
// Make newform 363.3.b.i in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_363_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_363_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_363_3_b_i();".
// 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_363_3_b_i();".
// 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 := [2, -8, 9, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [8, -2, 2, 0], [-15, 32, -6, 4], [-16, 34, -6, 4]];
    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_363_b();"
function MakeCharacter_363_b()
    N := 363;
    order := 2;
    char_gens := [122, 244];
    v := [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_363_b_Hecke();"
function MakeCharacter_363_b_Hecke(Kf)
    N := 363;
    order := 2;
    char_gens := [122, 244];
    char_values := [[-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, 1, 0], [-1, 0, 0, 1], [0, 0, 0, 1], [0, 1, 0, 0], [0, 0, 0, 0], [0, -3, 0, 0], [0, 0, 8, 0], [0, -2, 0, 0], [0, 0, 0, -11], [0, 0, 8, 0], [30, 0, 0, 0], [-10, 0, 0, 0], [0, 0, 16, 0], [0, 2, 0, 0], [0, 0, 0, 13], [0, 0, 0, -15], [0, 0, 0, 12], [0, 13, 0, 0], [-42, 0, 0, 0], [0, 0, 0, 23], [0, -10, 0, 0], [0, -3, 0, 0], [0, 0, 8, 0], [0, 0, 0, -22], [74, 0, 0, 0], [0, 0, 56, 0], [34, 0, 0, 0], [0, 0, 16, 0], [0, -9, 0, 0], [0, 0, 0, -42], [0, -9, 0, 0], [0, 0, 88, 0], [0, 0, 0, 56], [0, 12, 0, 0], [0, 0, 8, 0], [0, 9, 0, 0], [86, 0, 0, 0], [-14, 0, 0, 0], [0, 0, -32, 0], [0, 0, -72, 0], [0, 0, 0, 112], [-262, 0, 0, 0], [0, 0, 0, -21], [0, -20, 0, 0], [0, 0, 88, 0], [222, 0, 0, 0], [0, 52, 0, 0], [-46, 0, 0, 0], [0, 0, -80, 0], [118, 0, 0, 0], [0, 0, -32, 0], [0, 0, 16, 0], [0, -20, 0, 0], [0, 0, 0, 18], [0, 0, 0, 34], [0, 0, -176, 0], [0, 0, 0, -131], [0, 45, 0, 0], [0, -3, 0, 0], [0, 0, 8, 0], [0, -24, 0, 0], [0, 0, 8, 0], [0, -20, 0, 0], [0, 0, 0, -9], [398, 0, 0, 0], [0, 0, 0, -147], [178, 0, 0, 0], [0, 34, 0, 0], [0, 0, 184, 0], [0, 9, 0, 0], [0, 0, 0, 44], [0, 0, 96, 0], [-142, 0, 0, 0], [0, 57, 0, 0], [-350, 0, 0, 0], [0, 0, 0, 183], [0, 0, 0, 155], [442, 0, 0, 0], [0, 0, 0, -186], [0, 30, 0, 0], [0, 0, 0, 242], [-266, 0, 0, 0], [0, 0, -208, 0], [738, 0, 0, 0], [0, 57, 0, 0], [0, 0, 0, -42], [0, 0, 0, -180], [0, 90, 0, 0], [0, 0, -264, 0], [882, 0, 0, 0], [0, 0, 0, 188], [0, 0, -256, 0], [350, 0, 0, 0], [0, 0, 96, 0], [-98, 0, 0, 0], [0, 0, 192, 0], [0, 0, 0, 57], [0, 0, 0, -54], [0, 2, 0, 0], [0, -69, 0, 0], [0, -90, 0, 0], [0, 0, -168, 0], [0, 0, -208, 0], [0, 0, -424, 0], [0, -108, 0, 0], [310, 0, 0, 0], [0, 0, 0, -340], [0, 0, -88, 0], [0, 0, 0, 309], [0, 78, 0, 0], [0, -113, 0, 0], [0, -79, 0, 0], [0, 0, 0, -154], [-838, 0, 0, 0], [254, 0, 0, 0], [0, 0, 0, -42], [-438, 0, 0, 0], [0, 0, 0, 117], [0, 0, 0, 221], [0, 0, 176, 0], [-394, 0, 0, 0], [0, 118, 0, 0], [0, 0, -344, 0], [0, 0, 0, 154], [426, 0, 0, 0], [0, 0, 104, 0], [-42, 0, 0, 0], [0, 0, 0, -153], [1102, 0, 0, 0], [0, -65, 0, 0], [0, 140, 0, 0], [0, 0, 96, 0], [350, 0, 0, 0], [294, 0, 0, 0], [0, 0, -296, 0], [0, 112, 0, 0], [0, 0, 0, 145], [0, -42, 0, 0], [0, 0, 0, -87], [0, 0, 96, 0], [0, 130, 0, 0], [0, 0, 184, 0], [1042, 0, 0, 0], [0, 0, -472, 0], [1178, 0, 0, 0], [0, 0, 0, -361], [0, 79, 0, 0], [0, 0, 0, 0], [706, 0, 0, 0], [0, 0, 0, -175], [0, -13, 0, 0], [0, 0, 0, -528], [86, 0, 0, 0], [0, 0, 96, 0], [-966, 0, 0, 0], [0, 0, 0, 513], [0, -53, 0, 0], [0, 0, 0, 540], [0, -168, 0, 0], [0, 0, 184, 0], [0, 0, 0, -154], [0, 0, -608, 0], [0, -141, 0, 0], [0, 0, 0, 90], [0, 0, 0, -510], [0, 0, 0, 243], [574, 0, 0, 0], [0, 177, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_363_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_363_3_b_i();".
// 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_363_3_b_i(:prec:=4)
    chi := MakeCharacter_363_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_363_3_b_i();".
// 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_363_3_b_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_363_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![7, 0, 1]>,<5,R![8, 0, 1]>,<7,R![-56, 0, 1]>],Snew);
    return Vf;
end function;