
// Make newform 845.2.l.e in Magma, downloaded from the LMFDB on 14 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_845_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_845_l_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_845_2_l_e();".
// 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_845_2_l_e();".
// 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 := [4, -8, 8, -24, 20, 20, -8, 16, 4, -8, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-748, -48, 48, -144, 146, 94, -22, 18, -28, 4, -1, 1], [612, 148, -148, 444, -454, -286, 64, -44, 94, -20, 5, -5], [10, -10, 306, 34, -84, 322, -197, -123, 82, -32, 32, -10], [-12, 12, -1140, -50, 110, -414, 264, 166, -103, 43, -43, 12], [-156, -156, -436, 1060, 734, -330, 482, 220, -328, 94, -83, 39], [22, -22, 1676, 84, -194, 736, -461, -289, 185, -75, 75, -22], [-32, 32, -38, 86, 74, -12, 46, 4, -23, 5, -5, 3], [476, 524, -524, -3856, -1774, -846, -1734, 344, 666, -36, 101, -101], [1060, -1060, 1294, -2790, -2510, 92, -1700, -250, 941, -195, 195, -117], [612, -612, 732, -1636, -1424, 184, -906, -94, 469, -100, 100, -60], [748, 748, 2560, -5552, -3050, 174, -3250, -116, 1338, -216, 293, -187]];
    Rf_basisdens := [1, 460, 460, 230, 230, 460, 230, 20, 460, 460, 230, 460];
    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_845_l();"
function MakeCharacter_845_l()
    N := 845;
    order := 6;
    char_gens := [677, 171];
    v := [3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_845_l_Hecke();"
function MakeCharacter_845_l_Hecke(Kf)
    N := 845;
    order := 6;
    char_gens := [677, 171];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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 := [[1, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 1, 0], [0, 0, 0, 0, 1, -1, 0, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, -3], [3, 3, 3, 0, 0, 0, 0, 3, 0, -3, 3, 0], [0, 0, 0, -1, 0, -1, 3, 0, 1, 0, 0, -3], [-1, -1, -3, 0, 0, 0, 0, -1, 0, 3, -1, 0], [0, 0, 0, 0, -2, 0, 0, 0, 2, 0, 0, 2], [0, 0, 0, -2, 2, 0, -1, 0, 0, 0, 0, 0], [3, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, -2, 0, 0, -4, 0, 0, 0], [0, 0, 0, 2, -3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 3, -1, 0], [3, -5, -1, 0, 0, 0, 0, 3, 0, 1, -5, 0], [0, 0, 0, 4, -1, 0, 2, 0, 0, 0, 0, 0], [9, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], [7, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, -3, 0, 0, -4, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 6, 0, -4, 6, 0], [2, 2, 8, 0, 0, 0, 0, 2, 0, -8, 2, 0], [0, 0, 0, -1, 0, -1, -6, 0, 0, 0, 0, 6], [0, 0, 0, 0, -2, -6, 0, 0, 2, 0, 0, -7], [0, 0, 0, 6, 0, 6, 0, 0, -4, 0, 0, 0], [0, 0, 0, -3, -2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 4, -5, 0, 0, -4, 0, 0, 0], [-2, -6, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4, 0, 6, 6, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 2, -8, 0], [0, 0, 0, -4, -2, 0, 4, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, -2, 4, 0, -3, 0, 0, -4], [0, 0, 0, 3, 0, 3, 1, 0, 4, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, -5, 0, -3, 1, 