
// Make newform 8015.2.a.a 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_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_8015_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_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], [-3], [1], [1], [5], [-1], [3], [2], [6], [5], [2], [-8], [6], [-4], [-1], [6], [0], [10], [8], [12], [-10], [9], [4], [12], [11], [-12], [13], [6], [7], [12], [-10], [14], [16], [-10], [2], [-9], [-6], [-8], [-19], [11], [12], [6], [3], [4], [-10], [-4], [17], [-9], [-21], [1], [4], [-15], [18], [-18], [-22], [-26], [-28], [-6], [-26], [9], [-31], [-21], [-21], [24], [11], [18], [-8], [8], [22], [2], [21], [-24], [17], [32], [4], [-16], [31], [15], [25], [-4], [-16], [-11], [21], [34], [-14], [18], [-9], [22], [-14], [-14], [-21], [-36], [-8], [-13], [-5], [-21], [-18], [-18], [-16], [43], [-8], [-38], [-20], [38], [-20], [-1], [-4], [45], [-33], [-28], [27], [-10], [-36], [-8], [25], [-18], [-7], [0], [14], [7], [-32], [-44], [-33], [44], [-30], [-9], [-19], [-10], [32], [-19], [-29], [16], [15], [2], [30], [-26], [31], [-7], [-39], [27], [38], [45], [-4], [-14], [-28], [-30], [-34], [-38], [-48], [-40], [38], [14], [2], [16], [12], [-20], [23], [16], [17], [-12], [24], [-14], [22], [-42], [-32], [53], [-8], [19], [-5], [26], [-36], [28], [-21], [4], [6], [0], [33], [-5], [16], [-24], [14], [-6], [28], [18], [0], [-30], [56], [35], [-11], [58], [-6], [54], [-65], [36], [28], [65], [-45], [-40], [-3], [-27], [26], [-62], [-10], [6], [-10], [3], [-46], [-44], [6], [22], [48], [-54], [-48], [7], [-54], [24], [28], [22], [-24], [4], [22], [50], [-1], [-4], [33], [-50], [-51], [-45], [23], [-12], [-6], [2], [-32], [43], [35], [-17], [32], [2], [3], [4], [-56], [-70], [-19], [-38], [67], [-28], [-8], [42], [-51], [-48], [34], [0], [-68], [-48], [-62], [-51], [-26], [-4], [51], [43], [-28], [78], [4], [-9], [-18], [52], [39], [16], [-10], [-18], [44], [-62], [-79], [0], [-26], [-71], [40], [22], [11], [-10], [-35], [43], [-24], [-8], [-21], [51], [24], [-38], [8], [26], [23], [-53], [-48], [4], [-48], [46], [40], [-61], [14], [-9], [67], [-36], [-11], [-44], [53], [56], [-48], [-66], [-56], [24], [84], [61], [-62], [72], [-2], [-80], [10], [39], [49], [-15], [44], [20], [70], [20], [-22], [-28], [79], [-20], [90], [-34], [18], [42], [-28], [21], [60], [-77], [35], [-38], [40], [-23], [-16], [52], [-4], [43], [-66], [-4], [-16], [-86], [42], [77], [40], [-26], [69], [91], [65], [46], [-6], [-32], [36], [-24], [64], [16], [92], [-23], [-48], [94], [43], [2], [-93], [-71], [2], [44], [-27], [-42], [36], [24], [15], [43], [-70], [67], [-68], [10], [30], [-54], [-84], [81], [55], [-47], [56], [63], [91], [27], [-9], [28], [69], [90], [-28], [84], [33], [30], [86], [102], [-92], [33], [-56], [74], [8], [27], [0], [-46], [99], [50], [78], [-80], [8], [12], [-57], [8], [78], [-69], [1], [-94], [-82], [-42], [-9], [-69], [-74], [-106], [72], [-48]];
    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_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_8015_2_a_a(: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_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_8015_2_a_a( : 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![3, 1]>],Snew);
    return Vf;
end function;