
// Make newform 770.2.n.i in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_770_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_770_n_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_770_2_n_i();".
// 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_770_2_n_i();".
// 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, -24, 222, -55, 289, 192, 141, -34, 61, -21, 11, -3, 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], [-8921537346326, 79065916975275, 33168479495342, 46794157201284, -15346562559640, 23389434344437, -9123230230654, 5469795070560, -1678425495360, 577035333512, -98998119348, 27218910292], [128492152255425, 13883151413978, -117665859569765, -134890772634922, 21366121084050, -57563605946245, 14365224307255, -10184226264031, 2545013921522, -944387310072, 25030916843, -57839182555], [-8905487495815, 131281096073141, 7050671601780, 304517437522731, 168424612102574, 104264275203845, -11784244862477, 41264955531779, -13080407782797, 7508583879638, -1988591540098, 756356413694], [-6099055983892, 146761805411417, -127203131028240, 201352647451952, 138886238703404, 99400462679386, -24692296093752, 44692827542572, -15727724577708, 8258213919829, -2315285952880, 758466237704], [-384461798009, -1226787297395784, 134095431662108, -1901513418048192, -1270419211586650, -835274597635020, 162675075909756, -356314690439704, 119821961741903, -64774329869932, 17538701713972, -6099055983892], [-135257465459680, 1256468172102788, -392701453761660, 1722380650183657, 1192625942025354, 840198641188894, -199015931288349, 368695713608312, -127090557148760, 66840471646303, -18315988768907, 6115105834403], [82450603826419, -1984068895672710, 185284374747094, -2742110498393109, -1814117874400325, -1243889492047438, 261521619325931, -530154329461918, 179506653532477, -95971770913867, 25960106073751, -8905487495815], [127689837420424, -3072754072474749, 181061306906547, -4267710431577244, -2742920657581274, -1851245080789213, 360503848319742, -777570728389875, 260730882276699, -140869867980898, 37908100871573, -13210691808625], [210612483311877, -3145716928102473, 819051642603431, -3862593773139443, -2512038381582250, -1896145924731036, 471106367849297, -819088036978075, 282641481035258, -147445262229447, 40254045460811, -13325947510218], [-335824354765506, 8081111410953742, -1642213775295809, 11143248277518324, 7466421883844421, 5168262871535759, -1134634026352864, 2230223922339278, -760410883625436, 404321849282105, -110083091955905, 37373296056781]];
    Rf_basisdens := [1, 1, 83041551747559, 83041551747559, 83041551747559, 83041551747559, 83041551747559, 83041551747559, 83041551747559, 83041551747559, 83041551747559, 83041551747559];
    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_770_n();"
function MakeCharacter_770_n()
    N := 770;
