
// Make newform 546.2.j.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_546_j();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_546_j_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_546_2_j_e();".
// 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_546_2_j_e();".
// 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 := [16, 76, 345, 164, 233, 36, 110, 14, 15, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [438544, -66732, -301041, 1003571, -156870, 19422, -67528, 8466, -2241, -503], [3620228, 274164, 1236807, 579550, 644490, -79794, -28346, -34782, 9207, -3064], [-3118196, -3391749, -3196425, -478621, -1723536, -625302, -334006, -12699, -10833, -3977], [-2149816, -9757146, -3491505, -6543167, -967302, -3082518, -375260, -413376, 54315, -27409], [-1074620, -3963994, -2349679, -2887781, -440214, -1521222, -200758, -216152, 29941, -14545], [165664, 16896855, 5556402, 10649492, 744876, 5037264, 561428, 695481, -96426, 45928], [5377120, 29957226, 17738031, 20509619, 4653846, 10555542, 1662752, 1399728, -178473, 97465], [742892, 14145267, 5109630, 11604142, 367938, 5520312, 484042, 805587, -117222, 51344]];
    Rf_basisdens := [1, 1, 2044008, 1022004, 1022004, 2044008, 340668, 1022004, 2044008, 1022004];
    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_546_j();"
function MakeCharacter_546_j()
    N := 546;
    order := 3;
    char_gens := [365, 157, 379];
    v := [3, 1, 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_546_j_Hecke();"
function MakeCharacter_546_j_Hecke(Kf)
    N := 546;
    order := 3;
    char_gens := [365, 157, 379];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 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 := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 1], [2, 1, 0, 0, 1, 2, -1, 1, 0, 0], [0, -1, 0, -1, 0, 0, 0, -1, 0, 0], [0, 0, -1, -1, 0, 0, 0, 0, -1, 1], [0, -2, -2, 0, -1, -1, 2, 0, -2, 1], [0, 0, -1, -1, -1, 0, -1, 0, -2, 1], [0, 0, 0, -1, 0, 0, -1, -1, 1, 0], [0, 1, 1, -1, 1, 2, -1, -1, 1, -1], [1, 0, 0, -1, 0, 0, 0, 0, -1, 1], [0, 0, 0, 1, 1, 2, -1, 1, 1, -1], [1, -3, 0, 0, -1, 1, 0, -1, -1, -1], [-1, -1, 0, 0, 2, -1, -1, 1, 1, 1], [0, -1, -1, 1, -1, 3, 2, 1, -2, 1], [0, 0, -4, -2, 1, 0, 1, 0, -1, 2], [0, 1, 1, 0, -1, -3, 0, 0, 0, 1], [1, 0, 0, 0, 1, 1, 0, 2, 1, 1], [4, 3, 0, 0, 2, 4, -2, 0, 0, 0], [0, -3, -3, -3, 1, -2, 0, -3, 0, -1], [-2, -1, 0, 0, -2, -2, 2, 0, 0, 0], [0, 0, 0, 1, 1, 0, 1, 0, 0, 1], [-3, 0, 7, 2, -1, 0, -1, 0, 4, -5], [-2, 0, 0, 0, -1, -2, -1, -2, -2, -2], [-4, 1, 0, 0, -4, -4, 2, -2, -2, -2], [3, 3, 0, 0, 1, 3, 0, 1, 1, 1], [1, 0, 2, 0, 0, 0, 0, 0, 3, -3], [0, 0, 0, 2, 0, -5, 2, 2, -2, 0], [-6, 2, 0, 0, -1, -6, 1, 1, 0, 0], [0, -1, -1, 0, -2, 4, 3, 0, -3, 2], [-5, 1, 0, 0, -1, -5, -1, 2, -2, -2], [1, 