
// Make newform 35.3.i.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_35_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_35_i_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_35_3_i_a();".
// 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_35_3_i_a();".
// 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 := [9, 0, -135, 0, 1980, 0, -669, 0, 180, 0, -15, 0, 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], [-7767, 0, -9, 0, 132, 0, -144, 0, 12, 0, -1, 0], [-1611, 0, 23628, 0, -7884, 0, 2148, 0, -179, 0, 12, 0], [0, 1611, 0, -23628, 0, 7884, 0, -2148, 0, 179, 0, -12], [-15069, 0, 111777, 0, -37971, 0, 9767, 0, -811, 0, 53, 0], [-7269, 0, -111858, 0, 39159, 0, -10958, 0, 919, 0, -62, 0], [36, 0, -23313, 0, 7884, 0, -2148, 0, 179, 0, -12, 0], [0, -15939, 0, 108207, 0, -36471, 0, 9627, 0, -801, 0, 53], [0, -58743, 0, -757746, 0, 259653, 0, -71826, 0, 6003, 0, -404], [0, 168489, 0, -1179567, 0, 396681, 0, -105477, 0, 8781, 0, -583], [0, -86229, 0, -1179918, 0, 401829, 0, -110778, 0, 9249, 0, -622]];
    Rf_basisdens := [1, 1, 1575, 1575, 1575, 3150, 3150, 315, 1350, 9450, 9450, 9450];
    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_35_i();"
function MakeCharacter_35_i()
    N := 35;
    order := 6;
    char_gens := [22, 31];
    v := [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_35_i_Hecke();"
function MakeCharacter_35_i_Hecke(Kf)
    N := 35;
    order := 6;
    char_gens := [22, 31];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 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 := 3;
    raw_aps := [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1], [0, 1, 0, 0, -2, 0, -1, 0, 1, -1, 0, 0], [0, -1, 0, 0, 1, 0, 0, 0, -1, 3, 0, 0], [1, 0, 0, 1, 0, -1, 2, 0, 0, 0, 0, 0], [0, 2, 0, 0, -1, 0, 0, 0, -1, 0, -4, 0], [0, 8, 0, 0, -2, 0, 0, 0, 0, 3, 0, 1], [2, 0, -1, 1, 0, 0, -3, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, -3, -3, 3, -3], [0, 0, -1, 0, 0, 3, 3, 0, 0, 0, 0, 0], [-6, 0, 2, 6, 0, 2, 0, -1, 0, 0, 0, 0], [0, -19, 0, 0, 0, 0, 0, 0, -2, -2, -3, 3], [-3, 0, 1, -6, 0, 4, -4, -2, 0, 0, 0, 0], [0, 11, 0, 0, -2, 0, 0, 0, -2, 4, 7, 14], [0, -5, 0, 0, 10, 0, 0, 0, 13, -13, -2, -2], [0, -9, 0, 0, 15, 0, 0, 0, -22, 11, -4, -2], [22, 0, -18, -22, 0, 2, 0, 9, 0, 0, 0, 0], [-12, 0, -4, -6, 0, 0, 11, -4, 0, 0, 0, 0], [0, 1, 0, 0, -14, 0, 0, 0, 14, -7, 12, 6], [16, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 0, 6, 0, 0, 0, 0, 9, 0, -5], [0, 