
// Make newform 6160.2.a.q in Magma, downloaded from the LMFDB on 16 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_q();".
// 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_q();".
// 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], [6], [2], [-2], [6], [0], [0], [-4], [-8], [4], [4], [8], [4], [14], [-8], [-12], [2], [-6], [0], [-2], [12], [-18], [0], [12], [-16], [18], [-16], [-6], [18], [2], [-8], [2], [10], [-8], [-8], [2], [-12], [6], [-8], [18], [22], [0], [-12], [-8], [-4], [-6], [22], [30], [16], [-20], [-12], [-24], [-14], [12], [10], [14], [20], [22], [-16], [-24], [-20], [12], [-4], [14], [4], [-22], [-12], [34], [-8], [14], [12], [8], [-30], [-22], [14], [-8], [-20], [-10], [-14], [-24], [-36], [24], [-18], [38], [34], [14], [22], [40], [-34], [-28], [-20], [-16], [22], [-18], [0], [-36], [12], [-42], [36], [10], [-32], [-12], [6], [18], [-4], [28], [-16], [-26], [-2], [4], [8], [-30], [2], [-24], [-24], [16], [-10], [-26], [-6], [12], [8], [12], [50], [-32], [12], [-22], [20], [8], [-52], [48], [0], [12], [-34], [4], [-14], [-10], [-30], [-28], [-42], [12], [34], [-24], [-6], [-14], [-4], [38], [-2], [-30], [-4], [24], [28], [12], [-22], [-10], [-2], [46], [-20], [34], [-16], [36], [-30], [16], [-16], [-30], [22], [42], [48], [-10], [-42], [-26], [-40], [-30], [-20], [6], [-56], [-42], [2], [-14], [-16], [-42], [-38], [-54], [30], [34], [10], [32], [-8], [36], [-52], [30], [-52], [56], [46], [40], [-26], [-24], [-28], [64], [22], [64], [8], [54], [8], [-36], [2], [40], [2], [-70], [54], [46], [24], [-10], [-28], [-56], [-34], [-56], [32], [-36], [62], [-22], [12], [-20], [-64], [12], [32], [24], [-36], [-22], [-30], [-30], [54], [-48], [30], [10], [60], [16], [-60], [-38], [-76], [-74], [54], [-28], [40], [-4], [-44], [-24], [-22], [-68], [-14], [-14], [14], [60], [26], [-32], [14], [-26], [-72], [44], [30], [2], [-68], [60], [66], [-10], [52], [-30], [6], [-80], [-48], [6], [22], [60], [-8], [-16], [2], [6], [-40], [-68], [0], [6], [-76], [0], [-58], [66], [-48], [0], [70], [18], [22], [22], [-6], [18], [8], [-14], [6], [24], [-50], [30], [70], [4], [-20], [10], [28], [-22], [-60], [-18], [0], [-26], [-38], [-44], [-28], [-40], [56], [70], [40], [-18], [46], [-78], [78], [14], [18], [54], [-44], [-34], [48], [30], [-52], [12], [84], [-16], [-56], [42], [-30], [94], [-10], [-48], [-8], [66], [-30], [-8], [36], [20], [-38], [-48], [-12], [-42], [62], [-74], [50], [-80], [-32], [10], [-24], [66], [-6], [2], [-46], [56], [50], [60], [82], [-74], [14], [-52], [-18], [-36], [16], [-100], [-24], [28], [20], [56], [-6], [38], [44], [-2], [98], [-26], [44], [38], [-26], [84], [80], [22], [-4], [72], [-50], [70], [60], [38], [-24], [58], [52], [54], [-28], [-28], [22], [6], [88], [-16], [50], [28], [20], [78], [58], [92], [-50], [-22], [40], [-64], [-74], [-90], [50], [64], [76], [96], [-2], [-58], [70], [42], [68], [-10], [56], [-66], [-60], [-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_q();".
// 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_q(: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_q();".
// 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_q( : 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![-6, 1]>,<17,R![-2, 1]>,<19,R![2, 1]>,<23,R![-6, 1]>],Snew);
    return Vf;
end function;