0, 5, 2, -1, 0, -1, 0, 2, -3], [1, 0, 0, 0, 0, 1, 1, -3, 1, 1], [0, 2, 2, -4, -2, -9, 1, -4, -1, 2], [0, 6, 6, 2, 4, 2, -4, 2, 4, -4], [0, -4, -4, -3, 0, -3, -1, -3, 1, 0], [0, -3, -3, 1, -1, -10, 0, 1, 0, 1], [0, -2, -2, -2, -5, -2, 2, -2, -2, 5], [0, 1, 1, 1, 0, -3, 4, 1, -4, 0], [-6, 1, 0, 0, -3, -6, 1, -1, -2, -2], [-9, 0, -1, 2, 0, 0, 0, 0, 2, -2], [0, -4, -4, 0, 2, 5, 1, 0, -1, -2], [5, -4, 0, 0, -2, 5, 1, 2, -1, -1], [0, 1, 1, -2, -2, -5, 5, -2, -5, 2], [2, 0, -5, -1, 2, 0, 2, 0, -2, 4], [0, -3, -3, 0, -2, 4, -1, 0, 1, 2], [0, -6, -6, -3, 0, 5, 0, -3, 0, 0], [5, 0, 5, -1, 1, 0, 1, 0, -1, 2], [-1, -5, 0, 0, 2, -1, 1, 0, 3, 3], [-5, 4, 0, 0, -1, -5, 2, 0, 1, 1], [-1, 0, 0, -5, 0, 0, 0, 0, -4, 4], [-7, 0, 1, 3, -4, 0, -4, 0, 2, -6], [-6, 1, 0, 0, 0, -6, -3, 0, -3, -3], [17, 0, -3, 0, -1, 0, -1, 0, -4, 3], [0, -4, 0, 0, 2, 0, 0, 4, 2, 2], [-7, 0, 5, 1, 2, 0, 2, 0, 2, 0], [1, 0, -3, -1, 1, 0, 1, 0, -4, 5], [-9, 0, 2, -2, 4, 0, 4, 0, 2, 2], [-4, 0, -2, -3, 2, 0, 2, 0, 5, -3], [1, -3, 0, 0, 0, 1, -3, 0, -3, -3], [5, -9, 0, 0, -4, 5, 3, -3, -1, -1], [11, 0, -6, -5, 1, 0, 1, 0, -1, 2], [0, 0, 0, 5, 5, 13, -4, 5, 4, -5], [-8, 8, 0, 0, 2, -8, -1, 1, 1, 1], [12, 3, 0, 0, -2, 12, 0, 2, -2, -2], [0, 0, 0, 3, 0, -1, 3, 3, -3, 0], [8, 0, 0, -4, -2, 0, -2, 0, -7, 5], [10, 0, 1, 0, 4, 0, 4, 0, 0, 4], [0, 0, 0, 0, 9, 4, -5, 0, 5, -9], [-1, 5, 0, 0, 4, -1, -4, -1, 0, 0], [-8, -4, 0, 0, 0, -8, 3, -2, 3, 3], [10, -3, 0, 0, 0, 10, 0, 1, 0, 0], [0, 10, 10, 6, 4, 17, -7, 6, 7, -4], [0, 10, 10, 0, 3, -3, -7, 0, 7, -3], [0, -6, -6, -1, 3, 17, -2, -1, 2, -3], [0, 2, 2, 2, 1, -5, 1, 2, -1, -1], [0, 2, 2, 1, -4, 14, 0, 1, 0, 4], [-9, 0, -7, -6, -2, 0, -2, 0, -3, 1], [-9, 0, 3, 3, -2, 0, -2, 0, -5, 3], [0, 4, 4, 1, 9, 11, -8, 1, 8, -9], [-16, 0, 0, 4, -3, 0, -3, 0, 3, -6], [0, 0, 0, 0, -3, 0, 0, -1, -3, -3], [-14, 3, 0, 0, 9, -14, -5, -3, 4, 4], [4, 0, 5, 4, -3, 0, -3, 0, 2, -5], [-4, -2, 0, 0, 1, -4, -2, -3, -1, -1], [2, 10, 0, 0, -1, 2, 1, 7, 0, 0], [8, 0, 8, 5, 4, 0, 4, 0, 3, 1], [-23, -1, 0, 0, 3, -23, -1, 0, 2, 2], [-8, 0, -10, 2, 6, 0, 6, 0, -3, 9], [-13, 0, 0, 0, -1, -13, 2, -4, 1, 1], [2, -2, 0, 0, -2, 2, 1, 6, -1, -1], [5, 0, 0, -5, -2, 0, -2, 0, -7, 5], [0, 0, 0, 4, -4, 8, 2, 4, -2, 4], [4, -8, 0, 0, -8, 4, 4, -3, -4, -4], [-1, 8, 0, 0, 7, -1, -7, 0, 0, 0], [-6, 0, 0, -1, -1, 0, -1, 0, -6, 5], [0, -4, -4, 2, 1, -17, 1, 2, -1, -1], [-10, 0, 10, -1, 0, 0, 0, 0, 6, -6], [-3, -5, 0, 0, 0, -3, 3, -4, 3, 3], [8, 0, 7, 5, 1, 0, 1, 0, 3, -2], [5, -9, 0, 0, -7, 5, 6, -4, -1, -1], [-7, 0, -1, -1, -2, 0, -2, 0, -2, 0], [-9, 0, -5, -6, -4, 0, -4, 0, -3, -1], [0, 4, 4, -6, -2, -3, 2, -6, -2, 2], [0, -4, -4, 0, -9, 6, 7, 0, -7, 9], [0, 0, 0, -1, 2, 23, 2, -1, -2, -2], [5, 7, 0, 0, 4, 5, -3, 1, 1, 1], [0, -5, -5, 0, 3, 33, -1, 0, 