
// Make newform 8470.2.a.h 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_h();".
// 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_h();".
// 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], [6], [6], [8], [-6], [-8], [-4], [10], [4], [-10], [0], [-4], [-8], [-6], [8], [10], [2], [-4], [6], [16], [4], [10], [4], [-2], [-2], [20], [6], [-18], [-2], [6], [-10], [-10], [-22], [8], [14], [0], [-22], [-8], [10], [18], [-20], [-20], [-10], [-8], [-6], [-18], [-10], [10], [-12], [12], [4], [-18], [-20], [-10], [0], [24], [-22], [-16], [-12], [28], [-8], [-4], [2], [-12], [-12], [-4], [-14], [2], [18], [16], [-38], [-38], [-2], [-18], [30], [-20], [-34], [18], [4], [-28], [-6], [-34], [-34], [-16], [32], [-36], [-20], [-20], [32], [36], [0], [6], [30], [16], [-30], [-28], [-18], [36], [-40], [-36], [32], [-12], [-14], [24], [-42], [-8], [6], [-38], [-20], [-8], [-30], [-16], [46], [-24], [-24], [-26], [34], [-30], [42], [28], [22], [34], [-40], [-6], [-34], [20], [28], [8], [44], [46], [22], [46], [-4], [22], [-16], [26], [22], [44], [-20], [38], [-24], [22], [-6], [-4], [-4], [-46], [2], [54], [32], [-26], [0], [-10], [-34], [-22], [12], [-58], [-34], [20], [28], [-42], [42], [32], [22], [8], [-22], [-24], [62], [-46], [-18], [-36], [-30], [40], [-34], [16], [28], [4], [26], [44], [-30], [-12], [14], [-2], [-4], [-64], [28], [-60], [-20], [-52], [-34], [-68], [0], [-8], [24], [-2], [24], [38], [28], [22], [-6], [-12], [50], [44], [4], [-60], [-4], [30], [26], [52], [-36], [-36], [46], [68], [-18], [16], [-16], [2], [-24], [18], [-48], [12], [-70], [60], [24], [40], [-56], [24], [-74], [2], [-12], [16], [-48], [-58], [6], [-24], [-12], [38], [22], [6], [-10], [14], [8], [-22], [52], [20], [-28], [-2], [-38], [-36], [-6], [62], [4], [-38], [-68], [34], [-54], [36], [-36], [40], [-22], [4], [44], [2], [30], [60], [-54], [-14], [18], [-76], [-30], [62], [58], [10], [34], [-52], [10], [56], [-32], [64], [-36], [80], [-48], [66], [46], [16], [-2], [30], [-68], [-26], [42], [-30], [-72], [-52], [10], [42], [-40], [6], [46], [2], [-6], [-64], [34], [-18], [30], [-24], [-58], [40], [-58], [-8], [-22], [56], [-14], [20], [66], [28], [-84], [10], [-34], [-64], [-24], [54], [14], [32], [-36], [-48], [38], [-26], [-88], [16], [-48], [-88], [-12], [-22], [30], [-50], [-80], [20], [18], [32], [24], [-12], [48], [78], [-58], [-24], [-90], [14], [-18], [66], [-16], [-78], [-26], [-56], [62], [-66], [-32], [-38], [22], [-72], [88], [-28], [74], [-50], [-16], [-8], [-20], [20], [-56], [22], [56], [-32], [92], [38], [-50], [66], [6], [46], [6], [72], [-78], [-42], [6], [-96], [58], [44], [-18], [-78], [-82], [-28], [36], [72], [-34], [44], [30], [-24], [102], [60], [66], [88], [-72], [-24], [100], [-16], [-52], [-60], [20], [-46], [78], [42], [68], [-94], [78], [-50], [46], [-20], [0], [22], [-62], [-32], [-102], [84], [-18], [-54], [-4], [28], [4], [-46], [100], [-84], [-54], [10], [-62], [16], [-34], [-18], [16], [84], [-42], [82], [102], [-22], [-18], [-4], [56], [-14]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : 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![-6, 1]>,<19,R![-6, 1]>],Snew);
    return Vf;
end function;