
// Make newform 2100.2.f.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2100_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2100_f_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_2100_2_f_h();".
// 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_2100_2_f_h();".
// 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, 3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [1, 3, 1, 1], [0, -4, 0, -2], [2, -3, 1, -1]];
    Rf_basisdens := [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_2100_f();"
function MakeCharacter_2100_f()
    N := 2100;
    order := 2;
    char_gens := [1051, 701, 1177, 1501];
    v := [2, 1, 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_2100_f_Hecke();"
function MakeCharacter_2100_f_Hecke(Kf)
    N := 2100;
    order := 2;
    char_gens := [1051, 701, 1177, 1501];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [-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 := 2;
    raw_aps := [[0, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 0], [2, 1, 1, 0], [1, 1, 2, -1], [1, -1, 0, -1], [2, 2, 1, -2], [1, 1, -2, -1], [1, 3, 0, 3], [0, 0, 2, 0], [1, 1, 0, -1], [2, 2, 1, -2], [-8, 2, 0, 2], [1, 1, -2, -1], [-1, -1, 0, 1], [7, 1, 0, 1], [0, -4, 0, -4], [4, 4, 4, -4], [3, 3, 0, -3], [-5, -5, -2, 5], [9, -1, 0, -1], [-6, -2, 0, -2], [1, 1, -2, -1], [8, 2, 0, 2], [-3, -5, 0, -5], [2, -4, 0, -4], [13, -1, 0, -1], [7, -3, 0, -3], [-6, 4, 0, 4], [15, 1, 0, 1], [-9, -9, -4, 9], [2, 6, 0, 6], [-1, 1, 0, 1], [1, 1, -6, -1], [-6, -6, -2, 6], [0, 4, 0, 4], [13, -5, 0, -5], [-7, -7, -2, 7], [1, 1, 6, -1], [0, 0, 1, 0], [-1, -1, -8, 1], [-2, -2, 0, 2], [-1, -1, 2, 1], [0, 0, -3, 0], [23, 1, 0, 1], [-1, -1, -6, 1], [-8, 0, 0, 0], [5, 3, 0, 3], [1, 1, -10, -1], [8, 8, 6, -8], [-13, 5, 0, 5], [5, 5, 8, -5], [-10, -10, -8, 10], [2, -10, 0, -10], [-2, -2, 5, 2], [-11, -5, 0, -5], [-24, 2, 0, 2], [9, 9, 4, -9], [6, 6, 9, -6], [6, 6, 4, -6], [-7, 7, 0, 7], [0, 0, -7, 0], [7, 5, 0, 5], [-10, -10, 0, -10], [-11, 3, 0, 3], [-5, 5, 0, 5], [-12, -4, 0, -4], [0, 0, -1, 0], [-17, -7, 0, -7], [4, 4, 2, -4], [-6, -6, 3, 6], [13, 13, 4, -13], [-9, -7, 0, -7], [2, 2, 7, -2], [-6, 2, 0, 2], [1, 1, -6, -1], [2, 2, 2, -2], [-3, 3, 0, 3], [4, 4, -8, -4], [0, 0, -6, 0], [28, 0, 0, 0], [28, 2, 0, 2], [-1, -1, -2, 1], [17, 7, 0, 7], [-5, -5, -6, 5], [5, -13, 0, -13], [-8, -8, 2, 8], [4, 4, 11, -4], [-20, -2, 0, -2], [-1, -1, -12, 1], [-15, -15, -6, 15], [-4, -4, 0, -4], [-1, -1, 0, 1], [9, 9, -6, -9], [-6, 10, 0, 10], [-7, -7, -2, 7], [14, 4, 0, 4], [10, -8, 0, -8], [-15, 3, 0, 3], [-26, -8, 0, -8], [3, 3, -4, -3], [3, 5, 0, 5], [-13, -13, -12, 13], [-8, -8, -2, 8], [-20, 12, 0, 12], [-3, -13, 0, -13], [3, 3, 8, -3], [-2, -2, -13, 2], [-7, -7, 4, 7], [-14, -14, 0, 14], [11, 13, 0, 13], [-10, -10, -5, 10], [7, -15, 0, -15], [-3, -3, -2, 3], [0, 4, 0, 4], [-6, -6, -16, 6], [-31, 3, 0, 3], [9, 9, 10, -9], [3, -11, 0, -11], [-1, -1, -20, 1], [-12, -12, -8, 12], [-4, -4, 1, 4], [0, 0, 3, 0], [-27, 3, 0, 3], [-5, -5, -4, 5], [4, 4, 8, -4], [14, 4, 0, 4], [24, -8, 0, -8], [3, -3, 0, -3], [33, 7, 0, 7], [-18, -10, 0, -10], [9, 7, 0, 7], [-44, 0, 0, 0], [14, 14, 5, -14], [0, 14, 0, 14], [-6, -6, 6, 6], [0, 0, 9, 0], [23, -3, 0, -3], [-12, -12, -5, 12], [10, 10, 10, -10], [3, 3, 4, -3], [-12, -12, 4, 12], [11, 11, 12, -11], [-33, -3, 0, -3], [-10, -10, -6, 10], [4, 