
// Make newform 11.19.b.a in Magma, downloaded from the LMFDB on 16 May 2024.
// 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_19_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_19_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)
    N := 11;
    order := 2;
    char_gens := [2];
    char_values := [[-1]];
    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 := 19;
    raw_aps := [[0], [-20870], [-3063526], [0], [-2357947691], [0], [0], [0], [3592942977890], [0], [26636854831058], [-10400449085350], [0], [0], [-2113576298457550], [-1367378362647430], [-12373964663300278], [0], [37614783844431290], [91404547650956162], [0], [0], [0], [623295746335388018], [-1511803192851761470], [0], [2373115488942794690], [0], [0], [-5699180055764896990], [0], [0], [22986158593075686290], [0], [0], [0], [56049250986007652810], [-151048026623916596230], [0], [0], [324603134467754584538], [358776403363251390458], [420323003319338024978], [0], [0], [44253831881125468802], [0], [-2246832740431032682030], [0], [3422666504706625321562], [0], [0], [0], [-6980136967493176433398], [2572346996323815060290], [0], [5188654804501107708842], [0], [0], [0], [0], [0], [0], [-54402385869046911604318], [41061177557294111874290], [63131421556475453969930], [25732892023538661674858], [0], [0], [0], [59783438271590872141250], [0], [3919101574862500160690], [0], [-322643718526522189226038], [15501272195347811293970], [-301702306713166763919718], [-303102261975687551594710], [478476187857185098425698], [0], [-219821927639796787462342], [230786441232322324331738], [0], [-274946074955208315872350], [0], [1313844113131387625215370], [858422137941156075117698], [0], [0], [652603589928966505669490], [-665414092224812785646950], [0], [-1788973712574209618962750], [0], [-1687886701318657061879398], [0], [1119361263048067540678922], [-5471323677179321580566062], [0], [0], [0], [0], [0], [0], [0], [-11591951632579390033372030], [-12973763369512024377129430], [0], [13344730105707709222281698], [0], [0], [0], [-19536906388580967468888430], [-24956548990305196935486742], [25713727245884791477499042], [-24077960797496268813925822], [12240146148615680689943930], [31734221077934837149730690], [16888059639021741058652330], [0], [5369601473911421111031482], [0], [0], [43284748905122581838392490], [-67323501043006491018069478], [0], [-86925650608734421452690022], [83324630686264373074118258], [-111226097426843535852668830], [0], [0], [0], [-8974801809203621142027598], [-66015669936127300292554630], [0], [0], [-72387592517367430824535270], [0], [-149505556101660010670148790], [0], [0], [0], [-315801998083815145361435230], [0], [368805165983241020293990538], [393115247699508874734756482], [0], [0], [418692588985650860032999178], [-407782529113450235026372270], [0], [239393369004751223047542242], [646497658275327015666789530], [0], [611392784176031619606934250], [-345438237829752808929267982], [0], [1017118459322139558047376962], [0], [0], [953397329063626215179871770], [0], [0], [576019479350646970515116138], [298148396719684995215098850], [309555844129750044223454690], [-1648374493892974044963576622], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_11_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_11_19_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_19_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, 19));
    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_19_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_19_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_11_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,19,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;