
// Make newform 100.3.d.b in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_100_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_100_d_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_3_d_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_100_3_d_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 := [1, 0, -1, 0, 1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 2, 0, 0], [-1, 0, 0, 0, 0, 0, 4, 0], [-1, 0, 4, 0, 0, 0, 0, 0], [1, 0, 0, 0, 4, 0, 0, 0], [0, 0, 0, 2, 0, 0, 0, -2], [0, -2, 0, 4, 0, -2, 0, 4]];
    Rf_basisdens := [1, 1, 1, 1, 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_100_d();"
function MakeCharacter_100_d()
    N := 100;
    order := 2;
    char_gens := [51, 77];
    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_100_d_Hecke();"
function MakeCharacter_100_d_Hecke(Kf)
    N := 100;
    order := 2;
    char_gens := [51, 77];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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 := 3;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -3, -1, 0, 0, 0, 1, 1], [0, 0, 0, -3, -2, -1, 0, 0], [0, 1, 2, 0, 0, 0, 0, 1], [0, 4, -3, 0, 0, 0, 0, 4], [0, 0, 0, -2, 0, -2, 0, 0], [0, 3, 1, 0, 0, 0, -3, -1], [2, 0, 0, -2, 0, 2, 0, 0], [0, 0, 0, 3, 10, -7, 0, 0], [0, -5, 2, 0, 0, 0, 0, -5], [-28, 0, 0, -3, 0, 3, 0, 0], [0, -6, -2, 0, 0, 0, -3, 2], [0, 9, 3, 0, 0, 0, 13, -3], [0, 5, 22, 0, 0, 0, 0, 5], [0, 0, 0, 0, -12, 12, 0, 0], [32, 0, 0, -13, 0, 13, 0, 0], [0, 30, 10, 0, 0, 0, -11, -10], [0, 0, 0, 11, 2, 9, 0, 0], [0, 16, -33, 0, 0, 0, 0, 16], [0, 0, 0, 2, 20, -18, 0, 0], [0, 18, 6, 0, 0, 0, 13, -6], [22, 0, 0, -20, 0, 20, 0, 0], [0, -6, -33, 0, 0, 0, 0, -6], [82, 0, 0, 4, 0, -4, 0, 0], [0, -33, -11, 0, 0, 0, -23, 11], [0, -36, -12, 0, 0, 0, 17, 12], [32, 0, 0, 37, 0, -37, 0, 0], [0, -10, -63, 0, 0, 0, 0, -10], [0, -57, -19, 0, 0, 0, -23, 19], [0, 0, 0, 7, 42, -35, 0, 0], [0, 6, -13, 0, 0, 0, 0, 6], [0, 0, 0, -32, -4, -28, 0, 0], [-48, 0, 0, -19, 0, 19, 0, 0], [0, 0, 0, -49, -62, 13, 0, 0], [0, 27, 42, 0, 0, 0, 0, 27], [0, -84, -28, 0, 0, 0, 45, 28], [0, 27, 9, 0, 0, 0, -15, -9], [0, 5, 102, 0, 0, 0, 0, 5], [0, 0, 0, 44, 76, -32, 0, 0], [122, 0, 0, 8, 0, -8, 0, 0], [0, 0, 0, -7, -58, 51, 0, 0], [0, 8, -73, 0, 0, 0, 0, 8], [0, -35, -8, 0, 0, 0, 0, -35], [0, 0, 0, -48, -64, 16, 0, 0], [0, 0, 0, 35, 42, -7, 0, 0], [0, -105, -35, 0, 0, 0, 13, 35], [0, -24, -8, 0, 0, 0, 1, 8], [-118, 0, 0, 10, 0, -10, 0, 0], [0, -70, -3, 0, 0, 0, 0, -70], [0, 0, 0, 38, 36, 2, 0, 0], [-68, 0, 0, -15, 0, 15, 0, 0], [0, 0, 0, 13, -26, 39, 0, 0], [0, -2, -133, 0, 0, 0, 0, -2], [0, -87, -29, 0, 0, 0, -79, 29], [192, 0, 0, -35, 0, 35, 0, 0], [0, 0, 0, 49, 86, -37, 0, 0], [0, 45, -128, 0, 0, 0, 0, 45], [-28, 0, 0, 5, 0, -5, 0, 0], [0, 72, 24, 0, 0, 0, -15, -24], [0, -23, -38, 0, 0, 0, 0, -23], [0, -102, -34, 0, 0, 0, -11, 34], [0, 0, 0, -39, -10, -29, 0, 0], [0, 88, 87, 0, 0, 0, 0, 88], [0, -23, 132, 0, 0, 0, 0, -23], [0, 0, 0, -57, -70, 13, 0, 0], [0, -32, -63, 0, 0, 0, 0, -32], [0, -114, -38, 0, 0, 0, -91, 38], [242, 0, 0, 54, 0, -54, 0, 0], [0, 72, 7, 0, 0, 0, 0, 72], [0, 0, 0, 16, 16, 0, 0, 0], [0, 51, 17, 0, 0, 0, 157, -17], [0, -77, -128, 0, 0, 0, 0, -77], [0, 0, 0, 46, 32, 14, 0, 0], [0, 171, 57, 0, 0, 0, -43, -57], [-48, 0, 0, 77, 0, -77, 0, 0], [0, 47, 62, 0, 0, 0, 0, 47], [282, 0, 0, 36, 0, -36, 0, 0], [272, 0, 0, -75, 0, 75, 