
// Make newform 2960.2.a.o 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_o();".
// 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_o();".
// 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
        poly := [-8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
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, 0], [-2, 0], [1, 0], [2, -1], [0, 1], [2, -2], [-2, -1], [2, 0], [0, -2], [-2, 3], [6, -1], [1, 0], [2, 1], [-4, -1], [2, 2], [-2, 1], [-6, -2], [-2, -1], [2, -2], [0, 2], [2, 2], [-2, 2], [-6, 2], [10, 0], [10, -3], [2, -4], [-4, 4], [-10, 4], [-10, 3], [14, 1], [10, 2], [10, -2], [10, 2], [12, -3], [2, -4], [20, 0], [14, 1], [-16, -1], [12, -4], [6, 5], [6, 6], [-6, 4], [-22, 1], [2, 0], [14, 4], [18, -2], [4, 3], [-2, 1], [8, 1], [-10, 0], [-14, 6], [6, -7], [10, -6], [2, -4], [6, -8], [10, -3], [10, 0], [0, 2], [-6, 2], [18, 2], [8, -4], [-2, -5], [10, 4], [-2, 1], [10, -4], [22, 1], [-10, -4], [-14, 2], [-4, 8], [-22, 0], [-6, -5], [4, 8], [-14, 3], [-18, -4], [8, 0], [0, 4], [-2, 5], [-2, 8], [10, 0], [-14, 6], [0, -4], [6, -4], [-2, 1], [-14, 8], [10, 1], [-10, 8], [18, 0], [2, 5], [-2, -11], [-8, 4], [0, -9], [-10, 2], [4, 10], [12, -8], [-2, 8], [-16, -4], [-18, 2], [-10, 11], [-28, 0], [30, 0], [-36, -1], [2, 2], [4, -11], [-6, -8], [4, 7], [6, -4], [-4, -13], [10, -6], [-4, -4], [-2, -1], [16, -10], [6, 11], [2, -10], [-20, -3], [-18, 11], [-26, -7], [0, -1], [0, 4], [14, -4], [-12, 0], [-26, 9], [-22, 4], [-42, 0], [-12, 1], [8, -9], [14, -12], [30, 3], [28, 2], [48, 0], [-26, -1], [20, 5], [2, 1], [-4, -2], [-10, 16], [2, 1], [26, 6], [6, -9], [22, 0], [-14, -4], [-38, -4], [-36, -4], [-26, -2], [6, 10], [24, 9], [-10, -9], [16, 4], [22, 4], [-46, 5], [-14, 10], [-10, -13], [46, 1], [-2, 1], [8, 1], [22, 1], [0, -12], [-38, 6], [26, -10], [-6, 11], [10, 8], [18, 0], [0, -11], [26, 6], [28, 0], [-4, 7], [-22, 3], [-14, -5], [-2, -7], [6, 6], [46, 3], [-10, -4], [6, -8], [-2, 7], [14, 5], [-6, -2], [8, -6], [-14, 0], [-6, 12], [50, -4], [-30, 3], [-18, -2], [-40, 2], [-6, 6], [2, 16], [-30, 13], [-6, 15], [-10, -4], [-2, -9], [36, -5], [-6, -10], [-12, 8], [-2, 12], [-14, 2], [-14, -8], [-18, 18], [-6, -12], [-54, -4], [-22, 8], [-6, 10], [42, 8], [-44, 10], [6, 8], [-24, 12], [14, -15], [-10, 3], [8, -1], [10, -14], [16, -10], [-10, 6], [42, 0], [44, 3], [22, -19], [6, -9], [32, 4], [-34, 4], [-6, 15], [-2, 1], [32, 4], [-30, 6], [-52, -4], [-26, 19], [-38, -4], [-16, 0], [-26, 1], [-8, 16], [10, -10], [22, 0], [10, 6], [-20, 14], [38, -1], [46, 4], [-26, 5], [24, -8], [4, -8], [-38, -4], [6, -20], [-22, 13], [-14, -4], [-6, 18], [-38, -2], [14, 10], [-4, -9], [18, 10], [6, 2], [-42, -7], [-14, -8], [-58, 1], [4, 6], [62, 2], [-36, -5], [68, -4], [26, -12], [30, 12], [10, -5], [34, 6], [-50, 8], [-4, -13], [-26, 2], [-6, 12], [38, 5], [2, 7], [0, 6], [-4, 20], [6, -16], [22, -7], [-18, -12], [8, 15], [58, -2], [74, 2], [-26, 14], [18, -4], [6, 0], [32, -15], [-2, 12], [-22, -1], [-30, 0], [-2, -7], [-22, 8], [46, 0], [34, -9], [-42, -10], [14, -22], [22, 1], [-38, 5], [38, 14], [8, 4], [4, -10], [22, -3], [-42, 9], [2, 14], [-22, -12], [46, 1], [-44, 8], [10, 2], [44, 5], [-18, 1], [-34, -8], [-68, 2], [-34, -16], [6, -12], [-2, 8], [18, 5], [38, -4], [36, -4], [-4, 0], [38, 0], [-2, 15], [-52, 1], [-18, -8], [0, -14], [-30, 10], [-30, 14], [-46, 8], [54, 9], [-34, -14], [-40, 16], [2, 22], [28, -1], [14, 2], [-14, 6], [34, -2], [-18, 2], [50, 8], [62, -7], [22, -7], [-22, -6], [-22, 12], [-8, -1], [-48, -11], [-8, 28], [-18, -1], [-18, -4], [-2, -20], [22, 11], [4, -8], [34, -4], [6, -2], [30, 14], [10, 14], [42, 0], [-10, 2], [-34, 12], [-70, -4], [-50, 4], [2, 15], [14, -10], [-30, 6], [38, -12], [-10, -4], [14, -7], [14, -19], [-8, -9], [-14, -6], [22, 3], [-60, 0], [58, -6], [-22, 14], [2, -1], [54, -16], [2, 4], [-28, 0], [-54, 14], [10, -8], [-20, -10], [50, -6], [-6, -12], [34, -17], [-22, 14], [-44, -8], [-30, 24], [34, 10], [10, -4], [10, 13], [-30, 18], [-22, 1], [-42, 17], [16, -12], [-16, 8], [34, -10], [50, 8], [42, -18], [22, 15], [54, -12], [8, -2], [42, -8], [-34, -8], [-6, 21], [32, -4], [-10, -6], [-60, 13], [20, 24], [26, 5], [-10, 14], [-14, -20], [-48, 9], [-32, -2], [-22, 12], [10, 18], [-30, -9], [-8, -9], [70, 2], [50, -10], [26, 13], [24, 10], [-6, 12], [-18, 0], [8, -32], [14, -22], [-6, 27], [14, 18], [-16, 5], [2, 18], [6, 15], [-24, -21], [-54, 16], [34, -21], [50, -4], [-32, -16], [-38, 13], [18, 0], [8, -10], [-10, -16], [-58, -12], [-26, 10], [40, 13], [74, -12], [22, 5], [14, 10], [2, -25], [-64, 8], [-38, 25]];
    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_o();".
// 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_o(:prec:=2)
    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_o();".
// 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_o( : 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![-6, -3, 1]>,<13,R![-32, -2, 1]>],Snew);
    return Vf;
end function;