
// Make newform 2960.2.a.u in Magma, downloaded from the LMFDB on 16 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_u();".
// 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_u();".
// 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 := [10, -8, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-6, 1, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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, 0], [0, 0, 1], [-1, 0, 0], [0, 1, 0], [-4, 1, -1], [0, 0, 2], [0, 1, -1], [0, 0, 1], [0, 2, -2], [-2, 1, 1], [0, -3, 0], [-1, 0, 0], [-2, -3, 1], [4, -1, -1], [0, -2, 1], [6, 3, -1], [-4, -2, 1], [2, 1, 1], [0, -2, 1], [-8, 2, 2], [-6, 2, -2], [8, 2, 1], [0, -2, -3], [6, 0, 0], [-6, -3, 1], [-4, 2, -4], [8, 0, -2], [-4, 4, -1], [6, 1, -3], [0, 3, 1], [4, -2, 3], [-8, 2, -1], [2, 6, -2], [-4, 5, -3], [4, 2, -4], [0, 0, 2], [2, -1, -1], [4, 7, -3], [0, -4, -2], [-2, -9, 3], [-8, 6, -3], [8, 2, 0], [8, -5, 8], [-10, 4, -4], [14, 0, -4], [-4, 6, -3], [-12, 3, -5], [12, -1, -2], [-12, 1, -5], [2, 0, -8], [10, -6, -2], [-4, 3, 4], [10, -2, -6], [0, 0, -3], [8, -6, -4], [4, -5, -4], [4, 2, -4], [24, -2, -2], [-16, 0, 2], [-6, 2, 6], [4, -4, 6], [-6, -3, 1], [12, -4, 1], [16, -5, -2], [2, 4, -4], [-14, -1, 3], [8, 0, 3], [-6, -6, 2], [-4, 0, 4], [-4, -6, -4], [14, -9, 11], [8, 4, 2], [12, -3, -2], [2, -4, -8], [-12, 8, -10], [0, 0, 4], [-10, 7, 3], [-26, 4, 4], [2, 4, 0], [-18, 2, 6], [-4, -12, 2], [22, 0, 0], [-16, 3, 2], [22, -4, 4], [4, 3, -2], [-12, -4, -1], [6, -4, -4], [18, 5, 1], [-14, -5, 3], [-8, 4, -4], [-4, -9, -3], [12, 2, -5], [0, 2, -12], [4, 8, 0], [-4, -4, 5], [32, 0, -4], [-2, 6, -2], [16, -3, -1], [4, 0, -8], [6, -4, 4], [4, 7, -1], [0, 4, -2], [4, 13, -7], [10, -8, 8], [-4, -1, 11], [-8, 6, -8], [-12, 11, 3], [-2, -2, 10], [-16, 8, 10], [28, -3, 3], [0, 10, -6], [18, -11, 5], [18, 2, -2], [-12, 5, -3], [12, -7, 2], [-28, -1, -3], [-12, 7, 1], [24, 4, -4], [2, -8, -8], [4, 0, -4], [10, -1, -1], [18, -4, 0], [2, 0, 8], [-12, -7, 9], [-4, -9, 5], [-2, 8, -8], [-6, -3, -7], [-16, 10, -12], [-8, 0, 8], [-14, 9, 5], [-4, -11, -3], [24, 7, -2], [0, 2, -8], [-6, -12, 4], [-38, 1, 5], [2, 6, -6], [-2, 5, -3], [12, 8, -1], [0, 10, -12], [-30, 4, -4], [4, -12, -4], [-18, 10, -14], [24, 6, 3], [4, 1, 7], [18, 1, -3], [-16, 0, -4], [30, -4, -4], [-14, 13, -7], [4, 6, 5], [4, -11, 0], [26, -1, -1], [8, -5, -3], [-12, -7, 11], [0, -1, 12], [-20, -4, -6], [0, -10, 3], [-24, -2, -1], [6, -9, -5], [6, 8, -8], [0, -6, 8], [44, 5, -1], [2, -2, -2], [16, -8, -14], [-28, -1, 3], [34, 3, -1], [4, -19, 10], [12, 3, 0], [-6, 2, 2], [24, -11, 15], [-34, 0, 0], [4, 4, 13], [6, -3, -11], [-20, -1, 0], [-2, -2, -10], [32, -2, -2], [2, 0, -4], [12, 0, 11], [12, 2, 12], [8, -3, -4], [26, -6, 10], [32, -2, 6], [40, -6, -1], [4, 2, -16], [18, -3, 9], [-24, 9, -8], [-22, 8, -8], [22, -3, 5], [-4, 3, 7], [34, -10, 14], [-32, 4, -2], [-36, 10, -16], [16, -2, 13], [-20, 0, 3], [30, -10, -2], [4, -16, 5], [-26, -12, 4], [-26, 4, -8], [20, 0, -2], [30, -12, 8], [8, 10, -8], [-10, 20, 0], [-16, -8, -4], [-38, -1, -5], [12, 1, 7], [-4, 23, -11], [12, -4, 10], [-24, 6, -2], [-4, 2, 17], [-10, 8, 0], [-20, -13, -1], [20, 11, -3], [18, -15, 1], [8, 0, 8], [24, -4, 1], [20, -11, 8], [4, 7, 13], [8, -12, 12], [-26, 10, -14], [8, -12, 6], [10, 13, -7], [8, 10, 0], [-8, -4, 0], [4, -1, 13], [0, 