
// Make newform 1900.2.c.f in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1900_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1900_c_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_1900_2_c_f();".
// 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_1900_2_c_f();".
// 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, 18, 0, 9, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [1, 0, 5, 0, 1, 0], [10, 0, 8, 0, 1, 0], [0, 13, 0, 8, 0, 1], [0, 41, 0, 19, 0, 2]];
    Rf_basisdens := [1, 1, 3, 3, 3, 3];
    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_1900_c();"
function MakeCharacter_1900_c()
    N := 1900;
    order := 2;
    char_gens := [951, 77, 401];
    v := [2, 1, 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_1900_c_Hecke();"
function MakeCharacter_1900_c_Hecke(Kf)
    N := 1900;
    order := 2;
    char_gens := [951, 77, 401];
    char_values := [[1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, -1], [0, 0, 0, 0, 0, 0], [0, 2, 0, 0, -1, -1], [0, 0, 0, -1, 0, 0], [0, 1, 0, 0, 1, -1], [0, -2, 0, 0, -1, -1], [-1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 5, 0], [-2, 0, 2, 1, 0, 0], [-1, 0, 3, -1, 0, 0], [0, 0, 0, 0, 4, 1], [0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 5, -2], [0, -5, 0, 0, -1, -1], [0, -4, 0, 0, 1, 0], [2, 0, -2, 2, 0, 0], [-3, 0, 5, -1, 0, 0], [0, -4, 0, 0, -4, -3], [-1, 0, -2, 2, 0, 0], [0, 1, 0, 0, 4, 2], [1, 0, 0, -5, 0, 0], [0, -1, 0, 0, 10, 0], [-3, 0, -3, -2, 0, 0], [0, -1, 0, 0, -1, 5], [1, 0, -4, 5, 0, 0], [0, 8, 0, 0, -3, -3], [0, 3, 0, 0, -9, -2], [-4, 0, 5, 2, 0, 0], [0, -3, 0, 0, 6, 3], [0, 7, 0, 0, -3, -2], [6, 0, 3, 4, 0, 0], [0, 1, 0, 0, -10, 2], [9, 0, -2, -6, 0, 0], [-8, 0, 5, -4, 0, 0], [-10, 0, -3, 3, 0, 0], [0, 3, 0, 0, 7, -4], [0, -1, 0, 0, 3, -3], [0, -3, 0, 0, 0, 2], [0, 2, 0, 0, -11, 5], [5, 0, -8, -2, 0, 0], [-7, 0, 0, 0, 0, 0], [-2, 0, -4, 1, 0, 0], [0, 1, 0, 0, -2, -2], [0, -6, 0, 0, -6, 1], [2, 0, 2, -9, 0, 0], [17, 0, 0, -4, 0, 0], [0, -6, 0, 0, -1, 8], [0, -1, 0, 0, 4, -4], [-8, 0, 3, 5, 0, 0], [0, 0, 0, 0, 6, -11], [13, 0, 5, 3, 0, 0], [-8, 0, 7, 0, 0, 0], [6, 0, 3, -9, 0, 0], [0, 12, 0, 0, -3, -1], [0, -9, 0, 0, 3, -1], [28, 0, -1, -1, 0, 0], [0, 0, 2, 5, 0, 0], [0, -4, 0, 0, 5, -4], [4, 0, 8, -2, 0, 0], [0, -5, 0, 0, -2, -8], [0, 11, 0, 0, -17, -2], [0, -7, 0, 0, -1, -10], [19, 0, -1, -1, 0, 0], [0, -6, 0, 0, 8, -3], [0, 1, 0, 0, 14, -10], [0, 0, -1, -11, 0, 0], [0, 5, 0, 0, -1, 3], [0, 6, 0, 0, -6, -9], [0, 0, -5, 5, 0, 0], [0, 11, 0, 0, 4, -1], [15, 0, 3, -7, 0, 0], [0, -13, 0, 0, -4, -8], [0, -5, 0, 0, -17, 1], [-7, 0, 17, -3, 0, 0], [0, 3, 0, 0, 15, -1], [13, 0, 2, -16, 0, 0], [0, 1, 0, 0, 6, -1], [9, 0, 0, 3, 0, 0], [-1, 0, -1, -4, 0, 0], [-2, 0, -1, 11, 0, 0], [7, 0, 1, 7, 0, 0], [-8, 0, 8, 1, 0, 0], [0, 1, 0, 0, -17, -4], [-10, 