
// Make newform 2100.2.q.k in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2100_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2100_q_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_q_k();".
// 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_q_k();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 1, 0], [0, 4, 0, 1], [0, 2, 0, -1]];
    Rf_basisdens := [1, 2, 2, 2];
    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_q();"
function MakeCharacter_2100_q()
    N := 2100;
    order := 3;
    char_gens := [1051, 701, 1177, 1501];
    v := [3, 3, 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_2100_q_Hecke();"
function MakeCharacter_2100_q_Hecke(Kf)
    N := 2100;
    order := 3;
    char_gens := [1051, 701, 1177, 1501];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 1, 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], [1, 1, 0, 0], [0, 0, 0, 0], [0, -1, 0, -1], [0, 0, 1, 1], [1, 0, 1, -2], [0, 0, 1, 1], [0, -7, 0, 0], [0, 0, 2, -1], [-6, 0, -1, 2], [1, 1, 2, 2], [0, 1, -2, 1], [0, 0, 1, -2], [1, 0, 1, -2], [0, 6, 0, 0], [0, 0, 2, 2], [6, 6, 1, 1], [0, -4, 4, -2], [-1, -1, -1, -1], [0, 0, 3, -6], [5, 5, 1, 1], [0, 11, 0, 0], [6, 0, -1, 2], [0, -6, -2, 1], [-8, 0, -2, 4], [12, 12, -1, -1], [0, 13, 2, -1], [0, 0, -6, 3], [1, 1, 2, 2], [18, 0, 0, 0], [7, 0, 1, -2], [0, 6, 4, -2], [0, 0, 1, 1], [-7, 0, 2, -4], [0, -12, 0, 0], [-14, -14, -2, -2], [2, 2, -2, -2], [0, -8, 0, 0], [-6, 0, 3, -6], [0, 12, 4, -2], [6, 6, 0, 0], [-13, 0, 0, 0], [0, 6, 0, 0], [11, 11, -1, -1], [12, 0, -1, 2], [-2, -2, -2, -2], [14, 0, -2, 4], [16, 0, 2, -4], [6, 6, -5, -5], [0, -7, 0, 0], [0, -6, -4, 2], [-6, 0, -4, 8], [4, 4, 0, 0], [-6, 0, 3, -6], [0, -6, -2, 1], [0, 0, -2, -2], [-6, -6, 2, 2], [0, 2, -12, 6], [5, 5, 3, 3], [12, 0, -4, 8], [11, 11, 3, 3], [-12, 0, -4, 8], [1, 0, 1, -2], [0, 0, -5, -5], [0, -11, -6, 3], [0, 12, 6, -3], [0, 17, 0, 0], [1, 0, -3, 6], [-24, -24, 0, 0], [-10, 0, 0, 0], [-6, -6, -1, -1], [0, 6, -2, 1], [23, 23, 3, 3], [0, -5, -6, 3], [5, 0, 6, -12], [0, 0, -12, 6], [0, 0, 5, 5], [0, 13, -2, 1], [0, -12, 4, -2], [-17, -17, 0, 0], [-6, 0, 2, -4], [23, 0, -2, 4], [6, 6, -6, -6], [-29, 0, 1, -2], [0, -10, 0, 0], [0, -12, -4, 2], [-6, 0, 0, 0], [0, -11, -10, 5], [6, 0, -3, 6], [-23, 0, -3, 6], [0, -6, 8, -4], [-12, -12, 0, 0], [-25, -25, -3, -3], [24, 0, 2, -4], [0, 11, 4, -2], [18, 0, -2, 4], [0, 6, -12, 6], [-12, -12, 6, 6], [0, 1, 2, -1], [0, -7, -16, 8], [-8, 0, 6, -12], [-30, -30, 0, 0], [-6, -6, 0, 0], [0, -12, -14, 7], [-5, -5, 8, 8], [17, 17, -1, -1], [-24, 0, 5, -10], [0, 18, -10, 5], [24, 24, 2, 2], [5, 0, 2, -4], [0, 25, 2, -1], [20, 20, -6, -6], [18, 0, -2, 4], [-5, -5, 4, 4], [8, 0, 0, 0], [-30, -30, -1, -1], [7, 0, -3, 6], [6, 6, 1, 1], [0, -6, -2, 1], [24, 0, -4, 8], [19, 19, -4, -4], [-17, 0, 3, -6], [0, 0, -6, 3], [12, 12, 5, 5], [0, -7, 16, -8], [30, 0, 5, -10], [0, -28, -4, 2], [0, 18, 4, -2], [-5, 0, -1, 2], [0, -23, 10, -5], [7, 7, 2, 2], [6, 0, 5, -10], [0, 11, 16, -8], [-26, 0, -4, 8], [0, 42, 2, -1], [5, 0, 0, 0], [-18, -18, 7, 7], [20, 20, 2, 2], [36, 0, 1, -2], [6, 6, -8, -8], [-22, 0, -8, 16], [0, -24, 14, -7], [-22, -22, -4, -4], [-18, 0, 6, -12], [-11, -11, 0, 0], [30, 0, 1, -2], [13, 0, -3, 