
// Make newform 2100.2.q.i in Magma, downloaded from the LMFDB on 16 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_i();".
// 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_i();".
// 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 := [49, 0, 7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 7, 7];
    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, 0, 1, 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, 0, 1, 0], [0, 0, 0, 0], [0, 1, 0, 0], [3, 1, 3, 0], [1, 0, 0, 0], [-1, 1, -1, 0], [0, -1, -2, -1], [0, 0, -2, 0], [2, 0, 0, -2], [-4, 0, -4, 0], [0, 2, 3, 2], [-3, 0, 0, -1], [-2, 0, 0, 2], [0, 3, -5, 3], [-5, 1, -5, 0], [-7, -3, -7, 0], [0, 4, -1, 4], [0, 3, 0, 0], [-2, 0, 0, -2], [1, 2, 1, 0], [0, -1, -4, -1], [1, 0, 0, -1], [0, -3, -1, -3], [9, 0, 0, -2], [3, 3, 3, 0], [0, 3, 8, 3], [0, 5, -3, 5], [1, -4, 1, 0], [2, 0, 0, -4], [-4, 0, 0, 1], [0, -4, -10, -4], [4, 2, 4, 0], [8, 0, 0, 3], [0, -3, -3, -3], [14, 1, 14, 0], [-17, 0, -17, 0], [0, 5, -4, 5], [1, 0, 0, 7], [0, -4, 10, -4], [18, -2, 18, 0], [4, 0, 0, 4], [0, -7, -5, -7], [-16, 0, -16, 0], [-14, 0, 0, 2], [2, 1, 2, 0], [-10, 0, 0, 3], [6, 0, 0, 3], [14, 2, 14, 0], [0, 2, -15, 2], [0, -5, 11, -5], [-20, 0, 0, 4], [-5, -2, -5, 0], [-5, 0, 0, -9], [0, 1, -15, 1], [-24, 0, -24, 0], [-7, 5, -7, 0], [0, -2, -8, -2], [25, 0, 25, 0], [-9, 0, 0, -5], [-12, -7, -12, 0], [6, 0, 0, -6], [14, 0, 0, -6], [8, -2, 8, 0], [0, -2, 14, -2], [0, -4, -2, -4], [0, -1, -6, -1], [-14, 0, 0, -2], [-7, -7, -7, 0], [10, 0, 0, -2], [9, -3, 9, 0], [0, 3, -17, 3], [-4, -6, -4, 0], [0, -4, 13, -4], [-18, 0, 0, 1], [0, -2, 4, -2], [19, 7, 19, 0], [0, -10, -6, -10], [0, -7, 17, -7], [3, 4, 3, 0], [29, 0, 0, 1], [13, 0, 0, 6], [12, -6, 12, 0], [2, 0, 0, -2], [0, 9, -16, 9], [0, -2, -12, -2], [-3, 0, 0, -11], [0, -8, -13, -8], [16, 0, 0, -2], [26, 0, 0, 3], [0, -2, -28, -2], [-23, 1, -23, 0], [14, -10, 14, 0], [-8, 0, 0, -10], [0, 9, 4, 9], [-22, 0, 0, 2], [0, 6, -10, 6], [0, -6, 0, 0], [0, 6, 26, 6], [0, -6, 11, -6], [16, 0, 0, -6], [22, 2, 22, 0], [13, 7, 13, 0], [0, -3, -3, -3], [2, 3, 2, 0], [-6, -4, -6, 0], [6, 0, 0, 0], [0, -4, -10, -4], [25, -3, 25, 0], [9, 0, 0, 2], [0, 1, -14, 1], [-30, 2, -30, 0], [-26, 0, 0, -6], [-30, -2, -30, 0], [-4, 0, 0, 1], [0, 0, 0, 0], [-30, 0, 0, 1], [11, -15, 11, 0], [0, -5, 3, -5], [11, 0, 0, 11], [-7, -2, -7, 0], [-3, 0, 0, 2], [0, -2, -44, -2], [9, 11, 9, 0], [0, -5, 2, -5], [-33, 0, 0, 3], [0, 8, 15, 8], [0, -5, -11, -5], [-2, 0, 0, 11], [0, -14, -5, -14], [32, -7, 32, 0], [-22, 0, 0, 4], [0, -1, -36, -1], [11, 0, 0, 0], [0, -1, -47, -1], [4, 0, 0, 0], [16, -4, 16, 0], [0, 17, 0, 0], [12, 0, 0, -6], [40, 2, 40, 0], [0, 0, 0, -3], [0, 13, 17, 13], [-2, -7, -2, 0], [-32, 0, 0, -4], [-3, -16, -3, 0], [-2, 0, 0, 8], [-2, 0, 0, 0], [-6, 8, -6, 