
// Make newform 5184.2.a.r in Magma, downloaded from the LMFDB on 26 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5184_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_5184_2_a_r();".
// 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_5184_2_a_r();".
// 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_5184_a();"
function MakeCharacter_5184_a()
    N := 5184;
    order := 1;
    char_gens := [2431, 325, 1217];
    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_5184_a_Hecke(Kf)
    return MakeCharacter_5184_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], [0], [0], [2], [3], [-2], [-3], [1], [-6], [-6], [-4], [4], [9], [1], [-6], [-12], [-3], [-8], [-5], [-12], [11], [-4], [-12], [6], [5], [0], [14], [-3], [16], [6], [2], [0], [-3], [19], [6], [-10], [4], [4], [-12], [6], [-12], [-14], [-18], [5], [12], [-10], [-20], [26], [-21], [-14], [3], [6], [-7], [-21], [-21], [18], [24], [20], [10], [6], [4], [-30], [7], [-18], [29], [18], [4], [-1], [-33], [16], [-21], [-18], [-28], [34], [-23], [0], [-18], [-20], [-27], [17], [12], [-20], [-30], [-7], [8], [-3], [9], [17], [30], [20], [15], [-42], [26], [15], [13], [0], [0], [-3], [-20], [4], [1], [30], [39], [45], [37], [11], [9], [6], [12], [-37], [-28], [16], [27], [-35], [-40], [-3], [-23], [18], [6], [36], [4], [-22], [-36], [9], [-8], [-30], [4], [-36], [26], [-14], [-47], [6], [8], [-2], [42], [2], [-18], [4], [12], [33], [7], [18], [-16], [12], [10], [12], [-26], [-42], [-35], [24], [-8], [-42], [19], [0], [31], [48], [38], [-6], [14], [60], [-3], [9], [-22], [36], [-51], [0], [-16], [28], [2], [0], [3], [40], [36], [11], [38], [30], [1], [-54], [-28], [-8], [38], [9], [22], [27], [54], [0], [16], [-56], [29], [60], [-1], [15], [-59], [-36], [-33], [3], [47], [4], [-9], [6], [30], [-58], [-56], [35], [-21], [66], [-64], [27], [27], [-17], [-67], [-30], [62], [-36], [24], [-55], [68], [21], [66], [-48], [34], [-40], [-27], [38], [-51], [22], [-6], [12], [-34], [-24], [-26], [52], [-22], [-9], [16], [42], [-70], [18], [57], [-42], [45], [-20], [-4], [-38], [-33], [-48], [-28], [-45], [-68], [-54], [22], [27], [12], [26], [42], [39], [16], [28], [0], [-43], [32], [15], [4], [-14], [-18], [55], [12], [75], [-41], [30], [28], [-68], [35], [50], [23], [20], [-36], [10], [-25], [27], [-18], [-58], [-30], [-26], [49], [-18], [26], [-48], [-28], [63], [24], [-39], [-69], [-51], [64], [-66], [50], [-54], [36], [-68], [-7], [18], [-16], [36], [-8], [2], [-60], [40], [-6], [-80], [-48], [72], [66], [16], [-12], [11], [-33], [12], [-37], [6], [-35], [29], [-84], [2], [18], [41], [25], [-71], [-60], [-30], [-80], [-12], [-16], [-21], [-47], [-84], [-2], [63], [23], [20], [-32], [6], [12], [20], [18], [33], [52], [-17], [-72], [78], [28], [89], [78], [-28], [76], [-42], [-84], [-60], [-6], [6], [22], [15], [-72], [-45], [-53], [-25], [72], [26], [-19], [-60], [85], [36], [36], [38], [46], [-87], [-54], [-10], [69], [26], [-42], [-15], [-40], [-45], [-65], [60], [-10], [-80], [-80], [-12], [-1], [-36], [-33], [97], [66], [-58], [32], [-51], [-41], [72], [-14], [-39], [-76], [42], [-18], [20], [94], [69], [-59], [-36], [71], [102], [33], [-17], [62], [-6], [66], [14], [21], [0], [-60], [-20], [-36], [-51], [-70], [60], [39], [-54], [52], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5184_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5184_2_a_r();".
// 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_5184_2_a_r(:prec:=1)
    chi := MakeCharacter_5184_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_5184_2_a_r();".
// 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_5184_2_a_r( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5184_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![-2, 1]>,<11,R![-3, 1]>],Snew);
    return Vf;
end function;