6], [0, 0, 1, 1], [0, -22, 0, 0], [0, 30, -4, 2], [0, 10, 0, 0], [24, 0, -4, 8], [7, 0, 3, -6], [0, 6, -10, 5], [-43, -43, 3, 3], [30, 0, -5, 10], [0, 17, 8, -4], [0, 24, 6, -3], [-11, 0, -7, 14], [18, 18, 7, 7], [0, -18, 10, -5], [36, 0, 2, -4], [31, 0, -1, 2], [0, 36, 8, -4], [-30, -30, 5, 5], [-12, -12, -7, -7], [19, 19, -8, -8], [-13, -13, -3, -3], [-19, 0, -2, 4], [0, 24, 18, -9], [-24, -24, 2, 2], [-4, 0, 0, 0], [0, -30, 2, -1], [-1, -1, 13, 13], [1, 1, -4, -4], [-12, 0, 12, -24], [-7, 0, 2, -4], [-6, -6, 10, 10], [-20, 0, -4, 8], [0, -1, -16, 8], [0, -47, 6, -3], [6, 0, 9, -18], [-11, 0, 1, -2], [0, 24, 12, -6], [6, 6, -8, -8], [36, 36, 2, 2], [14, 14, -2, -2], [29, 29, 3, 3], [0, -1, 4, -2], [-12, -12, -1, -1], [0, -11, -6, 3], [30, 0, 1, -2], [0, -40, 0, 0], [0, -12, 0, 0], [0, 0, -9, -9], [12, 12, 1, 1], [-23, -23, 0, 0], [0, 28, 12, -6], [-18, 0, 6, -12], [0, 48, 6, -3], [30, 30, -2, -2], [29, 0, 6, -12], [0, 22, 12, -6], [28, 28, -6, -6], [0, 0, -11, 22], [30, 30, -6, -6], [0, -16, -16, 8], [0, 18, 16, -8], [-60, 0, 2, -4], [34, 34, 0, 0], [0, 31, -6, 3], [-6, 0, 5, -10], [16, 0, -2, 4], [0, 12, -8, 4], [18, 18, 5, 5], [0, -19, -20, 10], [-16, -16, 4, 4], [-24, -24, -4, -4], [0, 0, -12, 6], [42, 0, 4, -8], [0, -1, 16, -8], [29, 0, 4, -8], [0, -48, 6, -3], [0, 16, -24, 12], [6, 0, 10, -20], [8, 0, 10, -20], [0, 60, 0, 0], [-6, -6, -6, -6], [0, 43, -6, 3], [0, 36, -2, 1], [5, 5, -5, -5], [-20, -20, 12, 12], [-49, 0, 2, -4], [60, 60, 3, 3], [13, 0, -11, 22], [-6, -6, 1, 1], [0, 5, 12, -6], [0, -54, 6, -3], [-12, 0, 0, 0], [0, 0, -3, 6], [18, 18, 3, 3], [0, -19, -24, 12], [-31, -31, -7, -7], [0, -40, 8, -4], [0, 0, -6, 12], [0, -12, -4, 2], [-5, 0, -7, 14], [-36, -36, 1, 1], [10, 10, -8, -8], [36, 0, 8, -16], [25, 0, 9, -18], [0, -48, 8, -4], [18, 18, 12, 12], [-25, 0, -2, 4], [-48, -48, -2, -2], [0, 54, -2, 1], [-35, -35, -8, -8], [-1, -1, 7, 7], [42, 0, 3, -6], [0, -2, -24, 12], [-19, -19, -9, -9], [-24, 0, -10, 20], [-5, -5, 4, 4], [-29, 0, -7, 14], [18, 18, 6, 6], [0, -1, -16, 8], [36, 0, -1, 2], [18, 0, -8, 16], [22, 0, 0, 0], [42, 42, 3, 3], [0, -31, 20, -10], [-34, -34, 0, 0], [-19, -19, 1, 1], [0, -19, -20, 10], [10, 0, 2, -4], [0, -8, -28, 14], [0, 18, -18, 9], [-11, -11, 8, 8], [0, 17, -24, 12], [0, 0, 0, 0], [24, 24, -10, -10], [13, 13, 2, 2], [-6, 0, 0, 0], [-7, 0, -14, 28], [0, -8, 28, -14], [0, -12, -30, 15], [-31, -31, -5, -5], [6, 0, 9, -18], [55, 55, -2, -2], [-24, 0, -11, 22], [48, 48, -2, -2], [-12, -12, 10, 10], [0, 6, -34, 17], [6, 0, -9, 18], [-23, 0, -1, 2], [-18, -18, -12, -12], [0, -7, 4, -2], [24, 0, -13, 26], [0, -36, -10, 5], [-29, 0, 3, -6], [0, -14, 0, 0], [0, 42, -4, 2], [-26, -26, -4, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2100_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2100_2_q_k();".
// 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_q_k(:prec:=4)
    chi := MakeCharacter_2100_q();
    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_q_k();".
// 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_q_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2100_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![324, 0, 18, 0, 1]>,<13,R![-17, -2, 1]>],Snew);
    return Vf;
end function;