
// Make newform 165.3.l.a in Magma, downloaded from the LMFDB on 19 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_165_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_165_l_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_165_3_l_a();".
// 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_165_3_l_a();".
// 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 := [25, 0, 0, 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, 5, 5];
    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_165_l();"
function MakeCharacter_165_l()
    N := 165;
    order := 4;
    char_gens := [56, 67, 46];
    v := [2, 1, 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_165_l_Hecke();"
function MakeCharacter_165_l_Hecke( : Kf := false)
    N := 165;
    order := 4;
    char_gens := [56, 67, 46];
    char_values := [[-1, 0, 0, 0], [0, 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 := 3;
    raw_aps := [[0, 1, 0, 0], [-3, 0, 0, 0], [-4, 0, -3, 0], [0, 0, 0, -2], [0, 2, -9, -2], [0, 2, 0, 0], [0, -14, 0, 0], [0, -4, 0, 4], [-7, 0, -7, 0], [0, -6, 0, -6], [20, 0, 0, 0], [7, 0, 7, 0], [0, 22, 0, -22], [0, 14, 0, 0], [-43, 0, 43, 0], [17, 0, 17, 0], [22, 0, 0, 0], [0, -30, 0, -30], [-47, 0, -47, 0], [0, 0, 120, 0], [0, 14, 0, 0], [0, 2, 0, -2], [0, 0, 0, 38], [100, 0, 0, 0], [43, 0, 43, 0], [0, -44, 0, 44], [7, 0, -7, 0], [0, -26, 0, 0], [0, 68, 0, -68], [53, 0, 53, 0], [0, 0, 0, -26], [0, 40, 0, -40], [83, 0, -83, 0], [0, -4, 0, 4], [0, -30, 0, -30], [0, -36, 0, -36], [67, 0, 67, 0], [-137, 0, 137, 0], [0, 130, 0, 0], [0, 0, 0, -22], [-320, 0, 0, 0], [80, 0, 0, 0], [0, 0, 258, 0], [0, 50, 0, 0], [0, -146, 0, 0], [0, 0, 120, 0], [0, 78, 0, 78], [73, 0, -73, 0], [0, -98, 0, 0], [0, 0, 180, 0], [0, 0, 0, 2], [0, 120, 0, 120], [0, 48, 0, 48], [0, 0, 60, 0], [23, 0, -23, 0], [0, 0, 0, -106], [-182, 0, 0, 0], [0, -84, 0, -84], [0, 0, 0, -98], [0, 106, 0, -106], [0, -178, 0, 0], [0, 0, 0, 146], [0, 0, 0, -98], [0, 0, -240, 0], [127, 0, -127, 0], [-127, 0, 127, 0], [200, 0, 0, 0], [0, 0, 0, -230], [0, 58, 0, 0], [0, 80, 0, -80], [-313, 0, -313, 0], [0, -24, 0, -24], [97, 0, 97, 0], [0, 146, 0, 0], [0, 0, -60, 0], [413, 0, 413, 0], [298, 0, 0, 0], [73, 0, 73, 0], [0, 0, 258, 0], [0, -202, 0, 202], [-200, 0, 0, 0], [-400, 0, 0, 0], [0, -50, 0, 50], [193, 0, -193, 0], [0, -118, 0, 118], [-463, 0, -463, 0], [-260, 0, 0, 0], [0, 0, 0, 58], [0, 64, 0, -64], [-527, 0, 527, 0], [347, 0, -347, 0], [0, -12, 0, -12], [-257, 0, -257, 0], [0, -32, 0, 32], [0, 0, 540, 0], [0, 0, 0, -214], [742, 0, 0, 0], [0, 0, 942, 0], [0, 50, 0, 0], [0, 138, 0, 138], [0, 0, 0, 334], [0, 406, 0, 0], [0, 0, 0, 74], [0, -192, 0, -192], [0, -162, 0, -162], [43, 0, 43, 0], [-553, 0, 553, 0], [0, 0, 0, 230], [-860, 0, 0, 0], [0, -24, 0, -24], [0, 0, 0, -206], [0, -70, 0, 0], [-757, 0, 757, 0], [0, 0, -162, 0], [158, 0, 0, 0], [0, 0, -942, 0], [757, 0, -757, 0], [257, 0, -257, 0], [263, 0, 263, 0], [0, 126, 0, 126], [-1078, 0, 0, 0], [0, -226, 0, 0], [0, 70, 0, 0], [-397, 0, -397, 0], [362, 0, 0, 0], [0, -8, 0, 8], [0, 0, -1038, 0], [940, 0, 0, 0], [517, 0, 517, 0], [0, 398, 0, 0], [0, -280, 0, 280], [0, 0, 0, 26], [-922, 0, 0, 0], [607, 0, 607, 0], [0, -362, 0, 362], [0, -454, 0, 454], [-703, 0, -703, 0], [0, 0, 0, -122], [-373, 0, 373, 0], [0, 312, 0, 312], [0, -366, 0, -366], [0, -116, 0, 116], [487, 0, -487, 0], [0, 82, 0, 0], [0, 0, -162, 0], [-422, 0, 0, 0], [0, 86, 0, 0], [0, 298, 0, 0], [0, 0, 678, 0], [1013, 0, 1013, 0], [0, 0, 0, -410], [0, 0, -1080, 0], [517, 0, -517, 0], [0, -206, 0, 0], [727, 0, 727, 0], [0, 0, -738, 0], [0, -22, 0, 22], [-458, 0, 0, 0], [0, 0, 0, 526], [0, 412, 0, -412], [953, 0, -953, 0], [0, 0, 0, 662], [0, 0, 0, 286], [0, 0, 342, 0], [-163, 0, 163, 0], [827, 0, 827, 0], [-940, 0, 0, 0], [0, 0, 0, -302]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_165_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_165_3_l_a();".
// 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_165_3_l_a(:prec:=4)
    chi := MakeCharacter_165_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_165_3_l_a();".
// 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_165_3_l_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_165_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![25, 0, 0, 0, 1]>,<23,R![98, 14, 1]>],Snew);
    return Vf;
end function;