
// Make newform 15.8.a.b in Magma, downloaded from the LMFDB on 16 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_b();".
// 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_b();".
// 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 := [[-13], [-27], [-125], [1380], [-3304], [8506], [-9994], [41236], [84120], [132802], [-55800], [228170], [-139670], [-755492], [836984], [1641650], [-989656], [-1658162], [-4523844], [-389408], [5617330], [3901080], [-9394116], [2803746], [5099426], [15172266], [4705268], [2631204], [-4300594], [3982334], [28017676], [-8199192], [-1666458], [-58745596], [-19369742], [-53395208], [20452922], [-733588], [16892520], [11818554], [-31374640], [-58355530], [4061656], [-133220566], [13077138], [-69850232], [32853500], [69519356], [-230778860], [146157054], [311907438], [227310296], [-198482654], [-132535968], [-35864226], [-479640488], [-70876414], [407490256], [-79347702], [-87006846], [277611876], [246490482], [384964564], [46443456], [210558002], [-960971366], [399922812], [269184538], [-821868004], [648353902], [652666326], [-877430856], [286988988], [-1770132542], [1463113060], [-990456408], [-796901166], [795584194], [-2016318342], [-54802966], [110924536], [-2120637986], [-2485326480], [29995642], [-1048734680], [204679116], [2639623418], [1837383370], [3576778602], [1280677668], [-4199840036], [2482019880], [-1080653044], [-947786672], [-2076919244], [2732654896], [4524697154], [-279532670], [1463762068], [-2721936690], [-673048220], [-7199944134], [5801136324], [-1670688894], [5701386956], [2430634946], [-3333782076], [3422803438], [-928661192], [-6569738374], [-3507897644], [3639701194], [7825585182], [8351725844], [-21681968], [-7853610750], [-8818195092], [-8719969024], [-665755238], [-9995126680], [-2891603978], [-5832356070], [-12252424782], [-9692933868], [3332853076], [-2731988702], [344619886], [3357493728], [-13201695988], [4301789378], [-18241424636], [-236847232], [-14432560904], [16493357962], [-2968454046], [8154002498], [-2780590398], [-8209818596], [-54791606], [2049730330], [-15947121708], [9659231274], [3903976844], [-13222440964], [12048232350], [31060309000], [25215704442], [22531444814], [4078626820], [-11064318272], [36059458650], [-26295355606], [2788729388], [-18296966616], [-1212451716], [-20918089408], [-18154477040], [-33922505414], [36402256938], [50711742490], [23099415492], [-4163677114], [-52046779324], [-364572720], [-18178540474], [-18577791360], [18252005552], [-1245342558]];
    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_b();".
// 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_b(: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_b();".
// 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_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_15_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![13, 1]>],Snew);
    return Vf;
end function;