
// Make newform 169.4.b.a in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_169_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_169_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_169_4_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_169_4_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 := [1, 0, 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_169_b();"
function MakeCharacter_169_b()
    N := 169;
    order := 2;
    char_gens := [2];
    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_169_b_Hecke();"
function MakeCharacter_169_b_Hecke(Kf)
    N := 169;
    order := 2;
    char_gens := [2];
    char_values := [[-1, 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 := 4;
    raw_aps := [[0, 5], [-7, 0], [0, 7], [0, -13], [0, -26], [0, 0], [-77, 0], [0, 126], [96, 0], [-82, 0], [0, -196], [0, -131], [0, -336], [201, 0], [0, -105], [-432, 0], [0, -294], [-56, 0], [0, -478], [0, -9], [0, 98], [1304, 0], [0, 308], [0, -1190], [0, -70], [-420, 0], [-588, 0], [-684, 0], [0, -373], [-1734, 0], [-1892, 0], [1435, 0], [0, -1776], [-1869, 0], [0, -2466], [0, -3323], [-2730, 0], [0, -544], [0, 1624], [336, 0], [3029, 0], [28, 0], [422, 0], [0, 492], [0, -2991], [70, 0], [2851, 0], [0, -217], [0, 2576], [0, 455], [-3061, 0], [0, 3477], [0, -1610], [-1008, 0], [-6041, 0], [-3708, 0], [8344, 0], [0, -1617], [3820, 0], [0, -6214], [5292, 0], [0, -903], [0, 2114], [-3402, 0], [-10689, 0], [0, 7054], [0, -9704], [10449, 0], [-621, 0], [0, 12481], [0, 1400], [0, -4968], [8722, 0], [10012, 0], [0, 3372], [0, 847], [-11314, 0], [0, 1862], [0, 6820], [0, 12992], [-7343, 0], [0, 5059], [0, -3243], [-11599, 0], [17374, 0], [989, 0], [0, -14474], [0, 1594], [0, 5915], [0, -11072], [-1260, 0], [0, -12033], [0, 2280], [-16767, 0], [0, -12840], [-2198, 0], [0, 17066], [2583, 0], [18620, 0], [0, -16833], [-8615, 0], [0, 8535], [4641, 0], [4793, 0], [5563, 0], [0, -24038], [0, 21224], [0, 4354], [7310, 0], [-7595, 0], [-826, 0], [0, -14590], [0, -4888], [0, -11004], [0, -4975], [-3950, 0], [0, 3682], [-10402, 0], [-31680, 0], [21940, 0], [0, -31374], [-18013, 0], [-10640, 0], [0, -9336], [0, -4200], [-9872, 0], [0, 28450], [-32718, 0], [22834, 0], [0, -7875], [0, -2140], [0, -31971], [7432, 0], [20176, 0], [0, -9478], [0, 12096], [0, -17941], [0, 6664], [1442, 0], [30207, 0], [0, -21140], [0, -569], [8538, 0], [0, -32702], [21154, 0], [0, -2184], [0, 36687], [-36806, 0], [4900, 0], [0, 13697], [0, -6239], [-133, 0], [26003, 0], [-31248, 0], [38253, 0], [36374, 0], [-27648, 0], [0, -756], [20846, 0], [0, 41321], [0, 54966], [44553, 0], [0, 27907], [-16443, 0], [0, 45414], [0, -8981], [-17414, 0], [-23702, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_169_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_169_4_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_169_4_b_a(:prec:=2)
    chi := MakeCharacter_169_b();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_169_4_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_169_4_b_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_169_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<2,R![25, 0, 1]>],Snew);
    return Vf;
end function;