0, 3, 32, 0, -24, 12, -3, 0, 0, 0, 0], [0, -27, 0, 0, 12, 0, 0, 0, -37, 0, 2, 0], [76, 0, 8, 38, 0, 0, 9, 8, 0, 0, 0, 0], [0, 10, 0, 0, 22, 0, 0, 0, -16, 0, -4, 0], [-42, 0, -2, 42, 0, -11, 0, 1, 0, 0, 0, 0], [0, -4, 0, 0, 8, 0, 0, 0, -4, 4, -9, -9], [0, 0, 0, 0, 0, 0, 0, 0, 23, 23, 8, -8], [-76, 0, 0, -76, 0, -1, 2, 20, 0, 0, 0, 0], [0, 50, 0, 0, -82, 0, 0, 0, 15, -30, -17, -34], [0, -40, 0, 0, 57, 0, 0, 0, -25, 50, -8, -16], [18, 0, -4, 9, 0, 0, -11, -4, 0, 0, 0, 0], [0, -28, 0, 0, 14, 0, 0, 0, -20, 10, 36, 18], [60, 0, 0, 120, 0, 4, -4, 0, 0, 0, 0, 0], [0, 0, 12, 50, 0, 30, -15, -12, 0, 0, 0, 0], [52, 0, 0, 52, 0, -26, 52, -3, 0, 0, 0, 0], [0, -7, 0, 0, -1, 0, 0, 0, 0, -44, 0, 35], [0, -20, 0, 0, 0, 0, 0, 0, 7, 7, -13, 13], [0, 1, 0, 0, -23, 0, 0, 0, 26, 0, 2, 0], [0, -39, 0, 0, 78, 0, 0, 0, 8, -8, 27, 27], [-25, 0, 0, -25, 0, -5, 10, 17, 0, 0, 0, 0], [-22, 0, 19, -44, 0, -15, 15, -38, 0, 0, 0, 0], [0, 0, -38, -12, 0, 40, -20, 38, 0, 0, 0, 0], [0, 50, 0, 0, 34, 0, 0, 0, 60, -30, -40, -20], [0, 2, 0, 0, -51, 0, 0, 0, 46, -92, -3, -6], [6, 0, -12, -6, 0, 60, 0, 6, 0, 0, 0, 0], [-91, 0, -21, 0, 0, -25, -25, 0, 0, 0, 0, 0], [0, -6, 0, 0, -96, 0, 0, 0, 53, 0, -37, 0], [0, 10, 0, 0, 16, 0, 0, 0, 0, 1, 0, 41], [-120, 0, -18, -60, 0, 0, 16, -18, 0, 0, 0, 0], [0, -76, 0, 0, 0, 0, 0, 0, -7, -7, 37, -37], [60, 0, 29, 0, 0, -24, -24, 0, 0, 0, 0, 0], [-25, 0, 16, 25, 0, 7, 0, -8, 0, 0, 0, 0], [57, 0, 31, 114, 0, -27, 27, -62, 0, 0, 0, 0], [0, 88, 0, 0, -176, 0, 0, 0, 34, -34, -68, -68], [0, -33, 0, 0, -34, 0, 0, 0, 16, -8, 82, 41], [-76, 0, -60, 76, 0, -33, 0, 30, 0, 0, 0, 0], [336, 0, 9, 168, 0, 0, 12, 9, 0, 0, 0, 0], [0, -102, 0, 0, 110, 0, 0, 0, -162, 81, 42, 21], [143, 0, 12, 0, 0, -5, -5, 0, 0, 0, 0, 0], [0, 98, 0, 0, -106, 0, 0, 0, 0, -65, 0, -49], [0, 76, 0, 0, -69, 0, 0, 0, 70, 0, -75, 0], [0, 97, 0, 0, 38, 0, 0, 0, 87, 0, 28, 0], [-84, 0, -30, 84, 0, 38, 0, 15, 0, 0, 0, 0], [0, 48, 0, 0, -96, 0, 0, 0, 68, -68, 66, 66], [0, 74, 0, 0, 0, 0, 0, 0, -55, -55, -63, 63], [0, 0, 21, -259, 0, -50, 25, -21, 0, 0, 0, 0], [0, 80, 0, 0, -22, 0, 0, 0, -25, 50, 33, 66], [0, 49, 0, 0, -32, 0, 0, 0, 94, -47, -4, -2], [-158, 0, -9, -316, 0, -25, 25, 18, 0, 0, 0, 0], [0, -84, 0, 0, 76, 0, 0, 0, 0, 47, 0, 21], [0, 0, 63, -118, 0, 16, -8, -63, 0, 0, 0, 0], [0, 30, 0, 0, -45, 0, 0, 0, 0, 43, 0, -103], [0, -48, 0, 0, 0, 0, 0, 0, 0, 0, 52, -52], [163, 0, -5, 0, 0, -35, -35, 0, 0, 0, 0, 0], [0, -91, 0, 0, 182, 0, 0, 0, -161, 