
// Make newform 770.2.n.e 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_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_770_2_n_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 := [121, 88, 100, 28, 19, 2, 5, -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, 1, 0, 0, 0, 0, 0, 0], [440220, 319181, -332708, 71401, 33439, -15725, 2098, 528], [528473, 501961, 104068, -61888, 195125, -30517, -9973, 5794], [-701074, 18601, -77439, -58164, -171974, 183537, -59487, 7409], [2809884, 2179908, 380907, 521745, 200474, -106264, 16927, 8817], [-2305776, -1085964, -979996, -102852, -228404, 58156, 3536, -11971], [665808, -742824, 6060, -266949, 65418, -113608, 57068, -13790]];
    Rf_basisdens := [1, 1, 1168519, 1168519, 1168519, 1168519, 1168519, 1168519];
    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], [1, 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, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -1, 0, 0, 0, 0], [2, 1, 0, 1, 1, -1, -1, -2], [2, 0, -1, 2, 0, 0, -1, 0], [0, 1, -2, 2, 1, 0, 0, -1], [0, 1, 0, 1, 2, 0, -2, -1], [-2, 0, 0, 0, 0, 0, 0, 0], [-1, 1, -1, -4, 1, 1, -1, 1], [6, 0, -4, 6, 0, 0, 0, 0], [2, -2, -2, 0, 2, 2, 2, -2], [1, -2, -1, 1, -3, 0, 3, -1], [1, -3, -5, 0, 1, 3, 0, -2], [1, -6, -1, -2, -1, 0, 1, 2], [4, 0, -4, 4, 0, -2, -2, 0], [7, 0, 1, 7, -1, -1, 0, -7], [0, -2, 2, -4, 0, 2, 2, 8], [-4, 2, 1, 0, -1, -2, 0, -1], [0, 1, 5, -6, 2, 1, 1, 5], [-1, 0, -4, 3, 4, 4, 0, 1], [-2, 0, 5, -2, 0, 0, 1, 0], [0, 3, -4, 4, 3, 0, 0, -4], [-5, 1, 7, 0, 1, -1, 0, 6], [11, 0, -8, 11, 0, 0, 0, 0], [-2, 0, 12, -2, 0, 2, -2, 0], [1, 1, 3, 8, -3, -3, -1, -1], [-9, 4, 9, -5, -1, 0, 1, 5], [-2, 0, 8, 0, 4, 0, 0, 8], [-10, -7, 10, -7, -4, 0, 4, 7], [0, -2, -6, 4, 0, 2, 2, -10], [-10, 0, 7, 0, -5, 0, 0, 7], [0, -2, -5, 4, -1, 1, 1, -3], [8, 4, 0, 8, 0, 0, -4, -8], [0, -1, 11, -10, -2, -1, -1, 7], [-7, -8, 7, -2, -5, 0, 5, 2], [-3, -4, 3, -8, -3, 0, 3, 8], [-3, 0, -8, -3, 0, 0, -5, 0], [0, 0, 12, 0, 0, 0, 0, 0], [3, 5, -3, -1, 6, 0, -6, 1], [-7, -3, 7, -3, -3, 0, 3, 3], [0, -6, -6, -2, 2, 8, 8, -4], [-4, 2, 5, -9, -5, -5, -2, 4], [0, 2, -6, 10, -2, -4, -4, -4], [-6, 6, 12, 0, -4, -6, 0, 6], [4, -6, -4, 0, 0, 6, 0, 2], [-8, 0, 7, -8, 0, 5, -4, 0], [4, -6, -4, -2, 0, 0, 0, 2], [-3, 0, 4, -12, -4, -4, 0, 3], [2, 0, 0, 2, 0, -4, -4, 0], [-2, 0, -8, -2, 0, 1, -1, 0], [5, 3, -5, 17, -2, 0, 2, -17], [0, 2, -1, 0, -7, -2, 0, -3], [-12, 0, 6, -12, 0, 0, 6, 0], [-3, 2, -2, -9, 2, 2, -2, 3], [-18, 4, 10, 0, 0, -4, 0, 6], [0, 10, -4, 8, 6, -4, -4, 2], [0, -8, -2, 0, 2, 2, 8, 0], [0, 0, -10, 0, 0, -2, -6, 0], [0, -6, 1, 0, -7, -1, -1, -11], [1, 10, -1, 8, 3, 0, -3, -8], [0, 0, 4, -14, -4, -4, 0, 0], [10, -12, -13, 0, 4, 12, 0, -1], [-22, 2, 22, -22, -2, 0, 2, 22], [0, -5, -6, -3, 4, 9, 9, -9], [14, 0, 8, 14, 0, 0, 2, 0], [-6, -7, 5, 0, -2, 7, 0, 12], [13, 4, -7, 7, 7, 7, -4, -13], [0, 3, 14, -7, -4, -7, -7, 13], [12, -10, -12, 10, -2, 0, 2, -10], [9, -8, -11, 0, 6, 8, 0, -3], [-2, -2, -3, -29, 3, 3, 2, 2], [6, 6, -7, 3, 7, 7, -6, -6], [8, -6, -8, 0, 2, 6, 0, -2], [0, -8, -11, 7, -4, 4, 4, 3], [0, 0, 5, 0, 0, -10, 5, 0], [6, 0, -5, -7, 5, 5, 0, -6], [-3, 2, -6, 0, 7, -2, 0, -8], [0, 12, -13, 21, 4, -8, -8, -5], [-12, 0, 16, -12, 0, 