
// Make newform 6160.2.a.g in Magma, downloaded from the LMFDB on 25 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_g();".
// 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_g();".
// 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], [0], [1], [1], [-1], [-6], [6], [4], [8], [-10], [4], [6], [-10], [-4], [4], [6], [0], [-6], [-4], [0], [6], [8], [-12], [10], [10], [10], [-4], [20], [-10], [2], [0], [-4], [-6], [-4], [6], [16], [14], [12], [24], [-6], [12], [14], [16], [10], [-18], [4], [28], [-12], [12], [14], [-6], [0], [-2], [8], [-14], [-24], [-10], [-24], [-2], [2], [20], [26], [-4], [-28], [-22], [30], [-28], [-14], [-4], [26], [-14], [0], [4], [14], [12], [28], [-26], [-18], [-30], [14], [-24], [-26], [24], [26], [32], [12], [18], [2], [-6], [16], [8], [24], [8], [12], [4], [16], [22], [18], [4], [30], [-36], [-10], [-12], [-30], [4], [26], [0], [-18], [48], [-26], [-32], [-10], [-38], [8], [40], [18], [-24], [20], [-18], [-44], [-42], [-30], [2], [12], [-8], [6], [-10], [52], [-12], [34], [-20], [-40], [-40], [-10], [-34], [-26], [-18], [-28], [-18], [42], [-28], [-26], [-16], [12], [14], [44], [-14], [22], [-40], [-24], [14], [26], [-28], [48], [-44], [8], [24], [-22], [6], [42], [4], [-14], [0], [24], [18], [12], [8], [-46], [-46], [46], [20], [-50], [0], [-54], [-4], [2], [28], [-42], [-64], [-14], [24], [20], [38], [-18], [24], [30], [22], [-8], [-46], [0], [-22], [-60], [-12], [30], [-44], [-38], [26], [-34], [14], [16], [14], [0], [-34], [-10], [-24], [46], [-20], [-36], [-6], [48], [34], [30], [-16], [8], [32], [-22], [-24], [54], [24], [-66], [-18], [0], [18], [40], [12], [-2], [66], [-32], [32], [-4], [-18], [44], [-16], [-22], [16], [8], [74], [-18], [-52], [-52], [-52], [-68], [4], [46], [-42], [56], [-20], [24], [52], [-16], [38], [-42], [-24], [10], [-14], [60], [-10], [-4], [-26], [14], [-24], [-36], [50], [10], [-62], [32], [78], [-14], [44], [-58], [30], [-20], [50], [-32], [-2], [-4], [28], [-34], [-38], [-28], [64], [-32], [-16], [-54], [76], [-8], [-14], [-2], [-76], [30], [-18], [-24], [-22], [76], [54], [42], [36], [30], [12], [-28], [-10], [-2], [-56], [28], [28], [-14], [-20], [62], [-24], [-34], [64], [-42], [-70], [-44], [64], [-34], [16], [-32], [-70], [58], [12], [-70], [42], [-56], [-46], [2], [52], [16], [24], [54], [46], [30], [-16], [-28], [-68], [-48], [-34], [22], [90], [-56], [86], [42], [-38], [-88], [-2], [12], [-50], [-28], [80], [-18], [68], [-38], [-50], [-56], [-26], [30], [12], [68], [-6], [80], [62], [38], [88], [-36], [-72], [-38], [90], [40], [10], [52], [-80], [16], [22], [-40], [-2], [-32], [-72], [70], [-2], [-2], [-18], [-6], [16], [26], [-44], [-20], [0], [86], [-76], [68], [34], [-2], [76], [-8], [40], [-22], [56], [74], [28], [-18], [-102], [-46], [-32], [58], [10], [-40], [78], [-14], [0], [8], [94], [86], [44], [36], [-22], [-18], [32], [100], [-30], [8], [-50], [-102], [-80], [4], [-38], [-26], [-84], [66], [-16], [-64]];
    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_g();".
// 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_g(: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_g();".
// 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_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<13,R![6, 1]>,<17,R![-6, 1]>,<19,R![-4, 1]>,<23,R![-8, 1]>],Snew);
    return Vf;
end function;