0], [7, 1, 7, 0, 0, 0, 0, 7, 0, -7, 1, 0], [0, 0, 0, 2, 4, 0, -8, 0, 0, 0, 0, 0], [-8, 6, 4, 0, 0, 0, 0, -8, 0, -4, 6, 0], [-3, 1, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 8, 0, -4, -4, 0], [-8, 10, 2, 0, 0, 0, 0, -8, 0, -2, 10, 0], [2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0], [0, -10, -8, 0, 0, 0, 0, 0, 0, 8, -10, 0], [-5, 11, -1, 0, 0, 0, 0, -5, 0, 1, 11, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 9, -5, 0], [0, 0, 0, 1, 0, 1, -1, 0, 2, 0, 0, 1], [0, 0, 0, 2, 0, 2, 8, 0, 6, 0, 0, -8], [0, 0, 0, 1, 0, 1, -7, 0, -3, 0, 0, 7], [0, 0, 0, -6, -4, 0, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, -14, 2, 0], [0, 0, 0, 0, -6, -6, 0, 0, 6, 0, 0, 4], [0, 0, 0, 0, -8, 3, 0, 0, 8, 0, 0, 4], [0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 10, 0], [0, 0, 0, 0, 7, -4, 0, 0, -7, 0, 0, 4], [0, 0, 0, -6, -2, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, -2, 6, 0, 0, 0, 0, -6], [0, 0, 0, 0, 6, 6, 0, 0, -6, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, -3, 0, 5, -3, 0], [-3, 7, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 11, 0, 11, -1, 0, 0, 0, 0, 1], [21, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 5, 7, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 9, 0, 4, 0, 0, -9], [0, 0, 0, -5, -6, 0, 8, 0, 0, 0, 0, 0], [0, 0, 0, 0, -14, 7, 0, 0, 14, 0, 0, -1], [-13, -1, -3, 0, 0, 0, 0, -13, 0, 3, -1, 0], [0, 0, 0, 0, 0, 0, 8, 0, 3, 0, 0, -8], [0, 0, 0, 0, -4, -1, 0, 0, 4, 0, 0, 8], [0, 0, 0, 3, 16, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, -5, 3, 0, 0, 5, 0, 0, -15], [0, 0, 0, 0, 0, 0, 0, 1, 0, 7, 7, 0], [-6, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -3, -3, 0], [0, 0, 0, 0, -6, -5, 0, 0, 6, 0, 0, -3], [0, 0, 0, -8, -2, 0, 6, 0, 0, 0, 0, 0], [6, 2, 10, 0, 0, 0, 0, 6, 0, -10, 2, 0], [0, 0, 0, -6, 0, -6, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 13, 3, 0, 0, -13, 0, 0, 9], [0, 0, 0, -4, 8, 0, -4, 0, 0, 0, 0, 0], [-14, 0, -10, 0, 0, 0, 0, -14, 0, 10, 0, 0], [0, 0, 0, -9, 0, -9, 18, 0, 2, 0, 0, -18], [0, 0, 0, -6, -6, 0, -8, 0, 0, 0, 0, 0], [2, 2, -16, 0, 0, 0, 0, 2, 0, 16, 2, 0], [0, 0, 0, 11, 0, 0, -17, 0, 0, 0, 0, 0], [11, -7, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 8, 0, -8, 0, 0, -8], [0, 0, 0, 0, 3, 4, 0, 0, -3, 0, 0, -4], [0, 0, 0, 0, 0, 0, 0, -1, 0, 1, -7, 0], [8, 6, -16, 0, 0, 0, 0, 8, 0, 16, 6, 0], [0, 0, 0, -2, 0, -2, -2, 0, 13, 0, 0, 2], [0, 0, 0, 0, -6, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, -4, -5, 0, 0, 4, 0, 0, 5], [19, -5, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 8, -5, 0, 0, -8, 0, 0, 14], [0, 0, 0, -9, 0, -9, 7, 0, 0, 0, 0, -7], [0, 0, 0, 8, 0, 8, -9, 0, -6, 0, 0, 9], [-5, -1, -7, 0, 0, 0, 0, -5, 0, 7, -1, 0], [0, 0, 0, 4, 2, 0, -3, 0, 0, 0, 0, 0], [17, -5, 5, 0, 0, 0, 0, 17, 0, -5, -5, 0], [-18, 8, -6, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-19, 5, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [29, 11, 21, 0, 0, 0, 0, 29, 0, -21, 11, 0], [-14, -2, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-16, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-22, 0, -4, 0, 