
// Make newform 605.2.b.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_605_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_605_b_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_605_2_b_h();".
// 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_605_2_b_h();".
// 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 := [1089, 4398, 6697, 3848, -67, -668, 516, -20, -123, 34, 1, -4, 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], [2640803187617808, 3214832172020702, 830338876364632, -1024544568879806, 451532147574218, 100455369195159, -144716933147575, 29463765100965, 4902239486888, -4797665688052, 963958706563, 4845953018], [156378153801710, 150711385091199, -2633367965615, -34901727549676, 20698222998830, -861459944397, -4422506999162, 1451122751103, -71811869159, -121107594570, 42281880539, -3079103815], [-625607220, -731111689, -125236496, 480592660, -237821503, -15490998, 62985308, -21496194, 29294, 2180609, -682376, 66674], [-48489496588954, 20581144429809, -21972965776276, 56194969010624, -17932503852766, -2252818110113, 6509800106364, -2752803664949, 116703163235, 240501508768, -108022004876, 17157797861], [993528750572394, 72155206793867, 178655400257203, 953521102895434, -469211901841258, 60092547218757, 88901551533944, -56284018568505, 7607666777144, 2869558436669, -1840864299167, 344078139086], [227364595507030, 283470208403922, 40017270005661, -197964026974916, 90363515007482, 5452626654420, -24844432486591, 9183041720844, -228508949095, -856308492056, 312787367384, -38582738806], [1354116402969555, 2783110159517131, 1009945784211140, 306364476596648, -293221491288314, 134603366679414, 30570924246736, -53716346622306, 11168136370855, 1528746626863, -1768122412744, 403353306232], [364041010452140, 1031755067491947, 862720765054912, 101050327463435, -222965796707818, 116252132958211, 13051921442207, -32910179910835, 7565116724057, 791158199715, -1055801231693, 231664336675], [-3609546395800872, -9870775459126630, -7809222367564952, -985435816509719, 2033937285480269, -1038270617338185, -126582594542236, 301382108617977, -68392001388256, -7553443773796, 9706814485894, -2127198780859], [-386058541069797, -1124029667366819, -980356433060835, -117080896543899, 249275773018073, -127645956272930, -14668544866693, 36235143420730, -8239329204561, -884228369606, 1158943341170, -255788416792], [-724073294811309, -2009074080357948, -1621686956953874, -186144655855487, 426935913231974, -223082461775778, -25443799656918, 63423307459104, -14631348692510, -1565285873757, 2049366765782, -446829251302]];
    Rf_basisdens := [1, 308731269222837, 34303474358093, 167498433, 34303474358093, 308731269222837, 34303474358093, 308731269222837, 34303474358093, 308731269222837, 34303474358093, 34303474358093];
    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_605_b();"
function MakeCharacter_605_b()
    N := 605;
    order := 2;
    char_gens := [122, 486];
    v := [1, 2];
    // 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_605_b_Hecke();"
function MakeCharacter_605_b_Hecke(Kf)
    N := 605;
    order := 2;
    char_gens := [122, 486];
    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]];
