
// Make newform 2960.2.a.j in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2960_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_2960_2_a_j();".
// 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_2960_2_a_j();".
// 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_2960_a();"
function MakeCharacter_2960_a()
    N := 2960;
    order := 1;
    char_gens := [2591, 741, 1777, 2481];
    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_2960_a_Hecke(Kf)
    return MakeCharacter_2960_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], [-3], [-4], [3], [-2], [-6], [3], [-5], [1], [3], [1], [-12], [3], [0], [-1], [4], [-6], [-16], [-8], [12], [-6], [17], [-6], [4], [-6], [11], [9], [-20], [0], [12], [13], [6], [-8], [-13], [-11], [-12], [15], [0], [2], [-3], [14], [-6], [16], [-5], [-17], [27], [14], [0], [9], [-28], [-18], [18], [-9], [-18], [-2], [-28], [24], [4], [21], [34], [9], [26], [-21], [-26], [-4], [12], [26], [-21], [36], [-35], [-22], [16], [24], [9], [-34], [6], [32], [36], [-10], [-15], [2], [-35], [-30], [6], [17], [33], [4], [-27], [-24], [-2], [12], [-14], [-12], [0], [-39], [-20], [2], [1], [12], [3], [-6], [31], [-34], [-27], [-36], [-12], [-19], [22], [29], [0], [-35], [7], [15], [13], [48], [30], [36], [5], [-10], [-42], [15], [19], [30], [35], [42], [28], [-31], [-11], [51], [-14], [-34], [27], [-40], [-39], [22], [-18], [-30], [16], [36], [40], [3], [47], [48], [-10], [-15], [40], [-39], [-13], [15], [-29], [-33], [28], [24], [16], [45], [38], [-18], [-33], [-60], [40], [-57], [-45], [-51], [-47], [-28], [-7], [42], [30], [23], [-3], [-28], [-14], [30], [-62], [30], [-11], [8], [46], [0], [-22], [33], [-39], [30], [-7], [-35], [-4], [12], [-34], [-24], [34], [0], [-42], [-18], [-58], [-28], [-18], [6], [-30], [-8], [-13], [-55], [27], [-48], [-2], [36], [-6], [-5], [-43], [15], [-8], [-42], [-57], [71], [-32], [-60], [-48], [21], [-58], [4], [39], [-56], [-36], [26], [-24], [30], [-35], [42], [-49], [-56], [-68], [6], [10], [63], [-58], [24], [-48], [0], [-39], [-20], [40], [29], [6], [9], [34], [0], [-5], [-36], [-58], [18], [-27], [8], [-66], [-21], [20], [-62], [39], [-25], [10], [-24], [50], [11], [30], [19], [-48], [72], [76], [42], [-22], [43], [-10], [31], [-22], [-65], [-42], [-46], [-79], [-24], [-24], [-23], [-9], [-28], [4], [6], [-43], [-21], [-8], [-6], [21], [60], [24], [-75], [-37], [-18], [10], [78], [-6], [-62], [41], [18], [52], [-12], [22], [83], [-57], [-22], [42], [-52], [-24], [42], [-57], [-32], [60], [20], [-57], [72], [-28], [0], [64], [-58], [-75], [7], [72], [38], [-29], [-17], [0], [-39], [-10], [54], [-17], [-12], [-38], [0], [-88], [0], [-58], [-56], [26], [90], [-90], [55], [60], [0], [-34], [-62], [57], [-81], [-5], [56], [75], [64], [-76], [-24], [-33], [18], [-54], [-36], [8], [-78], [-42], [-24], [70], [71], [-84], [88], [-34], [-24], [70], [39], [0], [-59], [11], [12], [72], [92], [78], [-28], [-33], [6], [-2], [-54], [-14], [63], [-80], [32], [43], [12], [-1], [-60], [-30], [19], [42], [14], [52], [33], [-5], [-60], [104], [-87], [-14], [-54], [90], [-92], [44], [21], [4], [9], [-28], [33], [9], [82], [-85], [102], [12], [55], [54], [90], [-6], [2], [72], [9], [74], [39], [-72], [-15], [16], [-51]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2960_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2960_2_a_j();".
// 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_2960_2_a_j(:prec:=1)
    chi := MakeCharacter_2960_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_2960_2_a_j();".
// 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_2960_2_a_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<7,R![-1, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;