
// Make newform 390.2.x.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_390_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_390_x_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_390_2_x_b();".
// 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_390_2_x_b();".
// 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 := [2197, -338, -1196, 190, 104, 154, 98, -134, 8, 34, -8, -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], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [81183629852, 64277898945, -33307875431, -8165110694, -5740233327, 4655211661, 6489146722, -4513282461, 97338749, 591783880, -163110633, -203419], [1269272187404, -833086363785, -554153957987, 414518069862, -272345430479, 135156205247, -372787906, -55879543947, 17147379373, 2300362460, -2097063441, 60968787], [618192439839, 98318215515, 28687118858, 8309942667, 35262141211, -18436259198, 29967292429, 2513656023, -7036618932, 2051209685, 454030419, -120243408], [20321135952, 44426935995, -3499453881, 3964105106, 1601173623, 4136894311, -511723478, 1065744289, 267354099, -226795620, 83789417, 16426431], [11867687, -4439955, -7481036, 77961, -940887, 1844066, 35557, -817691, 233294, 57555, -37023, 4036], [21764327, 2476201, -12957312, -7615595, -2792113, 3193024, -1390803, -291763, -91862, -18833, 28887, -20638], [-40106339, 8732837145, 3156976042, -1963928377, -1228577871, 1496635908, 32851611, -247897203, 110201552, -9226575, -7145579, 5665538], [1108736562037, 550784663520, -633066289511, -58320816489, -85505629812, -46377269759, 138431742257, -56431077566, -20291622981, 11881898255, -845264698, -680246389], [-1182431604497, 783010043855, 859741244016, -574822497041, 206023103197, -93428783546, -22116401667, 75140023171, -35058222714, -1280213355, 4870077513, -1202455216], [16572567908, 99935137380, -15605355549, -26424465301, -3795941083, -1904719281, 7168384238, -695335944, -2253760279, 556127095, 161812093, -78524401]];
    Rf_basisdens := [1, 1, 63907274600, 830794569800, 830794569800, 63907274600, 11796200, 47610004, 12781454920, 830794569800, 830794569800, 31953637300];
    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_390_x();"
function MakeCharacter_390_x()
    N := 390;
    order := 6;
    char_gens := [131, 157, 301];
    v := [6, 3, 5];
    // 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_390_x_Hecke();"
function MakeCharacter_390_x_Hecke(Kf)
    N := 390;
    order := 6;
