
// Make newform 11.5.b.a in Magma, downloaded from the LMFDB on 03 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_11_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_11_b_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_11_5_b_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_11_5_b_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_11_b();"
function MakeCharacter_11_b()
    N := 11;
    order := 2;
    char_gens := [2];
    v := [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_11_b_Hecke();"
function MakeCharacter_11_b_Hecke( : Kf := false)
    N := 11;
    order := 2;
    char_gens := [2];
    char_values := [[-1]];
    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 := 5;
    raw_aps := [[0], [7], [-49], [0], [121], [0], [0], [0], [167], [0], [-553], [-2113], [0], [0], [-1918], [-718], [4487], [0], [-7753], [7607], [0], [0], [0], [-6433], [-9793], [0], [14882], [0], [0], [20687], [0], [0], [32687], [0], [0], [0], [-3073], [-48238], [0], [0], [-57193], [3647], [-48313], [0], [0], [-79198], [0], [98567], [0], [82607], [0], [0], [0], [-74473], [-95998], [0], [-13678], [0], [0], [0], [0], [0], [0], [35042], [-125713], [-191953], [97847], [0], [0], [0], [141407], [0], [133847], [0], [-269593], [276647], [3167], [308882], [-311998], [0], [-282478], [196082], [0], [-374353], [0], [-194473], [-15073], [0], [0], [-427513], [419447], [0], [291287], [0], [-135598], [0], [495887], [124607], [0], [0], [0], [0], [0], [0], [0], [447167], [587762], [0], [-707998], [0], [0], [0], [-480478], [763847], [675047], [-269713], [-670873], [-517993], [824207], [0], [455567], [0], [0], [-308878], [239687], [0], [111887], [318647], [778967], [0], [0], [0], [-181273], [-1141198], [0], [0], [681842], [0], [1241807], [0], [0], [0], [-784633], [0], [-706993], [-1287433], [0], [0], [-902713], [-1304638], [0], [658847], [317522], [0], [1417202], [1501442], [0], [-808318], [0], [0], [42407], [0], [0], [-1878793], [-408433], [-1529353], [538562], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_11_b_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_11_5_b_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_11_5_b_a(:prec:=1)
    chi := MakeCharacter_11_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 5));
    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_11_5_b_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_11_5_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_11_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,5,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;