
// Make newform 8670.2.a.bf in Magma, downloaded from the LMFDB on 23 December 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8670_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_8670_2_a_bf();".
// 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_8670_2_a_bf();".
// 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, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0], [0, 2]];
    Rf_basisdens := [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_8670_a();"
function MakeCharacter_8670_a()
    N := 8670;
    order := 1;
    char_gens := [2891, 6937, 6361];
    v := [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_8670_a_Hecke(Kf)
    return MakeCharacter_8670_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 := [[-1, 0], [1, 0], [-1, 0], [2, 1], [0, 2], [2, -1], [0, 0], [0, 0], [-6, 0], [2, 2], [-2, 2], [-2, 0], [0, 1], [4, -3], [0, 2], [6, -2], [-8, -1], [6, 0], [0, -3], [-2, 1], [8, 1], [14, 0], [0, -6], [-6, -2], [-4, 3], [6, -1], [0, 0], [8, 4], [10, 2], [-4, -4], [-8, 2], [16, -2], [-10, 4], [0, 0], [2, -1], [0, 6], [6, 3], [-4, 0], [-14, -2], [18, -2], [-4, 1], [-2, -2], [-8, 4], [-4, -1], [-2, -2], [2, -6], [8, -4], [16, 4], [-4, -4], [-10, 0], [-12, -6], [0, -6], [4, 0], [4, -5], [18, 0], [-16, 4], [-14, -4], [-8, 6], [10, -8], [10, 2], [8, 6], [10, -8], [-4, 5], [-14, -5], [-4, -5], [-18, -2], [-28, -2], [32, 1], [12, -4], [-6, -6], [14, 4], [0, -10], [22, 3], [2, 1], [20, 0], [8, 4], [18, 3], [-10, 6], [16, -5], [-2, 4], [-4, 8], [-14, -8], [2, -1], [-2, -4], [18, 0], [-12, 2], [24, -5], [22, 0], [22, 5], [-40, 0], [-12, 0], [22, 5], [-2, -7], [28, 1], [-16, -4], [2, -4], [18, -5], [-16, -5], [24, 3], [22, 0], [-16, -2], [6, 2], [8, 2], [-2, 10], [12, 0], [14, 4], [12, -2], [-2, 8], [24, 6], [8, -6], [-10, 5], [22, -7], [-4, 10], [20, 0], [-8, 0], [24, -1], [-4, -12], [-16, -8], [-6, 8], [0, 13], [-6, -6], [-12, 3], [-26, -2], [12, -12], [16, 8], [-18, -3], [-6, -12], [-18, 5], [-16, -6], [-38, -3], [8, -2], [-6, -6], [-14, -4], [14, -1], [30, -6], [22, 4], [2, -10], [-24, 6], [-18, 0], [-24, 5], [12, 12], [-18, 4], [-2, 7], [24, 8], [30, -6], [6, 7], [34, 4], [-8, 10], [-4, 10], [24, -2], [26, 6], [4, -5], [-8, -9], [-10, 4], [-16, 6], [6, 1], [-24, -6], [0, 9], [2, -18], [-42, 0], [40, -4], [2, -8], [40, -2], [8, -7], [6, 4], [-34, 2], [2, -14], [-34, 4], [12, -12], [-18, 4], [-8, 11], [10, -4], [-10, -17], [-30, 6], [-16, 0], [12, 9], [4, -4], [-10, 10], [16, -6], [-26, 12], [-32, -4], [-24, 10], [-10, 6], [30, -12], [48, 6], [6, -9], [6, -12], [-28, 3], [44, -2], [42, -7], [-8, -13], [-20, -16], [24, 6], [-6, 1], [20, -8], [12, 12], [-8, 14], [14, -4], [12, -16], [0, 4], [-18, -8], [-10, -2], [-18, -3], [26, 8], [0, -9], [-38, 8], [24, -6], [36, 8], [-4, 7], [4, 6], [36, -3], [-6, -3], [-30, 3], [-36, -2], [14, -1], [28, -2], [-24, 0], [-6, 14], [30, 8], [-26, 14], [6, 0], [54, -6], [2, -8], [-34, -3], [-60, 2], [18, 0], [12, 8], [18, -15], [-16, 6], [-44, -4], [-46, 7], [20, -12], [24, 0], [6, -10], [0, -3], [-24, -2], [20, 18], [-18, 6], [-8, -2], [16, 10], [-36, 0], [-4, -18], [-16, 0], [-2, -6], [-12, -18], [34, -1], [6, -3], [-12, -4], [12, -18], [-40, 2], [-2, 3], [-36, 5], [-72, 0], [-32, -6], [18, -2], [12, 7], [22, -18], [-28, -4], [22, 