    char_gens := [131, 157, 301];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 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, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, -2, 1, 0, -1, 0, 1, 1, -1], [0, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0], [0, 0, -1, 0, 0, 1, 0, 0, -1, 1, 0, 0], [0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1], [-4, 3, -1, -2, -3, 2, 0, -1, -3, 2, 0, -1], [0, -2, 1, 0, -1, 0, 0, -1, 2, -1, 0, -2], [-1, 1, 1, -1, 0, -3, 0, 1, 0, -2, -2, 0], [2, -1, 0, -1, 2, 0, 1, 1, 1, -3, -2, 0], [1, -2, -1, -1, 2, -3, 1, -1, -1, -1, -2, 3], [4, -1, 2, 2, 0, -2, 3, 0, 1, 2, 1, 0], [-2, 1, -1, 0, -2, 1, 1, 0, -2, 1, -1, 0], [4, -1, -1, 1, 3, -1, 1, 2, 1, -2, 0, 1], [-1, -2, 0, 0, 5, -1, 1, 3, 1, -1, 1, -1], [-2, 0, 3, -1, 2, 1, -5, -1, 1, -5, -2, -1], [-4, 1, -1, -1, 7, -1, -3, 6, -1, 0, -2, 3], [0, -1, -1, -2, -1, -2, 0, 3, -1, 0, 0, 1], [2, -4, 0, 2, 2, -4, -2, -2, 2, -2, -2, 4], [0, -4, 0, 0, 8, -2, -2, 6, 4, -4, -2, 0], [0, -6, 0, 0, -3, -3, 1, -1, 5, -3, 1, -1], [3, -4, 3, 3, -4, -1, 1, -3, 3, -1, 4, -1], [-6, 0, 2, 2, 2, -2, 0, 0, 0, -2, 2, 0], [-2, 4, 0, -1, 0, -2, -2, 1, -4, -1, -5, 0], [0, 2, 0, 0, -4, 4, -8, 0, 2, 2, 4, -2], [0, 5, 1, -4, -2, 1, 5, 2, -4, 5, 1, -2], [-2, 0, 3, 1, 1, 2, -6, 0, 2, -4, -1, -2], [0, -2, 2, 2, -6, 2, 2, -2, -2, 0, 4, 0], [2, 2, -4, 0, -4, 6, 8, 0, -2, 2, 4, 0], [2, -4, -1, 0, 1, -2, -1, -1, 4, -5, -2, 0], [-4, 2, 2, 1, 8, 0, 2, -1, 0, -1, -1, 0], [2, -3, -1, -1, -8, 0, 1, -3, 2, 0, 0, -1], [0, 0, -5, 2, -3, 2, -1, 1, 0, 5, 0, 0], [0, -2, -3, 0, -5, 4, 6, 1, -2, 1, 4, 2], [-4, -4, -2, 1, 0, -5, -7, -5, 4, -6, -4, 6], [-4, 4, 2, -6, 0, 0, -2, 6, -2, 0, 0, -2], [0, 1, 3, -1, 1, -1, 0, -8, 1, -1, -1, -2], [0, -1, 3, -3, 5, -7, 7, 2, -1, -4, -4, 1], [7, 5, -3, -4, 0, 1, 12, 6, -6, 5, 1, 2], [4, 2, 1, -2, -1, 0, 2, -1, -2, 3, -2, 2], [0, 0, -5, 1, -2, -1, 0, -3, -3, 2, 1, 6], [8, -4, -4, -4, 7, -1, -1, 5, 5, -1, -5, 1], [0, 2, 0, 0, 0, 4, 2, -4, 2, 2, 4, -2], [0, 3, 7, 4, -12, 3, 3, -14, 2, 7, 7, -10], [2, 4, 2, -5, -2, 2, 6, -1, -2, 1, -1, -4], [0, 0, -4, 6, 0, 8, -6, -8, 0, 4, 2, 0], [2, -6, 4, -1, 2, -4, -4, -9, 6, -9, -7, 0], [0, 2, -4, -1, 2, 0, 2, 7, -4, 3, 1, 4], [0, 2, -2, 0, -10, 6, 0, 4, 2, 4, 6, -2], [0, 2, -2, 2, -4, 8, 2, -6, 0, 0, 4, -2], [-1, 0, -7, 1, -4, 3, 1, 7, -3, 5, 4, 3], [0, 2, 4, 0, 0, -4, 0, -8, 2, 4, 0, -4], [2, 8, 0, -4, -12, 4, 5, -8, -8, 8, 0, 0], [0, -1, 4, -1, 6, -3, -11, -3, -1, -5, -2, 1], [1, -2, -6, -1, 0, 4, 1, 1, -6, 5, 1, 0], [7, -3, 3, 2, -4, 1, -4, -2, -4, -1, 5, 0], [0, 6, 10, 0, -2, 0, -8, 2, 6, -4, 0, -6], [-9, 3, -1, -3, 2, -5, 6, 3, -6, -2, -6, 0], [8, -4, 5, 9, -1, -1, 3, -2, 4, 1, 8, -2], [2, -4, -4, 2, 1, -3, 1, 13, -1, 1, -1, 5], [-2, 0, -5, -1, -4, 3, 2, 1, 3, 4, -1, 0], [0, 1, -6, 4, 0, 4, 2, -4, 1, 7, 0, -1], [-2, 2, -4, -4, 14, 2, 2, 10, -4, 2, -2, -2], [-8, 0, -2, -2, 8, 0, 2, 6, -2, 0, 