
// Make newform 1900.2.i.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1900_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1900_i_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_i_c();".
// 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_i_c();".
// 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 := [9, -12, 19, -2, 5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [180, 47, 57, 8, 15, -3], [-240, 131, -76, 17, -20, 4], [300, -60, 95, -42, 25, -5], [-45, -323, -35, -85, 17, -20], [36, 358, 28, 68, 3, 16]];
    Rf_basisdens := [1, 83, 83, 83, 249, 83];
    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_i();"
function MakeCharacter_1900_i()
    N := 1900;
    order := 3;
    char_gens := [951, 77, 401];
    v := [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_1900_i_Hecke();"
function MakeCharacter_1900_i_Hecke(Kf)
    N := 1900;
    order := 3;
    char_gens := [951, 77, 401];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 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, 0, 0], [0, 0, 0, -1, 0, 1], [0, 0, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0], [2, 0, 0, -1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 2, 1, -1, 0, 0], [0, -1, -1, 1, 1, 1], [0, 1, 2, 1, 5, -2], [0, 2, 4, 2, 3, -3], [4, 1, -1, 0, 0, 0], [-1, 0, 0, 1, 0, 0], [-3, 0, 0, 3, -3, -3], [-3, -2, -1, 2, -3, -1], [0, 2, 4, 2, 3, -3], [0, -1, -2, -1, 4, 2], [3, 4, 2, 1, 3, -3], [0, 0, 0, 0, 3, 2], [0, 2, 4, 2, 0, -2], [3, 2, 1, 2, 3, -3], [-7, -2, -1, 1, -7, 0], [-10, 0, 0, -2, -10, 2], [10, -1, 1, 0, 0, 0], [0, 0, 0, 0, 2, 4], [4, 4, 2, 3, 4, -5], [0, 1, 2, 1, -2, 2], [5, -1, 1, 0, 0, 0], [7, 3, -3, -2, 0, 0], [-7, 2, 1, 0, -7, -1], [-6, 0, 0, -3, 0, 0], [0, 2, 4, 2, -6, -8], [11, 0, 0, -1, 11, 1], [0, -1, -2, -1, -9, 3], [0, -2, -4, -2, -11, 4], [-5, 2, 1, 3, -5, -4], [5, 0, 0, -6, 0, 0], [-6, -2, -1, 2, -6, -1], [2, 1, -1, 2, 0, 0], [0, 0, 0, 0, 11, 1], [1, -4, -2, -3, 1, 5], [7, 1, -1, -4, 0, 0], [0, 0, 0, 0, 11, 6], [12, -2, 2, 4, 0, 0], [-2, 6, 3, 5, -2, -8], [7, 0, 0, 1, 0, 0], [0, 1, 2, 1, 9, -1], [8, 8, 4, 0, 8, -4], [-2, -4, -2, -5, -2, 7], [4, 4, -4, -4, 0, 0], [0, -4, 4, -3, 0, 0], [-2, 6, 3, 4, -2, -7], [-8, -1, 1, 0, 0, 0], [0, -2, -4, -2, 12, 2], [0, 0, 0, 0, 0, -9], [0, 0, 0, 0, 8, 1], [14, 0, 0, 2, 14, -2], [1, -4, -2, 0, 1, 2], [-9, -2, -1, -3, -9, 4], [14, -2, 2, -4, 0, 0], [0, 1, 2, 1, 6, -3], [16, 10, 5, 0, 16, -5], [9, -1, 1, -4, 0, 0], [4, -8, -4, 9, 4, -5], [1, -2, 2, -1, 0, 0], [0, 3, 6, 3, -5, 0], [0, -1, -2, -1, -15, -3], [2, -2, 2, 13, 0, 0], [18, -4, -2, 0, 18, 2], [9, -6, -3, -12, 9, 15], [-10, 1, -1, -5, 0, 0], [2, -1, 1, -2, 0, 0], [14, -6, -3, -7, 14, 10], [0, -6, -12, -6, -5, 12], [-25, 1, -1, 5, 0, 0], [9, 1, -1, -1, 0, 0], [5, -6, -3, 8, 5, -5], [0, 2, 4, 2, 8, 1], [17, 4, 2, -5, 17, 3], [-3, -6, -3, 6, -3, -3], [0, -1, -2, -1, 19, -2], [8, -1, 1, 4, 0, 0], [2, 8, 4, 3, 2, -7], [0, -2, -4, -2, 