
// Make newform 2160.2.a.c in Magma, downloaded from the LMFDB on 29 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2160_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_2160_2_a_c();".
// 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_2160_2_a_c();".
// 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_2160_a();"
function MakeCharacter_2160_a()
    N := 2160;
    order := 1;
    char_gens := [271, 1621, 2081, 1297];
    v := [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_2160_a_Hecke(Kf)
    return MakeCharacter_2160_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], [-2], [0], [2], [3], [-5], [3], [6], [-5], [2], [-12], [-8], [-12], [3], [6], [-7], [-2], [12], [-16], [1], [-15], [12], [-16], [-12], [4], [-12], [-7], [-6], [-2], [18], [9], [4], [12], [16], [-4], [10], [-9], [-15], [12], [17], [-6], [-10], [9], [-8], [-11], [-26], [-3], [23], [-6], [6], [-13], [0], [21], [0], [18], [-29], [8], [-30], [4], [-3], [-2], [-30], [32], [-33], [28], [2], [12], [17], [-18], [30], [22], [8], [1], [-33], [-24], [38], [6], [-7], [0], [-1], [-18], [-34], [19], [-15], [6], [-28], [30], [22], [21], [24], [28], [-18], [31], [27], [-12], [12], [22], [14], [-8], [-30], [36], [30], [19], [-4], [15], [9], [18], [-19], [-32], [-16], [-33], [16], [1], [0], [-14], [3], [-21], [18], [14], [26], [18], [-39], [-35], [42], [2], [42], [-8], [32], [-47], [-24], [-23], [50], [-30], [29], [-21], [-44], [-39], [12], [-20], [18], [16], [-33], [-34], [-24], [-4], [27], [-23], [3], [-28], [-54], [-50], [27], [28], [6], [16], [12], [32], [-18], [-27], [-6], [-14], [-30], [54], [-9], [13], [-28], [-13], [-27], [36], [-13], [6], [-52], [-11], [6], [-17], [18], [-44], [26], [-32], [24], [-22], [6], [-3], [12], [-46], [40], [17], [42], [14], [63], [13], [24], [-39], [21], [5], [-64], [42], [24], [-12], [-44], [-58], [-37], [-42], [30], [-68], [-39], [12], [52], [26], [24], [40], [-12], [-60], [53], [-68], [-72], [27], [27], [-46], [-5], [18], [-20], [-27], [59], [69], [36], [22], [60], [8], [-29], [-32], [-18], [46], [57], [41], [33], [-30], [48], [36], [-17], [46], [17], [6], [-6], [-32], [18], [76], [-51], [50], [-54], [-75], [41], [-33], [-48], [-67], [-62], [-57], [38], [-20], [75], [23], [-4], [69], [-44], [24], [6], [-50], [66], [35], [-2], [-28], [43], [-40], [-14], [27], [-55], [-7], [12], [24], [31], [-33], [23], [-20], [18], [26], [6], [-35], [-24], [72], [-69], [-18], [6], [-28], [-36], [43], [-6], [24], [-26], [68], [-45], [16], [81], [49], [20], [-12], [-73], [-24], [20], [-27], [-60], [24], [46], [57], [11], [-72], [-12], [-46], [30], [-71], [56], [60], [-56], [51], [-34], [25], [64], [-72], [-27], [50], [-66], [40], [60], [16], [60], [-13], [27], [-58], [-56], [-16], [-57], [-54], [61], [90], [-84], [11], [28], [-36], [-27], [28], [38], [30], [-8], [74], [75], [54], [18], [-78], [63], [2], [18], [0], [18], [52], [74], [51], [-38], [98], [-66], [67], [-15], [-72], [-77], [32], [-30], [-30], [8], [-30], [-34], [36], [-51], [-74], [-9], [-68], [84], [1], [50], [-56], [48], [26], [87], [90], [-50], [-90], [-28], [-35], [30], [85], [36], [50], [-54], [-14], [90], [54], [7], [-34], [-54], [-92], [-66], [-10], [-63], [-9], [-5], [-46], [-30], [-6], [16], [-3], [57], [-84], [74], [-39], [-42], [-34], [-75], [27], [42], [-47], [-12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2160_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2160_2_a_c();".
// 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_2160_2_a_c(:prec:=1)
    chi := MakeCharacter_2160_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_2160_2_a_c();".
// 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_2160_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![0, 1]>,<13,R![-2, 1]>,<17,R![-3, 1]>],Snew);
    return Vf;
end function;