
// Make newform 3680.2.a.r in Magma, downloaded from the LMFDB on 11 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_3680_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_3680_2_a_r();".
// 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_3680_2_a_r();".
// 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 := [2, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-5, -1, 1]];
    Rf_basisdens := [1, 1, 1];
    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_3680_a();"
function MakeCharacter_3680_a()
    N := 3680;
    order := 1;
    char_gens := [1151, 1381, 737, 3041];
    v := [1, 1, 1, 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;

function MakeCharacter_3680_a_Hecke(Kf)
    return MakeCharacter_3680_a();
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, 1, 0], [1, 0, 0], [-1, -1, 0], [0, 1, 0], [1, 1, 1], [0, 1, -1], [-1, -1, -1], [-1, 0, 0], [6, 0, 1], [-5, -1, 0], [4, -2, 1], [4, 3, -1], [4, 0, 0], [-2, 2, -2], [6, -2, -1], [-4, 4, 1], [6, -5, -2], [-8, 2, 1], [6, 1, 3], [4, 2, 0], [8, 0, 0], [-2, -2, -1], [4, -4, -2], [4, -3, 0], [-10, 0, -3], [-2, -3, 0], [14, 0, -1], [-9, -1, -1], [6, 2, -1], [8, -2, -2], [2, 0, -2], [9, 3, -1], [-8, 0, -5], [1, 1, 3], [-2, -5, 4], [18, 0, 1], [-3, -3, 3], [-4, 0, 4], [-2, -1, 2], [10, 2, 2], [7, -5, 1], [8, -2, 0], [4, -6, 2], [10, -7, -2], [6, 4, 2], [2, -6, -5], [-4, -4, 4], [2, -6, -2], [2, 8, 0], [-4, -2, 0], [0, 6, 5], [-4, -4, -4], [17, -7, -3], [-16, -2, -4], [-6, 5, 3], [2, -4, -1], [3, -5, 2], [2, 0, -4], [-2, -2, 4], [18, -2, 1], [10, 2, -3], [-4, -5, 0], [-8, 8, 4], [15, -1, -2], [13, 5, -1], [-4, -4, 5], [13, -1, -1], [8, -3, 0], [-6, -6, -3], [0, 0, 0], [-14, 0, -8], [8, -6, -1], [-4, 10, 2], [-12, -1, 0], [0, -2, -1], [-2, 9, 2], [6, -7, 2], [4, 2, 2], [-13, -1, -4], [-4, 0, 2], [-13, -1, 1], [-6, -6, -4], [-3, -7, 6], [-17, -3, -5], [3, 3, 1], [1, -7, 0], [-2, -2, 1], [0, -2, -2], [2, 4, 2], [0, 4, -7], [18, -8, -2], [0, 2, -2], [26, -4, 1], [-20, 10, 3], [-5, -9, 2], [-22, -2, 4], [-14, -4, 2], [14, 0, -2], [16, -4, 6], [7, -9, 3], [4, -2, -7], [4, 2, 3], [2, 4, 8], [20, 1, 0], [6, 0, 4], [-4, -3, -8], [0, 6, 2], [2, -7, -4], [-9, -1, -10], [8, 4, 2], [2, -4, -8], [-20, -7, -3], [-7, 1, 3], [0, -2, -6], [8, -4, 6], [-22, -4, -9], [18, 4, 10], [-15, 9, 7], [0, -4, 4], [-6, 7, -2], [-6, -8, -6], [-2, -10, 1], [21, 1, 7], [-26, 2, 2], [-7, -3, 1], [-14, 5, -2], [-7, 1, 6], [-14, 11, 3], [10, -10, 3], [4, 14, -3], [-9, -15, 3], [-16, -6, -4], [-8, 10, -1], [-13, 15, 0], [-22, 0, 6], [-28, 8, -2], [-20, -4, -11], [-8, -12, -1], [-28, 9, 1], [14, 4, 3], [2, 0, 4], [2, 12, 8], [-36, 0, 3], [-6, -10, -5], [18, -2, 6], [11, 11, 7], [14, 4, -4], [6, -20, -1], [-16, 2, -10], [-2, -7, -2], [-20, 4, -6], [-9, 3, -3], [-4, 4, 4], [2, 12, -1], [2, 6, 8], [4, 0, 6], [-34, -2, -7], [25, 3, -5], [-22, 7, 2], [35, -5, -1], [0, 9, 0], [20, -2, 4], [28, -5, 6], [-1, 7, 2], [6, 11, -3], [22, 7, 1], [10, -12, -2], [4, -2, -2], [-2, 1, -8], [-16, 8, 2], [42, 0, 7], [-14, 2, -8], [4, 15, 7], [50, -1, 0], [-2, 6, -8], [-38, -4, -6], [-12, 14, 7], [-6, 15, -1], [17, -11, 7], [0, -6, -10], [8, 7, 0], [21, 9, 7], [-30, 2, -4], [-30, -1, -5], [-39, -3, -5], [-19, -3, -3], [20, -14, -5], [20, -9, -1], [-10, -7, 7], [20, 6, 2], [-8, -4, -14], [-8, 4, 2], [12, -2, -3], [-22, -8, 6], [17, 1, 4], [6, -2, -4], [14, 0, 12], [6, 20, -8], [-2, 2, -4], [18, 11, -6], [51, -3, -1], [18, -7, 2], [8, 4, 12], [-20, -4, -6], [30, -4, 10], [26, -8, 6], [24, 7, 0], [-31, 9, 6], [18, -14, -2], [10, 6, -10], [-34, -8, 0], [7, 1, 11], [30, -2, 