
// Make newform 100.4.e.d in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_100_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_100_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_100_4_e_d();".
// 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_100_4_e_d();".
// 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 := [20, -100, 186, -190, 141, -58, 24, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [70, 20, -34, 29, -17, 3, -1, 0], [-620, 2760, -2756, 3213, -1190, 678, -98, 33], [2020, -5160, 5746, -4428, 1885, -783, 133, -33], [830, -2540, 2834, -2547, 1025, -487, 77, -22], [-6620, 19960, -21206, 19108, -7535, 3613, -563, 163], [-6920, 20560, -22016, 19543, -7790, 3658, -578, 163], [-615, 1880, -2092, 1876, -750, 356, -56, 16]];
    Rf_basisdens := [1, 10, 150, 150, 50, 150, 150, 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_100_e();"
function MakeCharacter_100_e()
    N := 100;
    order := 4;
    char_gens := [51, 77];
    v := [2, 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_100_e_Hecke();"
function MakeCharacter_100_e_Hecke(Kf)
    N := 100;
    order := 4;
    char_gens := [51, 77];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 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 := [[0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -5, 0, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, 0, -8], [0, 0, 0, -8, 0, 0, 8, 0], [0, 0, -24, 0, 0, -24, 0, 0], [0, -8, 0, 0, 0, 0, 0, 0], [0, 0, 0, 47, 0, 0, 47, 0], [0, 0, 0, 0, -46, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16], [0, 0, -64, 0, 0, -64, 0, 0], [-188, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -55, 0, 0, -55, 0], [0, 0, -87, 0, 0, 87, 0, 0], [0, 0, 0, -8, 0, 0, 8, 0], [0, 56, 0, 0, 0, 0, 0, 0], [72, 0, 0, 0, 0, 0, 0, 0], [0, 0, 267, 0, 0, -267, 0, 0], [0, 0, 0, 0, 0, 0, 0, -96], [0, 0, 0, 152, 0, 0, -152, 0], [0, -112, 0, 0, 0, 0, 0, 0], [0, 0, 0, -99, 0, 0, -99, 0], [0, 0, 0, 0, -726, 0, 0, 0], [0, 0, 136, 0, 0, 136, 0, 0], [1382, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -471, 0, 0, -471, 0], [0, 0, 245, 0, 0, -245, 0, 0], [0, 0, 0, 0, 784, 0, 0, 0], [0, 0, 0, -48, 0, 0, 48, 0], [0, 0, -341, 0, 0, 341, 0, 0], [0, 0, 0, 0, 0, 0, 0, -24], [0, 0, -144, 0, 0, -144, 0, 0], [0, 104, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 104, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 240], [0, 0, -784, 0, 0, -784, 0, 0], [0, 0, 0, -7, 0, 0, -7, 0], [0, 0, -33, 0, 0, 33, 0, 0], [0, 0, 0, -688, 0, 0, 688, 0], [0, 88, 0, 0, 0, 0, 0, 0], [4202, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -512], [0, 0, 0, 552, 0, 0, -552, 0], [0, 0, 56, 0, 0, 56, 0, 0], [0, -640, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -88], [0, 0, 0, 67, 0, 0, 67, 0], [0, 0, 649, 0, 0, -649, 0, 0], [0, 0, 0, 0, 3334, 0, 0, 0], [0, 0, 0, -328, 0, 0, 328, 0], [0, 464, 0, 0, 0, 0, 0, 0], [-3028, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 280], [0, 0, 736, 0, 0, 736, 0, 0], [0, 0, 0, -627, 0, 0, -627, 0], [0, 0, 0, 0, 5544, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 624], [0, 0, 416, 0, 0, 416, 0, 0], [-5308, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2189, 0, 0, -2189, 0], [0, 0, 0, 112, 0, 0, -112, 0], [0, 0, -405, 0, 0, 405, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1392], [0, 0, 0, 1472, 0, 0, -1472, 0], [0, 0, 936, 0, 0, 936, 0, 0], [0, 0, 0, 0, 0, 0, 0, -424], [0, 0, -1504, 0, 0, -1504, 0, 0], [0, 0, 1503, 0, 0, -1503, 0, 0], [0, 0, 0, 0, 5774, 0, 0, 0], [0, 0, 0, -128, 0, 0, 128, 0], [0, -944, 0, 0, 0, 0, 0, 0], [0, 0, 717, 0, 0, -717, 0, 0], [0, 0, 0, -368, 0, 0, 368, 0], [0, 248, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3931, 0, 0, 3931, 0], [0, 0, 0, 0, -3096, 0, 0, 0], [0, 0, -904, 0, 0, -904, 0, 