
// Make newform 8470.2.a.s 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_s();".
// 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_s();".
// 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], [-2], [-1], [-1], [0], [2], [2], [8], [-6], [8], [-8], [-10], [0], [4], [-6], [-2], [4], [-4], [2], [8], [-6], [4], [4], [-2], [-2], [-8], [-14], [20], [12], [-18], [8], [-20], [6], [4], [-4], [16], [10], [18], [0], [14], [20], [26], [20], [22], [-6], [-24], [12], [-2], [12], [6], [26], [20], [0], [8], [-22], [0], [18], [8], [-18], [12], [-12], [-6], [20], [-12], [-22], [-22], [-8], [22], [20], [4], [26], [-4], [-22], [-22], [-8], [18], [-2], [10], [10], [20], [24], [-6], [12], [-10], [-40], [26], [2], [-22], [-32], [-22], [-2], [-20], [-2], [-8], [-24], [24], [30], [42], [36], [20], [4], [-22], [36], [32], [16], [6], [-26], [-6], [8], [0], [32], [10], [22], [0], [48], [2], [14], [38], [30], [36], [30], [-18], [-6], [-18], [-8], [24], [-38], [-36], [-42], [-26], [20], [-40], [24], [2], [28], [16], [6], [28], [26], [-12], [40], [8], [54], [-12], [2], [-16], [14], [-54], [-40], [6], [-14], [-6], [2], [0], [2], [4], [-16], [18], [-42], [-24], [-14], [-6], [-56], [44], [-42], [42], [52], [-34], [-12], [-34], [12], [-34], [48], [58], [44], [-26], [56], [-26], [24], [-8], [-46], [16], [-14], [-26], [-54], [-46], [-30], [-38], [24], [-44], [-50], [-44], [4], [6], [-12], [-46], [12], [-2], [-10], [48], [-40], [68], [38], [-40], [40], [42], [8], [68], [-60], [-4], [26], [-42], [-18], [-2], [36], [-34], [-8], [-68], [-38], [-18], [-48], [68], [-2], [-18], [36], [60], [-18], [-4], [40], [-60], [26], [-60], [20], [-44], [-22], [32], [30], [30], [-40], [40], [10], [28], [-46], [-2], [6], [12], [-30], [-24], [28], [-32], [22], [20], [58], [22], [14], [28], [-2], [28], [34], [-66], [-56], [68], [-28], [2], [42], [52], [66], [22], [-36], [54], [-14], [-62], [-2], [56], [-50], [-58], [-38], [-80], [60], [-84], [32], [-76], [0], [40], [-84], [-36], [-34], [6], [-76], [44], [-66], [10], [58], [44], [-10], [32], [-36], [14], [-36], [-52], [-2], [-30], [8], [-66], [-64], [74], [-54], [50], [80], [18], [24], [42], [40], [-58], [72], [-40], [72], [-24], [-18], [-72], [60], [-58], [36], [34], [66], [-62], [-28], [6], [72], [-30], [16], [-30], [76], [-20], [52], [58], [-26], [54], [-70], [56], [-46], [58], [-16], [52], [62], [48], [-78], [70], [68], [-34], [64], [10], [94], [-16], [8], [66], [48], [-60], [-94], [30], [-2], [-12], [-10], [40], [10], [86], [-18], [-64], [20], [20], [-56], [72], [-84], [-12], [-30], [24], [-48], [-90], [20], [22], [34], [50], [80], [94], [-56], [-14], [96], [98], [-92], [14], [46], [6], [0], [-10], [-24], [-14], [-96], [-86], [36], [96], [-32], [6], [-88], [-38], [16], [-8], [26], [56], [74], [-84], [86], [78], [-54], [-80], [6], [78], [32], [-58], [-38], [-48], [-34], [-62], [50], [8], [34], [14], [-54], [72], [-16], [-60], [-10], [-48], [40], [54], [-54], [-18], [44], [106], [82], [-72], [-66], [30], [-4], [-16], [-100], [58], [28], [48], [106]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8470_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<13,R![-2, 1]>,<17,R![-2, 1]>,<19,R![-8, 1]>],Snew);
    return Vf;
end function;