
// Make newform 8470.2.a.x in Magma, downloaded from the LMFDB on 25 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_x();".
// 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_x();".
// 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], [-4], [4], [4], [-2], [-4], [2], [6], [-10], [-4], [0], [-10], [6], [-2], [14], [-12], [0], [8], [-16], [-14], [-14], [6], [0], [-4], [-8], [-2], [0], [4], [-18], [12], [4], [-20], [-10], [-2], [12], [-24], [-20], [-2], [12], [10], [-18], [-10], [-20], [20], [-12], [-26], [18], [20], [-10], [-2], [-18], [16], [22], [0], [10], [0], [-4], [-8], [16], [-22], [-22], [2], [-24], [-2], [12], [-18], [-14], [24], [32], [22], [-4], [-28], [22], [-2], [2], [-10], [-30], [6], [32], [14], [8], [34], [6], [-26], [26], [22], [4], [0], [10], [28], [-4], [-20], [6], [-10], [-16], [20], [-12], [-42], [4], [20], [16], [-38], [28], [-16], [-36], [22], [-12], [-6], [-18], [-10], [32], [-10], [4], [-24], [6], [24], [-6], [6], [20], [22], [18], [-12], [-46], [-10], [-16], [4], [-20], [-8], [-12], [34], [-26], [-22], [-14], [4], [-18], [36], [44], [48], [-26], [20], [18], [-14], [-32], [16], [6], [6], [26], [-58], [-46], [8], [54], [40], [-40], [6], [-28], [18], [-38], [14], [40], [-22], [38], [-48], [-8], [48], [52], [58], [4], [2], [36], [58], [34], [-50], [-20], [-54], [44], [62], [34], [-36], [34], [0], [18], [14], [22], [56], [44], [-48], [-50], [-60], [-32], [-54], [-12], [30], [-32], [54], [12], [-64], [-8], [-8], [22], [66], [-2], [-30], [14], [-44], [-20], [46], [50], [46], [22], [24], [-24], [26], [32], [-42], [-34], [14], [-52], [64], [-62], [-18], [48], [-40], [70], [-36], [40], [-24], [74], [-36], [-52], [52], [-24], [28], [22], [74], [76], [-62], [-2], [-52], [76], [50], [-24], [-8], [28], [42], [-20], [48], [42], [-42], [-26], [34], [-12], [36], [-38], [8], [-6], [-56], [-16], [-44], [-50], [-28], [-26], [-6], [-18], [30], [60], [54], [-14], [38], [54], [40], [28], [-52], [-30], [16], [52], [-60], [-76], [-72], [36], [46], [-80], [-48], [-34], [-6], [66], [42], [-30], [-8], [-14], [-12], [10], [-38], [-82], [-30], [28], [-60], [-36], [-6], [-52], [-6], [76], [-46], [2], [10], [-24], [38], [40], [42], [-12], [38], [64], [-86], [30], [64], [-2], [-36], [60], [-34], [54], [-14], [6], [-6], [-8], [44], [-32], [82], [-64], [2], [-56], [-12], [8], [-72], [-62], [80], [10], [-60], [50], [18], [-42], [4], [78], [-88], [-2], [2], [-16], [70], [-44], [42], [-34], [-44], [-32], [-4], [-26], [-92], [6], [78], [-22], [-62], [-42], [-68], [-48], [-26], [80], [-24], [28], [40], [50], [56], [28], [-96], [58], [22], [-28], [40], [-6], [-16], [-54], [66], [-72], [18], [12], [16], [64], [-22], [36], [12], [2], [-102], [-68], [76], [-48], [34], [56], [-30], [56], [-12], [90], [6], [-88], [58], [14], [0], [-86], [72], [20], [30], [-44], [62], [-38], [92], [94], [-22], [80], [16], [-30], [-40], [-98], [-8], [-2], [12], [-6], [2], [-74], [-36], [-22], [-24], [74], [20], [-64], [-84], [-30], [-42], [-96], [-54], [-58], [-16], [84], [58], [-72], [-12], [22], [-42], [44], [84], [86]];
    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_x();".
// 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_x(: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_x();".
// 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_x( : 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![4, 1]>,<17,R![-4, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;