
// Make newform 35.19.c.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_35_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_35_c_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_35_19_c_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_35_19_c_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_35_c();"
function MakeCharacter_35_c()
    N := 35;
    order := 2;
    char_gens := [22, 31];
    v := [1, 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_35_c_Hecke();"
function MakeCharacter_35_c_Hecke(Kf)
    N := 35;
    order := 2;
    char_gens := [22, 31];
    char_values := [[-1], [-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], [-39286], [1953125], [-40353607], [2637510122], [18822563674], [-54170520014], [0], [0], [-14623240000822], [0], [0], [0], [0], [261032556546466], [0], [0], [0], [0], [11592841444168322], [100714945072821154], [-126856003456085902], [367224638997840874], [0], [-1519865703804919214], [0], [-1839208847830884686], [0], [3675021243094383338], [0], [0], [0], [0], [0], [17401376712248529338], [-30846789089007121438], [85259517524927476666], [0], [159058658214731222386], [13205623039270782874], [-143692487252625251302], [0], [-499144010671382333038], [0], [0], [0], [1228383178888917383642], [1617198217764775990594], [273502402208168502346], [0], [0], [-2890208154533469762382], [0], [0], [7945981000650500869906], [0], [0], [0], [0], [-21817853316207846389518], [3569027908630394869594], [9158616934863553829194], [45903625264677302596906], [0], [6085556535064519029634], [0], [63061440020146678418282], [0], [0], [0], [166376877465901745110354], [167065955173822723091138], [-227434048190902734683294], [0], [-74006492861046642765622], [117288314124988770908674], [-370487708741613966355942], [487276241244210058901146], [-392166097765835645296798], [0], [0], [-162415911934858678384678], [-272905933058479643403598], [-90083737630297664547086], [0], [0], [1124570483226511736317058], [0], [0], [0], [-2077669235752177590817814], [0], [0], [2759673287679671762156522], [1840839488994607009064858], [-3286126791798848017019246], [0], [0], [-1669291697175770736464006], [5739129424754523112641482], [0], [0], [11249041428741546107005354], [8045687855666462139610418], [-10357242498456701616732598], [-3179677360454393735931374], [16218847788551208168956026], [-17659532924602741638365006], [12556294967327089660579298], [0], [-6205507886769377208382334], [0], [0], [0], [30979924763440481202074402], [-35324762606707252355651518], [27400570031627548257825034], [-39059953181443366960027214], [0], [-22802041440111201128970982], [0], [0], [-55367014158189554687710454], [0], [0], [-73413706949721019508248438], [-85417670890819032359313382], [0], [-22352815413406261331558894], [-102779693372382892856093126], [113732329189569475976584058], [0], [59452607260886134187496242], [0], [0], [0], [-147642313760311526139752246], [-118711553342094416725925654], [-213769755660766295342366534], [-152600804621995619500564462], [0], [-97170974184298562469620998], [0], [0], [0], [0], [-209653409406407349603716246], [-313309134124161815545698494], [0], [0], [0], [0], [0], [-384828368586089524278200174], [0], [774281773946832261281524082], [842990880549331655232096098], [0], [439246714073633869768132066], [0], [0], [0], [0], [0], [0], [1256123564849898536108356114], [-391781976651437081864520238], [1170711845690227933112362186]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_35_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_35_19_c_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_35_19_c_a(:prec:=1)
    chi := MakeCharacter_35_c();
    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_35_19_c_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_35_19_c_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_35_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,19,sign)));
    Vf := Kernel([<2,R![0, 1]>,<3,R![39286, 1]>],Snew);
    return Vf;
end function;