
// Make newform 43.9.b.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_43_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_43_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_43_9_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_43_9_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_43_b();"
function MakeCharacter_43_b()
    N := 43;
    order := 2;
    char_gens := [3];
    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_43_b_Hecke();"
function MakeCharacter_43_b_Hecke(Kf)
    N := 43;
    order := 2;
    char_gens := [3];
    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 := 9;
    raw_aps := [[0], [0], [0], [0], [10319], [-54721], [79967], [0], [540719], [0], [589679], [0], [-2262241], [3418801], [-6983806], [-13061761], [-7864606], [0], [-39816433], [0], [0], [73045634], [-93091441], [0], [162643199], [-184316161], [-91860913], [184486754], [-259285921], [0], [-308033521], [0], [0], [-30450961], [0], [0], [0], [0], [-720209233], [1469605154], [0], [-485677726], [0], [-912967201], [-2934519838], [0], [0], [0], [0], [4468108319], [0], [-6101010046], [0], [-767718961], [0], [0], [10162231967], [8038599887], [0], [-1392748033], [19953359], [11275652834], [994758959], [-18571441393], [0], [1797701567], [0], [-4837772161], [0], [0], [5178697439], [-16239146353], [-34123164286], [0], [-6373848913], [0], [0], [42723862562], [9331144127], [0], [0], [0], [20529680879], [0], [-53442669361], [-63268519198], [0], [0], [46781370914], [0], [0], [92910462287], [-91513270078], [0], [0], [0], [130889468447], [0], [0], [171266972447], [3918866639], [-58881350401], [89529486479], [-124705679521], [0], [0], [0], [0], [-245665190161], [0], [0], [-68873817886], [-289847665441], [-156699542686], [0], [0], [233068977314], [0], [0], [-316663800241], [198842555519], [0], [0], [-287847067633], [0], [-259215209566], [-223796504641], [-210925231486], [0], [0], [0], [0], [0], [0], [0], [-614691591166], [0], [765561599522], [-604856972638], [-856161482686], [0], [145230648287], [558556783919], [-249161840926], [0], [0], [-977864994913], [-925594433086], [0], [0], [922813262207], [762544368479], [712688368079], [0], [1078378429199], [0], [-201221621233], [0], [0], [1027035696959], [1447841801039], [0], [-1475221448881], [-491958178513], [-203620983166], [0], [0], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_43_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_43_9_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_43_9_b_a(:prec:=1)
    chi := MakeCharacter_43_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 9));
    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_43_9_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_43_9_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_43_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,9,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;