
// Make newform 722.3.b.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_722_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_722_b_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_722_3_b_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_722_3_b_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 := [4, 0, -2, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 1, 0], [0, 4, 0, -1]];
    Rf_basisdens := [1, 2, 1, 2];
    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_722_b();"
function MakeCharacter_722_b()
    N := 722;
    order := 2;
    char_gens := [363];
    v := [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_722_b_Hecke();"
function MakeCharacter_722_b_Hecke(Kf)
    N := 722;
    order := 2;
    char_gens := [363];
    char_values := [[-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, 1, 0, 0], [0, -1, 1, 0], [-1, 0, 0, 0], [2, 0, 0, -2], [-10, 0, 0, 2], [0, -6, 5, 0], [-7, 0, 0, -2], [0, 0, 0, 0], [5, 0, 0, -9], [0, -18, -11, 0], [0, 0, -18, 0], [0, 18, -2, 0], [0, -42, 3, 0], [-19, 0, 0, 23], [35, 0, 0, 19], [0, -48, 7, 0], [0, -33, 7, 0], [-37, 0, 0, -16], [0, -63, -17, 0], [0, -9, 51, 0], [-49, 0, 0, 32], [0, -21, -21, 0], [-16, 0, 0, -6], [0, 6, 1, 0], [0, -6, -81, 0], [-67, 0, 0, -30], [0, 12, 56, 0], [0, 12, 56, 0], [0, 30, 15, 0], [0, 18, -62, 0], [0, -3, -75, 0], [95, 0, 0, -5], [-7, 0, 0, -2], [5, 0, 0, 15], [77, 0, 0, -46], [0, -36, -20, 0], [35, 0, 0, -96], [-142, 0, 0, 6], [0, 33, 79, 0], [0, 84, -107, 0], [0, -120, 84, 0], [0, -24, -1, 0], [50, 0, 0, -94], [0, 60, 77, 0], [296, 0, 0, 18], [-37, 0, 0, -121], [0, 99, 121, 0], [0, -9, -11, 0], [0, -108, -56, 0], [-196, 0, 0, -26], [-307, 0, 0, -34], [-184, 0, 0, -46], [0, 90, -7, 0], [365, 0, 0, 51], [0, 84, 63, 0], [11, 0, 0, 51], [0, 234, -41, 0], [83, 0, 0, -129], [260, 0, 0, -110], [0, -36, -61, 0], [-73, 0, 0, 31], [0, -66, -138, 0], [0, 51, 107, 0], [56, 0, 0, 66], [-31, 0, 0, -114], [0, 42, 165, 0], [0, 276, 56, 0], [0, 222, -85, 0], [341, 0, 0, 15], [-268, 0, 0, -130], [80, 0, 0, 162], [-49, 0, 0, -81], [359, 0, 0, 51], [0, -276, 144, 0], [0, 204, 104, 0], [0, -285, -89, 0], [-289, 0, 0, 156], [83, 0, 0, 72], [0, -258, -49, 0], [0, -264, 155, 0], [566, 0, 0, -42], [0, -348, 5, 0], [0, -111, -171, 0], [0, -192, -277, 0], [0, 195, 203, 0], [-7, 0, 0, 183], [0, 126, 98, 0], [326, 0, 0, 48], [-211, 0, 0, 166], [-40, 0, 0, -278], [218, 0, 0, 22], [575, 0, 0, -93], [0, -336, -12, 0], [527, 0, 0, 3], [617, 0, 0, 71], [89, 0, 0, 135], [0, -138, -351, 0], [0, 168, -376, 0], [0, -123, -189, 0], [-643, 0, 0, 102], [0, -135, 389, 0], [299, 0, 0, -184], [0, -120, -4, 0], [0, 234, 362, 0], [554, 0, 0, 114], [20, 0, 0, -206], [-685, 0, 0, 27], [-379, 0, 0, 78], [0, -111, 137, 0], [0, -150, -262, 0], [0, 168, -76, 0], [-379, 0, 0, -310], [737, 0, 0, 74], [-874, 0, 0, -102], [83, 0, 0, -1], [0, -420, 57, 0], [-7, 0, 0, 203], [-646, 0, 0, 62], [-88, 0, 0, 230], [0, -363, -33, 0], [0, 210, -159, 0], [0, 414, -190, 0], [0, -138, -270, 0], [0, -168, -42, 0], [614, 0, 0, -174], [233, 0, 0, 126], [-697, 0, 0, 84], [299, 0, 0, -357], [-121, 0, 0, -285], [-4, 0, 0, 330], [-343, 0, 0, 171], [0, 255, 129, 0], [0, -171, -203, 0], [-145, 0, 0, -168], [8, 0, 0, -498], [-145, 0, 0, -332], [0, -36, 299, 0], [0, 336, 540, 0], [0, -702, 126, 0], [-100, 0, 0, -414], [0, -159, 139, 0], [-151, 0, 0, -22], [59, 0, 0, 483], [0, 381, 349, 0], [0, 480, 232, 0], [0, 303, 141, 0], [-517, 0, 0, 312], [0, -288, -175, 0], [131, 0, 0, 359], [0, -168, -126, 0], [0, 354, 87, 0], [-484, 0, 0, -14], [221, 0, 0, -365], [0, -195, -705, 0], [0, -153, 287, 0], [0, -804, 204, 0], [-382, 0, 0, 66], [671, 0, 0, 228], [-193, 0, 0, 24], [0, 204, 179, 0], [725, 0, 0, 39], [0, 144, -163, 0], [881, 0, 0, 183], [0, 231, -517, 0], [0, 6, 506, 0], [0, -261, 407, 0], [0, 423, -673, 0], [-229, 0, 0, -84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_722_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_722_3_b_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_722_3_b_b(:prec:=4)
    chi := MakeCharacter_722_b();
    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_722_3_b_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_722_3_b_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_722_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<3,R![1, 0, 10, 0, 1]>],Snew);
    return Vf;
end function;