1, -3], [0, -1, -1, -5, -4, -12, 2, -5, -2, 4], [0, -8, -8, -1, -3, -15, 5, -1, -5, 3], [-6, 1, 0, 0, 1, -6, 1, 0, 2, 2], [10, -6, 0, 0, 0, 10, 4, 0, 4, 4], [0, 13, 13, 9, 4, -13, -3, 9, 3, -4], [-4, -11, 0, 0, -3, -4, 0, -4, -3, -3], [-20, 0, -2, 1, 3, 0, 3, 0, -2, 5], [-3, 8, 0, 0, -3, -3, 0, -3, -3, -3], [-6, -2, 0, 0, -7, -6, 7, 3, 0, 0], [4, 0, -13, -4, 6, 0, 6, 0, -2, 8], [-9, 10, 0, 0, 6, -9, -6, 1, 0, 0], [9, 3, 0, 0, 6, 9, -10, 1, -4, -4], [0, -4, -4, -3, -1, -9, -1, -3, 1, 1], [-12, 3, 0, 0, 3, -12, 1, -7, 4, 4], [-15, 0, 1, 7, -1, 0, -1, 0, 3, -4], [-21, 0, 1, -1, -1, 0, -1, 0, 0, -1], [5, 0, 2, 6, 0, 0, 0, 0, -3, 3], [0, -6, -6, 1, -5, -28, 4, 1, -4, 5], [0, 5, 5, 1, 4, -11, -8, 1, 8, -4], [-2, 0, -1, -6, -2, 0, -2, 0, -2, 0], [14, 1, 0, 0, -4, 14, 2, 4, -2, -2], [13, 7, 0, 0, 9, 13, -9, -1, 0, 0], [0, 5, 5, 6, 1, 8, -2, 6, 2, -1], [0, 0, 0, 8, 2, 0, 2, 0, -1, 3], [0, -2, -2, -3, -5, 5, 2, -3, -2, 5], [0, -7, -7, -1, -3, 13, 6, -1, -6, 3], [0, -2, -2, -1, 2, 23, -3, -1, 3, -2], [-3, 0, 3, 2, 3, 0, 3, 0, 14, -11], [-2, 0, 1, 4, 3, 0, 3, 0, 0, 3], [-7, -9, 0, 0, -7, -7, 8, 0, 1, 1], [15, 5, 0, 0, 4, 15, -3, -7, 1, 1], [3, 0, 12, -2, -3, 0, -3, 0, 7, -10], [24, 0, 1, -7, 1, 0, 1, 0, -6, 7], [-25, 0, 0, -4, 4, 0, 4, 0, 7, -3], [-10, 0, 0, 8, -2, 0, -2, 0, 0, -2], [6, 11, 0, 0, 9, 6, -7, 6, 2, 2], [4, 3, 0, 0, 3, 4, -4, -1, -1, -1], [10, 0, 5, 4, 1, 0, 1, 0, 1, 0], [0, 10, 10, 1, 4, 20, -3, 1, 3, -4], [-31, -6, 0, 0, -8, -31, 7, -6, -1, -1], [5, 10, 0, 0, 9, 5, -5, -4, 4, 4], [0, -15, -15, -1, -10, -7, 8, -1, -8, 10], [0, 11, 11, 3, 0, 11, -2, 3, 2, 0], [-11, 0, 6, 10, 0, 0, 0, 0, 10, -10], [-6, 0, -3, 2, 7, 0, 7, 0, 1, 6], [-10, -4, 0, 0, -2, -10, 0, -5, -2, -2], [13, 0, -9, -3, 3, 0, 3, 0, -9, 12], [0, -7, -7, -3, -1, 27, 1, -3, -1, 1], [0, -15, -15, -2, -2, 7, -1, -2, 1, 2], [2, 0, -1, -6, 2, 0, 2, 0, -1, 3], [0, -20, -20, -11, -4, 1, 4, -11, -4, 4], [-17, 0, 5, 10, -5, 0, -5, 0, 2, -7], [18, -1, 0, 0, 2, 18, -2, 4, 0, 0], [-4, 0, -8, -1, -5, 0, -5, 0, -7, 2], [0, 13, 13, 3, 10, 6, -5, 3, 5, -10], [23, -6, 0, 0, 4, 23, -7, -1, -3, -3], [0, -3, -3, 1, 3, 3, 4, 1, -4, -3], [12, -6, 0, 0, 0, 12, 5, -1, 5, 5], [-15, 0, 7, 0, -2, 0, -2, 0, 2, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_546_j_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_546_2_j_e();".
// 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_546_2_j_e(:prec:=10)
    chi := MakeCharacter_546_j();
    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_546_2_j_e();".
// 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_546_2_j_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_546_j();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -76, 345, -164, 233, -36, 110, -14, 15, 2, 1]>],Snew);
    return Vf;
end function;