
// Make newform 770.2.n.c in Magma, downloaded from the LMFDB on 21 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_c();".
// 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_c();".
// 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, -1, 1, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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], [1, 0, 0, 0], [0, 0, 0, -1]];
    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 := [[1, -1, 1, -1], [0, -1, 0, -1], [0, 1, 0, 0], [0, 0, 0, -1], [2, 2, -1, 0], [2, -4, 4, -2], [-1, 1, -1, 0], [0, -1, -1, -1], [-2, 0, 4, -4], [0, 0, 0, -4], [-2, -2, 2, 2], [-6, 6, 0, 0], [0, -1, 8, -1], [-3, 0, 1, -1], [0, -2, -8, -2], [0, -4, 4, 0], [1, -1, 0, 2], [0, 8, 0, 0], [2, 0, 5, -5], [4, 0, 4, 0], [-7, 7, 0, 7], [8, -2, 2, -8], [5, -8, 5, 0], [5, 0, -3, 3], [-7, -4, 4, 7], [-14, 16, -16, 14], [6, -6, 0, -10], [0, 5, 0, 5], [4, 0, 10, -10], [0, -11, 7, -11], [10, -8, 10, 0], [10, 0, -3, 3], [3, 8, 3, 0], [4, -4, 0, 4], [6, -2, 6, 0], [0, 10, 4, 10], [0, -10, 10, -10], [22, -21, 21, -22], [-10, 10, -10, 10], [0, -2, 8, -2], [0, 3, -13, 3], [18, -6, 18, 0], [0, 0, 0, 2], [-4, 10, -4, 0], [2, 0, -12, 12], [2, 0, 2, -2], [-25, 20, -20, 25], [0, -8, -10, -8], [-9, 9, 0, 6], [-4, -2, 2, 4], [-15, 8, -8, 15], [0, -6, 14, -6], [-12, 0, -3, 3], [-8, -8, 8, 8], [9, -9, 0, 11], [-12, 0, 0, 0], [0, -14, 0, 0], [-6, 6, 0, -10], [6, 12, -12, -6], [-11, 0, -11, 0], [0, 16, -16, 16], [6, -6, 0, -12], [7, 0, -17, 17], [0, -2, 2, -2], [7, 1, 7, 0], [-8, -8, 8, 8], [1, 0, -7, 7], [-15, 15, 0, -8], [-13, 19, -13, 0], [0, -4, 8, -4], [-4, 0, -13, 13], [-6, 6, 0, -16], [8, -8, 0, -24], [8, 0, 8, -8], [3, 12, 3, 0], [-8, 14, -14, 8], [0, 0, 0, -8], [14, 0, 30, -30], [-17, 0, -17, 0], [-14, -14, 14, 14], [-3, 0, -19, 19], [0, 4, 2, 4], [-16, 10, -10, 16], [-11, 11, 0, -8], [12, 0, 20, -20], [0, 1, -28, 1], [-16, -3, 3, 16], [-9, 20, -9, 0], [28, 0, 18, -18], [-22, 0, -14, 14], [8, -12, 8, 0], [-18, 6, -18, 0], [0, -32, 18, -32], [29, -29, 0, -9], [-1, 1, 0, 1], [0, 6, -22, 6], [0, -10, 22, -10], [9, -9, 0, -32], [29, -11, 29, 0], [-24, 34, -34, 24], [0, -5, 35, -5], [20, -20, 0, 8], [-27, 0, 0, 27], [0, -25, 21, -25], [12, 0, 24, -24], [-7, -13, -7, 0], [-1, 1, 0, 10], [-12, 0, -1, 1], [0, -24, 0, 0], [9, -9, 0, -1], [34, -34, 34, -34], [0, 26, 6, 26], [20, 0, -11, 11], [0, -9, 35, -9], [2, -2, 0, 22], [0, -7, 36, -7], [-9, 8, -9, 0], [-2, 18, -18, 2], [0, 0, 0, 6], [18, 0, 5, -5], [18, 0, 32, -32], [-12, 9, -9, 12], [8, -2, 8, 0], [-28, 0, -28, 28], [10, -21, 21, -10], [0, -10, 12, -10], [2, -36, 2, 0], [-20, 20, 0, -18], [-8, 0, -28, 28], [-2, 2, 0, 18], [14, 7, -7, -14], [0, 22, 0, 0], [0, -12, -30, -12], [-10, -16, 16, 10], [-16, 31, -31, 16], [18, 0, 12, -12], [0, 20, 4, 20], [-13, 7, -13, 0], [-4, -28, -4, 0], [9, -41, 9, 0], [0, 11, 32, 11], [38, -38, 0, -26], [-28, -10, 10, 28], [-14, -5, 5, 14], [8, -8, 0, 2], [0, -14, 12, -14], [-6, 46, -6, 0], [31, 0, 27, -27], [42, 0, 3, -3], [-36, 18, -36, 0], [0, -18, 8, -18], [9, 0, 41, -41], [0, -1, 36, -1], [0, 0, 0, 22], [-11, -3, -11, 0], [10, 16, -16, -10], [2, 18, 2, 0], [7, -31, 7, 0], [8, 25, -25, -8], [-32, -2, -32, 0], [26, 0, 41, -41], [-3, 3, 0, -29], [-26, 0, 0, 0], [0, -28, 24, -28], [-34, 38, -38, 34], [2, -2, 0, -30], [-60, 0, -2, 2], [-18, 18, 0, -24]];
    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_c();".
// 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_c(:prec:=4)
    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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_770_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 3, 4, 2, 1]>],Snew);
    return Vf;
end function;