    order := 5;
    char_gens := [617, 661, 211];
    v := [5, 5, 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_770_n_Hecke();"
function MakeCharacter_770_n_Hecke(Kf)
    N := 770;
    order := 5;
    char_gens := [617, 661, 211];
    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, 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 := [[0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, -1, 0, 0, 0, 1, 1, -1, 0, -1, -1], [-1, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 1, 0, -2, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 1, 0, -1, 0, -1, 0, 0], [-1, 0, -1, 0, 2, -1, -1, -1, 1, 0, 0, 0], [0, -1, -1, 0, 0, 0, 2, 1, 0, 0, -1, -1], [-2, 2, -2, -2, 2, -2, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 1, 0, 3, 1, -2, 1, 1, 1], [0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, -1], [0, 3, -3, -2, 0, -3, 5, 2, -5, 0, -2, 0], [2, 2, -1, -1, 1, 0, 4, 0, -2, 1, 0, 0], [4, 3, -3, -3, 2, -2, 5, 5, 0, 1, -1, 0], [-4, 0, 0, -2, 2, 0, 2, -2, 4, 2, 2, 2], [-2, 1, 0, 0, 1, 1, 0, -2, 3, 0, 0, 2], [1, 0, -3, 0, -1, -1, -1, 0, 0, -1, -1, -1], [0, -2, 1, 2, -2, 1, -2, -3, 3, 0, -3, -2], [-1, -2, 2, -2, -4, 4, -2, -2, 0, 1, -1, 0], [0, -1, -1, 1, -4, -1, -1, 1, -1, 0, 0, -1], [0, 1, 4, 2, 0, -1, -1, 1, 1, 0, 2, 0], [-3, 1, 0, 0, 1, -1, 0, -3, 4, 1, 0, 5], [4, -1, 0, 1, 3, 0, 3, -2, 2, 0, 0, -1], [-4, 5, -5, -4, 4, -4, -4, -4, 0, 2, -2, 0], [4, -3, 0, 0, -3, 1, 0, 4, -2, 1, 0, -3], [-3, -1, 0, 0, -1, -1, 0, -3, -6, 1, 0, 1], [2, -1, -2, -2, -2, -1, -2, 3, 0, -2, -4, -2], [2, -2, -2, 1, -1, 0, 10, 3, -2, -1, -3, -3], [-1, 2, -2, 1, 2, -2, 2, 2, 0, 2, -2, 0], [5, -3, -1, 5, -1, 0, -3, 2, -5, -5, -2, -2], [0, -3, -3, 3, 3, -3, -3, -6, 6, 0, 0, -3], [-2, 4, -4, -5, 3, -3, 1, 1, 0, 0, 0, 0], [-9, 4, 1, -4, 4, 1, -5, -1, 1, 0, -1, 4], [1, -4, 3, 4, -1, 3, 2, -1, -3, -1, 3, -1], [6, 0, 4, 0, -4, 4, 6, 6, -6, 0, -2, 0], [4, 0, -1, 0, -2, 0, -2, 3, -4, 0, -3, -3], [-2, -3, 7, 1, -4, 0, -10, -3, 2, -1, 3, 3], [1, 4, 0, 0, 4, -5, 0, 1, 4, 0, 0, 3], [-7, 0, 0, 0, 0, -3, 0, -7, -1, 0, 0, -2], [-14, 7, -1, -1, 5, 0, -7, -4, 14, 1, 4, 4], [2, 0, 0, 3, -6, 0, 2, 6, -2, -3, -6, -6], [6, 1, -6, -1, 0, -6, 7, 12, -12, 0, -1, 1], [-1, -6, 9, 3, 1, 7, -11, -4, 12, 1, 4, 1], [0, -6, 7, 6, -3, 7, -6, -8, 8, 0, -5, -6], [4, 10, -10, -2, 6, -6, 6, 6, 0, 2, -2, 0], [11, 4, -4, -5, 2, -2, 2, 2, 0, 2, -2, 0], [3, -5, 0, 0, -5, 0, 0, 3, 1, -1, 0, -3], [6, -10, 7, 4, -11, 0, -3, 4, -6, -4, -4, -4], [-3, 1, 4, -5, 3, 2, 9, -5, -6, 3, -2, 3], [-5, -5, 0, 0, -5, 0, 0, -5, 3, 1, 0, -2], [2, -1, 0, 0, -1, 6, 0, 2, -13, -1, 0, -2], [-5, 6, -1, -3, 5, 0, -5, -4, 5, 3, 4, 4], [0, -8, 8, 4, -3, 3, 2, 2, 0, -1, 1, 0], [-1, 3, 0, 0, 3, -9, 0, -1, -9, -4, 0, 1], [-2, 1, -5, -7, 2, 1, -1, 1, 3, 2, -5, 2], [7, 1, -1, 1, 3, -3, 13, 13, 0, 4, -4, 0], [16, -5, 0, 5, -4, 0, 11, 2, -2, 