
// Make newform 770.2.n.f in Magma, downloaded from the LMFDB on 22 December 2025.
// 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_f();".
// 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_f();".
// 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, 0, 66, 0, 16, 0, 1, 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, 1, 0, 0, 0, 0, 0, 0], [-363, 0, 629, 0, -37, 0, 7, 0], [-539, 0, -525, 0, 148, 0, -28, 0], [0, -539, 0, -525, 0, 148, 0, -28], [2761, 0, 750, 0, 73, 0, 40, 0], [0, -1672, 0, -646, 0, 38, 0, -61], [0, 3300, 0, 1275, 0, -75, 0, 68]];
    Rf_basisdens := [1, 1, 1991, 1991, 1991, 1991, 1991, 1991];
    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 := false)
    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], [-1, 0, -1, -1, 0, 1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0, 0], [0, 0, 0, -1, -1, 0, 0, -1], [0, 0, 1, 0, 0, 0, 0, 0], [1, 0, 1, 1, 0, -1, 0, 0], [0, 1, 0, 0, 2, 0, -1, 0], [0, 2, -1, -1, 1, -3, -1, -1], [0, 0, 3, 0, 0, 0, -1, -1], [1, 0, 0, 0, -2, -1, 0, -2], [0, -2, 0, 0, 0, 0, 0, 2], [0, 0, 0, -5, -1, 5, 1, 0], [0, 4, 3, 3, 3, 3, -1, -1], [-2, 0, -2, -7, -1, 7, 1, 0], [0, 1, 0, 6, -1, 0, -1, -1], [3, 3, 1, 0, -2, -1, 0, -3], [-4, -2, 0, -4, 2, 4, 2, 2], [0, 1, 4, 4, -1, -7, -2, -2], [1, -2, 1, 6, -2, -6, 2, 0], [3, 0, 5, 3, 0, 0, -1, -2], [1, -1, 2, 0, 1, -2, 0, 1], [3, 0, -2, 3, 0, 0, -1, -1], [-11, -5, -11, -4, -3, 4, 3, 0], [0, -1, 6, 6, -1, -5, 0, 0], [-6, 0, -1, -6, 0, 0, 3, 0], [-4, -3, -6, 0, -1, 6, 0, 3], [0, -5, 1, 1, -3, 2, 2, 2], [0, -1, 2, 2, -1, 1, 0, 0], [7, -3, 7, -1, -3, 1, 3, 0], [11, -1, 0, 6, 2, -11, 1, 2], [-5, -1, 0, 0, 4, 0, 0, 1], [-3, -5, 0, -1, 4, 3, 5, 4], [4, 0, 5, 4, 0, 0, 6, -3], [7, -2, 9, 0, 2, -9, 0, 2], [2, 0, 7, 2, 0, 0, -5, 0], [0, 4, 0, 8, 2, -8, -2, 0], [4, 0, 6, 4, 0, 0, -2, 2], [7, -2, 0, 0, -1, -7, 2, -1], [-3, -4, 0, 0, -1, 3, 4, -1], [0, -4, 10, 10, -1, -3, 3, 3], [0, 5, 3, 3, 8, -4, 3, 3], [-2, 1, 0, -1, -4, 2, -1, -4], [-6, -6, 0, -10, 7, 6, 6, 7], [2, 0, 3, 2, 0, 0, -4, -7], [-13, 9, -13, -10, 4, 10, -4, 0], [-4, 0, -2, -4, 0, 0, -4, 6], [12, 2, 4, 0, 0, -4, 0, -2], [11, 3, 2, 0, 2, -2, 0, -3], [0, 6, -12, -12, 1, 6, -5, -5], [0, 3, 0, 1, -6, 0, -3, -6], [4, -1, 4, -1, -1, 1, 1, 0], [0, -9, -3, -3, -10, -2, -1, -1], [0, -3, 0, 0, 2, -1, 5, 5], [7, -5, 0, 9, 7, -7, 5, 7], [-6, 6, -8, 0, 1, 8, 0, -6], [0, -8, 6, 6, -4, -4, 4, 4], [7, 7, 7, 8, 1, -8, -1, 0], [22, 1, 3, 0, -1, -3, 0, -1], [2, 0, -13, 2, 0, 0, -2, 3], [-16, 12, -16, -9, 9, 9, -9, 0], [0, 3, 3, 3, 8, -8, 5, 5], [-7, 0, -12, -7, 0, 0, 2, 6], [10, -4, 0, 12, -2, -10, 4, -2], [-13, -3, -13, -11, -3, 11, 3, 0], [-1, -2, -10, 0, -3, 10, 0, 2], [-6, 6, 0, -14, -8, 6, -6, -8], [-6, 0, -3, -6, 0, 0, -9, 3], [0, -4, -7, -7, -5, -1, -1, -1], [-8, -1, -4, 0, 9, 4, 0, 1], [6, -11, 6, 3, -5, -3, 5, 0], [8, 0, 7, 8, 0, 0, 3, 1], [-6, -6, 0, 2, -2, 6, 6, -2], [-1, -5, 6, 0, 3, -6, 0, 5], [-14, -4, -14, -12, 0, 12, 0, 0], [2, 10, 2, 3, 1, -3, -1, 0], [-5, 2, 11, 0, -3, -11, 0, -2], [-2, 0, -2, -2, 0, 0, 9, -7], [0, -8, 9, 9, 1, -11, 9, 9], [3, 3, 3, 0, -2, 0, 2, 0], [5, -3, 20, 0, -2, -20, 0, 3], [3, 0, 12, 3, 0, 0, 4, 6], [0, 0, -20, -20, 2, 10, 2, 2], [14, -10, 11, 0, 2, -11, 0, 10], [0, -6, 