
// Make newform 648.3.m.d in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_648_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_648_m_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_3_m_d();".
// 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_3_m_d();".
// 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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 4, 0, 0], [0, 0, 1, 0], [0, 0, 0, 2]];
    Rf_basisdens := [1, 1, 2, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_m();"
function MakeCharacter_648_m()
    N := 648;
    order := 6;
    char_gens := [487, 325, 569];
    v := [6, 6, 5];
    // 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_m_Hecke();"
function MakeCharacter_648_m_Hecke(Kf)
    N := 648;
    order := 6;
    char_gens := [487, 325, 569];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [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+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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 6, 0], [0, -1, 0, 1], [-10, 0, 10, 0], [0, 0, 0, 4], [2, 0, 0, 0], [0, -2, 0, 0], [0, -3, 0, 3], [22, 0, -22, 0], [-6, 0, 0, 0], [0, 6, 0, 0], [0, 0, -82, 0], [0, -12, 0, 12], [0, 0, 0, 11], [0, 13, 0, 0], [0, 0, 86, 0], [-2, 0, 2, 0], [0, 0, 0, 22], [82, 0, 0, 0], [0, 0, -10, 0], [0, 13, 0, -13], [0, 0, 0, 6], [0, 0, 94, 0], [0, -9, 0, 9], [134, 0, -134, 0], [0, 0, 0, -9], [10, 0, 0, 0], [0, -12, 0, 0], [106, 0, 0, 0], [0, -1, 0, 0], [0, 18, 0, -18], [78, 0, -78, 0], [0, 29, 0, 0], [0, 0, -218, 0], [86, 0, -86, 0], [-222, 0, 0, 0], [0, -30, 0, 0], [0, 33, 0, -33], [0, 0, 0, -27], [90, 0, 0, 0], [0, 48, 0, -48], [-2, 0, 2, 0], [0, 0, 0, 15], [250, 0, 0, 0], [-34, 0, 34, 0], [0, 0, 278, 0], [0, -39, 0, 39], [-58, 0, 58, 0], [0, 0, 0, -70], [0, -4, 0, 0], [0, 0, 30, 0], [0, 0, 0, 19], [0, 32, 0, 0], [0, 38, 0, -38], [0, 0, 0, -71], [266, 0, 0, 0], [0, 0, -346, 0], [0, 22, 0, -22], [46, 0, -46, 0], [0, -39, 0, 0], [-30, 0, 0, 0], [0, 102, 0, 0], [0, 0, -210, 0], [0, -27, 0, 27], [0, 0, -434, 0], [510, 0, -510, 0], [0, -27, 0, 0], [0, 0, -426, 0], [0, -8, 0, 8], [0, 0, 0, 78], [0, 0, 566, 0], [-218, 0, 218, 0], [-142, 0, 0, 0], [0, 0, 0, 0], [0, -97, 0, 97], [-310, 0, 0, 0], [0, 60, 0, 0], [270, 0, -270, 0], [0, -9, 0, 0], [0, 0, 454, 0], [0, 0, 0, -44], [706, 0, 0, 0], [0, 0, 486, 0], [0, -125, 0, 125], [0, 0, 0, 128], [0, 0, -338, 0], [0, 137, 0, -137], [-74, 0, 74, 0], [0, 0, 0, 141], [0, -56, 0, 56], [-134, 0, 0, 0], [0, 9, 0, 0], [30, 0, -30, 0], [0, 0, 0, 42], [0, -21, 0, 0], [0, 0, 0, -14], [-494, 0, 0, 0], [234, 0, 0, 0], [0, 0, -290, 0], [0, 0, 0, -31], [0, 43, 0, 0], [0, 110, 0, -110], [-402, 0, 402, 0], [98, 0, 0, 0], [0, -97, 0, 97], [0, 0, 0, -124], [0, -114, 0, 0], [0, 0, 398, 0], [-170, 0, 170, 0], [-1030, 0, 0, 0], [0, -186, 0, 0], [0, 0, 14, 0], [1114, 0, 0, 0], [0, 80, 0, -80], [798, 0, -798, 0], [0, 0, 0, -26], [0, -57, 0, 0], [0, 141, 0, -141], [-986, 0, 986, 0], [0, 0, -34, 0], [0, 71, 0, -71], [0, 0, 0, 23], [0, 0, -578, 0], [0, 0, 0, -219], [0, 0, 1222, 0], [0, 0, 0, 44], [0, 0, 870, 0], [214, 0, -214, 0], [-958, 0, 0, 0], [0, 178, 0, 0], [630, 0, -630, 0], [602, 0, 0, 0], [0, 194, 0, 0], [-770, 0, 770, 0], [0, 0, 0, -33], [-514, 0, 514, 0], [0, -125, 0, 0], [0, 0, 0, -134], [-1454, 0, 0, 0], [0, 171, 0, -171], [166, 0, -166, 0], [0, 0, 0, -173], [1258, 0, 0, 0], [0, -234, 0, 234], [0, 0, 742, 0], [0, -42, 0, 42], [1230, 0, -1230, 0], [0, 0, 0, 8], [822, 0, -822, 0], [0, 0, 0, -116], [962, 0, 0, 0], [0, -202, 0, 0], [0, 0, -1042, 0], [0, 284, 0, -284], [-614, 0, 0, 0], [0, -8, 0, 8], [-462, 0, 0, 0], [0, 63, 0, 0], [0, -123, 0, 123], [0, 0, 0, 270], [70, 0, -70, 0], [0, 0, 0, 111], [0, -180, 0, 0], [0, 114, 0, -114], [-854, 0, 0, 0], [0, 0, 518, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_648_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_648_3_m_d();".
// 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_3_m_d(:prec:=4)
    chi := MakeCharacter_648_m();
    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_648_3_m_d();".
// 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_3_m_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_648_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![1024, 0, -32, 0, 1]>,<7,R![36, -6, 1]>],Snew);
    return Vf;
end function;