
// Make newform 490.4.e.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_490_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_490_4_e_a();".
// 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_490_4_e_a();".
// 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_490_e();"
function MakeCharacter_490_e()
    N := 490;
    order := 3;
    char_gens := [197, 101];
    v := [3, 2];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_490_e_Hecke();"
function MakeCharacter_490_e_Hecke(Kf)
    N := 490;
    order := 3;
    char_gens := [197, 101];
    char_values := [[1, 0], [0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 4;
    raw_aps := [[0, -2], [-8, 8], [0, 5], [0, 0], [-12, 12], [58, 0], [66, -66], [0, -100], [0, -132], [-90, 0], [152, -152], [0, 34], [438, 0], [32, 0], [0, -204], [-222, 222], [420, -420], [0, 902], [1024, -1024], [432, 0], [362, -362], [0, 160], [-72, 0], [0, 810], [-1106, 0], [-258, 258], [0, -988], [0, 24], [-950, 950], [-1038, 0], [-124, 0], [0, 132], [1254, -1254], [2860, 0], [0, -750], [448, -448], [2246, -2246], [0, 568], [1524, 0], [0, 3702], [-3180, 3180], [2098, 0], [0, -4392], [2158, -2158], [-1074, 0], [2840, -2840], [-2668, 0], [-1772, 0], [-2784, 2784], [0, 350], [0, -1962], [-4320, 0], [-478, 478], [-2652, 0], [0, -2334], [3948, -3948], [1590, -1590], [0, 4952], [-1646, 1646], [-1158, 0], [6992, -6992], [258, 0], [8944, 0], [1392, -1392], [0, -5878], [0, -10326], [0, 4228], [1106, 0], [-9336, 9336], [11770, 0], [8322, -8322], [0, -10680], [-5884, 5884], [0, 2098], [3860, 0], [0, -9588], [13410, -13410], [0, -13114], [0, 5838], [9530, -9530], [-7260, 0], [12062, 0], [13608, -13608], [3838, 0], [0, 7400], [0, -8352], [10770, 0], [0, 6694], [3018, 0], [14492, 0], [0, 7776], [-13680, 13680], [-7916, 7916], [13932, 0], [0, 8260], [11148, 0], [0, -9690], [-16038, 16038], [0, 992], [0, -7142], [7616, 0], [10314, -10314], [-7128, 7128], [0, -2010], [23188, -23188], [22466, -22466], [-22776, 0], [0, -21198], [-15960, 15960], [-5882, 0], [0, 8516], [-8462, 8462], [-11094, 0], [2180, -2180], [-26848, 0], [-26322, 26322], [10168, 0], [-23604, 23604], [0, -16422], [-26100, 0], [-3058, 3058], [10802, 0], [0, -10674], [28608, -28608], [0, -2428], [-6618, 0], [0, -20510], [0, 31680], [-13196, 0], [0, 8102], [12580, -12580], [29892, 0], [0, 40408], [32366, 0], [0, -17238], [-10850, 0], [9102, -9102], [-25504, 25504], [-14166, 0], [-33210, 33210], [-39212, 0], [0, -6222], [-31172, 31172], [-264, 0], [-29050, 29050], [21720, 0], [6658, 0], [-13974, 13974], [0, 23780], [0, 12228], [0, -11606], [32958, 0], [8072, 0], [0, 15756], [-18776, 18776], [-20568, 0], [0, 6280], [0, -20430], [-8906, 0], [-17418, 17418], [0, 2544], [15402, 0], [-49444, 0], [0, -25188], [-2946, 2946], [15012, -15012], [5128, -5128], [-49714, 49714]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_490_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_490_4_e_a();".
// 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_490_4_e_a(:prec:=2)
    chi := MakeCharacter_490_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_490_4_e_a();".
// 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_490_4_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_490_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![64, 8, 1]>,<11,R![144, 12, 1]>],Snew);
    return Vf;
end function;