
// Make newform 1960.2.a.e in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1960_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_1960_2_a_e();".
// 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_1960_2_a_e();".
// 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_1960_a();"
function MakeCharacter_1960_a()
    N := 1960;
    order := 1;
    char_gens := [1471, 981, 1177, 1081];
    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_1960_a_Hecke(Kf)
    return MakeCharacter_1960_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], [-1], [1], [0], [2], [0], [-4], [2], [1], [9], [-4], [4], [-1], [9], [0], [-10], [10], [-9], [5], [14], [-12], [14], [-11], [15], [18], [-3], [13], [9], [-1], [2], [8], [8], [12], [2], [-5], [8], [-2], [-20], [-17], [-16], [12], [25], [18], [-14], [18], [4], [2], [4], [20], [10], [8], [-24], [22], [-8], [0], [-1], [21], [-22], [-28], [18], [4], [-12], [-21], [-26], [-16], [-16], [0], [-2], [-7], [19], [-14], [12], [-7], [-28], [-30], [-21], [26], [-6], [-17], [21], [-16], [-27], [14], [-30], [28], [-11], [-41], [8], [-2], [39], [7], [16], [-8], [-20], [-38], [23], [15], [42], [-28], [-13], [-35], [30], [45], [46], [-26], [-2], [12], [-18], [4], [-10], [-27], [-20], [-12], [34], [-2], [-19], [4], [-23], [36], [2], [3], [24], [24], [9], [-50], [9], [21], [42], [31], [26], [10], [27], [12], [-24], [6], [-34], [0], [-45], [-8], [13], [44], [30], [49], [-17], [-2], [-10], [-38], [-30], [-40], [-1], [6], [-17], [20], [-15], [7], [-6], [42], [13], [-4], [30], [5], [28], [55], [8], [42], [-3], [-40], [-10], [-11], [14], [-48], [29], [8], [-10], [-60], [15], [-6], [-30], [11], [-51], [37], [-56], [-30], [-6], [12], [35], [-8], [20], [-41], [-42], [20], [24], [6], [-17], [-36], [-42], [33], [34], [-26], [64], [45], [16], [38], [-61], [-16], [66], [6], [20], [9], [44], [-6], [3], [-61], [65], [62], [-2], [25], [13], [8], [-48], [-55], [46], [-39], [16], [13], [-17], [0], [62], [-37], [-12], [-16], [-30], [6], [41], [35], [-15], [33], [12], [14], [-42], [-12], [32], [32], [-11], [-16], [-78], [-47], [-12], [-40], [-16], [-8], [-15], [-69], [-51], [20], [-18], [-49], [-23], [18], [-34], [7], [-31], [-10], [-4], [24], [-70], [30], [34], [9], [-52], [1], [32], [70], [24], [-70], [48], [72], [21], [-59], [-26], [27], [80], [56], [21], [13], [-36], [-66], [-48], [20], [26], [41], [59], [-48], [-30], [-46], [-39], [22], [-42], [-62], [76], [-32], [-42], [10], [-15], [-68], [-64], [68], [-17], [-24], [-36], [5], [-26], [-1], [-23], [-3], [15], [-56], [-30], [82], [-1], [28], [56], [-33], [-31], [36], [31], [-88], [17], [-24], [-58], [-78], [56], [-74], [63], [18], [19], [-38], [78], [75], [16], [20], [12], [45], [50], [-2], [70], [59], [17], [84], [-80], [30], [-52], [-15], [77], [13], [-24], [76], [-78], [-48], [-91], [42], [-30], [-67], [20], [-23], [36], [48], [16], [-19], [-14], [-28], [-1], [42], [48], [28], [-90], [-30], [-28], [-22], [-62], [-11], [72], [13], [32], [64], [-39], [14], [-4], [93], [63], [-34], [24], [64], [60], [-30], [86], [22], [95], [44], [-75], [-103], [-94], [-13], [72], [-39], [98], [-12], [-17], [-81], [-48], [46], [-42], [4], [-88], [38], [-42], [-102], [-32], [-54], [9], [-5], [48], [15], [0], [-34], [-34], [28], [7], [-56], [18]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1960_2_a_e();".
// 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_1960_2_a_e(:prec:=1)
    chi := MakeCharacter_1960_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_1960_2_a_e();".
// 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_1960_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 1]>,<11,R![-2, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;