
// Make newform 5915.2.a.q in Magma, downloaded from the LMFDB on 04 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5915_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_5915_2_a_q();".
// 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_5915_2_a_q();".
// 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 := [-1, 8, 0, -6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-2, 0, 1, 0, 0], [0, -3, 0, 1, 0], [2, 3, -4, -1, 1]];
    Rf_basisdens := [1, 1, 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_5915_a();"
function MakeCharacter_5915_a()
    N := 5915;
    order := 1;
    char_gens := [2367, 5071, 1016];
    v := [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_5915_a_Hecke(Kf)
    return MakeCharacter_5915_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, 0, 0, 0], [0, -1, -1, 1, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [0, 0, 0, 2, 0], [0, 0, 0, 0, 0], [0, 1, 0, -3, -1], [0, -1, 0, 1, -1], [0, 0, 0, 0, 2], [-2, -1, -1, 3, 0], [2, 1, -1, -1, 2], [2, 3, -1, -1, 0], [2, -1, -2, -1, -1], [-2, 4, 0, -2, 0], [-2, -2, 0, 0, 4], [-2, 2, 2, 0, 0], [2, -1, 5, -3, -2], [-4, -2, 2, 2, 2], [-2, 1, 0, -3, -3], [0, 0, -2, -4, -2], [-6, -4, 4, -2, 0], [-6, 3, 0, -5, -1], [0, 2, -2, -4, -4], [0, 3, 5, -3, 0], [2, -4, 2, 2, -2], [-8, -4, 0, 0, 6], [-10, -3, -2, 3, -1], [0, 6, 0, -4, -4], [-6, 0, 4, 4, 6], [0, 2, 2, -2, -2], [8, 2, -4, 4, -4], [-2, -2, 6, 2, -2], [6, -7, -3, 5, 4], [6, 0, 0, -6, -6], [10, 4, -2, -4, 0], [4, -6, 0, 0, 2], [-14, -5, -1, 1, 6], [-2, 1, -1, -7, 0], [0, -4, -6, 2, 2], [-2, 1, 5, 3, -2], [10, 1, 0, 5, -3], [-6, 0, -2, -4, -6], [2, -9, -3, -1, 4], [-6, 1, -4, -5, 3], [-10, 1, 13, -3, 4], [-2, 0, 8, -2, -2], [-14, -1, 8, 3, 7], [-8, -4, -10, 10, 6], [4, 0, -6, 6, 2], [-2, -5, -10, 11, -1], [-10, -8, 2, -2, 2], [-8, 0, 10, -8, 2], [2, -3, -2, -3, -3], [-2, 2, 2, -8, -6], [8, -4, 6, 0, 2], [-4, -2, 6, -6, -2], [-2, -2, -8, 16, 10], [-14, 4, 2, 8, 2], [-10, -8, 8, -8, 0], [2, -2, 6, 4, 2], [-6, -1, 2, -3, 9], [0, -8, -10, 10, 0], [-18, 6, 6, -12, -2], [-12, 6, -2, -10, -4], [-6, 3, -9, 5, 6], [-10, -4, -8, 4, 4], [2, 2, 0, 2, -4], [-6, -2, -6, 6, 0], [0, -4, -8, 10, -6], [8, -5, 5, 9, 0], [6, -2, -8, 2, -6], [6, 4, -8, -8, 2], [12, 0, 8, -8, -8], [-4, -6, -10, 16, 2], [0, 4, 16, -8, -4], [4, 10, 10, -14, 0], [-2, 3, 0, 1, -11], [12, 4, -6, 8, 0], [6, 2, 6, -16, -10], [-10, 12, 4, -12, 0], [-12, 14, 12, -4, 2], [12, 2, 6, -10, -4], [14, 6, -8, 0, 4], [-4, 13, 4, 1, -1], [10, -8, -12, 16, 2], [-14, 8, 8, 2, 4], [14, 8, 8, -8, -14], [-2, 17, 5, -11, -8], [-12, 5, 11, -5, 0], [-14, -3, 16, 5, 9], [-18, 7, 6, -7, 1], [-6, -3, -5, 11, 14], [10, 1, -13, 