0], [0, 8, -20, 8], [0, 11, 23, 11], [0, 12, -13, 12], [50, 0, 0, -2], [26, 0, 0, -1], [0, 2, -36, 2], [14, 17, 14, 0], [-10, 0, 0, 16], [0, -20, -4, -20], [0, 1, 5, 1], [0, 0, 0, 0], [18, 12, 18, 0], [0, 2, 32, 2], [29, 0, 0, 7], [22, 0, 0, 3], [0, 21, 3, 21], [15, 15, 15, 0], [3, -1, 3, 0], [8, -6, 8, 0], [-17, 12, -17, 0], [-34, 0, 0, 6], [0, 18, -6, 18], [16, -16, 16, 0], [15, 0, 0, 2], [0, 9, -19, 9], [15, 2, 15, 0], [-14, 9, -14, 0], [18, 0, 0, -14], [2, 0, 0, -18], [-41, 7, -41, 0], [-40, 0, 0, 3], [0, -4, -16, -4], [0, 4, -28, 4], [-30, 0, 0, 4], [8, 0, 0, -4], [0, 3, -43, 3], [-47, -7, -47, 0], [-19, 7, -19, 0], [11, -6, 11, 0], [-28, -2, -28, 0], [0, 4, 0, 4], [5, -5, 5, 0], [0, 14, -11, 14], [13, 0, 0, -11], [0, 5, -24, 5], [0, 18, -8, 18], [-16, 8, -16, 0], [-10, -10, -10, 0], [1, 6, 1, 0], [0, 0, 40, 0], [-35, 0, 0, -7], [0, -4, -34, -4], [23, 5, 23, 0], [-26, 0, 0, 5], [0, 18, 14, 18], [40, 4, 40, 0], [-3, 0, 0, 23], [13, 5, 13, 0], [0, -13, -4, -13], [0, 1, -23, 1], [-10, 0, 0, 14], [8, 17, 8, 0], [0, -6, -23, -6], [29, 0, 0, 1], [6, 0, 0, -11], [0, -3, -45, -3], [-25, -7, -25, 0], [0, 6, -23, 6], [30, 11, 30, 0], [32, -2, 32, 0], [0, -4, 16, -4], [-8, 0, 0, 18], [0, -6, 50, -6], [-38, 0, 0, -13], [0, 8, -34, 8], [0, -4, -12, -4], [13, 0, 0, -17], [-11, 0, 0, 6], [0, -6, -52, -6], [17, 5, 17, 0], [0, 2, 18, 2], [0, 4, -2, 4], [-9, -8, -9, 0], [24, -16, 24, 0], [20, 0, 0, 16], [-8, 4, -8, 0], [-10, 0, 0, -9], [15, -13, 15, 0], [0, 26, -3, 26], [0, 17, 19, 17], [18, 0, 0, 4], [-10, 0, 0, 18], [-15, -3, -15, 0], [0, -16, 0, -16], [10, -11, 10, 0], [0, -24, -1, -24], [10, 0, 0, 10], [0, 13, 29, 13], [0, 0, 0, 8], [35, -1, 35, 0], [-20, 5, -20, 0], [-27, 0, 0, -3], [37, 0, 0, 6], [0, -12, -12, -12], [-30, -18, -30, 0], [9, 0, 0, 20], [33, 3, 33, 0], [0, 4, 10, 4], [-2, -12, -2, 0], [-44, -5, -44, 0], [-53, 0, 0, 5], [0, -12, 44, -12], [4, -13, 4, 0], [-4, 0, 0, -16], [-3, 16, -3, 0], [26, 0, 0, 2], [26, -6, 26, 0], [0, -13, -22, -13], [57, 0, 0, 3], [-19, 0, 0, 13], [8, 0, 0, -12], [-2, 4, -2, 0], [0, -18, -11, -18], [-20, -4, -20, 0], [-55, -4, -55, 0], [0, 21, -6, 21], [51, 0, 0, 0], [0, -1, -20, -1], [0, -14, -16, -14], [-38, -6, -38, 0], [0, -10, 45, -10], [0, -2, -64, -2], [37, -5, 37, 0], [10, -3, 10, 0], [37, 0, 0, 7], [1, 0, 0, 0], [0, -3, 26, -3], [0, 3, -27, 3], [-41, 12, -41, 0], [-41, 0, 0, -5], [-4, 21, -4, 0], [2, 0, 0, 10], [-49, 3, -49, 0], [46, -4, 46, 0], [0, -10, 52, -10], [-53, 0, 0, 5], [-37, 0, 0, 10], [-14, 8, -14, 0], [0, -2, 66, -2], [12, 0, 0, 18], [0, 1, 71, 1], [-10, 0, 0, -13], [0, -14, 18, -14], [0, -20, 20, -20], [-38, -3, -38, 0]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2100_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, -12, 34, -6, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;