
// Make newform 80.6.n.a in Magma, downloaded from the LMFDB on 21 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_80_n();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_80_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_80_6_n_a();".
// 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_80_6_n_a();".
// 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, 0, 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_80_n();"
function MakeCharacter_80_n()
    N := 80;
    order := 4;
    char_gens := [31, 21, 17];
    v := [2, 4, 1];
    // 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_80_n_Hecke();"
function MakeCharacter_80_n_Hecke( : Kf := false)
    N := 80;
    order := 4;
    char_gens := [31, 21, 17];
    char_values := [[-1, 0], [1, 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 := 6;
    raw_aps := [[0, 0], [0, 0], [-38, 41], [0, 0], [0, 0], [475, -475], [-1525, -1525], [0, 0], [0, 0], [0, -8564], [0, 0], [-475, -475], [-4952, 0], [0, 0], [0, 0], [-16475, 16475], [0, 0], [54948, 0], [0, 0], [0, 0], [-54475, 54475], [0, 0], [0, 0], [0, -140464], [-126475, -126475], [98002, 0], [0, 0], [0, 0], [0, -246486], [-181425, 181425], [0, 0], [0, 0], [-178575, -178575], [0, 0], [0, 47614], [0, 0], [432475, 432475], [0, 0], [0, 0], [-556575, 556575], [0, 0], [439902, 0], [0, 0], [702475, -702475], [-700575, -700575], [0, 0], [0, 0], [0, 0], [0, 0], [0, -976564], [1078475, -1078475], [0, 0], [1739048, 0], [0, 0], [682575, 682575], [0, 0], [0, 2351414], [0, 0], [1584475, 1584475], [-149152, 0], [0, 0], [412575, -412575], [0, 0], [0, 0], [2402375, -2402375], [2092375, 2092375], [0, 0], [-602375, -602375], [0, 0], [0, 975636], [517625, -517625], [0, 0], [0, 0], [1827625, -1827625], [0, 0], [0, 0], [0, -5286286], [3267625, 3267625], [1592002, 0], [0, -4584486], [0, 0], [-4184852, 0], [0, 0], [-4158475, 4158475], [0, 0], [0, 0], [0, 8489614], [-6237625, -6237625], [6865802, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, -5120764], [7274102, 0], [0, 0], [-8253798, 0], [0, 0], [-7072475, -7072475], [0, 0], [0, -3966586], [0, 0], [-9477625, -9477625], [0, 0], [9397425, -9397425], [0, 0], [9163248, 0], [0, 0], [-13023325, 13023325], [-12169325, -12169325], [0, 0], [0, 0], [14827048, 0], [0, 0], [0, 0], [4735325, -4735325], [0, 0], [-20753052, 0], [801325, -801325], [9427625, 9427625], [0, 0], [0, 0], [-15137252, 0], [0, 26471236], [0, 0], [0, 0], [8307625, -8307625], [0, 0], [0, 0], [0, 0], [14192675, 14192675], [19519802, 0], [0, -25772664], [13984475, -13984475], [0, 0], [-23779425, -23779425], [0, 23619736], [0, 0], [21463148, 0], [0, 0], [0, 0], [0, -7968186], [0, 0], [17136475, -17136475], [3037625, 3037625], [0, 0], [0, 0], [31328675, 31328675], [-13397152, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 0], [0, 47663314], [20790475, 20790475], [4855702, 0], [0, 0], [-38989425, 38989425], [0, 0], [0, 0], [-41758675, -41758675], [0, 0], [0, 0], [7532375, 7532375]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_80_n_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_80_6_n_a();".
// 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_80_6_n_a(:prec:=2)
    chi := MakeCharacter_80_n();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 6));
    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_80_6_n_a();".
// 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_80_6_n_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_80_n();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,6,sign)));
    Vf := Kernel([<3,R![0, 1]>],Snew);
    return Vf;
end function;