17, 4], [16, 4, 4, 0, 0], [-10, -2, 6, 2, -6], [8, 0, -8, 4, 0], [-10, -1, -2, 11, 3], [6, -14, -4, -4, -2], [-4, -8, 8, 0, 8], [10, 0, -18, 10, 0], [6, -6, 4, -10, -6], [-2, -11, -4, 7, -9], [-2, 5, -10, -1, -1], [-2, 21, 1, -3, 0], [2, 5, -5, 5, 12], [10, -4, 0, -2, -8], [-2, -6, -4, 18, 0], [12, 12, -8, -6, -2], [12, 4, -4, 4, 4], [-2, 10, -10, -4, -4], [-8, -1, 3, 9, 0], [2, 4, -8, 16, 0], [10, 7, -16, 1, 1], [-10, -5, 5, 9, 2], [-2, 22, 8, -6, 0], [2, -11, 1, 9, 12], [-12, 12, 4, 8, 4], [8, -13, 3, -11, -4], [-4, 0, -18, 22, 8], [2, -5, 0, 15, -1], [6, 1, -6, 1, 1], [-2, 6, -16, -8, 6], [24, 4, 2, 4, 2], [2, -11, -12, 17, -7], [2, 1, -1, -13, -10], [-14, -15, 8, 3, 3], [-4, 10, 14, -10, -8], [0, 10, -6, -16, -12], [8, 9, 5, -1, 12], [0, -2, 0, 6, 8], [-16, -4, -4, 2, 0], [2, -3, 8, -3, -7], [10, 1, -21, 21, 4], [-4, -6, -4, -8, 8], [0, -5, 5, -3, 12], [-2, -4, -16, 16, 0], [-10, 14, 14, -8, -8], [-24, 4, -2, 8, 10], [2, 7, -1, 5, -2], [10, -7, -16, 7, -5], [10, -4, -6, 12, 6], [24, -10, 0, 8, 10], [-22, 8, 2, 2, 6], [8, 20, 4, 0, -12], [-2, -12, -10, 16, -2], [-10, -12, -2, -4, 6], [-12, 10, 8, -20, -12], [22, 7, -9, -3, -2], [4, 16, 16, -10, -12], [-14, -1, 0, -1, -1], [22, -5, 0, -11, -3], [10, -4, 0, -4, -4], [2, -14, -8, 8, 10], [-8, -13, -1, 5, 12], [-16, -12, -6, 6, 16], [-2, -7, 16, -3, 1], [-34, -1, 21, -13, 8], [16, -5, -7, 21, 0], [-8, 1, 12, -7, -5], [14, -5, -6, 7, -9], [-14, -5, 8, 7, 15], [12, 24, 0, -6, -2], [18, 11, 9, -1, -16], [16, 0, 0, 6, 0], [-18, -7, 8, -13, 7], [-16, 6, -8, 4, -6], [22, -5, 8, -1, -5], [-32, 13, 16, -3, 3], [-24, -6, 10, -4, 8], [-8, 7, -20, -9, 5], [-14, 2, 0, -10, -4], [-16, 3, 1, 9, 0], [-8, 12, 8, -12, 4], [2, 0, -12, -12, 0], [14, 8, -4, 2, -6], [-8, 6, 10, -22, -6], [-2, 8, -4, -10, 6], [32, 0, 0, 2, 0], [-12, -17, -9, 13, 8], [2, -16, -16, 26, 4], [-6, -21, -11, 7, 12], [-10, -12, 10, 10, 0], [-18, 2, -6, -4, -12], [-24, 8, 6, -2, 4], [-12, -16, 4, 8, -8], [-6, -6, 16, 0, 10], [-2, 0, -14, -6, -14], [24, 0, -18, -8, -10], [4, 12, 0, -2, 8], [-10, -29, 9, -3, -6], [-12, 7, 4, -13, -7], [-6, 1, -1, -15, -20], [-14, 5, 3, -15, 8], [10, -1, 18, -13, -5], [2, -8, 8, -22, -20], [4, 0, 10, 16, -14], [12, 12, -16, 12, 0], [-32, 16, 4, -10, -2], [-36, 0, 8, 8, 14], [-8, 3, -17, 33, 16], [32, -14, -8, 18, 10], [4, 0, -14, 6, -14], [-12, 2, 14, -2, -6], [30, 10, -4, -2, -10], [28, 3, -24, -7, -17], [-4, -27, -4, 13, -5], [-4, 7, 4, -3, -13], [14, -16, -18, 6, -6], [4, 10, -4, -8, -2], [-20, -22, 8, -2, -2], [6, 4, -4, 4, -4], [6, -8, -6, 16, 22], [34, -8, 2, 2, -6], [20, 4, 4, -12, -24], [2, -8, 6, 4, 14], [36, -19, -13, 7, 4], [-26, -16, -6, 10, 14], [-22, 6, 