 0], [-4, -1, 0], [-8, 2, -2], [-2, 2, 2], [4, -4, 2], [-2, 0, -2], [12, -3, 0], [-2, 4, 0], [-2, 5, 2], [-6, 0, 0], [-13, -2, 1], [-6, 2, -2], [-2, -8, 2], [-4, -4, 0], [-1, -4, 3], [12, -2, 0], [-11, 2, -1], [16, -1, 0], [10, -2, 0], [-3, 2, -3], [-2, -4, 2], [2, -1, 0], [-10, 0, 2], [-1, 2, -3], [-14, 0, -2], [-2, 2, 0], [-6, -3, 0], [-4, -4, 0], [10, -4, -2], [4, -2, 4], [12, -4, -2], [-10, -4, 0], [2, 2, 0], [12, 0, 0], [4, 4, 0], [-1, -6, 5], [10, -6, 4], [-6, -1, -2], [-6, -6, 0], [5, -6, 1], [2, 8, -4], [-2, -6, -2], [0, -2, 2], [2, 8, 0], [0, -7, 0], [12, 0, 0], [1, 4, -3], [-13, 4, 1], [2, 0, -2], [-9, 4, -1], [4, 3, 4], [-12, 4, -6], [0, 0, -6], [-10, 6, 2], [-8, 8, -4], [16, 0, -2], [26, -1, 0], [-4, 4, 4], [-20, 5, 2], [10, -3, -4], [-10, -12, 2], [-5, 4, 1], [-4, -2, -2], [-5, -10, 5], [-20, 4, 0], [-5, -4, -1], [5, -2, -7], [-1, -6, -1], [-19, 8, 3], [-10, -2, -2], [20, 0, 2], [2, 0, 6], [4, -4, 2], [18, 4, 2], [-6, 0, -2], [-14, 0, -2], [6, -6, 2], [11, -8, -5], [6, 10, -4], [-18, 10, 2], [-4, 0, 2], [12, 2, 2], [-21, 2, 3], [-14, -12, 4], [6, 12, 0], [16, -6, 4], [-10, 9, -4], [8, -4, -2], [12, -1, 4], [18, -4, 4], [-16, 5, 0], [-13, 0, -3], [-26, 4, -2], [0, -2, 2], [-4, 3, -4], [31, -6, 1], [4, 2, -4], [12, -2, 8], [-2, 2, 8], [8, -2, -8], [19, 4, -3], [-4, 6, -10], [4, 7, -6], [2, -10, -4], [0, 6, 6], [21, 6, -3], [14, -8, -6], [-11, -2, 3], [-20, 5, -10], [-21, 6, 3], [-22, 5, -6], [-20, 8, 2], [24, 2, -4], [-9, 8, -5], [-10, 4, -2], [4, 0, -2], [-21, 8, 1], [-4, -4, 0], [2, -6, -4], [14, -2, -8], [-32, -4, 2], [-10, -3, -4], [6, 14, -6], [26, -6, -4], [10, -6, -6], [20, -12, 2], [10, 4, -8], [-16, 10, -8], [1, 4, -5], [20, -6, 2], [18, 12, -2], [-32, -8, 0], [2, -7, 8], [-16, 4, -8], [-5, -2, 11], [-26, -10, 2], [6, 8, -8], [-36, 0, 0], [8, -14, 4], [-18, 0, 0], [31, 0, 7], [8, -1, -6], [-13, 10, -5], [12, 9, 0], [-22, 2, 2], [-46, 3, -4], [1, 8, 5], [-30, 1, 2], [-32, -3, 0], [-2, 8, 0], [-10, -6, 2], [6, -3, 0], [12, 6, 6], [12, 14, -6], [18, 12, -6], [4, 9, 4], [-40, 3, 4], [6, 8, -2], [-6, -2, 6], [-22, 16, 4], [18, -11, 14], [25, -2, 3], [-14, 0, -6], [10, 15, -8], [3, 0, -7], [-10, -18, 8], [18, 9, -6], [-23, 6, 7], [-29, 12, 5], [-14, 2, -4], [42, -5, 4], [-8, -13, -4], [24, -4, 2], [24, -12, 12], [12, -6, -6], [-2, -16, 8], [12, 4, 2], [7, -8, -9], [-16, -4, 0], [48, -4, -2], [-6, 14, -2], [-24, 2, -8], [32, -5, -2], [13, -10, 1], [-30, -7, 0], [0, -8, 8], [8, 2, 6], [-10, -8, -8], [-6, 0, 6], [-8, -11, 0], [-7, 12, -1], [24, -10, 12], [-16, 10, 2], [-10, -12, 8], [-21, -8, -1], [-2, -2, 0], [-28, 0, -4], [16, -2, 12], [-2, -8, -2], [-14, 0, 4], [37, -12, 1], [-10, 16, 4], [-20, 16, -10], [12, -16, 4], [-26, 21, 4], [-31, 12, 5], [-20, 8, -16], [4, -16, -2], [-19, -24, 5], [9, -6, -3], [9, -20, 5], [-18, -20, 2], [1, -4, -9], [26, 1, 4], [-28, -2, 4], [18, 3, -12], [-8, -4, 2], [4, 0, 16], [-26, 4, -6], [-18, -4, -8], [14, 0, 2], [40, 5, -4], [-46, 6, -4], [36, 2, 0], [-2, 6, 6], [-8, 18, -6], [-12, 10, 2], [11, 22, -5], [-12, 22, -8], [1, -14, -9], [34, -1, -4], [-2, 