
// Make newform 800.6.a.c in Magma, downloaded from the LMFDB on 02 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_800_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_800_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_800_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_800_a();"
function MakeCharacter_800_a()
    N := 800;
    order := 1;
    char_gens := [351, 101, 577];
    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_800_a_Hecke( : Kf := Kf)
    return MakeCharacter_800_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]];
    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], [0], [0], [0], [0], [244], [808], [0], [0], [-2950], [0], [11292], [20950], [0], [0], [-40244], [0], [-18950], [0], [0], [-20144], [0], [0], [-51050], [-160808], [-98002], [0], [0], [246486], [244144], [0], [0], [-432808], [0], [47614], [0], [-371292], [0], [0], [-544244], [0], [439902], [0], [-907656], [-1020908], [0], [0], [0], [0], [-1251150], [-619856], [0], [477150], [0], [-2015392], [0], [2351414], [0], [-2451092], [-2643050], [0], [1624244], [0], [0], [-1915656], [-3515092], [0], [-3488608], [0], [4445050], [3787856], [0], [0], [-1504044], [0], [0], [5286286], [6275292], [1592002], [-4584486], [0], [5948850], [0], [-532344], [0], [0], [-8489614], [7216808], [-6865802], [0], [0], [0], [0], [0], [0], [0], [10509050], [-7274102], [0], [8253798], [0], [491092], [0], [3966586], [0], [15649008], [0], [17035856], [0], [-15155050], [0], [14896044], [6624608], [0], [0], [-14595250], [0], [0], [19399556], [0], [8605250], [-15796344], [23411492], [0], [0], [21165050], [-3987150], [0], [0], [-10512244], [0], [0], [0], [3004508], [19519802], [20284750], [-32863556], [0], [14964508], [-28779050], [0], [-32114750], [0], [0], [-7968186], [0], [7552044], [-27216808], [0], [0], [-38819292], [44084750], [0], [0], [0], [0], [0], [47663314], [-52606992], [4855702], [0], [31779944], [0], [0], [-35712808], [0], [0], [-51275292], [40797150], [-40783556], [0], [65464102], [0], [-61236544], [0], [-49875050], [0], [776302], [0], [-66496586], [0], [0], [-23544244], [46367392], [0], [-73154750], [-80692508], [0], [85566186], [0], [-83787856], [0], [0], [63933050], [0], [-34507456], [77524750], [28023556], [71135392], [0], [-101619350], [0], [29429108], [89685150], [0], [733108], [0], [0], [51597350], [0], [-119999592], [-77474750], [0], [0], [0], [-80747050], [0], [131469698], [0], [37264244], [-123939150], [0], [71804750], [0], [0], [-19833314], [135979856], [0], [0], [0], [-148624156], [0], [0], [-91538650], [0], [0], [-156425214], [105608156], [0], [0], [0], [0], [0], [136044650], [190027456], [0], [0], [0], [0], [0], [189666892], [25568002], [0], [-75936486], [-16143844], [0], [65710398], [0], [73932508], [169008408], [0], [0], [-216819150], [-181447956], [110880808], [0], [227996650], [-239330102], [0], [126951844], [163079702], [0], [-240587456], [0], [-247647392], [0], [0], [151181050], [-272754650], [0], [0], [0], [0], [-227879698], [0], [0], [111635944], [-304428908], [0], [280823714], [314684650], [0], [254635856], [0], [-22960644], [139302886], [0], [315735844], [0], [0], [-346852344], [-329746892], [0], [0], [0], [-194721208], [0], [361773450], [0], [-131767356], [0], [89565250], [-254055598], [0], [0], [-254259450], [0], [0], [-267083456], [-212328686], [0], [-409200408], [218589050], [0], [-41859944], [433053150], [0], [0], [0], [-97240244], [29133450], [-183107692], [0], [0], [0], [0], [473366086], [-367819856], [68012550], [0], [-404935356], [-227854792], [255744014], [0], [368680444], [0], [-530307150], [0], [0], [167555492], [0], [-335918792], [418508650], [0], [246205250], [230347456], [0], [0], [-227520808], [0], [-335531292], [2974298], [0], [0], [0], [-149516256], [-230892308], [0], [-418410650], [0], [0], [0], [420383614], [0], [-357116308], [0], [0], [-89523744], [296693986], [-267280808], [621272102], [179900056], [0], [727775392], [0], [0], [0], [318186892], [0], [0], [-226913214], [-613408444], [0], [0], [0], [-735140744], [0], [-358827050], [0], [477645250], [-788809114], [623860744], [0], [791758792], [-819689714], [0], [805460908], [-297841502], [0], [0], [-526131744], [269770892], [0], [0], [871455392], [849502198], [0], [0], [901054792], [0], [-543936986], [-915181308], [0], [0], [850276344], [899916308], [0], [-535008086], [0], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_800_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_800_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_800_6_a_c(:prec:=1)
    chi := MakeCharacter_800_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(2999) - 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_800_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_800_6_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_800_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![0, 1]>,<11,R![0, 1]>,<13,R![-244, 1]>],Snew);
    return Vf;
end function;