0, 2, 0], [9, 1, 5, 0, 5, -1, 0, 4], [13, 7, -13, 11, 4, 0, -4, -11], [-11, 0, 14, -11, 0, 7, -1, 0], [-21, 0, -2, -22, 2, 2, 0, 21], [-8, 2, 2, 0, 4, -2, 0, 0], [7, -2, -7, 5, -7, 0, 7, -5], [1, 0, 5, 1, 0, -4, 6, 0], [0, 8, 9, -6, 5, -3, -3, 11], [-8, 6, -4, 0, -2, -6, 0, -10], [-14, 2, 0, 0, -10, -2, 0, -2], [0, -7, -5, 0, -2, 5, 5, -15], [0, 16, -4, 6, 14, -2, -2, -4], [-6, 2, 6, -18, 4, 0, -4, 18], [-24, -3, -1, -11, 1, 1, 3, 24], [-13, -2, -2, -9, 2, 2, 2, 13], [-11, 1, 11, -5, 2, 0, -2, 5], [-8, 10, 8, -2, 6, 0, -6, 2], [-1, 0, -3, 15, 3, 3, 0, 1], [0, -3, -12, 18, -9, -6, -6, -17], [0, 0, 7, 0, 0, 2, -9, 0], [-16, 11, 16, 7, 6, 0, -6, -7], [-6, -4, 4, -6, -4, -4, 4, 6], [-3, 0, -6, -3, 0, 0, 4, 0], [21, 11, -21, 7, 7, 0, -7, -7], [-11, 5, 12, 0, -7, -5, 0, 7], [0, 3, 14, -6, -5, -8, -8, 9], [-7, -8, 4, 16, -4, -4, 8, 7], [1, 5, -5, 0, 0, -5, 0, -10], [0, -18, -6, -5, -7, 11, 11, 10], [0, -7, 11, -11, -11, -11, 7, 0], [-1, 0, -2, -1, 0, 5, 3, 0], [-10, -4, 10, 2, -4, 0, 4, -2], [0, 8, 9, 0, -5, -8, 0, 1], [-2, -3, 2, 1, -8, 0, 8, -1], [-5, 1, 3, 14, -3, -3, -1, 5], [16, 11, -16, -2, 9, 0, -9, 2], [0, -15, 12, -12, -15, 0, 0, 4], [5, 0, -6, 5, 0, -1, -17, 0], [-4, -4, -6, -26, 6, 6, 4, 4], [9, 0, -11, 0, 6, 0, 0, -11], [4, 2, 16, 0, -12, -2, 0, 14], [-15, 0, 10, -15, 0, 12, 1, 0], [0, 0, 8, -7, -1, -1, -1, -12], [6, 8, 12, 0, -2, -8, 0, 4], [7, 0, 6, 7, 0, -8, -13, 0], [4, 4, -4, 0, 14, 0, -14, 0], [0, -3, 3, -8, 2, 5, 5, 33], [8, 0, -16, -12, 16, 16, 0, -8], [-11, 2, -2, 0, 15, -2, 0, -4], [12, -12, -1, -7, 1, 1, 12, -12], [1, 0, 14, 1, 0, 6, 3, 0], [0, 10, -10, 16, 4, -6, -6, -4], [21, -14, -21, -6, -9, 0, 9, 6], [-4, 0, -4, -4, 0, 8, -10, 0], [11, 0, 16, 11, 0, 2, -7, 0], [-8, 14, 16, 0, -6, -14, 0, 2], [-5, -1, 5, -25, 0, 0, 0, 25], [0, 6, -1, 3, 4, -2, -2, 2], [0, -15, -9, 10, -16, -1, -1, 15], [0, 0, -11, 11, 0, 0, 0, -8], [9, 4, -9, 15, 9, 0, -9, -15], [15, -5, -17, 6, 17, 17, 5, -15], [-12, 0, 12, -12, 0, -4, 16, 0], [-19, 0, 42, -19, 0, -12, 9, 0], [8, 10, 2, -6, -2, -2, -10, -8], [34, -6, -34, 40, 6, 0, -6, -40], [0, -9, -17, 12, -4, 5, 5, -7], [-23, -7, 25, 0, -7, 7, 0, 32], [-6, 4, 29, 0, -7, -4, 0, 25], [0, -12, -2, -12, 2, 14, 14, 2], [-16, 14, 16, 0, 4, 0, -4, 0], [-9, 15, 11, 0, -3, -15, 0, -4], [-7, -14, 7, -13, -5, 0, 5, 13], [-5, -1, 13, -4, -13, -13, 1, 5], [0, -17, 4, -7, -14, 3, 3, 13], [7, 0, -6, 7, 0, -7, 5, 0], [0, -16, -14, 1, -3, 13, 13, -4], [0, -3, -6, 11, -8, -5, -5, -17], [11, 0, -17, 11, 0, 0, -6, 0], [0, -2, 12, -8, -6, -4, -4, 16], [8, -6, -23, 0, -5, 6, 0, -17], [-10, 1, 5, 5, -5, -5, -1, 10], [18, 6, -14, 0, 8, -6, 0, -20], [16, -2, -16, 34, 6, 0, -6, -34], [2, 0, 6, 2, 0, 14, 6, 0], [1, -1, -13, -18, 13, 13, 1, -1], [11, 4, -20, 0, 9, -4, 0, -24], [-13, 1, 1, 24, -1, -1, -1, 13]];
    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_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_770_2_n_e(:prec:=8)
    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_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_770_2_n_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_770_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![121, -99, 20, 29, 49, 19, 10, 1, 1]>],Snew);
    return Vf;
end function;