9, -3], [0, 2, 4, 2, 16, -3], [-8, -8, -4, 7, -8, -3], [0, 0, 0, 0, -3, -3], [11, 3, -3, -4, 0, 0], [-4, 1, -1, -4, 0, 0], [3, 8, 4, 5, 3, -9], [17, 3, -3, -2, 0, 0], [-13, 0, 0, 8, 0, 0], [0, -3, -6, -3, -12, -3], [2, 1, -1, -4, 0, 0], [15, -2, -1, -5, 15, 6], [-9, 2, 1, 10, -9, -11], [0, 4, 8, 4, -12, -9], [0, 1, 2, 1, -10, -2], [18, -3, 3, 3, 0, 0], [0, -6, -12, -6, -6, 7], [0, 1, 2, 1, -5, -11], [0, -1, -2, -1, 22, -6], [0, 5, 10, 5, 3, 3], [13, -2, 2, -4, 0, 0], [0, -4, 4, 2, 0, 0], [-9, -6, 6, 10, 0, 0], [-14, 3, -3, -6, 0, 0], [-12, 6, 3, 0, -12, -3], [0, -2, -4, -2, -18, 6], [0, -2, -4, -2, -17, 2], [30, 2, -2, -6, 0, 0], [-7, -7, 7, 0, 0, 0], [-5, 10, 5, -9, -5, 4], [0, -5, -10, -5, 7, 2], [24, 0, 0, 1, 0, 0], [0, 6, 12, 6, 2, -6], [-2, 10, 5, -7, -2, 2], [-13, -12, -6, -9, -13, 15], [-3, 7, -7, 1, 0, 0], [4, -4, 4, -9, 0, 0], [0, -9, -18, -9, -21, 12], [0, -2, -4, -2, -9, -1], [-4, -4, 4, 9, 0, 0], [-3, -3, 3, 3, 0, 0], [-4, -2, 2, 3, 0, 0], [3, 4, -4, -7, 0, 0], [0, -4, -2, -7, 0, 9], [0, 4, 8, 4, -13, -9], [-3, -8, -4, -8, -3, 12], [29, 10, 5, -2, 29, -3], [6, 8, -8, -11, 0, 0], [4, 18, 9, 3, 4, -12], [9, -2, -1, 10, 9, -9], [0, -6, -12, -6, -11, 7], [-9, 0, 0, -8, -9, 8], [-13, -1, 1, 13, 0, 0], [0, -1, -2, -1, -2, -2], [0, 1, 2, 1, 30, -3], [0, -3, 3, 8, 0, 0], [8, -2, 2, 9, 0, 0], [-32, -1, 1, -6, 0, 0], [0, -4, -8, -4, 19, 16], [0, -5, -10, -5, -4, -5], [0, 0, 0, 0, 16, 0], [0, 2, 4, 2, -6, 3], [-1, -7, 7, 11, 0, 0], [4, 18, 9, -5, 4, -4], [-3, -10, -5, -3, -3, 8], [-25, -8, -4, -9, -25, 13], [0, 7, 14, 7, -13, -15], [19, -3, 3, 7, 0, 0], [3, 12, 6, 13, 3, -19], [22, -4, 4, -6, 0, 0], [0, 3, 6, 3, -39, -5], [0, 1, 2, 1, -14, -1], [-10, 0, 0, 0, -10, 0], [-33, 4, -4, 4, 0, 0], [16, -5, 5, 9, 0, 0], [3, 6, 3, 12, 3, -15], [0, 1, 2, 1, 16, -3], [0, 2, 4, 2, 6, -12], [-18, 10, 5, 10, -18, -15], [-24, 0, 0, 9, -24, -9], [10, 8, 4, 4, 10, -8], [32, -6, -3, -10, 32, 13], [-13, 5, -5, -2, 0, 0], [-9, 14, 7, -4, -9, -3], [-42, 0, 0, 2, -42, -2], [0, -2, -4, -2, -7, 14], [-8, 0, 0, 6, -8, -6], [0, -3, -6, -3, 7, -1], [-3, -5, 5, 1, 0, 0], [-15, -10, -5, 10, -15, -5], [-35, -4, -2, -6, -35, 8], [19, -2, 2, 7, 0, 0], [0, 7, 14, 7, -6, -10], [0, 1, 2, 1, 9, 12], [0, 2, 4, 2, 14, -6], [-21, -18, -9, -3, -21, 12], [7, 9, -9, 3, 0, 0], [-11, -4, -2, -1, -11, 3], [0, 1, 2, 1, 8, 8], [-9, -3, 3, 3, 0, 0], [0, 4, 8, 4, 4, 0], [2, 16, 8, -9, 2, 1], [14, -6, 6, -10, 0, 0], [3, -9, 9, 0, 0, 0], [0, 1, 2, 1, -49, -1], [1, 16, 8, 3, 1, -11], [16, -1, 1, 4, 0, 0], [0, -7, 7, -1, 0, 0], [0, -2, -4, -2, 34, -3], [0, 0, 0, 0, 11, 1], [4, -2, 2, 15, 0, 0], [8, -12, -6, -10, 8, 16], [0, -7, -14, -7, 21, 15], [0, 2, 4, 2, -5, 2], [0, -2, -4, -2, 9, 8], [10, -4, -2, -2, 10, 4], [-6, -4, 4, -4, 0, 0], [-2, -1, 1, 21, 0, 0], [0, 9, 18, 9, 20, -2], [0, -5, -10, -5, 18, 2], [-38, 12, 6, 14, -38, -20], [-13, 6, 3, -8, -13, 5], [14, -4, -2, 4, 14, -2], [0, 8, 16, 8, -2, -10], [0, -7, -14, -7, -26, 