0, -2], [-8, -2, -6, 6, -4, 4, -18, -2, 0, 4, 4, 2], [8, -6, 7, 7, 12, -9, -3, 1, 9, -9, 4, 3], [-12, -2, 2, 4, 6, 0, -6, 6, 2, 0, 4, -2], [-4, 2, -4, -6, -1, 1, -1, 7, -5, 1, 1, 3], [-8, -10, 8, 10, -8, 0, -4, -8, 10, -2, 10, -10], [2, -2, 12, 6, -4, 0, 0, -6, 2, -6, 8, 0], [4, 3, -3, 0, 0, 1, 7, 6, -4, 5, 3, 2], [-2, -6, -4, -4, 6, -10, -6, -2, -4, -2, -6, 10], [0, 2, 6, 4, -2, 2, -2, -4, 2, -2, 2, 0], [8, -8, 8, 3, 14, -3, 1, 11, 8, 0, 0, -2], [-4, 0, -4, -1, 20, 2, 4, 1, -2, 3, -1, 0], [0, -5, -12, 0, 2, 2, 1, -4, -5, 7, 2, 5], [-10, -3, -4, -4, 6, -9, -2, 4, 5, -9, -6, 2], [0, -5, 4, -5, -2, -7, 11, 9, -5, -9, -2, 5], [6, -1, -1, 1, -2, 3, -5, -1, -8, 2, 2, 0], [8, -5, 0, 0, -12, 2, 6, -10, 5, -5, 2, -9], [12, -4, 10, 2, 12, -6, 8, 18, 6, 0, -2, -4], [-8, 0, 0, -14, 7, -3, -9, 3, -7, -11, -7, 7], [-4, 2, 2, 2, -14, 2, 2, -2, 2, 0, 0, 0], [0, 3, 1, 0, -13, 0, 0, -13, -3, 4, 0, 3], [-14, 8, -2, -2, -2, 10, -6, 10, -4, 4, 6, -8], [-4, -6, -4, -4, 6, -12, -10, 10, -4, 0, -6, 10], [0, -4, -3, 2, 25, -6, -6, 19, 4, -7, -4, 8], [-10, 5, -7, 2, -8, 5, -5, -4, -6, 7, 7, 2], [-12, -2, -2, -7, 0, -1, -7, -1, 2, -4, -8, 0], [-16, 4, 2, 2, -20, 8, 2, -10, -6, 8, 4, -2], [2, 6, 2, 6, -8, 10, 6, 2, 4, 6, 8, -10], [-6, -2, -10, 0, -16, 10, 8, 0, -4, 10, 2, 0], [0, -4, -1, -6, -1, -10, -12, 11, -4, -3, -4, 4], [6, 2, -2, -3, -4, 6, 8, -7, 0, -5, -1, -4], [-2, -2, 0, -4, 4, -10, -4, -4, -2, 2, -4, 4], [4, -4, 3, 8, -3, 0, 2, -3, 4, -1, 8, -4], [-2, 1, -6, -7, 22, -8, 1, 7, -3, -1, -8, 0], [-10, 2, 1, 1, -2, 7, 1, -1, -3, 7, 2, -1], [-6, 4, -11, -1, -4, 9, 2, 1, -7, 10, -5, 0], [-2, -2, -6, 8, -5, 9, -7, -11, 1, 1, 5, 1], [10, 4, 2, 6, -6, 14, 20, -16, 4, -2, 6, -8], [0, -7, 5, 1, 16, -11, -7, 19, 4, -2, -6, 6], [0, 2, -8, -6, -4, 2, 2, -2, -2, -6, -4, -2], [4, 4, 14, -5, 2, 2, 8, 7, 4, 1, -1, -16], [-4, 6, -9, -9, -13, -4, 0, -2, -6, -4, -9, 0], [-16, 0, 0, 0, -29, 9, 7, -11, -7, 9, 7, -7], [-8, 12, 6, -8, -20, 6, 4, -24, -6, 10, 4, -12], [5, 2, -5, -5, -21, 6, 4, -6, -6, 6, -1, -4], [-8, 10, -6, -6, -20, 8, 2, -6, -12, 8, -4, -2], [7, 2, 2, 6, -12, 2, 9, -14, 0, 8, 8, -4], [20, 6, -5, -10, -1, -2, 8, -3, -6, 1, -12, 10], [-4, 1, 4, 0, 8, 0, -3, 18, 1, 2, 1, -4], [0, 4, 9, 2, 3, -6, 7, 3, 4, -5, -8, -4], [2, 4, 3, -7, 1, -6, 10, -10, -2, 4, -1, -2], [-4, -2, 4, 4, -20, 4, 2, -16, 2, 2, 8, -10], [0, -1, -4, 8, -12, 12, -12, 0, -1, 3, 4, 1], [0, 2, -2, -2, -2, 6, 14, -4, 2, 4, 8, -2], [8, -4, 2, -5, 16, -12, -4, 5, 12, -7, -1, 0], [-10, 2, 4, -1, 4, -6, 8, 1, -8, -5, -3, 0], [0, 1, -9, 5, -11, 9, 1, 2, 1, 10, 4, -1], [4, -3, 3, 2, 4, 1, -1, -2, 12, -1, 5, 0], [2, 5, 1, 2, -2, 3, -7, -2, -8, 1, -3, 0], [4, -10, 8, 0, 14, -16, -6, -10, 