
// Make newform 648.4.i.k in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_648_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_648_i_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_648_4_i_k();".
// 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_648_4_i_k();".
// 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 := [1, -1, 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_648_i();"
function MakeCharacter_648_i()
    N := 648;
    order := 3;
    char_gens := [487, 325, 569];
    v := [3, 3, 2];
    // 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_648_i_Hecke();"
function MakeCharacter_648_i_Hecke(Kf)
    N := 648;
    order := 3;
    char_gens := [487, 325, 569];
    char_values := [[1, 0], [1, 0], [0, -1]];
    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+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 := 4;
    raw_aps := [[0, 0], [0, 0], [0, 14], [24, -24], [-28, 28], [0, 74], [-82, 0], [92, 0], [0, 8], [-138, 138], [0, -80], [30, 0], [0, 282], [-4, 4], [240, -240], [130, 0], [0, 596], [218, -218], [0, 436], [-856, 0], [-998, 0], [32, -32], [-1508, 1508], [246, 0], [-866, 866], [270, -270], [0, 1496], [1692, 0], [406, 0], [0, 786], [1744, 0], [0, 652], [1530, -1530], [0, -516], [0, 1342], [424, -424], [0, -262], [-2292, 0], [0, -1896], [-2874, 2874], [1188, 0], [-3474, 0], [192, -192], [0, -4802], [-1518, 0], [5128, 0], [0, -1084], [-688, 688], [4812, -4812], [0, -2494], [-698, 0], [0, -6320], [6510, -6510], [-628, 0], [0, -4862], [5816, -5816], [-3526, 0], [-256, 0], [-142, 142], [8842, -8842], [0, 7180], [0, 7374], [1500, 0], [0, -7608], [4758, -4758], [4374, -4374], [7804, -7804], [0, -5106], [0, -4716], [-7302, 7302], [-4382, 4382], [-7224, 0], [-1408, 1408], [0, 1714], [884, 0], [0, 10368], [398, -398], [-5098, 0], [0, 10002], [0, 9270], [0, -6516], [2626, -2626], [4304, 0], [11794, 0], [5544, -5544], [-3788, 3788], [13342, 0], [4390, -4390], [5798, -5798], [0, 14656], [-8412, 0], [14848, -14848], [18568, 0], [0, -14364], [0, -21660], [17112, 0], [0, 11478], [-13114, 0], [-4508, 0], [22950, 0], [6580, -6580], [-7046, 0], [0, 8252], [-6838, 6838], [0, -23316], [-10558, 0], [1028, -1028], [-1202, 0], [0, -3576], [-8650, 8650], [0, -12656], [-3298, 0], [0, 5370], [16220, -16220], [-20360, 0], [14498, -14498], [0, -21612], [-12184, 0], [0, -28122], [-5700, 5700], [0, 29458], [-19810, 19810], [-10450, 10450], [-23300, 0], [14212, -14212], [15978, 0], [8866, -8866], [-7760, 0], [-13080, 13080], [0, -16934], [-7060, 0], [0, -12520], [0, 9792], [13166, 0], [0, -23222], [0, 39934], [17106, 0], [0, 9956], [0, -9130], [-11482, 0], [4612, 0], [-35010, 35010], [0, -13688], [-11668, 0], [-29306, 0], [-2664, 2664], [-26030, 26030], [44202, -44202], [0, 32748], [-45344, 0], [0, 8778], [4142, 0], [22076, 0], [0, -40376], [26396, -26396], [24368, -24368], [-5096, 0], [-18494, 18494], [-33222, 0], [0, 27846], [41052, -41052], [-5706, 0], [0, 39352], [33180, 0], [0, -4014], [20328, -20328], [11728, 0], [-50974, 50974]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_648_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_648_4_i_k();".
// 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_648_4_i_k(:prec:=2)
    chi := MakeCharacter_648_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_648_4_i_k();".
// 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_648_4_i_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_648_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![196, -14, 1]>],Snew);
    return Vf;
end function;