0, -1, -5], [5, -7, -2, 4, -5, 2, 1, 9, -6, -5, -1, -5], [-7, -2, 0, 0, -2, 1, 0, -7, -3, 4, 0, 0], [-6, -1, -1, 1, -1, -1, -7, -7, 7, 0, 1, -1], [12, -14, 4, 6, -10, 0, -2, 6, -12, -6, -6, -6], [0, 3, -7, -1, 0, -3, -2, 7, 2, 0, -1, 0], [7, 8, -8, -6, 11, -11, 6, 6, 0, 7, -7, 0], [-4, -4, 2, 0, 4, 0, -6, -6, 4, 0, 6, 6], [-4, -3, 1, 3, 4, 1, -7, -10, 10, 0, 2, -3], [-13, 9, 0, 0, 9, 4, 0, -13, 14, 4, 0, -2], [-8, -5, 5, -2, -6, 6, -18, -18, 0, 0, 0, 0], [10, -23, 11, 13, -10, 13, 1, -2, -11, -10, 3, -10], [-9, -6, 7, 6, 2, 7, -15, -13, 13, 0, -2, -6], [-10, -4, 10, -2, -6, 0, -12, -4, 10, 2, 4, 4], [5, 6, -6, 2, 2, -2, 8, 8, 0, 1, -1, 0], [-4, 0, 2, -2, 4, 4, 0, 4, 4, 4, 2, 4], [8, -19, 11, 10, -8, 11, 1, 4, -9, -8, 2, -8], [8, -9, 9, 2, -9, 9, -5, -5, 0, -2, 2, 0], [-16, 2, 10, -2, -3, 10, -14, -26, 26, 0, 5, 2], [1, -1, 0, 0, -1, 4, 0, 1, -12, -6, 0, -4], [-4, 12, -4, -9, 4, -8, -1, -3, 5, 4, -5, 4], [3, -7, 7, -1, -8, 8, -2, -2, 0, -1, 1, 0], [7, -2, -3, 2, 1, -3, 5, 14, -14, 0, -7, -2], [-5, 4, 0, 0, 4, -2, 0, -5, 13, 5, 0, -3], [13, -11, 11, -1, -5, 5, 3, 3, 0, 3, -3, 0], [10, -2, 2, 4, -8, 0, -12, 6, -10, -4, -6, -6], [8, -4, 0, 0, -4, 4, 0, 8, -18, -8, 0, -2], [-6, 5, -1, -6, 6, 1, 4, -11, 2, 6, 0, 6], [-5, 12, -12, -11, 8, -8, 6, 6, 0, 6, -6, 0], [-4, -2, 4, -1, -3, 0, -20, -1, 4, 1, 1, 1], [3, -6, 0, 0, -6, 6, 0, 3, 1, 0, 0, 6], [4, -1, -7, 1, 8, -7, 3, -4, 4, 0, -1, -1], [7, 9, -9, 2, 5, -5, 16, 16, 0, 3, -3, 0], [11, -9, 9, 1, -8, 8, 12, 12, 0, 1, -1, 0], [-1, 5, 2, -5, 8, 2, 4, -11, 11, 0, 10, 5], [-28, 7, -6, -7, 1, -6, -21, -7, 7, 0, -3, 7], [4, -1, 7, 1, -2, 0, -8, -5, -4, -1, 5, 5], [0, -7, 2, 5, 0, 7, -4, -2, 4, 0, 5, 0], [-8, 5, 5, -6, 8, 3, 8, 6, 0, 8, 2, 8], [-4, -8, 6, 2, -6, 0, -4, 0, 4, -2, 0, 0], [-13, 0, -1, -1, 1, 0, 3, 0, 13, 1, 0, 0], [7, 0, -2, 4, -7, -7, -2, -3, -5, -7, -3, -7], [-7, 2, -9, -2, -5, -9, -5, 4, -4, 0, -8, 2], [-8, -3, 0, 0, -3, 10, 0, -8, 19, -1, 0, -1], [-8, 1, 7, -2, 3, 0, -11, -10, 8, 2, 10, 10], [0, -1, 2, 0, 0, 1, 12, 21, -12, 0, 0, 0], [10, 3, 0, 0, 3, -14, 0, 10, -17, 5, 0, 8], [-10, -6, 11, 5, -8, 0, -6, -3, 10, -5, 3, 3], [6, 1, -1, 6, -2, 2, 6, 6, 0, -7, 7, 0], [1, 2, -9, -2, 6, -9, 3, -1, 1, 0, -8, 2], [0, -3, 2, -4, 0, 3, -11, -4, 11, 0, -4, 0], [9, -2, 2, 3, -7, 7, -8, -8, 0, -1, 1, 0], [10, 1, -2, -1, -10, -2, 11, 4, -4, 0, -1, 1], [-5, 17, -8, -6, 5, -12, 8, 13, -3, 5, -1, 5], [5, 7, 0, 0, 7, -8, 0, 5, 0, 8, 0, 6], [4, 12, -3, -8, 11, 0, 11, -8, -4, 8, 8, 8], [3, 9, -9, 2, 7, -7, 9, 9, 0, -1, 1, 0], [-6, 11, -7, -10, 10, 0, 12, -3, 6, 10, 3, 3], [6, -2, 8, 8, -6, -4, 4, 18, -10, -6, 2, -6], [-16, 18, -9, -3, 8, 0, -5, 1, 16, 3, -1, -1], [-5, -4, 4, 4, 9, 4, -9, -11, 11, 0, 6, -4], [0, -3, 0, 0, -3, 13, 0, 0, -13, 4, 0, -8], [6, -10, -1, 2, -6, 4, -9, -15, 3, -6, -4, -6], [-4, 13, -13, -7, 0, 0, 16, 16, 