4, -12, -14, 6], [0, 0, 0, 8, 8, 4, -8, -12, 0, 0, -4, 0], [-10, 0, 8, 5, -6, 2, 10, -5, -4, -3, 5, 0], [2, -3, 0, 0, 29, -10, -11, 9, 14, -10, -11, 11], [8, -11, 5, 4, 19, -6, -2, 13, 11, -6, -2, 1], [0, -6, -14, -6, -2, -2, 26, 4, -6, 8, 4, 6], [-10, 4, -3, -9, -1, -2, -10, 12, -6, 4, 1, 2], [16, -6, 5, 5, 0, -1, -1, -3, 7, -1, 4, 1], [4, -8, -8, -9, -8, -10, 4, 9, 2, -1, -1, 0], [3, 1, -7, -11, 18, -3, 4, 27, -6, -6, -10, 10], [-11, 5, -11, -9, 0, 5, -6, 1, -8, -4, -4, 6], [-4, 6, -2, -1, 24, 0, -2, 1, 4, 1, -7, 0], [0, -4, 1, 10, 1, -2, -2, -1, 4, -3, 8, 0], [-3, 5, -3, -3, 26, -3, -2, 3, 2, 0, -8, 0], [0, 4, 7, -4, 9, -4, -14, -5, 4, -3, 0, -4], [0, -5, -7, 5, -5, 1, 7, 4, -5, 2, -4, 5], [-12, 6, -8, -16, 8, 0, -6, 8, -6, -2, -16, 6], [0, 5, -7, 2, 12, 5, 5, 36, -6, 7, 7, 2], [4, 4, 15, -7, 7, -4, 8, 8, 4, -10, -7, -8], [2, -6, 3, -5, -2, -13, 4, 5, 3, -8, 1, 0], [4, -18, 11, 10, -5, -6, -4, -11, 18, -7, 4, -6], [6, -2, 8, 8, 8, 6, 0, 0, 2, 6, 8, 0], [4, -7, -9, -2, 6, -9, -3, -4, -2, -7, -9, 18], [16, -8, 2, 4, 30, -8, 0, 22, 8, -6, -4, 8], [-4, -1, 4, 4, 0, -17, -8, -6, 9, -17, -4, 8], [1, 2, -3, 3, -5, -4, 4, -2, 0, 14, 5, -2], [-24, 6, 7, 7, -33, 8, 4, -18, -10, 8, 11, -4], [1, -2, 1, 1, 8, -7, -1, 3, 3, -7, 0, 1], [0, -8, -4, -1, 6, 3, -7, -9, -8, -4, 4, 8], [32, -13, 5, 17, -6, -11, 19, -21, 10, 2, 4, 6], [8, -9, -4, 12, 1, -4, 3, 1, 4, 2, -1, 5], [-1, -5, -9, -2, -14, 5, 6, 2, 4, 7, 3, 0], [4, -8, -9, 1, 2, 11, 4, -1, -3, 10, 9, 0], [12, 2, 0, 0, 14, -4, -2, 6, 0, -4, -2, 2], [0, 0, 0, 4, 0, -4, -8, 4, 0, 0, -8, 0], [12, 1, 13, 16, -13, 4, 10, -9, -1, 14, 20, -7], [10, 0, -4, 2, -3, -11, 21, -3, -1, 17, 3, 1], [0, -2, -4, 4, -2, 4, -2, 10, 0, 0, 2, 2], [-14, 14, -12, -4, -12, 20, 0, 0, -10, 4, 10, -8], [-10, 3, -4, -2, -8, 0, -5, -8, -3, -1, -2, 3], [-16, 12, 1, 1, 5, -4, -4, 0, -8, -4, -3, 4], [0, 14, 19, 2, 11, -4, -8, -7, 14, -5, -6, -14], [7, -8, -10, -3, 10, -10, -1, 1, -2, -9, -11, 20], [7, 6, -4, -4, -21, 3, 3, -7, -9, 3, -1, -3], [1, -3, -19, 9, -8, 5, -2, -13, -4, -2, 6, 14], [32, 1, -2, -4, -12, 2, 18, -10, -1, -1, -2, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_390_x_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_390_2_x_b();".
// 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_390_2_x_b(:prec:=12)
    chi := MakeCharacter_390_x();
    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_390_2_x_b();".
// 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_390_2_x_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_390_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![1024, 512, 1664, 64, 1648, 112, 708, -20, 205, 6, 19, -2, 1]>],Snew);
    return Vf;
end function;