
// Make newform 6160.2.a.k in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6160_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_k();".
// 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_6160_2_a_k();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_6160_a();"
function MakeCharacter_6160_a()
    N := 6160;
    order := 1;
    char_gens := [5391, 1541, 3697, 5281, 5601];
    v := [1, 1, 1, 1, 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;

function MakeCharacter_6160_a_Hecke(Kf)
    return MakeCharacter_6160_a();
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 := 2;
    raw_aps := [[0], [2], [-1], [-1], [1], [2], [-6], [-2], [6], [0], [-8], [-4], [12], [4], [-12], [0], [0], [2], [-8], [-12], [2], [-14], [-12], [6], [8], [-6], [16], [-12], [-16], [-6], [-8], [18], [-6], [-14], [0], [-14], [14], [-8], [0], [-18], [12], [2], [24], [2], [6], [16], [4], [-8], [-24], [-10], [6], [-18], [20], [0], [24], [-24], [-18], [-20], [-22], [6], [16], [-6], [4], [-24], [8], [-12], [28], [14], [-12], [14], [-24], [-30], [-8], [14], [28], [24], [-6], [38], [30], [-4], [0], [-10], [-30], [-4], [28], [-24], [-18], [-10], [-18], [22], [6], [-24], [-2], [36], [4], [0], [18], [30], [-44], [20], [28], [6], [24], [-6], [-32], [8], [30], [18], [12], [32], [40], [14], [-18], [16], [40], [-30], [34], [24], [24], [0], [-22], [-10], [-42], [-36], [-20], [12], [-22], [0], [-20], [14], [4], [0], [4], [-16], [-12], [-40], [42], [16], [-42], [30], [-38], [12], [-50], [-12], [38], [-48], [-10], [42], [16], [-18], [-10], [-30], [-20], [-48], [-20], [36], [34], [54], [38], [18], [12], [-30], [-8], [48], [42], [0], [-32], [38], [14], [6], [0], [50], [30], [-10], [16], [18], [28], [30], [-32], [2], [-62], [-30], [-16], [30], [-54], [54], [-34], [10], [-46], [-48], [20], [12], [4], [-6], [-12], [-60], [14], [56], [-66], [-48], [60], [16], [2], [-52], [60], [-30], [40], [-36], [42], [52], [2], [30], [-26], [-18], [24], [-10], [4], [-36], [6], [72], [32], [28], [-66], [-38], [-24], [-52], [-36], [-12], [-8], [72], [-4], [-38], [10], [-30], [22], [72], [14], [-6], [36], [24], [-60], [34], [4], [38], [-18], [-60], [40], [-72], [4], [-72], [26], [0], [42], [2], [-78], [-36], [50], [28], [-6], [38], [64], [12], [2], [-10], [-72], [16], [66], [-18], [76], [-6], [50], [64], [-52], [70], [14], [-20], [-24], [-40], [-70], [-42], [-24], [28], [0], [50], [16], [24], [38], [-30], [-32], [-36], [-42], [42], [-42], [6], [-70], [78], [-8], [-66], [-66], [4], [-34], [6], [-2], [36], [76], [2], [60], [14], [60], [-34], [24], [-18], [18], [52], [-12], [20], [-12], [-66], [68], [6], [-26], [-22], [66], [70], [18], [-10], [-44], [-50], [72], [54], [20], [12], [-44], [-84], [-8], [-78], [74], [-90], [14], [-32], [-16], [-78], [54], [-8], [-60], [-12], [38], [-80], [-12], [18], [70], [-10], [-78], [-8], [8], [-6], [0], [42], [-6], [-78], [2], [-12], [-78], [60], [58], [-58], [-6], [-92], [38], [-12], [28], [-84], [48], [76], [-52], [-84], [-6], [62], [24], [62], [-66], [-66], [-68], [6], [-2], [36], [16], [50], [-92], [-24], [-82], [-78], [-60], [-98], [0], [-70], [52], [-90], [76], [-60], [50], [-42], [40], [-12], [-42], [-20], [-52], [54], [-38], [0], [-10], [-78], [72], [-32], [-10], [-30], [-54], [64], [-24], [24], [54], [-70], [30], [90], [-16], [-54], [24], [54], [-56], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6160_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_k();".
// 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_6160_2_a_k(:prec:=1)
    chi := MakeCharacter_6160_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_6160_2_a_k();".
// 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_6160_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<13,R![-2, 1]>,<17,R![6, 1]>,<19,R![2, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;