    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 := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, -1, 0, 0, 1], [0, -2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [3, 0, -2, 0, 2, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0, 2, 0, 0, 0], [0, 1, 0, 3, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1], [-1, 0, -2, 0, 2, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -2, 0, 0, 0, 1, 0, 0, 2], [11, 0, -2, 0, 2, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, -4, 0, 0], [0, -4, 0, -4, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, -1, 1, 0], [-9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -2, 0, -2, 0, 0, 0, 1, 0, 0, -3], [0, 4, 0, 3, 0, 2, 0, -3, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, 3, 0, 2, 3, 0], [0, -3, 0, -3, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, -3, 0, -3, 0, 0, 0, -2, 0, 0, 2], [0, 0, 0, 0, 0, 0, 0, 1, 0, -2, -5, 0], [0, 1, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0], [0, 0, -3, 0, -3, 0, 0, 0, 5, 0, 0, 3], [0, -3, 0, 2, 0, 1, 0, 1, 0, 0, 0, 0], [0, 1, 0, 3, 0, -3, 0, 1, 0, 0, 0, 0], [0, -3, 0, 4, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4], [0, 0, -5, 0, -5, 0, 0, 0, 4, 0, 0, 5], [0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 4, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0], [-6, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0], [9, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 2, 2, 0], [0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 4, 0], [0, 0, -2, 0, 2, 0, 2, 0, 0, 0, 0, 0], [0, 1, 0, 2, 0, 1, 0, -1, 0, 0, 0, 0], [0, 0, 5, 0, 5, 0, 0, 0, -4, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, -1, 0, 3, 2, 0], [-3, 0, -2, 0, 2, 0, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -9, 0, 1, 3, 0], [0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0], [0, -3, 0, -1, 0, -1, 0, 2, 0, 0, 0, 0], [13, 0, -4, 0, 4, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 2, 0, 3, -1, 0], [-2, 0, -4, 0, 4, 0, 1, 0, 0, 0, 0, 0], [0, -2, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -2, -6, 0], [0, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -3, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 5, -1, 0], [0, 0, 0, 0, 0, 0, 0, -8, 0, -5, -3, 0], [17, 0, -2, 0, 2, 0, 5, 0, 0, 0, 0, 0], [0, 0, 2, 0, 2, 0, 0, 0, -7, 0, 0, 3], [0, 0, 2, 0, 2, 0, 0, 0, 6, 0, 0, -2], [-3, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -5, 7, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, -9, 0, 0], [0, 8, 0, -4, 0, -2, 0, -3, 0, 0, 0, 0], [0, 0, -2, 0, -2, 0, 0, 0, 3, 0, 0, 5], [0, 11, 0, 6, 0, -1, 0, -5, 0, 0, 0, 0], [0, 0, 4, 0, 4, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 7, 3, 0], [24, 0, -4, 0, 4, 0, 4, 0, 0, 0, 0, 0], [0, 0, 7, 0, 7, 0, 0, 0, -3, 0, 0, -3], [-8, 0, 2, 0, -2, 0, -5, 0, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0, 8, 0, 0, 4], [9, 0, -10, 0, 10, 0, 6, 0, 0, 0, 0, 0], [0, 5, 0, 1, 0, 1, 0, -3, 0, 0, 0, 0], [-15, 0, 2, 0, -2, 0, 3, 0, 0, 0, 0, 0], [-24, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, -4, 0, 0, 0, -4, 0, 4, 0, 0, 0, 0], [0, 0, 2, 0, 2, 0, 0, 0, 5, 0, 0, 1], [0, -2, 0, -14, 0, -2, 0, 2, 0, 0, 0, 0], [0, 0, 3, 0, 3, 0, 0, 0, -4, 0, 0, 3], [0, 0, -6, 0, 6, 0, 3, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, 7, 3, 0], [0, 2, 0, 3, 0, 2, 0, -2, 0, 0, 0, 0], [0, 0, -1, 0, -1, 0, 0, 0, 2, 0, 0, -5], [0, 0, 2, 0, 2, 0, 0, 0, -9, 0, 0, 0], [0, -12, 0, -12, 0, -4, 0, 8, 0, 0, 0, 0], [0, 0, 3, 0, 3, 0, 0, 0, -3, 0, 0, -1], [0, 2, 0, 12, 0, -2, 0, 0, 0, 0, 0, 0], [-4, 0, 2, 0, -2, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, 6, 5, 0], [-7, 0, 7, 0, -7, 0, -4, 0, 0, 0, 0, 0], [-3, 0, 1, 0, -1, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, -7, -3, 0], [0, -6, 0, -9, 0, 0, 0, 3, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -2, 0, 