0, -4, 13, 0, 0], [0, -3, 0, 0, 18, -6], [-3, 0, 9, 7, 0, 0], [0, -1, 0, 0, 23, -1], [8, 0, 13, -1, 0, 0], [0, 9, 0, 0, -10, -1], [0, 9, 0, 0, -12, 2], [12, 0, -6, 6, 0, 0], [0, 5, 0, 0, -22, 8], [4, 0, -1, -8, 0, 0], [13, 0, 6, -7, 0, 0], [0, 10, 0, 0, -13, -7], [8, 0, -11, 12, 0, 0], [-24, 0, -6, 6, 0, 0], [0, -4, 0, 0, -3, 11], [-8, 0, 4, 13, 0, 0], [0, 7, 0, 0, 6, 5], [0, 10, 0, 0, -25, -7], [0, 4, 0, 0, 2, 1], [4, 0, -10, -14, 0, 0], [-12, 0, 11, 2, 0, 0], [0, -6, 0, 0, -11, 9], [0, 0, 0, 0, -36, 3], [0, -11, 0, 0, -7, 11], [2, 0, -8, 1, 0, 0], [-22, 0, 3, -2, 0, 0], [0, 7, 0, 0, 30, 0], [0, -5, 0, 0, 16, -11], [0, 14, 0, 0, -8, -10], [6, 0, -11, -2, 0, 0], [28, 0, -5, 2, 0, 0], [13, 0, -4, 0, 0, 0], [0, 0, 0, 0, -22, 0], [0, 5, 0, 0, -5, 10], [0, -1, 0, 0, -11, 6], [24, 0, -15, 3, 0, 0], [0, 0, -1, -1, 0, 0], [0, -7, 0, 0, 9, 12], [0, 0, 0, 0, -12, 9], [0, 10, 0, 0, 11, -9], [6, 0, -4, -16, 0, 0], [14, 0, 7, -7, 0, 0], [25, 0, 3, -15, 0, 0], [-13, 0, 1, 11, 0, 0], [0, 4, 0, 0, 21, 5], [0, -10, 0, 0, -9, 2], [-8, 0, 21, -9, 0, 0], [0, -11, 0, 0, 11, 8], [-13, 0, -3, -10, 0, 0], [0, -5, 0, 0, 3, -2], [-10, 0, 13, -14, 0, 0], [-6, 0, -14, 0, 0, 0], [0, 1, 0, 0, -7, -4], [0, 1, 0, 0, 21, -5], [0, 11, 0, 0, 16, 6], [-5, 0, 11, -9, 0, 0], [10, 0, -23, 7, 0, 0], [4, 0, 5, 6, 0, 0], [0, 3, 0, 0, 8, -3], [0, -17, 0, 0, -16, -7], [-10, 0, 2, 4, 0, 0], [3, 0, -21, 7, 0, 0], [0, -14, 0, 0, 16, -3], [0, 5, 0, 0, -26, 5], [-12, 0, 19, 8, 0, 0], [0, -6, 0, 0, 39, 2], [0, 16, 0, 0, -12, 4], [23, 0, -5, -6, 0, 0], [0, 14, 0, 0, 9, -7], [0, -10, 0, 0, 1, 17], [0, -3, 0, 0, 19, 6], [-11, 0, -7, -11, 0, 0], [10, 0, 15, 7, 0, 0], [0, 0, -9, 18, 0, 0], [0, 18, 0, 0, -29, -9], [26, 0, -11, 2, 0, 0], [0, 17, 0, 0, 1, -14], [0, -9, 0, 0, 18, 0], [0, 1, 0, 0, -30, 3], [0, 0, -6, -8, 0, 0], [0, -11, 0, 0, -1, 7], [0, -4, 0, 0, 25, 4], [-6, 0, -13, -16, 0, 0], [0, -13, 0, 0, -13, -4], [-20, 0, 12, 9, 0, 0], [0, 12, 0, 0, 18, -4], [-1, 0, 19, -7, 0, 0], [5, 0, 18, -20, 0, 0], [14, 0, -23, 7, 0, 0], [0, 0, 0, 0, 14, -6], [-19, 0, -19, -12, 0, 0], [11, 0, -17, -1, 0, 0], [-6, 0, 20, -8, 0, 0], [12, 0, 9, 9, 0, 0], [0, 10, 0, 0, -20, -1], [-12, 0, -8, 20, 0, 0], [0, 8, 0, 0, -13, -10], [0, 0, -27, -6, 0, 0], [0, -7, 0, 0, -3, 13], [0, -7, 0, 0, -29, -1], [0, -12, 0, 0, 15, 1], [33, 0, -12, 3, 0, 0], [0, 8, 0, 0, 26, -13], [0, -10, 0, 0, 24, 9], [-19, 0, -4, -7, 0, 0], [2, 0, -8, 17, 0, 0], [0, 0, 0, 0, -19, 12], [0, -12, 0, 0, 36, 1], [-14, 0, -17, 16, 0, 0], [33, 0, 0, -8, 0, 0], [0, 4, 0, 0, 23, 8], [0, -10, 0, 0, -26, 6], [21, 0, -16, 14, 0, 0], [0, -7, 0, 0, -3, 0], [0, 10, 0, 0, 8, -13], [0, 2, 0, 0, 4, 0], [33, 0, 9, 7, 0, 0], [-30, 0, 11, 5, 0, 0], [0, -6, 0, 0, -23, 10], [-26, 0, 9, -5, 0, 0], [-14, 0, 29, -15, 0, 0], [0, -14, 0, 0, -16, 21], [11, 0, 5, 17, 0, 0], [0, 