3, -14], [-18, 18, 0], [10, 2, 2], [13, -8, 9], [-12, 8, 4], [-2, -8, -12], [36, -6, 0], [16, 5, -10], [18, 0, 0], [11, 12, -7], [-8, -14, 10], [14, -5, 10], [-20, -16, 2], [-4, -6, -14], [8, 12, 2], [-2, -10, -4], [-10, 21, 8], [-24, -1, -2], [4, -22, -4], [32, 6, 2], [1, 0, 7], [16, -20, -10], [-10, -4, 0], [41, -12, -7], [-10, -2, -2], [-4, 18, -2], [-62, -2, -2], [16, -7, 8], [-6, -8, 12], [-4, -4, -12], [-43, 6, -1], [-4, -18, -4], [10, 6, 10], [-6, -3, -6], [-42, 16, 8], [-9, 18, -1], [-8, -17, 0], [-30, -2, 2], [-28, 10, -2], [16, 1, 4], [-10, -27, 8], [6, -24, 12], [22, 6, 4], [-2, 0, 4], [-21, -26, 5], [-18, 3, 4], [16, 19, -6], [-8, 0, 4], [-22, 3, -4], [-26, 6, -14], [-8, 15, 8], [25, 4, -3], [28, -6, -2], [-44, 8, 8], [-20, 12, -8], [-17, -22, 5], [10, -10, -8], [28, 9, 4], [33, -6, 15], [-6, -10, -14], [-58, 3, -8], [-22, 20, -6], [23, 6, -7], [-44, 12, -8], [-12, -1, -4], [-23, -12, 1], [-18, -2, -10], [-56, -12, 4], [15, -2, 11], [-26, -22, 2], [14, 3, -4], [14, -1, 0], [-52, -7, 0], [28, -12, -2], [48, 6, -8], [-48, 11, 4], [20, 8, 0], [13, 2, -1], [53, -6, -3], [19, 12, 7], [-14, 10, -12], [18, -10, 0], [6, 27, 0], [16, -20, 16], [-11, 14, 5], [28, -10, 2], [9, 14, 1], [48, -9, 18], [18, -10, -2], [22, 5, -8], [30, -9, -6], [-54, 4, -8], [-4, 3, -16], [18, -22, 4], [-26, -18, 10], [-42, 2, -8], [-18, -4, -14], [26, -6, 4], [-11, -10, 13], [0, -20, -4], [-48, 2, -2], [8, 0, 12], [-70, 0, 0], [-6, 30, 0], [-10, -13, 6], [-50, 8, 8], [10, -10, -4], [-38, -6, 10], [5, -2, 19], [-12, -12, 6], [-40, 14, -12], [-32, -24, 10], [-26, 0, 4], [6, 10, 2], [-54, 3, -12], [0, 25, 0], [34, -8, 8], [18, 5, 4], [24, -4, 4], [-10, -14, 10], [-22, 0, 14], [10, 34, -10], [-6, -8, -10], [-12, 21, -6], [0, 10, 8], [5, -16, 7], [48, -12, -12], [2, 12, -10], [0, 5, -8], [-2, 10, -6], [16, -14, 14], [-8, 9, -2], [-16, 7, 4], [-10, -6, -14], [42, -22, 4], [0, -10, 10], [-44, 12, 4], [1, -10, -11], [-17, -4, -11], [68, -4, -6], [59, 4, -5], [-24, 2, -8], [29, -24, 11], [54, -2, 2], [-4, 0, -8], [-26, 16, -6], [-25, 0, 11], [16, 8, -16], [-42, 12, -6], [27, -10, 13], [-18, -4, 16], [-10, -10, -8], [0, 10, 14], [12, -8, 8], [17, -12, 5], [-16, -24, 2], [24, 17, 8], [54, 6, 0], [27, 8, -11], [40, 14, 2], [30, -6, -6], [60, 8, 6], [-18, 5, -8], [-22, 8, 0], [-46, 13, -8], [11, -8, -5], [32, 2, 6], [-29, 18, 7], [-48, -8, -8], [32, 23, -12], [-53, 0, -5], [-20, 17, -22], [-10, -4, -12], [48, -6, 0], [14, -6, 2], [22, -30, 0], [-54, -4, -8], [6, 22, 8], [-9, -12, 9], [10, -22, 22], [20, 13, -8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5290_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5290_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_5290_2_a_p(:prec:=3)
    chi := MakeCharacter_5290_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_5290_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_5290_2_a_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5290_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -7, 1, 1]>,<7,R![-36, -15, 3, 1]>,<11,R![-6, 1, 5, 1]>],Snew);
    return Vf;
end function;