0, 0, 0, -22, 0, 4, 0, 0], [0, 0, 0, 8, -10, 0, -5, 0, 0, 0, 0, 0], [-16, 12, 2, 0, 0, 0, 0, -16, 0, -2, 12, 0], [0, 0, 0, 0, 0, 0, 0, 16, 0, -2, -6, 0], [0, 0, 0, 16, 0, 16, -6, 0, 3, 0, 0, 6], [0, 0, 0, 5, 17, 0, 11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -12, 0], [0, 0, 0, 0, 0, 0, 0, -13, 0, -11, -5, 0], [0, 0, 0, 0, -8, 3, 0, 0, 8, 0, 0, 0], [0, 0, 0, 0, -4, 11, 0, 0, 4, 0, 0, -5], [0, 0, 0, 0, 0, 0, 0, -6, 0, 10, -14, 0], [0, 0, 0, 0, -18, 4, 0, 0, 18, 0, 0, -4], [0, 0, 0, 0, -12, -1, 0, 0, 12, 0, 0, -11], [0, 0, 0, -14, 0, -14, 2, 0, -2, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, -15, 0, -1, 1, 0], [0, 0, 0, 0, -1, 11, 0, 0, 1, 0, 0, 9], [-6, 18, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 4, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, -2, -14, 0], [0, 0, 0, 8, 0, 8, -21, 0, -8, 0, 0, 21], [-2, -12, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 11, -7, 0, 0, -11, 0, 0, 19], [27, -3, -1, 0, 0, 0, 0, 27, 0, 1, -3, 0], [10, -2, 10, 0, 0, 0, 0, 10, 0, -10, -2, 0], [0, 0, 0, 0, -8, -12, 0, 0, 8, 0, 0, -22], [0, 0, 0, -17, -6, 0, 7, 0, 0, 0, 0, 0], [0, 0, 0, 0, -18, -3, 0, 0, 18, 0, 0, 7], [0, 0, 0, 0, 0, 0, 0, 13, 0, -19, -3, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -1, 15, 0], [0, 0, 0, -5, 2, 0, -1, 0, 0, 0, 0, 0], [25, -7, 5, 0, 0, 0, 0, 25, 0, -5, -7, 0], [0, 0, 0, 3, 0, 3, -5, 0, 7, 0, 0, 5], [0, 0, 0, 0, 2, 2, 0, 0, -2, 0, 0, 22], [0, 0, 0, -3, 2, 0, 4, 0, 0, 0, 0, 0], [27, -3, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0], [3, 13, 15, 0, 0, 0, 0, 3, 0, -15, 13, 0], [0, 0, 0, -11, 9, 0, -13, 0, 0, 0, 0, 0], [-15, -3, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 2, 2, 0, 4, 0, 0, -2], [-20, -10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [13, 17, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 12, 0, 14, 8, 0], [5, -1, -3, 0, 0, 0, 0, 5, 0, 3, -1, 0], [0, 0, 0, -6, 0, -6, 11, 0, 12, 0, 0, -11], [0, 0, 0, 0, 14, 5, 0, 0, -14, 0, 0, 4], [0, 0, 0, 0, 20, -3, 0, 0, -20, 0, 0, -8], [-4, 14, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -12, 0, -8, 8, 0], [0, 0, 0, -18, 4, 0, 10, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 2, 8, 0, 22, 0, 0, -8], [0, 0, 0, 19, 0, 19, 1, 0, -6, 0, 0, -1], [-23, -5, 13, 0, 0, 0, 0, -23, 0, -13, -5, 0], [0, 0, 0, 0, 8, 0, 16, 0, 0, 0, 0, 0], [3, 15, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -18, 0], [23, -5, -3, 0, 0, 0, 0, 23, 0, 3, -5, 0], [15, 17, 23, 0, 0, 0, 0, 0, 0, 0, 0, 0], [14, -8, 14, 0, 0, 0, 0, 14, 0, -14, -8, 0], [0, 0, 0, -20, -2, 0, 16, 0, 0, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_845_l_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_845_2_l_e();".
// 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_845_2_l_e(:prec:=12)
    chi := MakeCharacter_845_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_845_2_l_e();".
// 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_845_2_l_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_845_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![25, -5, 16, -7, 10, -3, 1]>],Snew);
    return Vf;
end function;