10], [0, -10, -20, -10, -1, 13], [11, 14, 7, -2, 11, -5], [-23, 5, -5, -2, 0, 0], [-23, -14, -7, -9, -23, 16], [0, 5, 10, 5, -39, -15], [-30, 5, -5, 10, 0, 0], [0, -3, -6, -3, 16, 14], [0, -7, 7, 14, 0, 0], [0, 1, 2, 1, 46, 4], [-13, -6, -3, 12, -13, -9], [-32, 0, 0, 7, 0, 0], [-23, -6, 6, 10, 0, 0], [-23, 8, 4, -3, -23, -1], [0, -6, -12, -6, -5, 13], [4, 10, -10, -9, 0, 0], [17, 0, 0, 5, 17, -5], [20, -6, -3, -3, 20, 6], [-12, -14, -7, -8, -12, 15], [0, -5, -10, -5, -12, 8], [41, -5, 5, -6, 0, 0], [42, 0, 0, -3, 42, 3], [40, -2, -1, -6, 40, 7], [10, -5, 5, -4, 0, 0], [0, 3, 6, 3, -25, 8], [0, -7, -14, -7, -23, 19], [35, -3, 3, -9, 0, 0], [-26, -2, 2, 11, 0, 0], [33, -1, 1, 1, 0, 0], [24, -2, -1, -2, 24, 3], [1, 4, -4, -9, 0, 0], [-1, -6, 6, -4, 0, 0], [-27, 8, 4, -16, -27, 12], [0, -3, -6, -3, 8, 10], [-10, 18, 9, 5, -10, -14], [-8, 2, -2, -17, 0, 0], [0, 0, 0, 0, 3, 13], [0, -2, -4, -2, 36, -1], [-21, -6, -3, -6, -21, 9], [6, 6, -6, -21, 0, 0], [0, 7, 14, 7, -4, -16], [0, 0, 0, 0, -28, 4], [32, -8, -4, 11, 32, -7], [0, 7, 14, 7, 0, -9], [20, -8, 8, 8, 0, 0], [7, -6, -3, 4, 7, -1], [-12, 4, -4, -23, 0, 0], [0, -7, -14, -7, 31, 25], [9, -10, -5, -4, 9, 9], [0, 5, 10, 5, 1, -16], [0, -9, -18, -9, -32, 10], [-7, 4, -4, -13, 0, 0], [-57, -10, -5, -1, -57, 6], [0, -9, -18, -9, 19, 9], [0, 2, 4, 2, -8, -6], [11, -18, -9, -25, 11, 34], [53, -2, -1, -7, 53, 8], [-17, -6, -3, 8, -17, -5], [0, -7, -14, -7, -12, 9], [37, 11, -11, -14, 0, 0], [6, -9, 9, -9, 0, 0], [-11, -5, 5, -10, 0, 0], [0, -5, -10, -5, 35, 20], [0, -1, -2, -1, 7, -1], [-16, -8, 8, -2, 0, 0], [18, -7, 7, 19, 0, 0], [3, 26, 13, 15, 3, -28], [5, -9, 9, 12, 0, 0], [0, 6, 12, 6, -1, 5], [28, 18, 9, -4, 28, -5], [-49, -5, 5, -3, 0, 0], [-42, -22, -11, 1, -42, 10], [0, -10, -20, -10, -15, 5], [0, -3, -6, -3, 20, 7], [29, 11, -11, -14, 0, 0], [36, 1, -1, 8, 0, 0], [0, -11, -22, -11, -2, 17], [27, 10, -10, -7, 0, 0], [2, -6, -3, 12, 2, -9], [0, 2, 4, 2, -13, -17], [-22, -11, 11, 2, 0, 0], [0, 3, 6, 3, -17, -1], [22, -2, -1, -20, 22, 21], [14, 5, -5, 4, 0, 0], [24, -8, 8, 25, 0, 0], [38, 2, -2, -11, 0, 0], [0, -2, -4, -2, 24, 18], [10, 8, -8, 9, 0, 0], [12, 28, 14, 11, 12, -25], [-38, -3, 3, 16, 0, 0], [0, -6, -12, -6, 47, 9], [10, -14, -7, -22, 10, 29], [-14, -32, -16, -13, -14, 29], [49, 7, -7, -11, 0, 0], [33, -10, -5, 12, 33, -7], [12, 14, 7, 17, 12, -24], [0, -4, -8, -4, -3, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1900_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1900_2_i_c();".
// 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_i_c(:prec:=6)
    chi := MakeCharacter_1900_i();
    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_i_c();".
// 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_i_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1900_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![9, 24, 67, -2, 9, 1, 1]>],Snew);
    return Vf;
end function;