
// Make newform 2960.2.a.n in Magma, downloaded from the LMFDB on 25 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_n();".
// 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_n();".
// 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], [3], [-1], [5], [3], [-4], [-4], [4], [4], [-8], [2], [1], [-5], [10], [-7], [3], [-4], [2], [12], [1], [-11], [-14], [-9], [-14], [-2], [-5], [-18], [8], [10], [6], [-13], [18], [6], [20], [15], [8], [-19], [-4], [22], [-13], [-12], [-19], [12], [4], [13], [-10], [27], [9], [12], [-17], [10], [-14], [0], [-4], [-22], [-3], [-6], [13], [10], [-6], [32], [-10], [-15], [12], [-30], [-18], [32], [-3], [-28], [-18], [18], [-15], [32], [-5], [-5], [-18], [0], [37], [6], [12], [-37], [-24], [-24], [-29], [-12], [-13], [10], [-18], [-10], [14], [-6], [-28], [8], [8], [-10], [12], [-27], [39], [-2], [-14], [8], [0], [-32], [-18], [7], [32], [-14], [-39], [-39], [-10], [12], [33], [27], [27], [-2], [-25], [-16], [38], [-6], [45], [50], [1], [15], [-32], [40], [36], [4], [11], [-26], [-11], [39], [-13], [25], [0], [-7], [-20], [21], [-11], [-48], [4], [-49], [37], [24], [-4], [40], [0], [8], [-10], [-32], [56], [-10], [-22], [8], [27], [2], [30], [-16], [-26], [15], [-42], [-28], [-33], [26], [-20], [28], [-49], [-26], [36], [-15], [44], [54], [12], [4], [-23], [-11], [-6], [36], [2], [49], [-14], [32], [58], [-64], [-24], [-16], [-35], [-41], [-14], [-34], [-5], [-16], [32], [-40], [21], [57], [55], [-20], [30], [-19], [42], [-28], [-19], [54], [43], [-32], [28], [-48], [4], [24], [-21], [38], [-12], [46], [-12], [42], [-61], [4], [44], [-12], [34], [-5], [55], [-33], [24], [31], [-22], [30], [-5], [25], [-72], [-23], [59], [-13], [-39], [15], [-13], [-54], [66], [-30], [66], [-10], [10], [13], [12], [18], [-50], [34], [40], [-37], [20], [-50], [-34], [32], [44], [60], [27], [-3], [12], [51], [74], [-50], [42], [42], [66], [2], [-7], [61], [-10], [17], [2], [34], [16], [-20], [10], [-42], [8], [-20], [-12], [35], [14], [-39], [48], [48], [-11], [20], [47], [2], [38], [-76], [0], [48], [50], [39], [57], [11], [75], [69], [42], [28], [17], [38], [7], [23], [78], [-38], [64], [8], [-56], [-65], [-26], [71], [-75], [-27], [-59], [8], [32], [-75], [-72], [23], [-28], [-46], [17], [-90], [-40], [-69], [60], [-4], [62], [-66], [14], [57], [42], [44], [-24], [58], [-84], [-43], [29], [58], [-36], [-49], [1], [76], [-54], [24], [80], [-31], [-1], [-18], [33], [-60], [17], [-76], [36], [22], [-10], [74], [42], [-15], [6], [-10], [-63], [66], [18], [-67], [-34], [48], [40], [40], [-30], [0], [-32], [54], [-18], [-21], [-8], [-21], [2], [6], [7], [60], [-66], [-62], [-30], [0], [63], [39], [-72], [14], [-20], [3], [-44], [6], [80], [-59], [30], [28], [1], [-65], [-30], [33], [64], [16], [-56], [40], [-61], [-10], [94], [-36], [63], [-71], [-27], [42], [-72], [38], [-42], [24], [-25], [6], [48], [-14], [38], [72], [-35], [-30], [-83], [-36], [45], [92], [64]];
    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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<7,R![-5, 1]>,<13,R![4, 1]>],Snew);
    return Vf;
end function;