6, -2, -8], [2, -9, 1, 27, 4], [-34, -1, 0, -11, 1], [18, 1, -12, 3, -9], [4, 5, 20, -25, -19], [20, -8, -24, 32, 0], [18, -5, 12, -17, -9], [-6, -17, 6, 9, 13], [18, -7, -3, -23, -4], [-14, 7, 11, -23, -6], [2, -9, -19, 19, 8], [-6, -13, -2, 5, 1], [16, 0, 0, 4, -4], [-18, -10, 22, -4, 4], [24, -18, -8, -4, 2], [12, 10, 0, -4, 14], [-14, 3, -17, 27, 24], [-2, 15, -10, -23, 1], [2, 8, -14, 16, -4], [6, -16, -16, 0, 0], [34, -9, -1, 3, -8], [10, 9, -12, 1, -19], [0, 10, -6, 12, 12], [-10, 5, 27, -7, 0], [10, -20, 26, 12, 0], [-8, 1, 9, 3, -4], [-14, -6, -8, 0, 4], [-14, -8, 6, -6, -14], [8, 18, 16, -16, 2], [-8, 10, 26, -38, -4], [4, -21, 24, -3, -5], [28, -14, -16, 4, -2], [18, -10, -22, 18, 0], [50, 13, 0, -1, -17], [22, -17, -14, 27, 11], [16, 0, -24, 0, -8], [-28, 2, 16, 12, 16], [36, -20, -6, -12, -2], [8, -18, -22, 30, 4], [-18, 9, 8, -13, -5], [-22, 0, -22, 30, 12], [8, -1, -16, 7, -3], [-2, -8, -12, 10, 4], [2, 0, 20, -16, -4], [6, -14, 10, 8, -4], [-16, -13, -27, 21, 8], [-2, -3, -15, 19, 2], [-28, 24, 26, -4, 4], [6, -22, -10, 14, -6], [20, -10, 2, 0, -8], [6, 3, 18, 3, 7], [6, -13, -11, 27, -4], [28, -20, -18, 8, 8], [2, -4, 14, -6, 6], [-2, 15, 0, -1, 7], [-8, -8, 10, 22, 16], [-34, -1, -12, -1, -1], [-28, 15, 12, 7, -7], [-18, -18, 18, 10, 10], [26, 3, 0, 15, 23], [24, -16, -10, -6, 6], [0, -6, 30, 4, 4], [8, -12, -22, 30, 6], [-30, 9, 18, -5, 11], [-22, 20, 0, -2, -10], [-36, -11, 1, 15, -8], [6, 16, 24, -24, 8], [-10, -14, 12, -4, 12], [-22, 9, 4, -7, -11], [-2, -30, 8, 14, 10], [-6, 36, 0, -20, -12], [-8, 11, -20, 13, 15], [52, 8, -26, 0, -16], [-22, -8, 16, -12, 8], [16, 0, -36, 8, -12], [14, -3, -3, -7, -8], [-24, 23, 8, -31, -1], [-22, -8, -10, 32, -6], [4, -18, 8, 28, -8], [-8, -6, 0, 38, -10], [-28, -5, -28, 19, 9], [26, 14, -10, 10, 18], [-10, -22, 22, -4, -2], [-8, -1, 4, 19, -19], [2, 1, 0, -17, -17], [14, -15, -21, -3, -4], [2, -12, -6, -20, 18], [-14, -17, -8, -5, 15], [-6, -19, -8, -1, 15], [10, 10, 12, -2, -14], [16, 6, -16, 12, 16], [32, 6, -20, 6, -18], [-4, -10, 10, -6, -6], [18, 17, 26, -17, 3], [-44, -18, 14, -12, 0], [30, 13, -32, -9, -13], [-10, 16, -4, -4, 12], [-54, -21, 21, -9, 20], [-4, -18, -4, 8, 8], [42, -4, -14, -12, -10], [20, 14, 12, 4, -14], [-28, 2, 6, 20, -14], [-2, 15, -12, 17, 5], [-4, -10, 6, 14, 0], [-2, 13, 32, -17, 3], [16, -2, -8, -14, 12], [34, -25, -3, 19, 8], [30, 27, -13, -9, 0], [-34, -2, -8, 12, -10], [36, -2, -20, 18, -10], [-10, 10, 4, -2, 0], [10, -26, -20, 28, 18], [-10, -10, -14, 36, 36], [6, -34, -10, 10, 6], [-18, 22, -28, -8, 6], [-6, 18, 0, 8, -8], [-12, 14, -6, -10, -12], [-18, 20, 2, -10, 8], [-8, 22, 0, -16, -2], [40, -8, -16, -14, -8], [-12, -6, 