
// Make newform 1296.3.q.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1296_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1296_q_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_1296_3_q_b();".
// 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_1296_3_q_b();".
// 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_1296_q();"
function MakeCharacter_1296_q()
    N := 1296;
    order := 6;
    char_gens := [1135, 325, 1217];
    v := [6, 6, 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_1296_q_Hecke();"
function MakeCharacter_1296_q_Hecke(Kf)
    N := 1296;
    order := 6;
    char_gens := [1135, 325, 1217];
    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 := 3;
    raw_aps := [[0, 0], [0, 0], [0, 0], [2, -2], [0, 0], [0, 22], [0, 0], [-26, 0], [0, 0], [0, 0], [0, -46], [26, 0], [0, 0], [-22, 22], [0, 0], [0, 0], [0, 0], [-74, 74], [0, 122], [0, 0], [-46, 0], [-142, 142], [0, 0], [0, 0], [-2, 2], [0, 0], [0, 194], [0, 0], [-214, 0], [0, 0], [-146, 0], [0, 0], [0, 0], [0, -22], [0, 0], [-286, 286], [0, 118], [262, 0], [0, 0], [0, 0], [0, 0], [314, 0], [0, 0], [0, 382], [0, 0], [-386, 0], [0, -166], [338, -338], [0, 0], [0, -26], [0, 0], [0, 0], [286, -286], [0, 0], [0, 0], [0, 0], [0, 0], [-242, 0], [-122, 122], [0, 0], [0, 458], [0, 0], [358, 0], [0, 0], [142, -142], [0, 0], [362, -362], [0, -482], [0, 0], [502, -502], [0, 0], [0, 0], [-718, 718], [0, -698], [694, 0], [0, 0], [0, 0], [362, 0], [0, 0], [0, -626], [0, 0], [358, -358], [0, 0], [-862, 0], [-94, 94], [0, 0], [0, 0], [814, -814], [0, 0], [0, -526], [0, 0], [0, 0], [-962, 0], [0, 0], [0, 26], [0, 0], [0, 0], [0, 0], [982, 0], [1034, 0], [506, -506], [0, 0], [0, 0], [0, 0], [0, -886], [962, 0], [0, 0], [0, 0], [0, 0], [526, -526], [0, -814], [-1126, 0], [0, 0], [-214, 214], [-674, 0], [0, 0], [0, 314], [0, 0], [0, 0], [0, 0], [0, -122], [-1154, 1154], [0, 0], [0, 0], [-1318, 1318], [0, 0], [934, -934], [0, 0], [482, -482], [0, -1034], [1222, 0], [0, 0], [0, 1202], [-838, 0], [0, 0], [0, 1534], [0, 0], [0, 1562], [0, 0], [0, 0], [-1514, 0], [0, 0], [0, 1058], [0, 0], [458, 0], [0, 0], [-1658, 1658], [0, 0], [0, 1418], [0, 0], [0, 598], [0, 0], [1702, 0], [0, 0], [-214, 214], [0, 0], [-866, 0], [0, 0], [-1198, 0], [0, 0], [0, 0], [0, 0], [0, -1534], [0, 0], [0, 0], [0, 0], [46, 0], [1894, -1894], [1826, 0], [0, 0], [0, 0], [0, 1846], [0, 0], [-1298, 1298], [0, 626], [0, 0], [1994, -1994], [0, 0], [1342, 0], [-1706, 1706], [-2158, 2158], [0, 0], [0, 1702], [0, 0], [0, 0], [0, 0], [-118, 0], [-1222, 1222], [0, 814], [0, 0], [1538, 0], [0, 0], [-1754, 0], [0, 0], [0, 0], [0, 0], [0, 2398], [-74, 74], [0, 0], [0, 0], [0, 0], [2162, -2162], [0, -2426], [2302, -2302], [0, 0], [0, 0], [1774, 0], [0, 0], [0, 0], [0, 554], [-478, 0], [0, 0], [2018, -2018], [0, 0], [0, 0], [2158, -2158], [0, -2638], [0, 0], [0, 0], [0, 0], [0, -2714], [0, 1826], [0, 0], [2446, 0], [0, 0], [-506, 506], [0, 0], [0, 0], [2882, -2882], [0, 0], [0, 2902], [-218, 0], [0, 2834], [0, 0], [-1366, 1366], [0, 0], [0, 1822], [0, 0], [0, 0], [0, 0], [0, 0], [-1034, 0], [0, -382], [746, 0], [0, 0], [0, 0], [1198, 0], [0, 0], [0, -2134], [0, 0], [0, 694], [0, 0], [0, 0], [-146, 146], [0, 0], [0, 0], [-2566, 0], [3146, -3146], [0, 0], [1586, 0], [1298, -1298], [0, 0], [0, -2138], [3338, 0], [0, 0], [698, -698], [0, 0], [0, 0], [0, -1846], [0, 0], [0, 2326], [-2906, 0], [3406, -3406], [0, -814], [0, 3358], [2782, 0], [0, 0], [-3146, 3146], [1874, 0], [0, 0], [0, 0], [0, 962], [0, 0], [-3674, 3674], [0, -598], [0, 0], [2978, 0], [0, 0], [-3742, 3742], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [22, -22], [0, 0], [1874, -1874], [0, 0], [0, 0], [-2374, 2374], [0, -914], [0, 0], [1294, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1296_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1296_3_q_b();".
// 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_1296_3_q_b(:prec:=2)
    chi := MakeCharacter_1296_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1296_3_q_b();".
// 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_1296_3_q_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1296_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![4, -2, 1]>],Snew);
    return Vf;
end function;