
// Make newform 1960.2.a.f in Magma, downloaded from the LMFDB on 19 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_f();".
// 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_f();".
// 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], [3], [1], [-5], [-6], [0], [-5], [2], [-4], [-2], [10], [-9], [6], [-6], [-12], [-2], [0], [-14], [1], [12], [0], [-9], [-18], [-1], [2], [-19], [-4], [-6], [-16], [4], [16], [18], [9], [-2], [-8], [-5], [15], [-20], [-10], [-25], [24], [-6], [22], [-11], [11], [-17], [2], [-24], [-15], [4], [-12], [14], [26], [-16], [28], [4], [27], [1], [-29], [25], [22], [13], [10], [-20], [32], [36], [2], [3], [0], [23], [-24], [-32], [32], [-15], [11], [13], [34], [-22], [13], [3], [-2], [28], [18], [-33], [18], [12], [40], [-3], [-4], [-28], [-3], [7], [-39], [40], [-26], [0], [15], [20], [-12], [-20], [2], [36], [9], [-20], [-3], [-17], [22], [13], [22], [-12], [-44], [-37], [-10], [-9], [-20], [-14], [-33], [44], [-40], [-3], [-36], [12], [27], [27], [24], [32], [-51], [-33], [54], [-31], [24], [24], [-34], [-23], [-45], [39], [-3], [40], [5], [26], [-8], [-10], [-52], [-22], [-38], [-20], [-18], [30], [0], [36], [32], [20], [16], [17], [-42], [43], [-12], [-26], [-34], [-50], [-14], [-18], [-35], [36], [15], [9], [-30], [-4], [14], [-5], [-24], [14], [36], [-37], [1], [44], [8], [16], [16], [2], [-6], [-44], [-16], [10], [-13], [21], [-24], [6], [-6], [-39], [42], [32], [-39], [-37], [-22], [-41], [-5], [30], [10], [-58], [10], [-6], [-17], [60], [24], [26], [-32], [10], [-68], [26], [-49], [-30], [-4], [-68], [46], [46], [-44], [42], [-2], [9], [42], [7], [38], [51], [23], [19], [32], [-48], [-30], [-40], [9], [35], [43], [-34], [4], [9], [-8], [-4], [20], [39], [-26], [45], [-2], [56], [66], [-7], [-18], [54], [26], [-22], [-28], [-26], [-5], [38], [36], [-11], [45], [-46], [-4], [70], [-15], [-50], [8], [45], [6], [-66], [-54], [14], [2], [-45], [-16], [-46], [39], [-48], [62], [-43], [-30], [59], [5], [-24], [-76], [71], [67], [38], [-18], [62], [-66], [-31], [49], [-66], [-30], [-38], [-34], [0], [-39], [24], [-41], [7], [-72], [79], [-36], [-43], [16], [82], [36], [-44], [-22], [-12], [51], [2], [-58], [-34], [-56], [34], [-17], [-89], [17], [38], [10], [24], [-10], [-42], [38], [49], [20], [78], [-2], [-82], [-82], [4], [41], [20], [87], [25], [-6], [-86], [-81], [-84], [-22], [-16], [-57], [68], [-36], [-64], [8], [64], [-57], [14], [-32], [53], [-77], [5], [78], [-70], [40], [-22], [-24], [-28], [50], [-62], [-61], [-92], [78], [7], [2], [-31], [-1], [72], [-14], [-69], [50], [54], [20], [35], [-47], [-96], [73], [-2], [36], [-22], [12], [14], [79], [11], [-47], [-64], [73], [73], [-49], [-19], [-8], [71], [-48], [-16], [90], [-41], [-42], [-48], [66], [68], [-9], [-36], [-6], [-36], [-41], [-20], [8], [-73], [-78], [-48], [68], [4], [-64], [-85], [-24], [18], [-57], [5], [58], [-78], [-40], [65], [7], [-70], [-70], [-46], [-88]];
    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_f();".
// 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_f(: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_f();".
// 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_f( : 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![-3, 1]>,<13,R![-1, 1]>],Snew);
    return Vf;
end function;