7, -3, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 10, 4, 0], [0, 0, 0, 0, 0, 0, 0, 5, 0, -7, 4, 0], [0, 13, 0, 9, 0, 3, 0, -8, 0, 0, 0, 0], [0, 8, 0, 16, 0, 4, 0, -6, 0, 0, 0, 0], [0, 0, -6, 0, -6, 0, 0, 0, 15, 0, 0, 5], [0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, -3], [0, 0, 0, 0, 0, 0, 0, -5, 0, -17, -1, 0], [-21, 0, 4, 0, -4, 0, 3, 0, 0, 0, 0, 0], [0, -12, 0, -12, 0, -2, 0, 7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 2, 0, -7, 5, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, -10, 2, 0], [0, 0, -6, 0, -6, 0, 0, 0, 7, 0, 0, 1], [6, 0, 6, 0, -6, 0, 2, 0, 0, 0, 0, 0], [12, 0, 10, 0, -10, 0, -4, 0, 0, 0, 0, 0], [-18, 0, -5, 0, 5, 0, 6, 0, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, -5], [0, 0, 1, 0, 1, 0, 0, 0, 6, 0, 0, 1], [0, 0, 4, 0, 4, 0, 0, 0, 2, 0, 0, -4], [0, 4, 0, -12, 0, 0, 0, -2, 0, 0, 0, 0], [12, 0, -2, 0, 2, 0, 11, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -4, 0, 11, -3, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, -2, 8, 0], [0, 0, 1, 0, 1, 0, 0, 0, -6, 0, 0, -1], [1, 0, -8, 0, 8, 0, -1, 0, 0, 0, 0, 0], [0, -4, 0, -12, 0, -2, 0, 3, 0, 0, 0, 0], [-3, 0, 7, 0, -7, 0, -8, 0, 0, 0, 0, 0], [3, 0, -6, 0, 6, 0, 9, 0, 0, 0, 0, 0], [0, 0, 5, 0, 5, 0, 0, 0, -10, 0, 0, 3], [0, 0, 0, 0, 0, 0, 0, 1, 0, 10, 6, 0], [0, -4, 0, 4, 0, -8, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, -12, 3, 0], [14, 0, 2, 0, -2, 0, 4, 0, 0, 0, 0, 0], [0, 0, 8, 0, 8, 0, 0, 0, -7, 0, 0, -1], [0, -4, 0, -6, 0, 2, 0, 1, 0, 0, 0, 0], [0, 6, 0, 8, 0, -4, 0, -1, 0, 0, 0, 0], [0, 0, -5, 0, -5, 0, 0, 0, -1, 0, 0, -7], [0, 0, 0, 0, 0, 0, 0, 5, 0, 8, -7, 0], [0, 0, 9, 0, 9, 0, 0, 0, -11, 0, 0, -13], [0, -4, 0, -18, 0, 0, 0, 2, 0, 0, 0, 0], [0, -5, 0, -14, 0, 3, 0, 1, 0, 0, 0, 0], [0, 3, 0, -9, 0, 5, 0, -4, 0, 0, 0, 0], [0, 0, 6, 0, 6, 0, 0, 0, -3, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -15, 0, 6, 1, 0], [-18, 0, 8, 0, -8, 0, -1, 0, 0, 0, 0, 0], [-8, 0, -4, 0, 4, 0, -2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 11, 0, -4, -6, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [27, 0, -12, 0, 12, 0, -7, 0, 0, 0, 0, 0], [0, 0, 6, 0, 6, 0, 0, 0, 5, 0, 0, -2], [0, 0, 0, 0, 0, 0, 0, 5, 0, -1, -7, 0], [2, 0, 4, 0, -4, 0, -7, 0, 0, 0, 0, 0], [0, 0, -5, 0, -5, 0, 0, 0, 19, 0, 0, 1], [0, 0, 0, 0, 0, 0, 0, -9, 0, -21, -4, 0], [0, 0, 1, 0, 1, 0, 0, 0, -14, 0, 0, 1], [6, 0, 8, 0, -8, 0, -6, 0, 0, 0, 0, 0], [0, 17, 0, 6, 0, 1, 0, -9, 0, 0, 0, 0], [-2, 0, 5, 0, -5, 0, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5, 0, -8, -8, 0], [0, -4, 0, 15, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, -3, 0, -3, 0, 0, 0, -7, 0, 0, -7], [0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 4, 0], [0, 0, 0, 0, 0, 0, 0, 8, 0, 11, 17, 0], [0, 0, 8, 0, -8, 0, -6, 0, 0, 0, 0, 0], [0, 0, 3, 0, 3, 0, 0, 0, -4, 0, 0, 2], [0, 0, -4, 0, -4, 0, 0, 0, 1, 0, 0, 2], [-9, 0, 6, 0, -6, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -7, 0, -1, 5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_605_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_605_2_b_h();".
// 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_605_2_b_h(:prec:=12)
    chi := MakeCharacter_605_b();
    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_605_2_b_h();".
// 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_605_2_b_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_605_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 0, 29, 0, 11, 0, 1]>,<19,R![-3888, 0, 1332, 0, -72, 0, 1]>],Snew);
    return Vf;
end function;