
// Make newform 8007.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8007_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_8007_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_8007_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_8007_a();"
function MakeCharacter_8007_a()
    N := 8007;
    order := 1;
    char_gens := [5339, 1414, 7855];
    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_8007_a_Hecke(Kf)
    return MakeCharacter_8007_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], [-1], [0], [2], [4], [2], [1], [-4], [-6], [-4], [0], [-2], [0], [4], [0], [-2], [4], [12], [-4], [8], [-4], [-10], [-16], [-6], [-12], [2], [8], [-14], [14], [6], [0], [14], [10], [2], [-6], [8], [1], [-14], [-20], [-6], [-4], [-24], [8], [10], [-22], [-4], [-2], [-4], [10], [6], [10], [24], [-16], [4], [14], [-24], [-32], [-20], [-2], [6], [-24], [-14], [-8], [-8], [6], [-22], [-12], [16], [4], [-26], [14], [4], [-10], [-38], [10], [-36], [6], [-20], [4], [-14], [-16], [10], [24], [18], [10], [-24], [-16], [10], [18], [8], [-12], [-30], [28], [0], [-30], [26], [-6], [32], [44], [8], [8], [14], [24], [-30], [-12], [10], [24], [18], [-32], [2], [-22], [-2], [-18], [40], [24], [-6], [-46], [40], [34], [12], [14], [-44], [18], [34], [6], [-42], [-34], [-30], [-40], [46], [-28], [36], [14], [-2], [-22], [14], [-26], [-22], [-6], [40], [46], [-10], [-54], [-8], [26], [30], [-22], [0], [44], [28], [8], [-44], [-8], [38], [-32], [-40], [-8], [-50], [22], [6], [14], [-34], [0], [-52], [2], [-46], [36], [40], [10], [14], [-16], [2], [20], [-34], [20], [-36], [28], [52], [-16], [-50], [-24], [54], [24], [18], [0], [26], [-32], [-20], [6], [8], [-16], [48], [24], [14], [-30], [0], [-12], [48], [42], [24], [50], [-16], [-22], [-62], [52], [10], [52], [12], [-38], [-4], [-48], [-6], [4], [4], [58], [-4], [-8], [30], [-8], [46], [-50], [-20], [18], [12], [12], [-2], [36], [42], [-28], [42], [-46], [24], [-32], [22], [-40], [-24], [62], [-8], [76], [68], [12], [-28], [-8], [54], [-14], [0], [-56], [48], [-20], [-40], [-14], [-22], [-8], [-2], [18], [60], [-50], [-28], [58], [-6], [36], [16], [-38], [36], [-18], [-52], [6], [14], [-36], [-6], [58], [-32], [-10], [12], [-74], [-48], [-28], [50], [62], [64], [-24], [-42], [64], [34], [40], [-40], [10], [60], [-4], [-50], [-6], [58], [-66], [-72], [10], [-48], [72], [54], [54], [24], [50], [-30], [0], [-6], [24], [60], [44], [-70], [-36], [-38], [78], [-4], [26], [-36], [-48], [10], [-44], [22], [-20], [-18], [-40], [28], [-14], [88], [36], [10], [4], [46], [84], [8], [32], [-54], [-54], [32], [-26], [2], [-26], [-66], [52], [48], [-94], [-26], [42], [-80], [46], [68], [42], [44], [-40], [-4], [-60], [-20], [-30], [-42], [-58], [-30], [-56], [-28], [-36], [-48], [-82], [-14], [-12], [18], [80], [-30], [98], [-64], [2], [36], [96], [44], [6], [-72], [52], [12], [-38], [-14], [-6], [-38], [66], [58], [10], [-46], [88], [18], [64], [-26], [8], [48], [52], [-12], [-20], [28], [56], [-10], [-8], [26], [14], [-58], [-36], [50], [-68], [62], [-70], [60], [-10], [-42], [104], [42], [28], [-58], [76], [-28], [-50], [90], [-30], [26], [14], [28], [54], [-32], [32], [92], [-10], [-62], [-74], [-82], [-84], [96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8007_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8007_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_8007_2_a_a(:prec:=1)
    chi := MakeCharacter_8007_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_8007_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_8007_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8007_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>],Snew);
    return Vf;
end function;