
// Make newform 6160.2.a.m in Magma, downloaded from the LMFDB on 12 May 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_m();".
// 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_m();".
// 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], [-4], [0], [4], [-6], [-6], [4], [2], [-6], [4], [-6], [-6], [0], [-10], [-2], [-12], [8], [16], [12], [-18], [-10], [-6], [10], [0], [-10], [-18], [4], [12], [6], [-20], [6], [-8], [14], [-2], [12], [-24], [0], [-22], [0], [20], [12], [-20], [4], [-26], [-12], [-10], [0], [0], [26], [12], [-6], [-12], [18], [16], [8], [6], [4], [-12], [16], [24], [26], [6], [4], [20], [-24], [-10], [18], [-24], [10], [-4], [-20], [18], [-30], [14], [30], [-10], [-12], [38], [24], [-34], [-8], [6], [18], [8], [-30], [10], [18], [0], [-2], [36], [-20], [-24], [-18], [6], [-8], [-10], [28], [-12], [-24], [-6], [4], [-10], [42], [0], [-12], [26], [-44], [20], [-18], [-20], [-20], [18], [10], [6], [-6], [12], [2], [-4], [36], [18], [4], [-6], [26], [0], [-2], [32], [28], [36], [-20], [38], [6], [26], [-30], [-32], [-18], [6], [-20], [-30], [-50], [-48], [-34], [-36], [-16], [-48], [4], [6], [-4], [-18], [-2], [12], [46], [-24], [40], [-18], [20], [18], [54], [-24], [4], [-48], [-42], [-30], [-32], [32], [-58], [-6], [60], [50], [-24], [-40], [-32], [-6], [52], [-42], [16], [-46], [58], [12], [-34], [-36], [-54], [42], [20], [-50], [26], [0], [38], [36], [-56], [42], [60], [-54], [-10], [26], [60], [12], [-54], [-8], [50], [26], [60], [-18], [52], [0], [-18], [-32], [-40], [54], [46], [-6], [-24], [14], [-32], [54], [-54], [66], [-22], [64], [-6], [-2], [-24], [-10], [-18], [-48], [-44], [-60], [38], [52], [-32], [30], [-2], [-48], [-22], [-24], [36], [36], [6], [-20], [10], [2], [12], [48], [58], [36], [-68], [12], [56], [6], [-30], [-58], [0], [-60], [50], [28], [54], [20], [76], [60], [-10], [44], [-6], [40], [-78], [18], [76], [24], [-34], [-2], [14], [40], [-4], [-2], [30], [-22], [-82], [36], [84], [-56], [-48], [-46], [4], [-48], [-58], [24], [-56], [6], [-30], [6], [24], [-12], [-16], [-6], [64], [-30], [-36], [16], [-4], [60], [16], [66], [-20], [-46], [30], [-82], [-60], [20], [24], [54], [-54], [-86], [-36], [-22], [84], [48], [62], [-30], [4], [-82], [78], [-62], [-72], [50], [-8], [10], [48], [-36], [50], [-90], [88], [-48], [28], [78], [-70], [-36], [26], [-8], [38], [-78], [-6], [64], [42], [-84], [26], [10], [0], [66], [4], [-70], [6], [-80], [2], [48], [-60], [-12], [-72], [30], [8], [-18], [6], [-60], [-2], [-82], [0], [-44], [-34], [24], [-44], [-12], [-66], [-80], [38], [12], [24], [92], [-42], [20], [30], [90], [-20], [12], [-68], [66], [52], [-58], [-32], [54], [50], [-54], [-12], [-38], [-12], [32], [16], [-54], [-56], [-78], [50], [54], [-32], [-6], [-6], [-56], [-10], [42], [-50], [0], [-28], [48], [30], [-44], [-100], [-78], [-48], [70], [42], [72], [-66], [-76], [-78], [-60], [-70], [66], [-42], [-30], [-20], [0]];
    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_m();".
// 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_m(: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_m();".
// 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_m( : 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![4, 1]>,<17,R![0, 1]>,<19,R![-4, 1]>,<23,R![6, 1]>],Snew);
    return Vf;
end function;