
// Make newform 1200.2.v.j in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1200_v();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1200_v_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_1200_2_v_j();".
// 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_1200_2_v_j();".
// 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, 0, 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, 1, 0, 1], [0, 1, 0, -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_1200_v();"
function MakeCharacter_1200_v()
    N := 1200;
    order := 4;
    char_gens := [751, 901, 401, 577];
    v := [4, 4, 2, 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_1200_v_Hecke();"
function MakeCharacter_1200_v_Hecke(Kf)
    N := 1200;
    order := 4;
    char_gens := [751, 901, 401, 577];
    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, 0, -1, 0], [0, 0, 0, 0], [-1, -1, 1, 1], [0, 2, 2, 0], [-1, 1, -2, 2], [1, -1, -2, 2], [0, -2, -2, 0], [-3, -3, 1, 1], [2, 0, 0, 4], [0, 0, 0, 4], [3, 3, -2, -2], [0, 0, -4, 0], [1, -1, -1, 1], [-5, 5, 1, -1], [-3, -3, 0, 0], [-4, 0, 0, 0], [6, 0, 0, 4], [3, 3, -5, -5], [0, -2, 6, 0], [-1, 1, 4, -4], [0, 2, 2, 0], [1, 1, 3, 3], [-10, 0, 0, 4], [-1, -1, 0, 0], [0, -4, 4, 0], [-7, 7, 1, -1], [11, -11, -1, 1], [0, 4, 0, 0], [1, 1, -6, -6], [7, 7, 1, 1], [0, -6, 2, 0], [-7, 7, -2, 2], [0, 6, -2, 0], [2, 0, 0, -8], [8, 0, 0, -4], [3, 3, 6, 6], [1, -1, -1, 1], [-5, 5, -3, 3], [5, 5, 4, 4], [-12, 0, 0, 0], [6, 0, 0, 8], [0, -10, -10, 0], [-9, 9, -4, 4], [9, -9, 4, -4], [0, 10, -6, 0], [-4, 0, 0, 12], [-7, 7, 1, -1], [-5, 5, 7, -7], [0, 20, 4, 0], [9, 9, -2, -2], [16, 0, 0, 0], [-6, 0, 0, -4], [0, 18, -6, 0], [1, -1, 6, -6], [-3, -3, -7, -7], [-6, 0, 0, -16], [-16, 0, 0, -4], [11, 11, 2, 2], [0, 0, 4, 0], [1, -1, 3, -3], [-11, -11, 4, 4], [3, 3, 7, 7], [0, 14, 6, 0], [-9, 9, 8, -8], [1, -1, -4, 4], [4, 0, 0, -4], [-1, -1, 0, 0], [-5, 5, -5, 5], [0, -4, 4, 0], [-7, -7, 6, 6], [-24, 0, 0, 8], [-9, -9, -3, -3], [-9, 9, 6, -6], [0, -10, 14, 0], [-11, -11, 1, 1], [18, 0, 0, 8], [-5, -5, 2, 2], [0, 16, 0, 0], [0, 16, -4, 0], [4, 0, 0, 24], [14, 0, 0, -12], [0, 6, -10, 0], [15, -15, 0, 0], [0, -6, -14, 0], [-7, -7, -5, -5], [-10, 0, 0, -8], [-1, -1, 0, 0], [0, -12, 12, 0], [-23, 23, 1, -1], [-5, 5, -13, 13], [0, 0, 0, -16], [-1, -1, -11, -11], [0, -30, -6, 0], [0, -2, -26, 0], [13, 13, 5, 5], [2, 0, 0, 4], [0, -24, 0, 0], [17, -17, 7, -7], [-18, 0, 0, 8], [3, 3, -9, -9], [-7, 7, 16, -16], [17, 17, 11, 11], [-34, 0, 0, 8], [-12, 0, 0, -12], [-9, -9, 4, 4], [-5, 5, 11, -11], [-7, -7, 6, 6], [-8, 0, 0, 24], [2, 0, 0, 12], [-9, -9, 17, 17], [-1, 1, -6, 6], [-15, 15, 2, -2], [0, 6, -10, 0], [8, 0, 0, 28], [0, 8, 20, 0], [1, -1, -21, 21], [-5, 5, -15, 15], [21, 21, -4, -4], [-12, 0, 0, -16], [-2, 0, 0, 4], [15, -15, -8, 8], [-7, 7, -20, 20], [-7, -7, 11, 11], [28, 0, 0, -4], [0, -4, 0, 0], [0, 4, -20, 0], [-16, 0, 0, 0], [-17, -17, -3, -3], [-9, 9, 10, -10], [0, -50, -2, 0], [-19, -19, 13, 13], [-16, 0, 0, -20], [11, 11, 18, 18], [0, -24, -8, 0], [0, -16, 24, 0], [5, 5, -4, -4], [-29, -29, -1, -1], [1, -1, -4, 4], [38, 0, 0, 4], [-28, 0, 0, 4], [0, 4, -24, 0], [-7, 7, 9, -9], [-5, 5, 23, -23], [0, 4, -8, 0], [-24, 0, 0, -8], [-1, 1, 18, -18], [25, -25, 6, -6], [0, -10, 