
// Make newform 550.3.d.a in Magma, downloaded from the LMFDB on 06 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_550_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_550_d_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_550_3_d_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_550_3_d_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
        poly := [2, 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_550_d();"
function MakeCharacter_550_d()
    N := 550;
    order := 2;
    char_gens := [177, 101];
    v := [2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_550_d_Hecke();"
function MakeCharacter_550_d_Hecke(Kf)
    N := 550;
    order := 2;
    char_gens := [177, 101];
    char_values := [[1, 0], [-1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := 3;
    raw_aps := [[0, -1], [-1, 0], [0, 0], [0, 6], [7, 6], [0, -6], [0, -18], [0, -18], [-17, 0], [0, -24], [17, 0], [-47, 0], [0, 6], [0, -12], [58, 0], [-2, 0], [-55, 0], [0, -60], [-89, 0], [-7, 0], [0, -90], [0, -24], [0, 24], [-97, 0], [121, 0], [0, -6], [82, 0], [0, -54], [0, 108], [1, 0], [0, 18], [0, -66], [-167, 0], [0, -66], [0, 114], [0, -204], [1, 0], [-110, 0], [0, -78], [0, -12], [209, 0], [119, 0], [-319, 0], [0, 12], [0, 72], [182, 0], [0, -84], [31, 0], [0, -66], [-73, 0], [0, 144], [0, 204], [0, 84], [65, 0], [-170, 0], [0, 222], [-430, 0], [0, 270], [0, 60], [0, -96], [0, 48], [0, -366], [0, -18], [-154, 0], [-95, 0], [-23, 0], [185, 0], [0, -6], [0, -150], [0, -186], [-167, 0], [0, -450], [607, 0], [0, 246], [-295, 0], [-377, 0], [-121, 0], [-578, 0], [-550, 0], [0, 420], [-370, 0], [338, 0], [0, 162], [25, 0], [0, 264], [-257, 0], [47, 0], [0, -192], [0, -348], [631, 0], [367, 0], [0, -330], [511, 0], [0, 24], [494, 0], [0, 468], [503, 0], [-745, 0], [0, -234], [0, -486], [0, -588], [0, -450], [0, 552], [0, 408], [0, -444], [385, 0], [-806, 0], [0, 108], [998, 0], [0, 534], [0, 264], [0, -252], [-314, 0], [1193, 0], [-439, 0], [671, 0], [487, 0], [-569, 0], [1081, 0], [0, 552], [1007, 0], [0, 456], [0, 618], [610, 0], [-1255, 0], [0, -54], [455, 0], [-223, 0], [535, 0], [0, 444], [0, 612], [0, -816], [-55, 0], [-2, 0], [0, -894], [0, -426], [1222, 0], [0, -636], [289, 0], [0, -162], [0, -540], [0, -420], [-305, 0], [0, 414], [119, 0], [-1615, 0], [0, -768], [0, 708], [593, 0], [-1190, 0], [0, -654], [263, 0], [394, 0], [0, 12], [418, 0], [-490, 0], [0, -432], [-742, 0], [0, -846], [0, 96], [-113, 0], [0, 18], [0, -240], [569, 0], [649, 0], [175, 0], [1334, 0], [0, -786]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_550_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_550_3_d_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_550_3_d_a(:prec:=2)
    chi := MakeCharacter_550_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_550_3_d_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_550_3_d_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_550_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1, 1]>],Snew);
    return Vf;
end function;