
// Make newform 546.2.i.h in Magma, downloaded from the LMFDB on 25 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_546_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_546_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_546_2_i_h();".
// 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_i_h();".
// 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 := [4, 0, 2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]];
    Rf_basisdens := [1, 1, 2, 2];
    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_i();"
function MakeCharacter_546_i()
    N := 546;
    order := 3;
    char_gens := [365, 157, 379];
    v := [3, 1, 3];
    // 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_i_Hecke();"
function MakeCharacter_546_i_Hecke( : Kf := false)
    N := 546;
    order := 3;
    char_gens := [365, 157, 379];
    char_values := [[1, 0, 0, 0], [-1, 0, -1, 0], [1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, -1, 0], [0, 0, 1, 0], [0, 1, 0, 0], [1, -1, 1, -2], [0, 1, -1, 1], [1, 0, 0, 0], [0, 3, 1, 3], [-3, 0, -3, 0], [2, 5, 2, 0], [5, 0, 0, 0], [0, 2, 4, 2], [-2, 5, -2, 0], [6, 0, 0, 1], [2, 0, 0, -2], [-9, 0, -9, 0], [0, 8, 1, 8], [0, -1, 9, -1], [1, 3, 1, 0], [0, -4, -9, -4], [-1, 0, 0, 0], [0, -5, 6, -5], [-2, -8, -2, 0], [-2, 0, 0, 4], [2, 3, 2, 0], [-2, 0, 0, -7], [0, 2, -4, 2], [2, -6, 2, 0], [-8, -4, -8, 0], [0, 4, -4, 4], [1, 0, 0, 5], [0, 0, 0, -7], [0, -7, 0, 0], [0, -5, -6, -5], [0, 0, 0, 15], [-6, -7, -6, 0], [0, 1, -9, 1], [0, -5, 15, -5], [-3, -10, -3, 0], [1, 0, 0, -8], [-9, -8, -9, 0], [0, 4, -18, 4], [-13, 0, 0, -1], [-8, 4, -8, 0], [0, 3, -20, 3], [-6, 0, 0, -11], [0, -1, 12, -1], [-10, 0, 0, 13], [3, 0, 0, -13], [0, -10, -12, -10], [10, -2, 10, 0], [-9, 9, -9, 0], [-11, 0, 0, -6], [0, -6, -8, -6], [10, 0, 0, -2], [2, 14, 2, 0], [0, -11, 8, -11], [0, 8, -15, 8], [-3, 13, -3, 0], [0, -1, -15, -1], [-4, 0, 0, -8], [0, 5, 10, 5], [-6, 0, 0, 9], [3, 0, 0, 16], [0, 1, -8, 1], [-4, -10, -4, 0], [2, -2, 2, 0], [14, 6, 14, 0], [23, 0, 0, -4], [0, -3, 10, -3], [4, 0, 0, -1], [0, 16, -10, 16], [-10, 18, -10, 0], [0, 0, 14, 0], [21, -7, 21, 0], [-28, 0, 0, 4], [2, -6, 2, 0], [0, 0, 23, 0], [24, -3, 24, 0], [-12, -5, -12, 0], [0, 1, -4, 1], [-2, 0, 0, 18], [12, 0, 0, 1], [0, 6, 6, 6], [-19, 0, 0, 0], [12, -1, 12, 0], [-10, -9, -10, 0], [0, 0, 0, 24], [-6, 15, -6, 0], [-8, 0, 0, 18], [-4, 0, 0, 24], [-4, 3, -4, 0], [0, -10, -17, -10], [0, -9, -11, -9], [2, 0, 0, 20], [30, 0, 30, 0], [16, 0, 0, -14], [4, -12, 4, 0], [0, 0, -14, 0], [18, -10, 18, 0], [14, -19, 14, 0], [-10, 0, 0, -3], [0, -12, 6, -12], [0, -6, 8, -6], [27, -1, 27, 0], [0, -4, 4, -4], [0, 0, -18, 0], [7, 0, 0, -3], [30, -1, 30, 0], [0, -19, 12, -19], [-15, 0, 0, 2], [-20, -5, -20, 0], [0, -14, -16, -14], [2, 0, 0, 6], [0, 18, 18, 18], [14, 0, 0, -16], [0, -2, 36, -2], [25, 0, 0, -8], [0, 16, 6, 16], [16, -22, 16, 0], [18, 0, 0, -9], [0, -11, 2, -11], [-4, 0, 0, -2], [-31, 2, -31, 0], [0, 18, -20, 18], [3, -10, 3, 0], [40, 0, 0, -6], [8, 17, 8, 0], [-6, 20, -6, 0], [-36, 0, 0, 2], [-12, 29, -12, 0], [0, 12, -10, 12], [3, 0, 0, -6], [40, -4, 40, 0], [39, 0, 0, 11], [32, -10, 32, 0], [-14, 0, 0, -5], [0, -28, 10, -28], [0, 2, -11, 2], [-20, 0, 0, 12], [0, 17, 11, 17], [-10, 0, 0, -26], [28, -13, 28, 0], [0, 7, -18, 7], [-19, 0, 0, -23], [0, 21, -11, 21], [11, 0, 0, 6], [12, 0, 0, 21], [0, -13, -33, -13], [-20, 14, -20, 0], [-4, 0, -4, 0], [-22, -1, -22, 0], [-10, 0, 0, 22], [-8, 0, 0, -5], [-30, -8, -30, 0], [0, 7, -42, 7], [54, 0, 0, 0], [6, 14, 6, 0], [10, -20, 10, 0], [1, 0, 0, -28], [0, -12, -4, -12], [-13, -3, -13, 0], [11, 0, 0, 15], [-9, 0, 0, -13], [-42, -11, -42, 0], [0, 6, -6, 6], [0, 2, -25, 2], [0, -8, 4, -8], [0, 21, -23, 21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_546_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_546_2_i_h();".
// 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_i_h(:prec:=4)
    chi := MakeCharacter_546_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(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_i_h();".
// 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_i_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_546_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 0, 2, 0, 1]>,<17,R![289, -34, 21, 2, 1]>],Snew);
    return Vf;
end function;