30, 0], [5, 5, 1, 1], [-13, -13, -18, -18], [0, -8, -12, 0], [33, -33, -1, 1], [-5, 5, 9, -9], [27, 27, -1, -1], [0, 38, -2, 0], [0, -6, 2, 0], [22, 0, 0, 24], [31, 31, 0, 0], [0, 4, -28, 0], [11, -11, -9, 9], [-7, -7, 6, 6], [-17, -17, 1, 1], [0, 34, -6, 0], [9, -9, 10, -10], [-3, -3, 21, 21], [32, 0, 0, -12], [11, 11, -14, -14], [0, 0, 24, 0], [5, 5, -20, -20], [28, 0, 0, -24], [-34, 0, 0, -16], [0, 30, -2, 0], [-1, 1, -28, 28], [0, 34, -6, 0], [-10, 0, 0, -12], [20, 0, 0, 12], [0, 12, -20, 0], [-7, 7, 13, -13], [0, -12, -8, 0], [7, 7, 9, 9], [0, 26, -6, 0], [-9, 9, -18, 18], [-39, 39, -2, 2], [-27, -27, -11, -11], [10, 0, 0, 4], [-21, -21, -6, -6], [-15, 15, -9, 9], [0, 48, 4, 0], [0, -42, 14, 0], [23, -23, 12, -12], [9, 9, -1, -1], [-4, 0, 0, -12], [0, 44, 8, 0], [11, -11, -5, 5], [-7, -7, 6, 6], [-6, 0, 0, -4], [-9, 9, 10, -10], [9, -9, 2, -2], [13, 13, -23, -23], [18, 0, 0, -12], [-16, 0, 0, -28], [-5, -5, 26, 26], [0, 0, 16, 0], [12, 0, 0, -8], [-31, 31, -4, 4], [0, 18, 10, 0], [17, 17, 15, 15], [6, 0, 0, -12], [4, 0, 0, 12], [-25, -25, -4, -4], [0, 20, 0, 0], [9, -9, -19, 19], [27, -27, 7, -7], [40, 0, 0, 16], [-14, 0, 0, 28], [23, 23, -19, -19], [0, -32, 8, 0], [-21, 21, -15, 15], [-19, -19, 0, 0], [-18, 0, 0, -12], [0, 26, 18, 0], [-50, 0, 0, -12], [-7, 7, 5, -5], [11, -11, 7, -7], [0, 12, 8, 0], [17, 17, 18, 18], [32, 0, 0, -8], [-17, -17, -19, -19], [0, 34, -14, 0], [-17, 17, 6, -6], [0, -2, 14, 0], [0, 0, 0, 44], [0, -16, 4, 0], [-15, 15, -5, 5], [-21, 21, 9, -9], [0, -8, -4, 0], [-3, -3, 16, 16], [-36, 0, 0, 0], [0, -34, -26, 0], [1, 1, -9, -9], [-12, 0, 0, -20], [-23, 23, -11, 11], [0, -20, 20, 0], [9, 9, 14, 14], [-8, 0, 0, 0], [7, 7, -3, -3], [0, -38, 2, 0], [0, 6, 6, 0], [21, 21, 9, 9], [-5, -5, 2, 2], [0, -32, 0, 0], [11, -11, -27, 27], [0, 0, -20, 0], [-11, -11, -4, -4], [-12, 0, 0, 16], [22, 0, 0, 32], [11, 11, 19, 19], [1, -1, -16, 16], [31, 31, -16, -16], [41, -41, 5, -5], [-5, 5, -5, 5], [0, 20, -12, 0], [15, -15, -2, 2], [17, -17, -18, 18], [0, 14, -26, 0], [18, 0, 0, 4], [0, 56, 0, 0], [-15, 15, 7, -7], [37, 37, 12, 12], [30, 0, 0, -24], [3, 3, -5, -5], [-9, 9, -24, 24], [0, 18, -30, 0], [7, 7, -4, -4], [9, -9, -19, 19], [11, -11, -1, 1], [0, -4, 20, 0], [50, 0, 0, -4], [0, 10, -30, 0], [-43, -43, -11, -11], [24, 0, 0, 20], [-5, 5, -11, 11], [38, 0, 0, -16], [-37, -37, -5, -5], [0, 6, -18, 0], [-9, 9, -12, 12], [9, -9, -20, 20], [0, 26, 2, 0], [-26, 0, 0, 0], [0, 28, -8, 0], [-21, 21, 3, -3], [1, 1, -38, -38], [0, 2, 2, 0], [55, -55, 2, -2], [26, 0, 0, -8], [32, 0, 0, 4], [21, 21, 20, 20], [28, 0, 0, -16], [35, 35, -1, -1], [-49, 49, -4, 4], [17, -17, 12, -12], [0, -30, 18, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1200_v_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1200_2_v_j();".
// 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_1200_2_v_j(:prec:=4)
    chi := MakeCharacter_1200_v();
    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_1200_2_v_j();".
// 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_1200_2_v_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1200_v();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, -8, 8, 4, 1]>,<11,R![16, 0, 24, 0, 1]>,<17,R![196, 56, 8, -4, 1]>],Snew);
    return Vf;
end function;