4, 0, 4], [1, -9, 0, -9], [2, 2, -11, -2], [-7, -7, -18, 7], [37, 7, 0, 7], [-2, -2, 13, 2], [10, -12, 0, -12], [11, 11, 0, -11], [-5, -5, 0, 5], [-1, -1, 12, 1], [-13, -13, 2, 13], [30, 2, 0, 2], [2, 8, 0, 8], [17, -1, 0, -1], [-4, -18, 0, -18], [23, 1, 0, 1], [3, -3, 0, -3], [5, 5, -14, -5], [-14, 6, 0, 6], [3, 21, 0, 21], [15, 15, 4, -15], [4, 16, 0, 16], [13, -13, 0, -13], [-4, 6, 0, 6], [-8, -8, 13, 8], [7, 7, 16, -7], [14, 14, 8, -14], [-5, -5, 10, 5], [0, 0, 5, 0], [-5, -5, -6, 5], [-14, -4, 0, -4], [40, -8, 0, -8], [-2, -2, 24, 2], [21, -13, 0, -13], [2, 2, 10, -2], [13, 13, 10, -13], [6, -14, 0, -14], [6, 6, -9, -6], [6, 6, 17, -6], [5, 19, 0, 19], [-18, -4, 0, -4], [-2, -2, 5, 2], [-3, -1, 0, -1], [2, -16, 0, -16], [-46, 2, 0, 2], [-43, -5, 0, -5], [-19, 7, 0, 7], [-32, 8, 0, 8], [26, 8, 0, 8], [3, 5, 0, 5], [14, 14, 23, -14], [-10, -8, 0, -8], [2, 2, 15, -2], [14, 14, 13, -14], [13, 13, -6, -13], [-12, -12, -26, 12], [1, 1, 24, -1], [21, 11, 0, 11], [-2, -2, 22, 2], [-16, -4, 0, -4], [-12, -12, -25, 12], [-5, -5, 6, 5], [1, 3, 0, 3], [6, 6, -18, -6], [-5, -5, -24, 5], [16, 16, -1, -16], [52, 2, 0, 2], [13, 13, -10, -13], [-7, -7, -26, 7], [4, -12, 0, -12], [-2, -2, 32, 2], [3, 3, -16, -3], [8, 14, 0, 14], [3, -3, 0, -3], [-17, -7, 0, -7], [-28, 10, 0, 10], [19, 19, 14, -19], [-2, -2, -18, 2], [5, 5, 30, -5], [-9, -9, 4, 9], [2, 4, 0, 4], [-22, -22, -17, 22], [5, 5, 4, -5], [-1, -11, 0, -11], [9, 9, -2, -9], [-18, -18, -1, 18], [-19, -19, -6, 19], [0, 12, 0, 12], [-6, -6, 8, 6], [5, 23, 0, 23], [3, 3, 24, -3], [-8, -8, 6, 8], [-41, 9, 0, 9], [7, 7, 24, -7], [34, 10, 0, 10], [-7, -13, 0, -13], [-1, -1, -20, 1], [33, -5, 0, -5], [-16, 6, 0, 6], [6, 6, 19, -6], [-48, -8, 0, -8], [15, -7, 0, -7], [-6, -2, 0, -2], [42, 10, 0, 10], [37, -1, 0, -1], [-22, -22, -11, 22], [-40, -2, 0, -2], [-5, 9, 0, 9], [-32, -32, -18, 32], [-12, -12, -11, 12], [-13, -13, 16, 13], [-8, 26, 0, 26], [-29, -11, 0, -11], [0, 0, -17, 0], [-27, 11, 0, 11], [-1, -1, -8, 1], [-37, 13, 0, 13], [8, 8, 22, -8], [-15, 7, 0, 7], [-2, -2, 5, 2], [-7, -7, 22, 7], [20, 20, 10, -20], [-22, -8, 0, -8], [-1, -1, 0, 1], [-13, -11, 0, -11], [-10, -10, -4, 10], [-11, -11, -2, 11], [-55, 7, 0, 7], [-30, 6, 0, 6], [45, 11, 0, 11], [-11, -13, 0, -13], [31, -3, 0, -3], [-18, -20, 0, -20], [-28, 14, 0, 14], [26, -10, 0, -10], [3, 3, -12, -3], [-20, -8, 0, -8], [-13, -13, -20, 13], [-10, -10, 4, 10], [15, 5, 0, 5], [3, 3, 10, -3], [12, 12, 13, -12], [23, 1, 0, 1], [15, 15, 10, -15], [-30, -24, 0, -24], [16, 16, 8, -16], [-11, -11, 18, 11], [-21, -3, 0, -3], [-30, 14, 0, 14], [-14, -14, -17, 14], [-18, 16, 0, 16], [-23, -23, -28, 23], [-8, -8, -27, 8], [-8, -28, 0, -28], [31, -3, 0, -3], [-15, 15, 0, 15], [-29, -3, 0, -3], [-10, 10, 0, 10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2100_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2100_2_f_h();".
// 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_2100_2_f_h(:prec:=4)
    chi := MakeCharacter_2100_f();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2100_2_f_h();".
// 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_2100_2_f_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2100_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![16, 0, 28, 0, 1]>,<13,R![-4, -2, 1]>,<23,R![-44, -2, 1]>,<41,R![44, 16, 1]>],Snew);
    return Vf;
end function;