
// Make newform 720.4.f.l in Magma, downloaded from the LMFDB on 23 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_720_f();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_720_f_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_720_4_f_l();".
// 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_720_4_f_l();".
// 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 := [64, 0, -15, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 39, 0, -1], [0, 7, 0, -1], [-60, 0, 8, 0]];
    Rf_basisdens := [1, 8, 4, 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_720_f();"
function MakeCharacter_720_f()
    N := 720;
    order := 2;
    char_gens := [271, 181, 641, 577];
    v := [2, 2, 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_720_f_Hecke();"
function MakeCharacter_720_f_Hecke(Kf)
    N := 720;
    order := 2;
    char_gens := [271, 181, 641, 577];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 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 := 4;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, -1, -1, 0], [0, 0, 0, -1], [0, 2, 1, 0], [0, 0, 0, 3], [0, 0, 31, 0], [84, 0, 0, 0], [0, 0, 70, 0], [0, -18, -9, 0], [-16, 0, 0, 0], [0, 0, 0, -11], [0, -20, -10, 0], [0, 0, 0, 16], [0, 0, -50, 0], [0, 0, -369, 0], [0, -58, -29, 0], [-358, 0, 0, 0], [0, 0, 0, -38], [0, -84, -42, 0], [0, 0, 0, 20], [-936, 0, 0, 0], [0, 0, 652, 0], [0, -64, -32, 0], [0, 0, 0, -34], [0, -58, -29, 0], [0, 0, 0, -31], [0, 0, -348, 0], [1142, 0, 0, 0], [0, 0, 39, 0], [0, 0, 0, 59], [0, -42, -21, 0], [0, 0, -531, 0], [-860, 0, 0, 0], [0, -122, -61, 0], [1376, 0, 0, 0], [0, 0, 0, 83], [0, 0, 0, -10], [0, 0, 1086, 0], [0, 0, 1717, 0], [0, 302, 151, 0], [-1678, 0, 0, 0], [0, -156, -78, 0], [0, 0, 0, 110], [0, 0, 263, 0], [-744, 0, 0, 0], [-1476, 0, 0, 0], [0, 0, 0, 37], [0, 0, 1100, 0], [-5246, 0, 0, 0], [0, 0, 421, 0], [0, -524, -262, 0], [2530, 0, 0, 0], [0, 606, 303, 0], [0, 0, 129, 0], [0, 0, -386, 0], [0, 234, 117, 0], [6392, 0, 0, 0], [0, 0, 0, 79], [0, 576, 288, 0], [0, 0, 0, 62], [0, 0, -687, 0], [0, 0, 0, -12], [0, -324, -162, 0], [0, 0, 0, 350], [0, 0, -4667, 0], [-1636, 0, 0, 0], [0, 0, 0, -8], [0, 0, -1920, 0], [-6682, 0, 0, 0], [0, 0, -1055, 0], [0, 168, 84, 0], [0, 0, 0, -433], [0, 0, 0, -519], [-6220, 0, 0, 0], [0, 0, 6950, 0], [0, 326, 163, 0], [0, 0, 0, -23], [0, -268, -134, 0], [-14822, 0, 0, 0], [0, 778, 389, 0], [4894, 0, 0, 0], [0, -880, -440, 0], [0, 0, 0, -166], [-9288, 0, 0, 0], [0, 0, -3684, 0], [0, -36, -18, 0], [0, 0, 0, -662], [0, 426, 213, 0], [0, 0, 0, 327], [0, 0, -2832, 0], [0, -908, -454, 0], [0, 0, 0, -519], [0, -42, -21, 0], [9204, 0, 0, 0], [0, 0, -630, 0], [0, -186, -93, 0], [0, 476, 238, 0], [0, 0, 0, -418], [9658, 0, 0, 0], [0, 0, 0, -40], [0, 0, 4533, 0], [0, 0, -784, 0], [0, 188, 94, 0], [13916, 0, 0, 0], [0, 0, 0, 364], [0, 0, 10404, 0], [0, 0, 10743, 0], [0, -2328, -1164, 0], [-250, 0, 0, 0], [0, 0, 0, -599], [0, 0, 0, -661], [0, 0, -7205, 0], [-21580, 0, 0, 0], [15160, 0, 0, 0], [0, -548, -274, 0], [0, 0, 0, 642], [0, 0, -5958, 0], [0, 0, 12323, 0], [0, 1610, 805, 0], [-14638, 0, 0, 0], [0, 0, 0, -1398], [0, 0, -3191, 0], [0, 0, 2272, 0], [-13564, 0, 0, 0], [0, -134, -67, 0], [17826, 0, 0, 0], [0, -1688, -844, 0], [0, 0, 0, 1485], [0, 0, 0, 1561], [-22940, 0, 0, 0], [0, 0, -13230, 0], [30160, 0, 0, 0], [0, 0, 0, 835], [0, -492, -246, 0], [-29406, 0, 0, 0], [0, 0, -15561, 0], [0, 0, 0, -1546], [0, 0, -6533, 0], [0, 3588, 1794, 0], [-14428, 0, 0, 0], [0, 1490, 745, 0], [0, 0, 0, -1137], [0, 0, -8296, 0], [-18234, 0, 0, 0], [0, 2816, 1408, 0], [0, 0, 0, -101], [0, 0, 12603, 0], [17540, 0, 0, 0], [0, 0, 16554, 0], [0, 0, 0, -1309], [0, 792, 396, 0], [0, 0, 0, 882], [0, 0, -4002, 0], [0, 0, 0, -60], [0, 2880, 1440, 0], [2832, 0, 0, 0], [0, 2868, 1434, 0], [0, 0, 0, -1156], [0, 634, 317, 0], [0, 0, -876, 0], [0, 0, -1925, 0], [0, 0, 0, 1099], [0, 198, 99, 0], [0, 0, 14991, 0], [0, 0, 17642, 0], [11528, 0, 0, 0], [0, 0, 0, 447]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_720_f_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_720_4_f_l();".
// 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_720_4_f_l(:prec:=4)
    chi := MakeCharacter_720_f();
    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_720_4_f_l();".
// 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_720_4_f_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_720_f();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<7,R![496, 0, 1]>,<11,R![-496, 0, 1]>],Snew);
    return Vf;
end function;