2], [16, -10, -7], [30, -10, -2], [8, 10, -6], [-38, 2, 3], [-3, 9, 4], [38, 0, 6], [14, -4, 11], [-10, 16, 6], [-48, 5, -3], [9, 9, 8], [4, 16, 8], [30, 12, 1], [-13, 1, -7], [-29, 3, -4], [-11, 3, -7], [24, -6, -5], [43, 7, -3], [-48, 13, 6], [-36, 0, 6], [-12, -5, 5], [-10, 24, 3], [12, -4, -4], [-14, -10, -10], [-6, 2, -4], [34, -4, 3], [10, -9, -1], [30, 4, 11], [-6, -22, -1], [22, -8, 0], [4, -6, -10], [6, -22, 6], [5, -1, -11], [14, -6, 6], [3, 19, -5], [-8, 19, 14], [-30, 13, -3], [-16, 2, -14], [8, 10, -8], [-27, 5, -8], [32, 8, 0], [2, 12, -2], [4, -20, 5], [50, -15, -4], [-4, 2, 4], [21, -7, -1], [-30, 16, 2], [22, 9, -5], [2, 14, 1], [28, -6, -5], [-46, -4, -3], [2, -4, 6], [-12, 17, -4], [6, 3, -8], [-30, -12, 0], [-8, -12, -5], [-1, -1, 5], [14, 12, -1], [-16, -2, 5], [19, -9, -12], [-12, -12, -14], [14, 0, 18], [0, -14, -2], [-12, 11, 6], [-18, -12, 7], [-10, -12, -8], [3, -13, -13], [28, -4, -10], [28, 12, 4], [-22, 13, 4], [-30, 4, 0], [3, 15, -1], [2, -11, -9], [26, 6, -5], [24, -14, 6], [-18, 13, 8], [44, 7, 8], [58, -4, 6], [12, 8, -1], [12, -2, -6], [-39, 5, -5], [30, -13, -6], [10, -1, 2], [2, -10, -8], [18, 3, -8], [-14, 6, -3], [16, -7, 0], [-1, 3, 6], [2, -4, 11], [48, 2, -2], [24, 8, 8], [1, 1, -11], [-4, 2, -4], [-40, 15, -2], [-3, 9, 5], [2, -2, -2], [30, 19, -8], [-26, 22, 12], [-13, -17, 5], [-12, -6, 14], [-52, 13, 8], [21, -27, -4], [-14, -6, 2], [10, 10, 11], [-15, -3, 6], [-2, 10, 3], [-24, -11, 4], [-24, -9, 2], [-8, 19, 1], [-22, 4, -4], [14, -6, 10], [52, 1, 7], [60, 0, 6], [-13, 19, -5], [29, -3, 13], [-33, 1, 3], [38, 4, 13], [44, -4, -9], [-22, -1, -8], [4, 22, 3], [57, -3, -1], [24, -24, -4], [-3, 9, 11], [-18, 7, 12], [10, 20, 0], [0, 7, 16], [-38, -3, -3], [-2, -4, -2], [4, 31, -8], [8, 24, -6], [-30, -10, -2], [6, -2, -7], [-16, -18, 3], [-6, -10, 7], [-15, 9, -9], [34, 4, 2], [30, 24, -7], [-2, -24, 5], [34, -20, -12], [-48, 20, 7], [-10, 17, 7], [18, -24, -12], [10, 14, 2], [24, -2, 0], [7, -5, 15], [-38, 0, 4], [-42, 4, 0], [20, -8, -3], [10, -10, -17], [-4, 18, -8], [-16, 27, 6], [-16, 15, -2], [-34, -8, -12], [-46, 29, 8], [-16, -14, -11], [2, 22, 10], [34, 6, 5], [-36, 24, 1], [-30, -8, -2], [18, 1, 2], [-8, -12, -2], [-21, -5, -7], [-2, 20, -9], [28, -18, 0], [16, 7, -9], [30, 6, 8], [-8, -16, 0], [-30, -3, -2], [12, 7, 16], [16, -18, 0], [-34, -16, 6], [-24, -16, -2], [12, -20, 6], [-1, -17, -12], [-47, 9, 9], [6, 2, -9], [35, 3, 12], [46, -24, -10], [47, -13, -3], [-12, 6, -5], [-16, -20, 8], [-22, 16, 0], [53, -15, 2], [-10, -4, -8], [-4, -16, -16], [49, 1, -3], [-6, 6, -13], [16, -24, -13], [-20, 22, 8], [68, -10, -5], [-17, 15, 10], [0, 18, 7], [38, -9, 3], [6, -8, 9], [-17, 17, 13], [42, -22, -6], [32, 4, -4], [-72, -2, 0], [-10, -19, 0], [-24, -12, 0], [-60, 9, 8], [-11, -11, -12], [10, 12, -12], [21, -1, 1], [-26, 6, -2], [-72, -7, 3], [-45, -23, 7], [50, -15, 6], [-6, -22, 8], [-34, -6, 2], [-6, 2, -10], [-2, 6, 12], [-10, -8, -2], [44, 4, -8], [19, -9, 6], [0, 10, 0], [6, 7, 11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3680_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3680_2_a_r();".
// 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_3680_2_a_r(:prec:=3)
    chi := MakeCharacter_3680_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_3680_2_a_r();".
// 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_3680_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3680_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, -7, -1, 1]>,<7,R![-7, -2, 4, 1]>,<11,R![2, -7, -1, 1]>,<13,R![32, -19, -3, 1]>],Snew);
    return Vf;
end function;