0, -12, 8, 0, 6, 8], [0, -2, 10, 10, -6, -2, -4, -4], [1, 4, 1, 3, 5, -3, -5, 0], [3, 5, -6, 0, -8, 6, 0, -5], [-5, 3, 0, 2, -6, 5, -3, -6], [0, 8, 13, 13, -2, -10, -10, -10], [6, 0, -3, 6, 0, 0, -3, -10], [11, 5, 2, 0, 0, -2, 0, -5], [-3, 13, -8, 0, -2, 8, 0, -13], [10, 0, 0, 10, 0, 0, -8, -6], [1, 0, 0, 1, 0, 0, -1, -3], [-7, -12, 0, -10, 5, 7, 12, 5], [4, 6, 4, 16, -1, -16, 1, 0], [4, -12, 4, -8, -3, 8, 3, 0], [-10, 2, 0, -2, 8, 10, -2, 8], [1, 3, 0, -7, 9, -1, -3, 9], [-10, 7, -10, -6, 10, 6, -10, 0], [-11, 0, -16, -11, 0, 0, -2, -1], [0, -4, 5, 5, -9, -7, -5, -5], [14, -9, 0, 19, 11, -14, 9, 11], [-21, 17, -21, -5, 7, 5, -7, 0], [0, -3, -20, -20, -2, -9, 1, 1], [3, -4, 0, 10, 2, -3, 4, 2], [-6, -12, 0, 0, 10, 0, 0, 12], [-4, 0, -3, -4, 0, 0, -7, 1], [-8, 1, -8, -17, 1, 17, -1, 0], [37, 5, 8, 0, 5, -8, 0, -5], [-16, 0, -21, -16, 0, 0, 0, -1], [-2, -14, -2, 2, -9, -2, 9, 0], [0, -6, -3, -3, -3, -17, 3, 3], [-18, -3, 0, 1, -2, 18, 3, -2], [18, -1, -5, 0, 8, 5, 0, 1], [-23, -2, 0, -26, 6, 23, 2, 6], [24, 16, 24, 16, 6, -16, -6, 0], [1, 2, 0, 13, -10, -1, -2, -10], [10, 0, 19, 10, 0, 0, -5, -10], [0, -7, -12, -12, -1, 19, 6, 6], [3, -9, 3, -20, 4, 20, -4, 0], [22, -8, 12, 0, -5, -12, 0, 8], [13, -8, 15, 0, 1, -15, 0, 8], [0, 15, -17, -17, 13, 3, -2, -2], [-23, 0, -6, -23, 0, 0, -9, -7], [24, -2, 2, 0, -6, -2, 0, 2], [0, -14, -3, -3, 2, -2, 16, 16], [7, -4, 0, -16, 5, -7, 4, 5], [-2, 0, -17, -2, 0, 0, -6, -9], [7, -3, 7, 18, 6, -18, -6, 0], [0, 6, 26, 0, -10, -26, 0, -6], [26, -2, 26, 12, 2, -12, -2, 0], [0, -3, 2, 2, -10, -16, -7, -7], [0, 0, 24, 0, 0, 0, 0, -6], [-1, 5, 0, 17, -3, 1, -5, -3], [0, -10, 1, 1, -13, -3, -3, -3], [0, -13, 6, 6, -6, -20, 7, 7], [-12, -2, -2, 0, -12, 2, 0, 2], [20, -1, 0, 15, 16, -20, 1, 16], [3, 0, -18, 3, 0, 0, 6, 7], [-19, 0, -7, -19, 0, 0, 3, 4], [-13, 0, -15, -13, 0, 0, 10, -8], [5, 4, 0, 7, -18, -5, -4, -18], [-27, -7, -27, -27, -11, 27, 11, 0], [0, 10, 19, 19, 9, -1, -1, -1], [0, 8, 12, 12, -3, -13, -11, -11], [8, -4, 8, -3, -3, 3, 3, 0], [-11, -1, 0, -3, 13, 11, 1, 13], [-23, 0, -7, -23, 0, 0, 9, 8], [-17, 13, -13, 0, -4, 13, 0, -13], [-9, -4, 7, 0, -10, -7, 0, 4], [13, 0, 26, 13, 0, 0, -5, -7], [-3, -11, 0, -7, -9, 3, 11, -9], [-4, -17, 8, 0, 9, -8, 0, 17], [-5, -1, 0, -4, 8, 5, 1, 8], [7, 7, 7, 14, 6, -14, -6, 0], [-12, 0, -13, -12, 0, 0, 7, 9], [0, -9, 1, 1, -2, -4, 7, 7], [4, 0, 15, 4, 0, 0, -10, 7], [-2, 0, -16, -2, 0, 0, 5, -10], [0, -12, -18, -18, 7, 25, 19, 19], [15, 0, -12, 15, 0, 0, -13, -5], [5, 21, -4, 0, -9, 4, 0, -21], [5, 17, 5, -7, 20, 7, -20, 0], [-23, -7, -20, 0, -4, 20, 0, 7], [6, 0, 0, 14, 0, -6, 0, 0], [0, 14, -2, -2, -2, 4, -16, -16], [7, -11, 7, 22, -2, -22, 2, 0], [-19, 8, 5, 0, 1, -5, 0, -8], [-7, 5, -7, 12, 4, -12, -4, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_770_n_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_770_2_n_f();".
// 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_f(: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_f();".
// 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_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_770_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, 0, 65, 80, 46, -8, 4, -2, 1]>],Snew);
    return Vf;
end function;