
// Make newform 8004.2.a.b in Magma, downloaded from the LMFDB on 16 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8004_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_8004_2_a_b();".
// 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_8004_2_a_b();".
// 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_8004_a();"
function MakeCharacter_8004_a()
    N := 8004;
    order := 1;
    char_gens := [4003, 2669, 3133, 553];
    v := [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_8004_a_Hecke(Kf)
    return MakeCharacter_8004_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], [-1], [-2], [2], [4], [-5], [5], [7], [-1], [-1], [8], [-3], [4], [7], [-8], [6], [9], [-2], [8], [-7], [-10], [11], [12], [-3], [-14], [-2], [0], [-12], [-16], [6], [-2], [0], [-23], [20], [-16], [17], [-7], [-8], [15], [-1], [11], [2], [25], [10], [7], [-12], [4], [-29], [6], [-1], [29], [15], [-22], [9], [3], [-16], [4], [4], [11], [16], [-4], [19], [8], [28], [20], [12], [6], [-3], [21], [-21], [-10], [-23], [-32], [36], [21], [6], [15], [10], [6], [-22], [20], [-19], [-6], [-14], [-25], [-4], [12], [2], [24], [23], [7], [4], [-8], [12], [-11], [41], [5], [-6], [-6], [44], [17], [-12], [21], [30], [44], [8], [-23], [17], [-32], [4], [-14], [-18], [10], [15], [42], [-13], [4], [-3], [-4], [33], [24], [-15], [42], [0], [36], [-8], [-20], [-9], [-32], [-22], [-50], [-27], [23], [-5], [25], [31], [14], [-42], [-42], [-26], [44], [33], [-54], [45], [0], [-28], [-34], [-18], [-20], [-24], [-25], [-35], [-11], [34], [29], [9], [-22], [27], [-10], [28], [-28], [48], [52], [-28], [2], [21], [32], [-40], [-6], [-58], [-12], [-5], [8], [51], [32], [-12], [35], [-42], [12], [-20], [-32], [-43], [10], [-43], [42], [16], [18], [-28], [-24], [-52], [1], [50], [-20], [-20], [-47], [-2], [-30], [-28], [-68], [21], [-19], [13], [20], [-47], [19], [43], [35], [-19], [-19], [-16], [-4], [7], [29], [-17], [48], [-60], [49], [-14], [22], [48], [48], [28], [0], [35], [33], [-10], [49], [-4], [-35], [12], [72], [64], [-67], [68], [-62], [36], [-9], [11], [-15], [-68], [-71], [35], [-51], [-10], [46], [-13], [-44], [-20], [-2], [33], [38], [-74], [-19], [10], [-54], [-48], [-54], [60], [-7], [-69], [-16], [69], [-63], [-22], [76], [-7], [-19], [15], [40], [-10], [-65], [82], [28], [-5], [2], [-4], [-48], [10], [-50], [6], [-56], [40], [-24], [-12], [42], [76], [-65], [57], [44], [45], [54], [-48], [-75], [-51], [0], [6], [27], [27], [-52], [18], [41], [54], [-32], [-30], [52], [37], [-18], [-56], [4], [47], [40], [54], [-80], [-69], [30], [12], [-46], [-60], [-18], [0], [64], [-32], [-19], [48], [14], [-46], [20], [63], [20], [76], [66], [1], [91], [54], [-75], [-76], [14], [-18], [4], [-18], [-60], [-74], [56], [-36], [-84], [36], [-86], [-68], [12], [-69], [55], [-35], [22], [-28], [-42], [-42], [51], [-38], [-52], [-53], [-92], [90], [16], [30], [-56], [8], [-8], [78], [43], [-18], [-45], [13], [-35], [-54], [1], [66], [-48], [45], [-40], [13], [7], [10], [44], [-52], [4], [-55], [-28], [14], [7], [61], [-73], [-6], [6], [52], [45], [-20], [70], [-15], [10], [-70], [-66], [62], [23], [-22], [38], [97], [60], [60], [-75], [-57], [13], [102], [0], [80], [51], [-58], [30], [-21], [61], [24], [-88], [-77], [90], [78], [-19], [66], [-90], [-50], [52], [-8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8004_2_a_b();".
// 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_8004_2_a_b(:prec:=1)
    chi := MakeCharacter_8004_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_8004_2_a_b();".
// 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_8004_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![-2, 1]>],Snew);
    return Vf;
end function;