
// Make newform 228.3.l.b in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_228_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_228_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_228_3_l_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_228_3_l_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 := [1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_228_l();"
function MakeCharacter_228_l()
    N := 228;
    order := 6;
    char_gens := [115, 77, 97];
    v := [6, 6, 5];
    // 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_228_l_Hecke();"
function MakeCharacter_228_l_Hecke(Kf)
    N := 228;
    order := 6;
    char_gens := [115, 77, 97];
    char_values := [[1, 0], [1, 0], [0, 1]];
    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, 1], [2, -2], [-1, 0], [16, 0], [18, -9], [-22, 22], [19, 0], [0, -40], [20, -10], [-29, 58], [9, -18], [-8, -8], [49, -49], [0, -46], [-56, 28], [-38, -38], [0, 97], [-30, 15], [-28, -28], [-35, 35], [-51, -51], [-146, 0], [-44, 22], [36, 36], [0, -28], [73, -146], [48, -96], [-44, -44], [-106, 212], [32, -16], [128, -128], [0, 116], [0, 175], [-148, 148], [-112, 224], [193, -193], [233, 0], [84, -42], [66, 66], [-48, 96], [56, -28], [70, 0], [-175, -175], [-2, 0], [0, -197], [-211, -211], [141, 141], [-86, 172], [83, 0], [80, -80], [58, 0], [-326, 163], [0, 266], [-216, 108], [164, -164], [-298, -298], [-14, 14], [206, 0], [116, -58], [-542, 542], [278, -556], [-92, -92], [322, 0], [0, -26], [-388, 194], [-127, 254], [-337, -337], [470, -470], [-295, 0], [262, 0], [464, -464], [0, -233], [-308, 616], [93, -186], [-332, -332], [0, 152], [499, -499], [58, 58], [456, -228], [628, 0], [372, 372], [-524, 262], [578, -289], [195, 195], [0, -730], [258, -516], [-397, 0], [-574, 574], [-367, 0], [424, 0], [0, 728], [156, -312], [-760, 760], [463, -463], [0, -130], [-612, 306], [-348, 696], [-370, 185], [0, -365], [-266, 133], [0, -640], [56, -112], [88, -176], [335, 0], [746, 0], [-178, 178], [0, -274], [-336, 168], [-589, 1178], [91, -182], [802, -802], [0, 764], [-127, 0], [0, -323], [-582, -582], [445, -445], [106, 0], [328, 0], [464, -232], [1104, -552], [-301, 602], [-254, 508], [-414, 828], [422, 0], [-370, 370], [0, 91], [1004, -1004], [295, -295], [-250, 0], [571, -571], [4, 4], [-1122, 561], [-239, 239], [1402, 0], [0, 103], [-92, 46], [109, -218], [-364, 728], [1054, 0], [1280, -640], [0, 974], [0, 634], [1540, -770], [171, -342], [930, 930], [439, -439], [354, 354], [0, -557], [-774, 1548], [-265, -265], [910, 0], [0, -5], [-1236, 618], [120, 120], [-452, 904], [-1225, 0], [290, -290], [0, 553], [1296, -648], [740, -740], [-724, -724], [-335, 335], [722, 722], [746, -1492], [-276, -276], [637, 637], [0, 541]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_228_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_228_3_l_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_228_3_l_b(:prec:=2)
    chi := MakeCharacter_228_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_228_3_l_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_228_3_l_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_228_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![4, -2, 1]>,<7,R![1, 1]>],Snew);
    return Vf;
end function;