
// Make newform 390.2.y.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_390_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_390_y_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_390_2_y_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_390_2_y_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 := [1, 0, -1, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_390_y();"
function MakeCharacter_390_y()
    N := 390;
    order := 6;
    char_gens := [131, 157, 301];
    v := [6, 3, 4];
    // 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_390_y_Hecke();"
function MakeCharacter_390_y_Hecke(Kf)
    N := 390;
    order := 6;
    char_gens := [131, 157, 301];
    char_values := [[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, 1, 0, 0], [0, -1, 0, 0], [0, 2, -1, -2], [2, -1, -1, 1], [1, 1, -1, -2], [2, -3, -2, 0], [4, -1, -2, 1], [0, 1, 3, 1], [3, -1, 3, 0], [-3, 2, 3, -4], [4, 0, 0, 0], [-4, 1, -4, 0], [4, -3, -4, 6], [-2, -5, 1, 5], [-5, 0, 10, 1], [1, 0, -2, 6], [0, 6, -2, 6], [0, -7, -2, -7], [-5, -1, -5, 0], [0, -1, -3, -1], [-5, 0, 10, 4], [-12, 0, 0, 0], [-1, 0, 2, 7], [2, 4, -2, -8], [0, 10, 0, -10], [-7, 4, 7, -8], [-1, 0, 2, 11], [1, 3, 1, 0], [-10, 0, 0, 0], [0, 9, 0, -9], [0, 4, 0, 0], [0, 16, 0, -8], [4, -15, -2, 15], [0, -4, -8, -4], [0, -4, 13, -4], [7, 14, 0, -7], [-6, 0, 12, 1], [4, -10, -2, 10], [8, 0, 8, 0], [24, 2, -12, -2], [13, -3, -13, 6], [-4, 14, 0, -7], [0, -2, 6, -2], [-5, 2, -5, 0], [-8, -4, -8, 0], [0, 11, -9, 11], [18, -2, -18, 4], [-6, -10, -6, 0], [-6, -1, 3, 1], [-8, 24, 0, -12], [6, 0, -12, 2], [-11, -18, 0, 9], [0, -6, -1, -6], [0, -2, 18, -2], [0, -3, 0, 0], [-7, 5, -7, 0], [0, -6, -6, -6], [8, 8, -8, -16], [20, 5, -10, -5], [0, -2, 0, 1], [5, 1, 5, 0], [-2, 12, 1, -12], [-1, 0, 2, 21], [-9, -14, 0, 7], [0, 0, 0, -14], [-1, 0, 2, -28], [0, -12, 8, -12], [-11, 0, 22, 8], [-10, -3, 5, 3], [-26, -4, 26, 8], [-14, 1, -14, 0], [-21, -14, 0, 7], [-15, 7, -15, 0], [4, 27, -2, -27], [2, -14, -2, 28], [-8, 20, 4, -20], [31, 8, 0, -4], [-20, 2, 10, -2], [-10, -11, 10, 22], [0, 4, 23, 4], [-10, -2, 10, 4], [-2, -22, 0, 11], [13, 3, -13, -6], [-6, -20, 3, 20], [15, 3, -15, -6], [20, 0, -40, 0], [0, 8, 6, 8], [-3, -14, -3, 0], [0, -4, -33, -4], [5, 0, -10, -15], [11, 0, -22, 3], [-16, 0, 16, 0], [34, -9, -17, 9], [-27, -3, 27, 6], [-30, 12, 0, -6], [6, 27, -3, -27], [-3, 16, 3, -32], [10, 34, 0, -17], [-17, 7, -17, 0], [4, 6, 0, -3], [3, 0, -6, -31], [5, -6, 5, 0], [0, 8, 0, -8], [32, 8, -32, -16], [-17, 10, 0, -5], [-15, 0, 30, 2], [8, 24, 8, 0], [6, 0, -12, 9], [12, 24, 0, -12], [-17, 0, 17, 0], [12, -38, -6, 38], [12, -1, 12, 0], [-32, 5, 16, -5], [-2, -44, 0, 22], [0, 8, 4, 8], [0, 11, 10, 11], [24, 12, -12, -12], [10, -6, 10, 0], [20, 2, 20, 0], [0, 2, -38, 2], [16, -13, -16, 26], [-1, -12, -1, 0], [20, 0, -40, 14], [8, -20, -4, 20], [-25, 9, 25, -18], [-10, 4, 0, -2], [0, -11, 18, -11], [0, 6, -22, 6], [-1, 0, 2, 15], [6, 0, -12, 35], [16, 8, -16, -16], [-2, 18, -2, 0], [-17, 11, 17, -22], [22, -2, 22, 0], [0, -4, -30, -4], [42, -2, -42, 4], [-40, 6, 20, -6], [-20, 2, 10, -2], [8, 8, -4, -8], [-24, -15, 24, 30], [-20, 16, 0, -8], [14, 14, -14, -28], [12, -30, -6, 30], [-12, 0, 24, -28], [30, 7, -30, -14], [0, -22, 10, -22], [4, 0, -8, -19], [-4, 0, 8, 3], [1, -50, 0, 25], [7, 0, -14, 9], [4, 13, -2, -13], [-30, -7, 30, 14], [-14, 0, 28, 6], [4, 36, 4, 0], [-4, 0, -4, 0], [6, -44, 0, 22], [0, 2, -54, 2], [0, -9, 24, -9], [11, 0, -22, -2], [2, -12, 0, 6], [-16, 0, -16, 0], [32, -10, -16, 10], [9, 0, -18, 1], [0, -4, 48, -4], [-14, -3, -14, 0], [-16, 0, 32, -4], [11, 11, -11, -22], [-8, 9, 4, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_390_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_390_2_y_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_390_2_y_c(:prec:=4)
    chi := MakeCharacter_390_y();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_390_2_y_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_390_2_y_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_390_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, -12, 14, -6, 1]>],Snew);
    return Vf;
end function;