
// Make newform 8008.2.a.a in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8008_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_8008_2_a_a();".
// 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_8008_2_a_a();".
// 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_8008_a();"
function MakeCharacter_8008_a()
    N := 8008;
    order := 1;
    char_gens := [6007, 4005, 3433, 4369, 4929];
    v := [1, 1, 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_8008_a_Hecke(Kf)
    return MakeCharacter_8008_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 := [[0], [-2], [-1], [1], [-1], [-1], [-6], [-7], [1], [-5], [-7], [-2], [6], [1], [7], [-13], [0], [6], [-14], [-6], [-3], [-11], [-13], [9], [17], [16], [8], [-4], [16], [1], [8], [-8], [-18], [0], [-20], [-10], [2], [-4], [-19], [-18], [-7], [-12], [0], [2], [-2], [-18], [-15], [-13], [24], [10], [15], [20], [17], [12], [28], [11], [24], [-8], [-15], [-32], [0], [-21], [11], [-4], [-8], [4], [26], [13], [16], [-3], [26], [2], [-8], [-18], [-24], [24], [26], [-7], [20], [-23], [32], [26], [30], [40], [-2], [-39], [12], [22], [30], [-16], [-36], [-27], [-14], [-24], [34], [30], [-33], [12], [10], [14], [7], [16], [-32], [27], [-31], [-14], [7], [39], [-27], [-18], [28], [-28], [14], [-28], [-22], [17], [-44], [12], [-14], [11], [-39], [37], [12], [40], [-21], [5], [28], [36], [-2], [-19], [-36], [36], [-39], [-7], [27], [-5], [26], [5], [2], [5], [-20], [38], [48], [4], [-40], [12], [-45], [-30], [-6], [-36], [18], [30], [0], [-8], [53], [-5], [0], [-29], [-2], [5], [-42], [5], [32], [30], [-8], [-35], [0], [28], [-36], [-2], [16], [-14], [-17], [34], [-36], [8], [24], [26], [40], [-28], [-20], [-16], [-26], [19], [44], [0], [-57], [-35], [22], [-27], [54], [-54], [-27], [63], [43], [-10], [-56], [41], [-27], [-66], [10], [-10], [13], [-40], [-33], [-12], [45], [9], [10], [-10], [-13], [-58], [0], [48], [-21], [-21], [5], [-60], [-32], [-6], [25], [-16], [-60], [34], [10], [-65], [-28], [15], [-4], [-32], [-2], [70], [-56], [-70], [10], [39], [34], [-60], [3], [66], [8], [-10], [-4], [-74], [17], [34], [63], [-39], [-30], [15], [-36], [2], [38], [-58], [-46], [-36], [3], [-28], [64], [-23], [25], [19], [53], [68], [78], [-30], [-6], [57], [-40], [-78], [20], [30], [-46], [39], [46], [-16], [-18], [14], [-40], [56], [10], [22], [80], [9], [52], [-27], [-5], [84], [20], [10], [7], [-72], [54], [37], [17], [-24], [-38], [-76], [6], [77], [-84], [46], [1], [-31], [-45], [-8], [52], [-46], [-70], [31], [-62], [24], [-78], [-61], [-78], [-48], [61], [56], [51], [-23], [-80], [-12], [48], [32], [-72], [86], [-80], [-1], [48], [26], [16], [71], [-60], [39], [-20], [-7], [-50], [-42], [-42], [-6], [-18], [22], [-26], [-12], [30], [-20], [28], [26], [-51], [84], [53], [52], [-30], [66], [38], [-40], [35], [-53], [-38], [-20], [-78], [66], [-31], [-62], [-32], [79], [55], [-48], [27], [-52], [-64], [-14], [15], [-10], [79], [56], [0], [-62], [-36], [34], [-33], [38], [-22], [-70], [-49], [-57], [-82], [92], [-70], [-42], [38], [-11], [48], [-16], [4], [77], [48], [-62], [43], [96], [51], [35], [48], [-43], [-5], [-38], [84], [82], [-9], [-1], [-96], [89], [75], [-4], [-33], [-74], [12], [40], [-30], [2], [7], [90], [78], [70], [106], [15], [-11], [-92], [-49], [10]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8008_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8008_2_a_a();".
// 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_8008_2_a_a(:prec:=1)
    chi := MakeCharacter_8008_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(2999) - 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_8008_2_a_a();".
// 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_8008_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8008_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![1, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;