0, 0], [0, 0, 0, -56, 28, -84, 0, 0], [-328, 0, 0, -57, 0, 57, 0, 0], [0, 0, 0, 85, 126, -41, 0, 0], [0, -30, 117, 0, 0, 0, 0, -30], [0, 0, 0, 40, -48, 88, 0, 0], [0, 132, 44, 0, 0, 0, -19, -44], [-528, 0, 0, -51, 0, 51, 0, 0], [0, 60, 277, 0, 0, 0, 0, 60], [-278, 0, 0, 40, 0, -40, 0, 0], [0, -3, -1, 0, 0, 0, -167, 1], [0, 348, 116, 0, 0, 0, 13, -116], [0, 0, 0, -102, -172, 70, 0, 0], [0, 237, 79, 0, 0, 0, -67, -79], [0, 0, 0, 45, 38, 7, 0, 0], [0, 0, 0, 102, -24, 126, 0, 0], [0, -321, -107, 0, 0, 0, 73, 107], [602, 0, 0, -70, 0, 70, 0, 0], [202, 0, 0, -180, 0, 180, 0, 0], [0, 156, 52, 0, 0, 0, 161, -52], [-438, 0, 0, -40, 0, 40, 0, 0], [0, -60, -20, 0, 0, 0, 1, 20], [0, -43, 352, 0, 0, 0, 0, -43], [0, -276, -92, 0, 0, 0, 89, 92], [112, 0, 0, -27, 0, 27, 0, 0], [0, 0, 0, 51, 122, -71, 0, 0], [0, 118, -93, 0, 0, 0, 0, 118], [0, 102, 34, 0, 0, 0, 153, -34], [0, -48, 47, 0, 0, 0, 0, -48], [0, 0, 0, -114, -12, -102, 0, 0], [172, 0, 0, 41, 0, -41, 0, 0], [0, -207, -69, 0, 0, 0, -251, 69], [0, 35, -328, 0, 0, 0, 0, 35], [0, 110, 27, 0, 0, 0, 0, 110], [0, 0, 0, 98, 200, -102, 0, 0], [0, 0, 0, -73, -198, 125, 0, 0], [212, 0, 0, 141, 0, -141, 0, 0], [0, 78, 26, 0, 0, 0, -271, -26], [0, 57, 19, 0, 0, 0, 117, -19], [0, -21, 232, 0, 0, 0, 0, -21], [0, 0, 0, -60, -92, 32, 0, 0], [-48, 0, 0, 171, 0, -171, 0, 0], [0, -130, 57, 0, 0, 0, 0, -130], [0, 153, -68, 0, 0, 0, 0, 153], [0, -66, -22, 0, 0, 0, 61, 22], [0, 0, 0, 41, 86, -45, 0, 0], [-568, 0, 0, 31, 0, -31, 0, 0], [-518, 0, 0, -158, 0, 158, 0, 0], [0, 0, 0, 6, -76, 82, 0, 0], [0, -231, -77, 0, 0, 0, -55, 77], [0, -223, -118, 0, 0, 0, 0, -223], [0, 0, 0, 8, 36, -28, 0, 0], [0, 201, 67, 0, 0, 0, 213, -67], [0, 0, 0, 199, 282, -83, 0, 0], [0, -67, -288, 0, 0, 0, 0, -67], [122, 0, 0, 172, 0, -172, 0, 0], [102, 0, 0, 164, 0, -164, 0, 0], [0, 71, 52, 0, 0, 0, 0, 71], [0, 174, 58, 0, 0, 0, 169, -58], [0, 105, 372, 0, 0, 0, 0, 105], [-218, 0, 0, -100, 0, 100, 0, 0], [0, 0, 0, -83, -122, 39, 0, 0], [232, 0, 0, -201, 0, 201, 0, 0], [0, -201, -67, 0, 0, 0, -115, 67], [0, 330, 110, 0, 0, 0, -267, -110], [352, 0, 0, -35, 0, 35, 0, 0], [0, 0, 0, 6, 220, -214, 0, 0], [0, 295, 132, 0, 0, 0, 0, 295], [0, -130, -213, 0, 0, 0, 0, -130], [0, 0, 0, -20, -52, 32, 0, 0], [0, 237, 79, 0, 0, 0, -147, -79], [0, 71, 142, 0, 0, 0, 0, 71], [-348, 0, 0, -131, 0, 131, 0, 0], [0, 96, 32, 0, 0, 0, -103, -32], [0, -27, -9, 0, 0, 0, -199, 9], [0, 156, 52, 0, 0, 0, -27, -52], [0, 0, 0, 275, 442, -167, 0, 0], [0, 0, 0, 146, 204, -58, 0, 0], [-968, 0, 0, -127, 0, 127, 0, 0], [0, -10, 147, 0, 0, 0, 0, -10], [1082, 0, 0, 80, 0, -80, 0, 0], [0, -210, -70, 0, 0, 0, -259, 70], [0, -96, -433, 0, 0, 0, 0, -96], [0, -273, -91, 0, 0, 0, 125, 91], [0, 0, 0, 75, -246, 321, 0, 0], [0, -34, -613, 0, 0, 0, 0, -34], [0, -183, -61, 0, 0, 0, 65, 61], [0, 0, 0, -129, -14, -115, 0, 0], [0, -173, -158, 0, 0, 0, 0, -173]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_100_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_100_3_d_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_100_3_d_b(:prec:=8)
    chi := MakeCharacter_100_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_3_d_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_100_3_d_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_100_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![80, 0, -20, 0, 1]>],Snew);
    return Vf;
end function;