16, 14, 20], [-8, -7, -9, 19, 8], [0, 15, 31, -27, 12], [-10, -25, -5, 15, 16], [20, 4, 10, 12, -12], [4, 13, 3, 19, 0], [-2, 13, 15, -7, 0], [-2, 16, -4, -40, 4], [-2, -17, -31, 19, -4], [-14, 16, 4, 22, 12], [4, 16, 8, 16, -8], [0, 3, 24, -15, -9], [0, -8, -6, 12, 18], [10, 9, -9, 11, -2], [-26, -24, -4, 8, -8], [-6, 16, -14, 28, 4], [-12, -7, -23, -1, -8], [-26, 17, 33, -39, -8], [14, -19, 17, 11, 6], [48, 11, -4, 5, 11], [-50, 15, 17, -11, -6], [-12, 8, -2, 30, 20], [-6, 1, -9, 25, 16], [-10, -5, 7, 15, -8], [6, 14, 4, 0, 22], [38, -6, -2, -16, -20], [20, 6, 16, -10, 10], [4, 21, -19, 15, 20], [-22, -10, -28, 8, 2], [-14, 6, 34, -16, -4], [14, 9, -16, -11, 9], [6, 3, 23, -13, -4], [-2, 13, 16, -31, -23], [-34, 28, 6, 16, 8], [-26, 9, 11, 13, 16], [-10, -24, 8, -4, -24], [0, 0, 2, -10, 22], [24, 36, -14, 4, 0], [16, -5, -12, -23, 3], [-10, 11, -12, -33, -1], [-30, 12, 20, -28, -8], [14, 0, -34, -6, -10], [-42, -7, 9, -9, -18], [-12, 7, -11, 41, 12], [-14, -16, -28, 32, 4], [-10, 17, 12, -35, -35], [-2, 27, 12, -11, -15], [-10, 17, 43, -13, -2], [8, 6, -26, -18, 0], [10, -22, -38, 32, -10], [0, 5, -28, 9, -13], [20, 0, 8, -16, -32], [4, 13, 25, -33, -28], [-12, -14, 18, -14, -8], [-2, -8, -28, 8, -12], [-14, -4, -4, 46, 18], [-28, 21, 13, -9, 4], [24, -2, -28, 30, -2], [0, -6, 20, -22, -16], [22, 23, 1, -27, -34], [-8, -16, -6, 18, -20], [-10, 25, 19, -19, -12], [-2, -4, 10, 10, 28], [18, -3, 2, 9, 21], [38, -24, -8, 2, -8], [38, 19, -11, -9, 0], [-6, 12, -6, -22, -14], [30, 21, -6, 13, -15], [-16, 6, 20, -18, -10], [-10, 7, 24, -47, -3], [-14, -20, -14, 4, 20], [-18, 0, -14, -6, 12], [2, -20, -2, -4, -10], [0, -8, -22, 36, -6], [-24, -24, 20, 8, -14], [12, -4, -18, 16, 16], [-14, 3, 8, -5, 7], [-6, 14, 8, 8, -18], [-8, 12, 30, -6, -12], [54, -20, -12, -24, -6], [-56, -28, 24, -4, 16], [8, -4, -2, -16, 4], [16, -17, 19, 5, 8], [-18, 20, -16, 12, 4], [28, 10, 18, -34, -6], [10, -19, -16, 29, -7], [32, -14, -16, 20, -2], [-24, -14, 28, 22, -4], [-18, -6, 8, -22, -2], [-4, 2, -14, 40, 34], [22, 16, -30, 6, 4], [6, -32, -10, 4, 10], [32, -2, 22, 0, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5915_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5915_2_a_q();".
// 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_5915_2_a_q(:prec:=5)
    chi := MakeCharacter_5915_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_5915_2_a_q();".
// 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_5915_2_a_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5915_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 8, 0, -6, 0, 1]>,<3,R![16, 11, -12, -7, 2, 1]>],Snew);
    return Vf;
end function;