0, 6, -6, 0], [-12, 11, -11, -11, 4, -4, 7, 7, 0, 2, -2, 0], [3, 1, 0, 0, 1, -3, 0, 3, 4, 6, 0, -6], [-7, 12, -8, -12, 13, -8, 5, -2, 2, 0, -1, 12], [2, -10, 10, 12, -2, 2, -12, -12, 0, -10, 10, 0], [2, -2, 0, 0, -2, -2, 0, 2, 10, -5, 0, 1], [5, 4, 2, -1, -6, 0, -10, 4, -5, 1, -4, -4], [3, 0, -2, 0, 4, -2, 3, 15, -15, 0, 5, 0], [4, 0, -6, 5, -4, -4, -19, 3, 15, -4, 1, -4], [-14, -12, 12, 14, 2, -2, -22, -22, 0, -10, 10, 0], [-2, -8, 14, 6, 2, 10, -12, 6, 14, 2, 8, 2], [-5, -2, 0, 0, -2, -11, 0, -5, 3, -9, 0, -1], [-12, 0, -6, 0, -4, -6, -12, 6, -6, 0, -2, 0], [15, 2, 1, -1, -9, 0, -5, 8, -15, 1, -8, -8], [-3, 5, 0, 0, 5, -10, 0, -3, 6, -2, 0, -2], [-9, 3, 0, 0, 3, 0, 0, -9, -3, 4, 0, -5], [12, -3, 3, 0, -6, 6, -12, -12, 0, 3, -3, 0], [-2, -13, 7, 7, -11, 0, -17, 4, 2, -7, -4, -4], [15, 6, -5, -6, 1, -5, 21, 10, -10, 0, -6, 6], [17, -11, 19, 11, -16, 19, 6, 4, -4, 0, 3, -11], [16, -10, -3, 10, -9, -3, 6, 7, -7, 0, -3, -10], [-15, 20, -16, -4, 20, 0, 3, -4, 15, 4, 4, 4], [-3, 7, 6, -5, 3, -4, -2, 6, 5, 3, -2, 3], [11, -16, 0, 0, -16, -3, 0, 11, -14, -7, 0, -12], [-9, 2, 0, 0, 2, -1, 0, -9, 6, -6, 0, 7], [-4, 5, 0, -5, 4, -1, 23, 16, -19, 4, -1, 4], [8, 3, -2, -1, 1, 0, -9, 1, -8, 1, -1, -1], [-23, -3, 15, 3, -6, 15, -26, -24, 24, 0, 7, -3], [-20, -6, 6, -3, -4, 4, 1, 1, 0, 4, -4, 0], [-24, 16, -16, -1, 15, -15, -7, -7, 0, -2, 2, 0], [25, 4, -8, -4, 5, -8, 29, 26, -26, 0, -3, 4], [-7, 9, -1, -2, -1, 0, -5, 2, 7, 2, -2, -2], [-8, 3, -3, 0, 2, -2, 2, 2, 0, -2, 2, 0], [2, 2, -10, -5, 9, 0, 14, 1, -2, 5, -1, -1], [12, -18, 15, 18, -12, 6, 9, 13, -21, -12, 6, -12], [-2, -3, 11, 3, -16, 11, -5, 0, 0, 0, 2, -3], [-9, 5, 0, 0, 5, -10, 0, -9, 11, -3, 0, 6], [6, 9, -18, -9, 4, -18, 15, 20, -20, 0, 3, 9], [13, -1, 1, 1, 2, 1, 12, 15, -15, 0, -13, -1], [-19, 1, 0, 0, 1, 6, 0, -19, 10, 5, 0, -3], [-2, 3, 3, -3, -2, 3, 1, -3, 3, 0, -4, 3], [-33, 8, -8, -12, 4, -4, -4, -4, 0, 5, -5, 0], [8, -14, 6, 9, -8, 6, -18, -9, 10, -8, 1, -8], [11, 17, -17, -7, 6, -6, 12, 12, 0, 9, -9, 0], [5, -6, -1, -5, -2, 0, 4, 3, -5, 5, -3, -3], [1, 5, 0, 0, 5, -1, 0, 1, 21, 2, 0, 3], [-2, -4, -9, 6, 2, 6, -13, -15, 15, 2, 8, 2], [10, -11, 11, 13, 2, -2, -7, -7, 0, -16, 16, 0], [1, 2, 6, 2, -1, -3, -25, -10, 24, -1, 1, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_770_n_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_770_2_n_i();".
// 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_770_2_n_i(:prec:=12)
    chi := MakeCharacter_770_n();
    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_770_2_n_i();".
// 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_770_2_n_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_770_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![841, -3625, 6897, -4649, 3749, -246, 459, -118, 87, -17, 14, -2, 1]>],Snew);
    return Vf;
end function;