
// Make newform 80.6.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_80_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_80_6_a_c();".
// 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_80_6_a_c();".
// 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_80_a();"
function MakeCharacter_80_a()
    N := 80;
    order := 1;
    char_gens := [31, 21, 17];
    v := [1, 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_80_a_Hecke(Kf)
    return MakeCharacter_80_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 := 6;
    raw_aps := [[0], [-6], [-25], [118], [-192], [1106], [762], [2740], [-1566], [5910], [6868], [-5518], [-378], [2434], [-13122], [-9174], [34980], [-9838], [-33722], [-70212], [21986], [-4520], [109074], [38490], [-1918], [77622], [46714], [1038], [206930], [139386], [-299882], [-7872], [-164238], [282100], [-388950], [97948], [-3718], [43234], [-186522], [-374454], [-272100], [-75418], [356988], [-438694], [-156798], [162520], [181648], [288274], [-1125522], [-415810], [770586], [595320], [273902], [-850752], [825402], [-1364646], [-113310], [849628], [438602], [-1456698], [120394], [-2642094], [1447558], [928068], [2295626], [2736522], [-3818792], [-2210878], [2327238], [-311290], [-3086574], [3530760], [-35762], [-1715254], [3101740], [-5319486], [1161450], [628562], [-2724318], [1780190], [-650580], [-3540598], [548748], [-1492414], [-4862120], [1861554], [3737190], [-6482758], [1509102], [-8684006], [-6964122], [5510520], [-5518082], [1512768], [1930420], [-6731046], [-556650], [10111002], [7037194], [-4231138], [-4440242], [-9014478], [9812874], [13315230], [-9958952], [4503722], [-625842], [-2503854], [756480], [-13856458], [-11377202], [-7001614], [7903002], [-4023620], [10022668], [6373902], [-5004566], [8719278], [-1584774], [-12640980], [-3615718], [11131346], [-235518], [-20583006], [9548248], [1293042], [-21272050], [-8317320], [4367398], [-4056454], [-768260], [-6187806], [-18169772], [19349402], [-30199158], [21585410], [3908946], [26508238], [10794042], [-11144550], [11486608], [30434682], [-4097726], [17065158], [-24761710], [-31652880], [28267106], [26086962], [33811060], [-22281726], [-34674838], [14260302], [37517674], [-40765722], [35711638], [21138948], [-18599600], [44511030], [-21941878], [-7777218], [-31719882], [-5602854], [20353198], [23430648], [-43041198], [47500314], [-20923052], [29633162]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_80_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_80_6_a_c();".
// 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_80_6_a_c(:prec:=1)
    chi := MakeCharacter_80_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_80_6_a_c();".
// 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_80_6_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_80_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![6, 1]>],Snew);
    return Vf;
end function;