161, -59, -59], [364, 0, 0, 364, 0, 8, -16, -30, 0, 0, 0, 0], [0, -34, 0, 0, 68, 0, 0, 0, 112, -112, -8, -8], [0, 0, -26, -3, 0, 116, -58, 26, 0, 0, 0, 0], [-68, 0, 42, 68, 0, -55, 0, -21, 0, 0, 0, 0], [-181, 0, 22, -362, 0, -83, 83, -44, 0, 0, 0, 0], [-70, 0, 36, 0, 0, 17, 17, 0, 0, 0, 0, 0], [38, 0, 0, 38, 0, -40, 80, 104, 0, 0, 0, 0], [0, -146, 0, 0, 28, 0, 0, 0, -181, 0, -7, 0], [-84, 0, 48, -42, 0, 0, -100, 48, 0, 0, 0, 0], [0, 264, 0, 0, 0, 0, 0, 0, 34, 34, -69, 69], [-127, 0, 75, 0, 0, 22, 22, 0, 0, 0, 0, 0], [0, -86, 0, 0, 0, 0, 0, 0, -8, -8, -20, 20], [274, 0, 22, 548, 0, 8, -8, -44, 0, 0, 0, 0], [0, 83, 0, 0, -139, 0, 0, 0, -16, 32, -72, -144], [0, 38, 0, 0, -76, 0, 0, 0, 111, -111, 98, 98], [-100, 0, -46, 100, 0, 82, 0, 23, 0, 0, 0, 0], [0, 76, 0, 0, -30, 0, 0, 0, 10, -5, -142, -71], [104, 0, -111, 0, 0, -6, -6, 0, 0, 0, 0, 0], [0, 0, -18, 230, 0, -60, 30, 18, 0, 0, 0, 0], [0, 65, 0, 0, 206, 0, 0, 0, -37, 0, 104, 0], [112, 0, -31, 56, 0, 0, -23, -31, 0, 0, 0, 0], [-55, 0, 176, 55, 0, -117, 0, -88, 0, 0, 0, 0], [0, 176, 0, 0, -352, 0, 0, 0, 33, -33, -133, -133], [0, 0, 35, -356, 0, -82, 41, -35, 0, 0, 0, 0], [0, -17, 0, 0, 100, 0, 0, 0, -71, 142, 12, 24], [0, 19, 0, 0, -57, 0, 0, 0, 84, -42, 46, 23], [0, 25, 0, 0, 94, 0, 0, 0, 0, 79, 0, 134], [0, 0, -120, 131, 0, -14, 7, 120, 0, 0, 0, 0], [388, 0, 0, 388, 0, 86, -172, 69, 0, 0, 0, 0], [0, -118, 0, 0, 226, 0, 0, 0, 0, 240, 0, 94], [0, -98, 0, 0, -48, 0, 0, 0, -137, 0, -87, 0], [0, 46, 0, 0, -92, 0, 0, 0, 181, -181, 121, 121], [-212, 0, 0, -212, 0, -44, 88, -42, 0, 0, 0, 0], [-32, 0, -16, -64, 0, 164, -164, 32, 0, 0, 0, 0], [0, 71, 0, 0, -142, 0, 0, 0, -31, 31, -87, -87], [0, -159, 0, 0, 489, 0, 0, 0, -186, 93, 132, 66], [0, 36, 0, 0, 62, 0, 0, 0, -12, 24, 86, 172], [9, 0, -100, -9, 0, -125, 0, 50, 0, 0, 0, 0], [-452, 0, 22, 0, 0, 30, 30, 0, 0, 0, 0, 0], [-111, 0, 0, -111, 0, -22, 44, -89, 0, 0, 0, 0], [0, 10, 0, 0, -68, 0, 0, 0, 17, 0, -61, 0], [0, -200, 0, 0, -17, 0, 0, 0, 0, -143, 0, -91], [0, -111, 0, 0, 0, 0, 0, 0, 105, 105, 160, -160], [188, 0, -15, 0, 0, -20, -20, 0, 0, 0, 0, 0], [50, 0, 56, -50, 0, -229, 0, -28, 0, 0, 0, 0], [0, -32, 0, 0, -74, 0, 0, 0, 144, -288, 38, 76], [0, 179, 0, 0, -358, 0, 0, 0, 91, -91, -170, -170], [0, -7, 0, 0, -84, 0, 0, 0, -20, 10, -6, -3], [360, 0, -42, 180, 0, 0, 182, -42, 0, 0, 0, 0], [766, 0, -32, 0, 0, 37, 37, 0, 0, 0, 0, 0], [0, 0, 103, -108, 0, -94, 47, -103, 0, 0, 0, 0], [100, 0, -28, 50, 0, 0, -192, -28, 