
// Make newform 441.6.a.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_441_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_441_6_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_441_6_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_441_a();"
function MakeCharacter_441_a()
    N := 441;
    order := 1;
    char_gens := [344, 199];
    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_441_a_Hecke(Kf)
    return MakeCharacter_441_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 := [[-11], [0], [0], [0], [76], [0], [0], [0], [4952], [-7282], [0], [-8886], [0], [11748], [0], [-24550], [0], [0], [69364], [2224], [0], [80168], [0], [0], [0], [0], [0], [64900], [-219582], [-123202], [-262064], [0], [353450], [0], [337018], [-261624], [0], [663100], [0], [0], [-584564], [0], [-1003048], [-385902], [-385814], [0], [-1097052], [0], [0], [0], [1279498], [-723536], [0], [0], [0], [-1531552], [0], [0], [-2551450], [-2547974], [0], [0], [0], [0], [0], [221714], [2971476], [-4159650], [-2298164], [0], [0], [-4268968], [0], [599302], [-5592732], [0], [1260118], [0], [-6032926], [0], [0], [2074774], [7338848], [0], [0], [5104900], [-1998082], [-5779686], [0], [2886200], [0], [0], [-2761464], [-8827324], [-11120364], [0], [0], [0], [0], [11126126], [-2236036], [-6239486], [0], [-10473782], [6339124], [0], [0], [0], [7177568], [0], [0], [17551402], [7148986], [0], [19978576], [11496274], [0], [0], [-10977250], [20223236], [0], [1384350], [0], [21314852], [0], [-4886002], [-14322486], [0], [0], [0], [-2648932], [-24950200], [25108752], [-17887014], [0], [0], [0], [0], [0], [-18370682], [0], [-35588102], [7086752], [36855236], [0], [0], [0], [0], [0], [-27614248], [-43342750], [0], [-39401052], [0], [44834700], [-48784976], [50671632], [0], [0], [0], [55000900], [9971302], [20022200], [0], [49981714], [0], [-57314424], [0], [10140482], [0], [-66219868], [0], [-62537024], [-48289450], [0], [0], [-39713476], [2026198], [0], [0], [27091264], [0], [-63482102], [0], [-47496352], [0], [-58404786], [0], [85256182], [0], [0], [85384948], [34118676], [0], [-27800564], [0], [-1911426], [73202098], [0], [0], [44765314], [0], [0], [0], [0], [0], [0], [116488900], [119298982], [0], [-76524514], [0], [114418152], [0], [0], [0], [-127551864], [0], [120031936], [-18796402], [-55731126], [0], [-147450818], [46075152], [0], [114564186], [0], [-129104032], [0], [150806876], [149528898], [0], [-31338648], [-29866474], [0], [0], [0], [-170861702], [-138753932], [0], [-87448348], [0], [0], [147535518], [0], [0], [0], [0], [49360564], [97017800], [191266350], [0], [195859600], [0], [0], [-21944164], [-93106726], [0], [0], [0], [-125747600], [-66263164], [0], [0], [0], [0], [193438082], [0], [245530252], [174394198], [0], [-193557300], [0], [-24726768], [0], [0], [-173244964], [263147214], [-95370726], [0], [0], [269973152], [0], [0], [0], [258410176], [263004498], [-249830086], [0], [0], [-111958526], [0], [-305858102], [0], [89612798], [0], [0], [0], [0], [0], [0], [-61377250], [224246968], [-331019852], [341612324], [362248350], [-81526964], [0], [-161113768], [-318379050], [0], [389154086], [372672674], [-170373300], [-306752336], [0], [0], [-45163448], [0], [-303531118], [0], [-170581050], [0], [-411153048], [369265798], [0], [291108332], [0], [260345536], [82320502], [-458917602], [400917614], [0], [0], [-115081252], [0], [51392882], [0], [0], [0], [-251527450], [332063114], [0], [-494148576], [-176667698], [-363235532], [0], [504471636], [0], [0], [0], [-506532886], [-532281598], [0], [505768582], [0], [0], [0], [-262347250], [501225800], [-569783786], [0], [-592396400], [201680468], [0], [357106950], [0], [-494698952], [-190300374], [-127310324], [0], [-252196648], [608381114], [0], [-654436114], [0], [-485047376], [0], [0], [0], [0], [-98187098], [192581664], [-377233186], [0], [0], [-679452024], [0], [-543979348], [0], [741148718], [0], [376695332], [0], [740748424], [399309702], [0], [0], [303792652], [554748326], [0], [-14959298], [-585305864], [0], [0], [0], [644057086], [-78344002], [0], [0], [0], [-753435386], [173675500], [858461076], [-871936714], [0], [284205064], [0], [0], [0], [-764531986], [0], [879225800], [0], [0], [0], [906937748], [896910586], [0], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_441_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_441_6_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_441_6_a_a(:prec:=1)
    chi := MakeCharacter_441_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_441_6_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_441_6_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![11, 1]>,<5,R![0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;