11, 0, 0, -14, -13], [-11, 0, 26, 3, 0, 0], [3, 0, 11, 12, 0, 0], [0, -17, 0, 0, 12, 9], [-13, 0, -14, 2, 0, 0], [0, -5, 0, 0, 13, 18], [0, 6, 0, 0, 39, 1], [-14, 0, 17, -2, 0, 0], [-30, 0, -7, 9, 0, 0], [0, 9, 0, 0, 34, -12], [21, 0, -24, 1, 0, 0], [0, -6, 0, 0, 18, -5], [0, 14, 0, 0, -17, -13], [-2, 0, -14, 13, 0, 0], [28, 0, 3, 1, 0, 0], [27, 0, -15, 11, 0, 0], [0, 33, 0, 0, -28, -9], [0, -7, 0, 0, -11, -8], [35, 0, -4, 0, 0, 0], [0, -14, 0, 0, -19, 18], [-9, 0, -9, -6, 0, 0], [0, -16, 0, 0, -22, -4], [-26, 0, -1, -1, 0, 0], [0, -17, 0, 0, -29, 7], [-19, 0, -1, -10, 0, 0], [32, 0, 9, -12, 0, 0], [-11, 0, -1, 26, 0, 0], [0, -10, 0, 0, -21, 8], [0, 11, 0, 0, 10, 7], [-9, 0, 22, 6, 0, 0], [0, -18, 0, 0, -6, 13], [4, 0, -13, 1, 0, 0], [42, 0, -3, -10, 0, 0], [0, 9, 0, 0, 18, -1], [1, 0, -8, -2, 0, 0], [0, -10, 0, 0, 6, -16], [-19, 0, -7, -9, 0, 0], [0, -21, 0, 0, 6, -3], [-15, 0, 15, -3, 0, 0], [0, 14, 0, 0, 20, 8], [-21, 0, 3, -1, 0, 0], [46, 0, -6, -5, 0, 0], [0, -14, 0, 0, 2, 14], [0, -7, 0, 0, 23, -1], [21, 0, -12, -16, 0, 0], [0, -7, 0, 0, -20, -14], [-16, 0, 20, 14, 0, 0], [0, -25, 0, 0, -5, 1], [13, 0, -19, 3, 0, 0], [37, 0, 1, -8, 0, 0], [0, -4, 0, 0, 11, 11], [0, -35, 0, 0, 23, 8], [0, 9, 0, 0, -17, 15], [0, 12, 0, 0, 17, -12], [0, 3, 0, 0, -9, 10], [36, 0, 16, 0, 0, 0], [0, 16, 0, 0, -20, -5], [0, 1, 0, 0, -61, 5], [-19, 0, -7, 11, 0, 0], [-6, 0, -3, 15, 0, 0], [-11, 0, 5, -1, 0, 0], [0, 7, 0, 0, -41, 13], [0, -16, 0, 0, -14, 15], [32, 0, 12, 3, 0, 0], [0, -8, 0, 0, 27, -8], [0, -13, 0, 0, 54, -1], [22, 0, -18, 0, 0, 0], [0, 4, 0, 0, -12, -3], [0, -23, 0, 0, 10, 1], [0, -4, 0, 0, 19, -12], [28, 0, 15, 8, 0, 0], [-13, 0, 3, -5, 0, 0], [45, 0, -18, 8, 0, 0], [0, 11, 0, 0, -20, 2], [30, 0, -4, -7, 0, 0], [0, -7, 0, 0, 13, 5], [-23, 0, -8, 3, 0, 0], [0, 0, 0, 0, 10, 6], [8, 0, -14, -2, 0, 0], [0, 6, 0, 0, 38, -25], [0, -21, 0, 0, 42, 10], [18, 0, 4, -1, 0, 0], [-14, 0, 2, 5, 0, 0], [10, 0, -4, -23, 0, 0], [0, 13, 0, 0, -10, -2], [0, -8, 0, 0, -7, 14], [-25, 0, 16, 12, 0, 0], [0, 12, 0, 0, 17, 1], [-27, 0, 27, -1, 0, 0], [-37, 0, 18, -12, 0, 0], [0, 4, 0, 0, 20, -30], [-19, 0, -17, 27, 0, 0], [0, 5, 0, 0, -4, 10], [0, 20, 0, 0, -39, -7], [0, 7, 0, 0, 44, -16], [-8, 0, 21, -7, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1900_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1900_2_c_f();".
// 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_1900_2_c_f(:prec:=6)
    chi := MakeCharacter_1900_c();
    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_1900_2_c_f();".
// 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_1900_2_c_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1900_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![81, 0, 61, 0, 14, 0, 1]>,<7,R![2209, 0, 549, 0, 42, 0, 1]>],Snew);
    return Vf;
end function;