
// Make newform 8470.2.a.z in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8470_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_8470_2_a_z();".
// 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_8470_2_a_z();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8470_a();"
function MakeCharacter_8470_a()
    N := 8470;
    order := 1;
    char_gens := [6777, 6051, 7141];
    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_8470_a_Hecke(Kf)
    return MakeCharacter_8470_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+1)]];
    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], [-1], [0], [6], [2], [4], [-4], [-6], [0], [-2], [6], [4], [4], [-2], [12], [2], [-8], [-8], [10], [8], [12], [10], [-6], [-14], [-4], [-4], [-14], [-6], [8], [12], [18], [20], [-6], [-8], [-2], [8], [0], [14], [20], [6], [8], [-6], [-18], [-24], [-4], [-4], [-4], [6], [10], [24], [-2], [20], [18], [-16], [-2], [0], [-26], [30], [20], [14], [20], [0], [10], [14], [-28], [-22], [28], [34], [-14], [16], [20], [-34], [-12], [-28], [6], [-34], [34], [-34], [12], [-26], [-8], [-6], [24], [16], [2], [26], [-6], [20], [0], [24], [-28], [-20], [28], [8], [-2], [-22], [12], [18], [-36], [6], [4], [-42], [-4], [-38], [8], [-30], [-40], [-26], [-40], [-34], [34], [4], [8], [-30], [32], [-12], [-42], [-36], [22], [10], [14], [24], [-36], [42], [22], [-40], [-12], [46], [12], [0], [-32], [6], [14], [14], [30], [-28], [-34], [46], [-52], [-30], [4], [12], [30], [-40], [6], [42], [-44], [-52], [-10], [2], [56], [48], [32], [-40], [40], [-30], [50], [-38], [24], [-22], [32], [-52], [-54], [-4], [-40], [-10], [-26], [-50], [20], [-2], [24], [-6], [-24], [26], [-20], [-58], [40], [10], [20], [-12], [-10], [-30], [36], [22], [54], [56], [-10], [-8], [-38], [36], [-28], [-2], [60], [-6], [-58], [-58], [-6], [-32], [-54], [-40], [46], [46], [36], [-18], [-32], [-60], [-34], [44], [-62], [-10], [36], [-8], [0], [-38], [32], [-10], [-52], [-10], [2], [48], [50], [-4], [-28], [58], [50], [-24], [-32], [-12], [-18], [28], [-72], [-66], [0], [48], [-30], [38], [-44], [72], [-72], [-28], [76], [-18], [42], [-48], [-28], [60], [-28], [24], [-50], [-26], [-36], [26], [-34], [44], [-10], [-76], [-26], [-62], [24], [-4], [-22], [-42], [2], [-60], [30], [10], [20], [-74], [78], [-8], [-30], [0], [42], [-12], [-16], [-70], [-50], [-36], [-8], [32], [-24], [18], [-44], [8], [58], [-58], [0], [54], [78], [-8], [-54], [-60], [78], [74], [8], [46], [36], [-12], [34], [22], [-16], [48], [20], [34], [48], [14], [-24], [30], [80], [-10], [-2], [-80], [16], [-26], [12], [0], [-86], [30], [44], [-30], [-78], [4], [-38], [-78], [20], [56], [-48], [78], [26], [-66], [24], [84], [-28], [48], [-34], [-6], [-22], [48], [22], [-38], [50], [32], [86], [-12], [-58], [72], [-88], [62], [20], [-22], [-2], [-16], [-14], [42], [48], [-12], [2], [36], [22], [30], [-68], [12], [-56], [34], [-2], [-16], [30], [-12], [20], [24], [-6], [64], [14], [28], [64], [-70], [-58], [2], [-34], [-6], [16], [-22], [-44], [-36], [0], [54], [-20], [12], [-62], [62], [36], [64], [88], [-38], [0], [42], [-68], [-62], [-6], [90], [32], [-6], [18], [40], [-34], [-82], [-24], [20], [58], [-42], [16], [100], [-14], [-18], [-16], [4], [66], [24], [94], [-82], [36], [44], [58], [-58], [72], [-42], [-92], [16], [10], [84], [-60], [-4], [62], [98], [-82], [-74], [48], [8], [104], [66], [10], [96], [6], [66], [-68], [24], [-46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8470_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8470_2_a_z();".
// 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_8470_2_a_z(:prec:=1)
    chi := MakeCharacter_8470_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(3169) - 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_8470_2_a_z();".
// 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_8470_2_a_z( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<13,R![-6, 1]>,<17,R![-2, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;