0], [-1778, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5804, 0, 0, 0], [0, -648, 0, 0, 0, 0, 0, 0], [1672, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 16], [0, 0, 0, -1928, 0, 0, 1928, 0], [0, 1424, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5435, 0, 0, -5435, 0], [0, 0, 0, 0, -2476, 0, 0, 0], [0, 0, -2624, 0, 0, -2624, 0, 0], [-13918, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1433, 0, 0, 1433, 0], [0, 0, -633, 0, 0, 633, 0, 0], [0, -1952, 0, 0, 0, 0, 0, 0], [0, 0, 221, 0, 0, -221, 0, 0], [0, 0, 0, 0, 0, 0, 0, 168], [0, -1560, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2929, 0, 0, 2929, 0], [0, 0, 0, 0, 1554, 0, 0, 0], [-5638, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1107, 0, 0, 1107, 0], [-16078, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3553, 0, 0, -3553, 0, 0], [0, 0, -384, 0, 0, -384, 0, 0], [0, 0, 0, -3937, 0, 0, -3937, 0], [0, 0, 0, 0, 24564, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2296], [0, 0, 4176, 0, 0, 4176, 0, 0], [0, 0, 1241, 0, 0, -1241, 0, 0], [0, 0, 0, 4112, 0, 0, -4112, 0], [0, 480, 0, 0, 0, 0, 0, 0], [2572, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4895, 0, 0, 4895, 0, 0], [0, 0, 0, -3048, 0, 0, 3048, 0], [0, 0, 1496, 0, 0, 1496, 0, 0], [0, -2648, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2224], [27492, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 4515, 0, 0, 4515, 0], [0, 0, 6453, 0, 0, -6453, 0, 0], [0, 0, 0, 6552, 0, 0, -6552, 0], [0, 2616, 0, 0, 0, 0, 0, 0], [-6048, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5288, 0, 0, 5288, 0], [0, 0, -2024, 0, 0, -2024, 0, 0], [0, 0, 0, 10161, 0, 0, 10161, 0], [0, 0, 0, 0, 0, 0, 0, 4568], [-14928, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -20966, 0, 0, 0], [0, 1872, 0, 0, 0, 0, 0, 0], [0, 0, 5019, 0, 0, -5019, 0, 0], [0, 0, 0, -5648, 0, 0, 5648, 0], [0, 2264, 0, 0, 0, 0, 0, 0], [0, 0, 0, -8085, 0, 0, -8085, 0], [0, 0, 0, 0, 0, 0, 0, -160], [0, 0, -4464, 0, 0, -4464, 0, 0], [-678, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -10546, 0, 0, 0], [0, 0, 0, -5848, 0, 0, 5848, 0], [0, 0, -6739, 0, 0, 6739, 0, 0], [0, 0, 5256, 0, 0, 5256, 0, 0], [0, 0, 0, 0, 34854, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1448], [9392, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 14087, 0, 0, 14087, 0], [0, 0, -10121, 0, 0, 10121, 0, 0], [0, 0, 0, 0, 43384, 0, 0, 0], [0, 1584, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5208, 0, 0, 5208, 0], [0, 0, -2224, 0, 0, -2224, 0, 0], [0, -5864, 0, 0, 0, 0, 0, 0], [0, 0, 0, 3943, 0, 0, 3943, 0], [0, 0, 1936, 0, 0, 1936, 0, 0], [22372, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -7369, 0, 0, -7369, 0], [0, 0, 7491, 0, 0, -7491, 0, 0], [0, 0, 10235, 0, 0, -10235, 0, 0], [0, 0, 0, 0, 0, 0, 0, -3248], [0, -4608, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -13996, 0, 0, 0], [0, 0, 6536, 0, 0, 6536, 0, 0], [-18078, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3517, 0, 0, 3517, 0, 0], [0, 0, 0, 9552, 0, 0, -9552, 0], [0, 0, 12177, 0, 0, -12177, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2424], [0, 0, 10936, 0, 0, 10936, 0, 0], [0, 0, 0, -1659, 0, 0, -1659, 0], [0, 0, 0, 0, 0, 0, 0, 4128], [0, 0, 10296, 0, 0, 10296, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_100_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_100_4_e_d();".
// 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_100_4_e_d(:prec:=8)
    chi := MakeCharacter_100_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_100_4_e_d();".
// 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_100_4_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_100_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![100, 0, 0, 0, 1]>],Snew);
    return Vf;
end function;