
// Make newform 69.4.c.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_69_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_69_c_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_69_4_c_b();".
// 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_69_4_c_b();".
// 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 := [36, -6, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-26, -5, 5, 1], [6, 0, 0, -1]];
    Rf_basisdens := [1, 1, 10, 5];
    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_69_c();"
function MakeCharacter_69_c()
    N := 69;
    order := 2;
    char_gens := [47, 28];
    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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_69_c_Hecke();"
function MakeCharacter_69_c_Hecke(Kf)
    N := 69;
    order := 2;
    char_gens := [47, 28];
    char_values := [[-1, 0, 0, 0], [-1, 0, 0, 0]];
    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 := [[-1, 0, -1, -1], [-1, 2, 1, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-35, 24, 4, 12], [0, 0, 0, 0], [0, 0, 0, 0], [23, 0, 0, 46], [-61, 0, -94, -28], [-173, -12, -2, -6], [0, 0, 0, 0], [-103, 0, -142, -64], [0, 0, 0, 0], [59, 0, -16, 134], [0, 0, 0, 0], [-170, 0, 0, -340], [0, 0, 0, 0], [0, 0, 0, 0], [173, 0, 392, -46], [-605, 96, 16, 48], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-260, 0, 0, -520], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [1345, 492, 82, 246], [353, 0, 932, -226], [0, 0, 0, 0], [319, 1356, 226, 678], [0, 0, 0, 0], [1837, 348, 58, 174], [0, 0, 0, 0], [685, -1596, -266, -798], [-818, 0, 0, -1636], [604, 0, 0, 1208], [-49, 0, -916, 818], [0, 0, 0, 0], [0, 0, 0, 0], [-2255, -1344, -224, -672], [-859, 0, -1834, 116], [0, 0, 0, 0], [2468, 0, 0, 0], [-4840, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-523, 0, -1918, 872], [1169, 0, 2456, -118], [0, 0, 0, 0], [0, 0, 0, 0], [-241, 0, 1094, -1576], [0, 0, 0, 0], [-1081, 0, -334, -1828], [8912, 0, 0, 0], [-605, 3768, 628, 1884], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-10420, 0, 0, 0], [1943, 0, 1952, 1934], [0, 0, 0, 0], [2332, 0, 0, 4664], [-6023, -12, -2, -6], [0, 0, 0, 0], [-1898, 0, 0, -3796], [-4559, 3624, 604, 1812], [1907, 0, 1142, 2672], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [7375, -1992, -332, -996], [0, 0, 0, 0], [3049, 6576, 1096, 3288], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [4231, 7044, 1174, 3522], [-3547, 0, -5932, -1162], [1000, 0, 0, 2000], [0, 0, 0, 0], [1517, 0, -1042, 4076], [-11680, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [9535, -3612, -602, -1806], [-4153, 0, -6892, -1414], [-8441, -6564, -1094, -3282], [0, 0, 0, 0], [-919, 0, 2654, -4492], [0, 0, 0, 0], [0, 0, 0, 0], [-407, -10488, -1748, -5244], [-10175, 5892, 982, 2946], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [11485, 7152, 1192, 3576], [-2599, 0, 716, -5914], [-2456, 0, 0, -4912], [-3410, 0, 0, -6820], [-1187, -12288, -2048, -6144], [8840, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [3071, 0, 9536, -3394], [-1057, 0, -7342, 5228], [0, 0, 0, 0], [0, 0, 0, 0], [17125, 3984, 664, 1992], [0, 0, 0, 0], [-6403, 0, -11788, -1018], [-6028, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1430, 0, 0, -2860], [0, 0, 0, 0], [0, 0, 0, 0], [-16181, -10884, -1814, -5442], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-4933, 0, -13042, 3176], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-7640, 0, 0, -15280], [3283, 19212, 3202, 9606], [8380, 0, 0, 16760], [10315, 18276, 3046, 9138], [0, 0, 0, 0], [-4966, 0, 0, 0], [0, 0, 0, 0], [24590, 0, 0, 0], [-8941, 0, -9106, -8776], [-25151, 276, 46, 138], [3743, 0, 13688, -6202], [-34090, 0, 0, 0], [0, 0, 0, 0], [-2860, 0, 0, 0], [5891, 0, 776, 11006], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-11119, 0, -16486, -5752], [0, 0, 0, 0], [0, 0, 0, 0], [8939, 0, 6404, 11474], [0, 0, 0, 0], [6325, 24828, 4138, 12414], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [9128, 0, 0, 0], [-60190, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_69_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_69_4_c_b();".
// 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_69_4_c_b(:prec:=4)
    chi := MakeCharacter_69_c();
    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_69_4_c_b();".
// 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_69_4_c_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_69_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![1, 0, 25, 0, 1]>],Snew);
    return Vf;
end function;