
// Make newform 15.8.a.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_15_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_15_8_a_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_15_8_a_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_15_a();"
function MakeCharacter_15_a()
    N := 15;
    order := 1;
    char_gens := [11, 7];
    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;

function MakeCharacter_15_a_Hecke(Kf)
    return MakeCharacter_15_a();
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 := 8;
    raw_aps := [[-22], [27], [-125], [-420], [-2944], [-11006], [-16546], [-25364], [-5880], [163042], [-201600], [120530], [-115910], [-19148], [841016], [501890], [-1586176], [-372962], [4561044], [1512832], [-1522910], [4231920], [-1854204], [-6888174], [3700034], [-18002454], [-5372068], [-11539764], [-1571794], [-25204954], [39408004], [14108208], [-8005122], [46071644], [72352498], [-37006208], [-78507982], [-46818412], [-25004280], [53067066], [42205400], [-100020490], [61761016], [-76741894], [-181031838], [61608448], [-169916860], [148128884], [-39612740], [-37181586], [-179591418], [-373328104], [-257021534], [-127344408], [130681926], [-267746792], [149432066], [-153184904], [654462402], [551492754], [-210200436], [542401698], [926477356], [-212976144], [363896258], [-317049014], [224556372], [-1238861078], [-583643116], [-469471298], [61855854], [-12353736], [-12551388], [-565993958], [1777759180], [1181946168], [602482194], [-186765014], [-996333222], [-2386442326], [2596441936], [2838495214], [-1528084320], [-2660612822], [2292224800], [-934042716], [-88501222], [-1210641590], [857427882], [-1946619588], [-959955164], [303578520], [436058564], [834813448], [-4502299364], [2416997824], [3512093474], [990013090], [445925932], [2847531150], [4341160300], [3001237914], [-7438857204], [-720170814], [-2301861244], [-7372565086], [736500396], [-5085392138], [4860225208], [6784663226], [2852501644], [7754672626], [-6340967562], [3198311084], [-3794592488], [6779638290], [-1132032588], [-5542646896], [9417645178], [7463900240], [5583091942], [3399326250], [-13892978478], [-7225755732], [-5833864724], [4374864898], [-3546849794], [-10654527792], [-9211691932], [-581769958], [143208364], [17601209792], [-21039792464], [-3910148702], [12024211074], [11894778338], [-7776143502], [24436520836], [-2269696214], [-16319960150], [6993929052], [4009493514], [18857165156], [16638566084], [-13722438210], [-656953880], [-8709973182], [1938246806], [-795332180], [2241718192], [19916852370], [-27690602566], [-16561617868], [6699035496], [3878144916], [-7158695008], [25415979160], [-23456596934], [9300700362], [-11311724390], [4462699548], [19087166174], [-13410184756], [46419105240], [39297221294], [-36936970560], [-10286750008], [-32443943142]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_15_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_15_8_a_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_15_8_a_a(:prec:=1)
    chi := MakeCharacter_15_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_15_8_a_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_15_8_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_15_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![22, 1]>],Snew);
    return Vf;
end function;