
// Make newform 720.3.l.c in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_720_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_720_l_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_3_l_c();".
// 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_3_l_c();".
// 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 := [9, 0, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, -1, 0, 1], [-2, 0, 1, 0], [0, 7, 0, -1]];
    Rf_basisdens := [1, 1, 1, 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_l();"
function MakeCharacter_720_l()
    N := 720;
    order := 2;
    char_gens := [271, 181, 641, 577];
    v := [2, 2, 1, 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_720_l_Hecke();"
function MakeCharacter_720_l_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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 0], [4, 0, 0, -1], [0, -1, 6, 0], [2, 0, 0, 1], [0, -4, 0, 0], [-8, 0, 0, -2], [0, -2, 6, 0], [0, 8, -6, 0], [-2, 0, 0, -2], [-34, 0, 0, -3], [0, -3, 24, 0], [-20, 0, 0, 2], [0, 14, 12, 0], [0, 10, 24, 0], [0, -17, -18, 0], [-10, 0, 0, -6], [76, 0, 0, 0], [0, -12, 12, 0], [38, 0, 0, -6], [-50, 0, 0, -6], [0, -2, 24, 0], [0, 21, 12, 0], [-106, 0, 0, 2], [0, -2, -54, 0], [28, 0, 0, -15], [0, 22, 0, 0], [26, 0, 0, 10], [0, 26, 0, 0], [-20, 0, 0, -9], [0, -11, -42, 0], [0, -6, -108, 0], [-98, 0, 0, 12], [0, -6, 30, 0], [10, 0, 0, -12], [-22, 0, 0, 1], [196, 0, 0, 10], [0, -38, -6, 0], [0, 34, -84, 0], [0, -5, -30, 0], [38, 0, 0, -10], [0, -26, 24, 0], [110, 0, 0, 0], [0, 4, 102, 0], [94, 0, 0, 8], [-38, 0, 0, 32], [64, 0, 0, 13], [0, -36, -120, 0], [-178, 0, 0, 8], [0, 36, -24, 0], [0, 50, -60, 0], [-220, 0, 0, -4], [0, -49, -30, 0], [0, 50, -36, 0], [0, 86, 42, 0], [0, -68, -6, 0], [-362, 0, 0, -12], [-10, 0, 0, 35], [0, -83, 0, 0], [16, 0, 0, 28], [0, 12, 6, 0], [-344, 0, 0, 12], [0, -2, 0, 0], [302, 0, 0, 32], [0, 52, -138, 0], [220, 0, 0, -18], [206, 0, 0, -22], [0, -24, 12, 0], [-94, 0, 0, -6], [0, 12, 108, 0], [0, 98, 72, 0], [-272, 0, 0, 3], [410, 0, 0, -31], [-182, 0, 0, 16], [0, 46, 204, 0], [0, 20, 54, 0], [14, 0, 0, 11], [0, -15, -48, 0], [-298, 0, 0, -4], [0, -17, -150, 0], [-202, 0, 0, -60], [0, -30, 252, 0], [-478, 0, 0, -38], [-50, 0, 0, 0], [0, 4, -228, 0], [0, 19, 180, 0], [-490, 0, 0, -6], [0, -42, 138, 0], [-224, 0, 0, 21], [0, 114, 84, 0], [0, -32, -180, 0], [-428, 0, 0, -19], [0, -37, 6, 0], [-332, 0, 0, -30], [0, 82, 12, 0], [0, -6, -66, 0], [0, 63, 84, 0], [580, 0, 0, -30], [470, 0, 0, 46], [616, 0, 0, 6], [0, -66, -264, 0], [0, -64, 132, 0], [0, 19, 96, 0], [-116, 0, 0, -82], [-94, 0, 0, -18], [0, 174, 24, 0], [0, 70, -132, 0], [0, -30, 396, 0], [20, 0, 0, -32], [316, 0, 0, -51], [-538, 0, 0, 15], [0, -132, -132, 0], [-422, 0, 0, 12], [-326, 0, 0, -2], [0, 123, -324, 0], [-440, 0, 0, -30], [0, -2, 60, 0], [0, 44, -330, 0], [0, -53, -258, 0], [-586, 0, 0, 6], [698, 0, 0, -30], [0, -88, 6, 0], [0, -62, 108, 0], [328, 0, 0, 6], [0, 94, 18, 0], [-430, 0, 0, 32], [0, 198, -120, 0], [-212, 0, 0, 111], [-46, 0, 0, 57], [-524, 0, 0, -78], [0, -42, -348, 0], [154, 0, 0, 102], [302, 0, 0, -81], [0, -67, -420, 0], [-304, 0, 0, -104], [0, -52, 246, 0], [880, 0, 0, 66], [0, 154, -180, 0], [0, -189, -132, 0], [-158, 0, 0, 40], [0, 88, 282, 0], [16, 0, 0, -37], [0, 104, 252, 0], [506, 0, 0, 18], [0, 18, -312, 0], [110, 0, 0, -69], [0, 136, -420, 0], [-206, 0, 0, 24], [0, -122, 330, 0], [1070, 0, 0, 7], [0, -3, 324, 0], [100, 0, 0, 46], [0, -266, -150, 0], [112, 0, 0, -102], [0, 360, 24, 0], [1546, 0, 0, -6], [0, 35, -396, 0], [146, 0, 0, 148], [0, 164, -30, 0], [0, 56, 564, 0], [0, -34, -36, 0], [136, 0, 0, -39], [0, -131, 558, 0], [0, 264, -324, 0], [0, 70, -132, 0], [-734, 0, 0, 24], [302, 0, 0, -63]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_720_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_720_3_l_c();".
// 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_3_l_c(:prec:=4)
    chi := MakeCharacter_720_l();
    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_720_3_l_c();".
// 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_3_l_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_720_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<7,R![-74, -8, 1]>],Snew);
    return Vf;
end function;