8], [42, -6], [-42, 9], [0, 20], [-26, -8], [-14, 16], [24, -8], [24, 12], [30, 3], [-10, -2], [-44, -4], [-30, 6], [26, -20], [4, 11], [6, 0], [32, 2], [-6, 18], [-48, 6], [0, -10], [26, 18], [-22, -8], [-16, 17], [32, -4], [14, -20], [22, -4], [26, 0], [0, -18], [-24, 14], [20, -1], [22, -20], [-56, 4], [30, -10], [-54, 6], [-36, -8], [18, -20], [24, -2], [-18, 18], [-6, -6], [0, -6], [14, -4], [12, 16], [36, 15], [26, -16], [-6, -14], [-54, 0], [-12, -12], [-28, 0], [-4, 25], [24, -22], [-50, 6], [-64, 4], [-42, -9], [26, 2], [2, -8], [-8, -7], [-48, 3], [24, 2], [-78, 0], [-12, -1], [-30, 1], [0, 3], [-18, -2], [12, 12], [-4, 21], [2, -1], [32, 4], [-8, 2], [-2, 8], [-40, 16], [-64, -6], [6, 16], [-2, -4], [-14, 6], [-18, 0], [-22, 6], [-24, -12], [36, 12], [0, 4], [42, -12], [-20, 14], [-56, 0], [-16, 2], [-30, -3], [-18, -20], [14, 2], [32, -18], [14, -10], [12, -24], [22, 18], [40, 15], [-14, 7], [30, -6], [4, 2], [-20, 11], [18, -13], [14, 15], [-34, -2], [6, -16], [-48, -10], [-12, -24], [-16, -4], [-32, 4], [30, 6], [-20, 13], [0, -10], [36, 4], [8, 17], [-30, 18], [-6, -16], [-8, 22], [-28, -6], [32, -11], [-8, 16], [10, 4], [-42, -5], [-24, 18], [58, -10], [12, -16], [62, 7], [18, 12], [-30, 12], [42, 6], [-30, 12], [10, -16], [-22, -21], [-12, 6], [-44, -20], [6, -2], [-32, -2], [-10, 11], [-28, 4], [-24, -20], [0, -6], [-34, -22], [0, 9], [-36, -21], [-24, -6], [4, -15], [-64, -4], [-54, -4], [-16, 8], [30, -17], [-50, 12], [62, 4], [-48, -12], [-24, 16], [26, 5], [-66, 12], [-10, -29], [-10, -14], [-12, -3], [-60, -8], [-8, -5], [-30, 4], [48, 2], [48, -12], [62, 6], [30, 12], [18, -13], [-58, -15], [24, -24], [32, 10], [6, -15], [-6, -6], [30, 12], [36, 3], [44, -21], [-54, 12], [0, 20], [44, -7], [72, -6], [-18, 13], [102, 0], [44, 23], [92, 0], [34, 10], [58, 6], [-50, -20], [-8, 0], [-22, 2], [-8, -18], [56, -14], [36, -24], [-60, -11], [6, -12], [32, 26], [48, 6], [6, 12], [32, 17], [-6, 20], [52, -18], [-42, -24], [28, -13], [-42, -5], [-8, -4], [10, 32], [-22, -18], [30, 19], [10, -14], [-72, 5], [2, -10], [20, 20], [-16, -12], [46, 18], [-36, 7], [-26, 2], [8, -17], [-70, 6], [-24, 12], [76, -4], [56, -13], [20, 22], [-50, 1], [-24, -6], [62, 13], [0, -24], [36, -24], [-74, -6], [-90, -2], [0, -6], [14, -22], [42, 6], [38, -18], [6, 8], [-52, 15], [-42, 2], [10, 15], [20, -14], [-14, -12], [40, -14], [-20, 32], [24, -30], [-46, 3], [0, 34], [16, 18], [-10, -4], [12, -12], [-6, 18], [0, 30], [-54, -12], [-46, -4], [40, -20], [78, -8], [-16, 2], [48, 6], [-62, -1], [6, -21], [42, -4], [48, 18], [-38, 12], [62, -17], [-16, -6], [48, -11], [48, 2], [94, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8670_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8670_2_a_bf();".
// 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_8670_2_a_bf(:prec:=2)
    chi := MakeCharacter_8670_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(3673) - 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_8670_2_a_bf();".
// 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_8670_2_a_bf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8670_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-4, -4, 1]>,<11,R![-32, 0, 1]>,<13,R![-4, -4, 1]>,<23,R![6, 1]>],Snew);
    return Vf;
end function;