
// Make newform 8015.2.a.f in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8015_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_8015_2_a_f();".
// 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_8015_2_a_f();".
// 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_8015_a();"
function MakeCharacter_8015_a()
    N := 8015;
    order := 1;
    char_gens := [3207, 4581, 4586];
    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_8015_a_Hecke(Kf)
    return MakeCharacter_8015_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 := [[2], [1], [1], [1], [-3], [3], [-1], [-6], [-2], [-3], [-2], [-4], [-6], [-8], [3], [2], [-8], [-10], [8], [12], [-2], [9], [-4], [12], [7], [8], [9], [14], [-1], [8], [-6], [-18], [-8], [-2], [-6], [15], [2], [4], [-15], [-9], [-12], [-6], [-13], [4], [10], [-20], [-23], [-13], [7], [-1], [-20], [25], [26], [30], [-30], [-18], [24], [22], [-2], [-23], [-3], [-9], [31], [20], [-17], [2], [24], [-28], [30], [2], [-15], [32], [-27], [-24], [12], [-24], [-33], [11], [-39], [-24], [0], [29], [5], [-38], [-14], [-6], [-1], [26], [14], [-10], [-1], [16], [-4], [19], [-13], [-33], [-26], [30], [8], [-29], [8], [-34], [-12], [6], [-20], [11], [36], [33], [-33], [8], [-1], [38], [-24], [44], [-7], [30], [-11], [32], [30], [-1], [32], [-12], [-5], [36], [-26], [-17], [13], [6], [-40], [-15], [-29], [16], [7], [-42], [22], [14], [-37], [37], [13], [27], [6], [-3], [4], [10], [-52], [-14], [-50], [26], [-8], [-48], [22], [2], [14], [8], [-24], [-36], [-9], [16], [-35], [28], [48], [14], [-34], [-14], [-12], [-39], [0], [-33], [-29], [-30], [44], [-8], [35], [-24], [46], [-24], [1], [-45], [-8], [40], [2], [14], [0], [50], [-24], [50], [-16], [7], [45], [-10], [10], [14], [15], [36], [-36], [37], [-61], [24], [57], [9], [-6], [-62], [-2], [38], [-42], [47], [-46], [-60], [6], [42], [32], [18], [-32], [-13], [-30], [0], [-48], [-6], [28], [-72], [14], [10], [-41], [-64], [61], [-26], [49], [11], [-13], [4], [38], [50], [0], [11], [-17], [27], [-72], [38], [-45], [-40], [-24], [54], [-31], [66], [-73], [-64], [8], [-38], [-59], [4], [58], [-12], [-12], [64], [-6], [5], [22], [60], [-65], [39], [16], [-34], [-4], [43], [-2], [-24], [15], [76], [-14], [6], [-36], [-58], [-75], [64], [78], [-19], [48], [6], [-37], [46], [-15], [67], [16], [28], [-25], [-13], [4], [42], [40], [-26], [-49], [-33], [24], [12], [-44], [58], [-28], [-17], [-70], [27], [-9], [0], [21], [-36], [29], [-36], [60], [50], [-72], [-32], [-68], [-59], [-30], [36], [-66], [88], [-6], [-25], [-19], [33], [72], [-28], [-66], [-36], [58], [-52], [47], [52], [54], [-86], [-14], [94], [40], [-7], [-4], [-17], [-61], [-62], [-68], [-27], [4], [68], [72], [-21], [6], [36], [-4], [-90], [-6], [33], [-96], [66], [-19], [31], [-55], [-58], [34], [12], [48], [52], [-80], [-56], [92], [73], [-56], [-58], [63], [34], [-53], [-63], [-2], [44], [5], [86], [-32], [72], [-73], [7], [-54], [-1], [-32], [-78], [26], [58], [-100], [21], [-25], [-11], [16], [3], [19], [-9], [31], [-52], [85], [30], [52], [-44], [81], [-74], [2], [66], [-16], [29], [0], [2], [-20], [-85], [56], [22], [-97], [2], [-42], [96], [-52], [-44], [-17], [16], [-66], [87], [81], [18], [78], [18], [75], [39], [74], [-18], [64], [-64]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8015_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8015_2_a_f();".
// 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_8015_2_a_f(:prec:=1)
    chi := MakeCharacter_8015_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_8015_2_a_f();".
// 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_8015_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8015_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-2, 1]>,<3,R![-1, 1]>],Snew);
    return Vf;
end function;