
// Make newform 46.3.b.a in Magma, downloaded from the LMFDB on 13 June 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_46_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_46_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_46_3_b_a();".
// 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_46_3_b_a();".
// 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 := [599, 0, 58, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 29, 0, 1], [29, 0, 1, 0], [0, 51, 0, 1]];
    Rf_basisdens := [1, 11, 11, 11];
    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_46_b();"
function MakeCharacter_46_b()
    N := 46;
    order := 2;
    char_gens := [5];
    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_46_b_Hecke();"
function MakeCharacter_46_b_Hecke(Kf)
    N := 46;
    order := 2;
    char_gens := [5];
    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]];
    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, 1, 0], [0, 1, 0, 0], [0, 0, 0, -1], [0, 0, 0, 1], [-1, 0, -8, 0], [0, 1, 0, -2], [0, -2, 0, 2], [13, 2, 7, -1], [21, 0, -10, 0], [29, 0, -3, 0], [0, 5, 0, 2], [5, 0, -6, 0], [0, -4, 0, -2], [-7, 0, 17, 0], [0, -4, 0, 6], [-46, 0, -14, 0], [0, 0, 0, -4], [0, 6, 0, -3], [-25, 0, 61, 0], [-37, 0, -60, 0], [0, 10, 0, -6], [0, -10, 0, -5], [0, 0, 0, 10], [0, 7, 0, 0], [8, 0, -66, 0], [0, -12, 0, -3], [0, 8, 0, 0], [0, -18, 0, -8], [0, -15, 0, 10], [117, 0, 57, 0], [-47, 0, 13, 0], [0, 2, 0, -12], [-71, 0, -71, 0], [0, 2, 0, 4], [-1, 0, 1, 0], [0, -2, 0, 14], [127, 0, 1, 0], [-90, 0, 2, 0], [64, 0, 50, 0], [93, 0, -131, 0], [0, -9, 0, -8], [0, -10, 0, 19], [-183, 0, -62, 0], [201, 0, -34, 0], [0, -6, 0, 5], [-74, 0, -2, 0], [-36, 0, 34, 0], [0, -20, 0, 19], [0, -10, 0, 12], [-37, 0, 224, 0], [-5, 0, 189, 0], [0, 47, 0, 0], [0, -26, 0, -13], [91, 0, 80, 0], [0, 50, 0, 12], [151, 0, -184, 0], [102, 0, 142, 0], [-295, 0, -62, 0], [0, 5, 0, 20], [0, -26, 0, -5], [0, 18, 0, 2], [34, 0, -218, 0], [121, 0, -243, 0], [0, -4, 0, -34], [-196, 0, 10, 0], [277, 0, 149, 0], [0, -24, 0, 58], [-436, 0, -102, 0], [315, 0, 116, 0], [411, 0, 24, 0], [0, 34, 0, -8], [0, -46, 0, -29], [0, -5, 0, -52], [0, 42, 0, -14], [0, -44, 0, 2], [0, 2, 0, 2], [103, 0, 308, 0], [0, 3, 0, -6], [307, 0, 36, 0], [0, 14, 0, -28], [0, 29, 0, -10], [0, -82, 0, -26], [0, -51, 0, -22], [499, 0, -239, 0], [-317, 0, -63, 0], [-258, 0, 128, 0], [0, 9, 0, 32], [-121, 0, -132, 0], [-252, 0, 298, 0], [0, -18, 0, 36], [0, 24, 0, -1], [-141, 0, -363, 0], [-63, 0, -227, 0], [-179, 0, 137, 0], [0, 78, 0, 32], [61, 0, -210, 0], [0, 6, 0, 46], [0, -24, 0, 5], [-573, 0, -104, 0], [-131, 0, 413, 0], [0, -32, 0, 6], [0, 60, 0, -18], [0, -19, 0, 46], [0, 70, 0, -27], [-483, 0, 342, 0], [-385, 0, 333, 0], [4, 0, -110, 0], [166, 0, 150, 0], [-481, 0, -140, 0], [546, 0, -102, 0], [0, -27, 0, -2], [0, -44, 0, 34], [0, -94, 0, 52], [0, 14, 0, 0], [0, 18, 0, -46], [0, 16, 0, 1], [583, 0, -271, 0], [-939, 0, -86, 0], [0, -64, 0, 115], [0, 90, 0, -80], [-721, 0, -52, 0], [0, 56, 0, -4], [413, 0, 237, 0], [390, 0, 434, 0], [0, 1, 0, -94], [0, -107, 0, 44], [-582, 0, -478, 0], [0, 112, 0, -63], [0, -45, 0, 54], [83, 0, -83, 0], [0, 74, 0, -131], [0, -86, 0, 44], [0, -121, 0, 90], [-567, 0, 150, 0], [0, 25, 0, 100], [0, -12, 0, -54], [0, 54, 0, -56], [0, -52, 0, -54], [956, 0, -142, 0], [767, 0, -291, 0], [-260, 0, 334, 0], [-841, 0, 29, 0], [0, 106, 0, -10], [224, 0, -134, 0], [0, 34, 0, 17], [532, 0, -150, 0], [-149, 0, 304, 0], [-1077, 0, 225, 0], [635, 0, 49, 0], [140, 0, 430, 0], [0, -123, 0, 40], [198, 0, 322, 0], [-357, 0, -487, 0], [0, 4, 0, -102], [0, -124, 0, 92], [0, -34, 0, 91], [-1091, 0, -286, 0], [0, -87, 0, -12], [0, -39, 0, 104], [-863, 0, 161, 0], [0, -52, 0, -110], [897, 0, -579, 0], [0, -116, 0, -6], [0, 45, 0, -8], [0, -32, 0, -22], [-506, 0, 10, 0], [300, 0, -118, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_46_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_46_3_b_a();".
// 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_46_3_b_a(:prec:=4)
    chi := MakeCharacter_46_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_46_3_b_a();".
// 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_46_3_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_46_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;