
// Make newform 441.6.a.o in Magma, downloaded from the LMFDB on 25 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_o();".
// 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_o();".
// 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
        poly := [-7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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 := [[0, 1], [0, 0], [0, 0], [0, 0], [0, -302], [0, 0], [0, 0], [0, 0], [0, -418], [0, 2036], [0, 0], [8886, 0], [0, 0], [-11748, 0], [0, 0], [0, -12364], [0, 0], [0, 0], [69364, 0], [0, 32098], [0, 0], [80168, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 86098], [219582, 0], [0, -91432], [-262064, 0], [0, 0], [0, 98648], [0, 0], [0, -160436], [261624, 0], [0, 0], [663100, 0], [0, 0], [0, 0], [0, -237050], [0, 0], [0, 39050], [385902, 0], [0, -385076], [0, 0], [1097052, 0], [0, 0], [0, 0], [0, 0], [0, -398168], [0, 608950], [0, 0], [0, 0], [0, 0], [0, -619618], [0, 0], [0, 0], [-2551450, 0], [0, -271448], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 1349876], [-2971476, 0], [4159650, 0], [0, 1456126], [0, 0], [0, 0], [0, -896686], [0, 0], [599302, 0], [5592732, 0], [0, 0], [0, -2205236], [0, 0], [0, -851752], [0, 0], [0, 0], [2074774, 0], [0, -896950], [0, 0], [0, 0], [0, -2454914], [0, 3139664], [5779686, 0], [0, 0], [2886200, 0], [0, 0], [0, 0], [2761464, 0], [0, -2274898], [11120364, 0], [0, 0], [0, 0], [0, 0], [0, 0], [11126126, 0], [-2236036, 0], [0, -5007476], [0, 0], [0, 4290736], [6339124, 0], [0, 0], [0, 0], [0, 0], [0, 6058514], [0, 0], [0, 0], [17551402, 0], [0, 6617776], [0, 0], [19978576, 0], [0, 6554152], [0, 0], [0, 0], [0, 7115636], [0, 3549050], [0, 0], [-1384350, 0], [0, 0], [0, -4475218], [0, 0], [0, -9660100], [14322486, 0], [0, 0], [0, 0], [0, 0], [-2648932, 0], [0, 6360914], [-25108752, 0], [17887014, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 12239536], [0, 0], [0, -5675900], [7086752, 0], [0, 5196674], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 12829718], [-43342750, 0], [0, 0], [39401052, 0], [0, 0], [-44834700, 0], [0, -4307798], [-50671632, 0], [0, 0], [0, 0], [0, 0], [0, 1751002], [0, 20856232], [20022200, 0], [0, 0], [0, 12320176], [0, 0], [57314424, 0], [0, 0], [10140482, 0], [0, 0], [0, -1173886], [0, 0], [0, 10342498], [-48289450, 0], [0, 0], [0, 0], [-39713476, 0], [0, -27707900], [0, 0], [0, 0], [27091264, 0], [0, 0], [-63482102, 0], [0, 0], [0, -24711082], [0, 0], [58404786, 0], [0, 0], [85256182, 0], [0, 0], [0, 0], [0, 13201618], [-34118676, 0], [0, 0], [0, 35158474], [0, 0], [1911426, 0], [73202098, 0], [0, 0], [0, 0], [0, -36275900], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 6928714], [0, 489136], [0, 0], [-76524514, 0], [0, 0], [-114418152, 0], [0, 0], [0, 0], [0, 0], [127551864, 0], [0, 0], [0, 25874926], [0, 52322732], [55731126, 0], [0, 0], [0, -8208464], [-46075152, 0], [0, 0], [-114564186, 0], [0, 0], [0, 33834086], [0, 0], [0, -20649298], [-149528898, 0], [0, 0], [31338648, 0], [0, -62800552], [0, 0], [0, 0], [0, 0], [0, -8273132], [0, 39679486], [0, 0], [0, -59915482], [0, 0], [0, 0], [-147535518, 0], [0, 0], [0, 0], [0, 0], [0, 0], [49360564, 0], [0, 65845286], [-191266350, 0], [0, 0], [0, 25376098], [0, 0], [0, 0], [0, 79293050], [-93106726, 0], [0, 0], [0, 0], [0, 0], [-125747600, 0], [0, 82028726], [0, 0], [0, 0], [0, 0], [0, 0], [0, 54635636], [0, 0], [245530252, 0], [0, 67730068], [0, 0], [193557300, 0], [0, 0], [24726768, 0], [0, 0], [0, 0], [0, -78265726], [-263147214, 0], [95370726, 0], [0, 0], [0, 0], [269973152, 0], [0, 0], [0, 0], [0, 0], [0, 59685802], [-263004498, 0], [0, 66308276], [0, 0], [0, 0], [0, 111336148], [0, 0], [0, 35716768], [0, 0], [89612798, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -132705748], [224246968, 0], [0, 52629082], [341612324, 0], [-362248350, 0], [0, -136397426], [0, 0], [0, 128184386], [318379050, 0], [0, 0], [0, -5556100], [0, 49600352], [170373300, 0], [0, -95821826], [0, 0], [0, 0], [0, -153831050], [0, 0], [0, 108778064], [0, 0], [170581050, 0], [0, 0], [411153048, 0], [0, -83400032], [0, 0], [291108332, 0], [0, 0], [0, -142260326], [0, 171352468], [458917602, 0], [0, 95122676], [0, 0], [0, 0], [-115081252, 0], [0, 0], [51392882, 0], [0, 0], [0, 0], [0, 0], [-251527450, 0], [0, 144179024], [0, 0], [494148576, 0], [0, 187178068], [0, -145452086], [0, 0], [-504471636, 0], [0, 0], [0, 0], [0, 0], [-506532886, 0], [0, -57036100], [0, 0], [505768582, 0], [0, 0], [0, 0], [0, 0], [0, 193139552], [0, -108782014], [-569783786, 0], [0, 0], [0, 761002], [0, -213458686], [0, 0], [-357106950, 0], [0, 0], [-494698952, 0], [190300374, 0], [0, 238819702], [0, 0], [0, -227342182], [0, -92820100], [0, 0], [-654436114, 0], [0, 0], [0, 181979302], [0, 0], [0, 0], [0, 0], [0, 0], [0, -266337632], [-192581664, 0], [0, 235244624], [0, 0], [0, 0], [679452024, 0], [0, 0], [-543979348, 0], [0, 0], [741148718, 0], [0, 0], [0, -248134414], [0, 0], [0, 72737698], [-399309702, 0], [0, 0], [0, 0], [303792652, 0], [0, -210825748], [0, 0], [0, -300552032], [-585305864, 0], [0, 0], [0, 0], [0, 0], [644057086, 0], [0, 312480200], [0, 0], [0, 0], [0, 0], [0, -154666676], [0, 319097086], [-858461076, 0], [-871936714, 0], [0, 0], [0, -318567050], [0, 0], [0, 0], [0, 0], [0, -188508100], [0, 0], [0, 111046298], [0, 0], [0, 0], [0, 0], [0, 114007718], [0, -130032200], [0, 0], [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_o();".
// 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_o(:prec:=2)
    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_o();".
// 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_o( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_441_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<2,R![-7, 0, 1]>,<5,R![0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;