0, 0, 0, 0], [0, -295, 0, 0, 101, 0, 0, 0, -340, 0, 56, 0], [0, 43, 0, 0, -86, 0, 0, 0, -94, 94, 165, 165], [-653, 0, 0, -653, 0, 61, -122, -54, 0, 0, 0, 0], [0, 7, 0, 0, 161, 0, 0, 0, -14, 28, 154, 308], [0, 0, -80, 84, 0, 132, -66, 80, 0, 0, 0, 0], [0, -220, 0, 0, 6, 0, 0, 0, 105, -210, -109, -218], [-470, 0, 98, -235, 0, 0, -89, 98, 0, 0, 0, 0], [19, 0, 72, 38, 0, 19, -19, -144, 0, 0, 0, 0], [0, 121, 0, 0, -57, 0, 0, 0, 0, 123, 0, -116], [0, -85, 0, 0, -16, 0, 0, 0, 0, -185, 0, 68], [0, -66, 0, 0, -186, 0, 0, 0, 200, 0, 80, 0], [-217, 0, 0, -217, 0, -118, 236, -22, 0, 0, 0, 0], [-99, 0, -157, -198, 0, 27, -27, 314, 0, 0, 0, 0], [0, 0, -12, 110, 0, 112, -56, 12, 0, 0, 0, 0], [0, -25, 0, 0, 40, 0, 0, 0, 174, -87, 224, 112], [0, 325, 0, 0, -576, 0, 0, 0, 120, -240, -131, -262], [372, 0, -208, -372, 0, 72, 0, 104, 0, 0, 0, 0], [354, 0, 12, 708, 0, -136, 136, -24, 0, 0, 0, 0], [0, 350, 0, 0, 211, 0, 0, 0, 73, 0, -66, 0], [0, 232, 0, 0, -2, 0, 0, 0, 0, 136, 0, 92], [-88, 0, -26, -44, 0, 0, 238, -26, 0, 0, 0, 0], [0, 105, 0, 0, 0, 0, 0, 0, -40, -40, -16, 16], [0, -237, 0, 0, 0, 0, 0, 0, 91, 91, 62, -62], [383, 0, 89, 766, 0, -30, 30, -178, 0, 0, 0, 0], [0, -176, 0, 0, 234, 0, 0, 0, 37, -74, 95, 190], [0, -238, 0, 0, 476, 0, 0, 0, -85, 85, 288, 288], [0, -347, 0, 0, 548, 0, 0, 0, -640, 320, 54, 27], [554, 0, -126, 0, 0, 40, 40, 0, 0, 0, 0, 0], [0, 0, 219, -226, 0, 28, -14, -219, 0, 0, 0, 0], [526, 0, -37, 263, 0, 0, -160, -37, 0, 0, 0, 0], [0, 360, 0, 0, -140, 0, 0, 0, 126, 0, -374, 0], [-108, 0, 144, 108, 0, 188, 0, -72, 0, 0, 0, 0], [0, -224, 0, 0, 0, 0, 0, 0, -19, -19, 74, -74], [0, -286, 0, 0, 6, 0, 0, 0, 212, -424, -68, -136], [0, 13, 0, 0, 68, 0, 0, 0, -71, 142, 10, 20], [66, 0, 104, 33, 0, 0, 231, 104, 0, 0, 0, 0], [0, -132, 0, 0, 286, 0, 0, 0, 32, -16, 296, 148], [0, -48, 0, 0, -183, 0, 0, 0, 0, -188, 0, -226], [236, 0, 0, 236, 0, 96, -192, -205, 0, 0, 0, 0], [0, 726, 0, 0, -128, 0, 0, 0, 0, 281, 0, 189]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_35_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_35_3_i_a();".
// 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_35_3_i_a(:prec:=12)
    chi := MakeCharacter_35_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_35_3_i_a();".
// 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_35_3_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_35_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([],Snew);
    return Vf;
end function;