24, -4], [0, 18, -15], [-14, 8, 16], [30, 2, 10], [16, -2, -4], [40, 1, 6], [8, -16, 7], [18, 11, -1], [-12, -8, -6], [32, -4, 10], [18, 12, 4], [-20, -20, 7], [-4, 11, 0], [-26, -8, 0], [-8, -4, 6], [56, -2, -1], [0, 10, -11], [-12, -1, 3], [36, 6, 5], [4, -10, 13], [46, -5, 7], [18, -4, -4], [-8, -21, 10], [-16, 14, 12], [-12, -18, 7], [28, 3, -1], [8, -16, 14], [-4, 10, -8], [-4, 18, -4], [-4, -3, 22], [42, 2, 6], [18, -4, -4], [36, -5, -5], [2, 2, -6], [24, 0, -5], [6, -1, -1], [-10, 3, -17], [16, -10, 2], [36, -12, 8], [-26, -4, -4], [-26, -5, 15], [-4, 10, 8], [12, 7, -7], [52, -4, -2], [-10, -14, -6], [8, -10, 15], [-44, 2, -8], [10, -16, 20], [-4, 9, -21], [-24, -2, 0], [16, 5, -14], [26, -12, -20], [8, 23, -8], [0, 0, -5], [-14, 0, -8], [-50, 3, 7], [4, 10, -9], [-8, 6, -13], [4, -5, -16], [-16, 3, -2], [-10, 10, 10], [4, -20, 14], [-16, -2, -4], [-12, 11, -19], [-14, 7, 15], [0, 6, 3], [14, 20, 0], [6, -5, 19], [20, -16, 8], [-18, 2, 2], [-20, -19, 5], [-10, 11, -5], [38, -16, 16], [24, 26, -4], [-26, 4, 0], [12, 0, 1], [-56, -4, -3], [22, -15, 13], [18, -12, 8], [48, -4, 6], [-4, 16, -8], [-4, 0, -7], [8, -7, 15], [-44, 9, 1], [-6, 8, -24], [16, 14, 6], [28, -8, -18], [-24, -6, 5], [4, 2, -4], [12, -1, 5], [4, -18, -7], [-16, 8, 4], [34, -6, 14], [-4, -1, -17], [20, 2, 7], [-30, 18, 2], [-30, 6, -10], [-52, 6, 15], [14, 0, 4], [14, 11, -21], [0, -9, 8], [10, -10, 6], [2, 12, 8], [-12, 7, 1], [-12, -3, -5], [32, 4, -12], [-26, 5, -23], [2, -8, -8], [6, -16, 16], [28, -7, 2], [-36, 8, 20], [-24, 0, -3], [44, -6, -3], [-22, -10, 6], [-10, 22, -22], [54, -8, -8], [16, -2, -1], [-30, 12, -16], [-38, 16, 8], [6, -24, 12], [4, -11, 28], [50, -10, -2], [12, -2, 17], [46, -20, 28], [-44, -8, 3], [40, -13, 2], [10, -21, 23], [20, 15, -11], [-10, 18, 6], [10, -11, 25], [16, -16, -2], [-40, 8, 6], [-10, 26, -6], [-8, 13, 6], [-16, 0, 3], [-34, 8, 8], [-24, -8, -10], [8, 32, -6], [46, 12, 4], [0, 22, -8], [16, -6, 7], [20, 0, 21], [18, 15, 11], [28, -8, -2], [56, -4, 6], [46, -8, -20], [16, -14, -13], [28, -12, 7], [-36, 3, -16], [28, -12, 6], [24, -21, 20], [26, -1, 11], [-68, -4, 2], [-32, -12, 0], [-22, -6, -6], [-18, -4, 12], [-14, -22, 2], [-42, -3, -7], [-28, 10, 0], [-24, 26, -18], [46, 16, 4], [12, -4, -3], [-52, 19, -20], [8, -16, 4], [-18, 10, -6], [12, 21, -11], [-40, 20, -14], [-6, 29, -7], [14, 2, 10], [-48, 8, -3], [12, 1, -1], [8, 14, 10], [2, 0, 16], [-8, 10, 15], [-34, 11, -17], [20, 23, -7], [-14, 26, -14], [-24, 12, 10], [-2, -23, -3], [-48, 2, 14], [14, -24, 0], [-22, 16, -4], [-32, 4, 4], [50, -2, -2], [-2, -17, 27], [12, -2, 25], [-12, -11, 11], [-14, 10, -14], [22, 21, 5], [36, -5, -7], [-72, -8, -3], [18, 11, 7], [0, 18, 16], [0, -16, 16], [20, -13, 12], [-26, 12, 12], [40, 2, 14], [58, 0, 4], [26, -28, 4], [-40, -2, 7], [28, 5, 3], [-14, -4, 4], [30, -1, -13], [16, -2, 15], [-54, 7, -5], [4, 16, -14], [-4, 3, 6]];
    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_u();".
// 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_u(:prec:=3)
    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_u();".
// 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_u( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2960_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, -10, 0, 1]>,<7,R![10, -8, -1, 1]>,<13,R![-32, -40, 0, 1]>],Snew);
    return Vf;
end function;