
// Make newform 2184.2.a.c in Magma, downloaded from the LMFDB on 12 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2184_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_2184_2_a_c();".
// 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_2184_2_a_c();".
// 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_2184_a();"
function MakeCharacter_2184_a()
    N := 2184;
    order := 1;
    char_gens := [1639, 1093, 1457, 1249, 2017];
    v := [1, 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_2184_a_Hecke(Kf)
    return MakeCharacter_2184_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], [1], [-4], [1], [-6], [4], [0], [6], [0], [6], [-6], [4], [0], [-2], [4], [-2], [4], [0], [-6], [8], [12], [10], [-14], [6], [16], [12], [14], [2], [-8], [12], [18], [4], [-10], [8], [-2], [20], [-8], [-2], [20], [-10], [-8], [2], [6], [16], [12], [0], [-4], [-26], [-22], [24], [2], [-28], [-6], [-16], [-2], [0], [-26], [18], [4], [-18], [-12], [-24], [-6], [30], [12], [-14], [12], [14], [18], [16], [-8], [-10], [-4], [16], [-18], [-34], [10], [10], [-20], [38], [-24], [18], [16], [-4], [10], [26], [-10], [32], [12], [-32], [8], [12], [-28], [-8], [-26], [34], [20], [30], [44], [-18], [-36], [10], [-28], [18], [-12], [-14], [-32], [26], [-8], [-26], [2], [-20], [-8], [2], [-28], [-8], [-26], [20], [38], [-30], [6], [12], [-28], [22], [6], [48], [0], [30], [-44], [-16], [-40], [38], [42], [-46], [46], [52], [30], [10], [28], [38], [-32], [-4], [30], [24], [38], [2], [36], [24], [-50], [-38], [12], [-24], [-12], [40], [16], [18], [42], [-10], [-12], [-6], [-24], [36], [10], [-56], [24], [-58], [34], [-26], [60], [-18], [-16], [-6], [40], [50], [60], [54], [-16], [-50], [-32], [28], [22], [42], [-24], [22], [-18], [20], [26], [48], [-46], [-36], [60], [22], [-12], [18], [-14], [-18], [50], [24], [30], [8], [22], [-46], [52], [-50], [-64], [-28], [2], [-28], [-46], [-58], [-32], [36], [8], [26], [-8], [58], [16], [-50], [22], [-8], [74], [-16], [-36], [22], [-30], [40], [-64], [44], [-66], [28], [16], [74], [4], [0], [34], [-10], [-68], [-40], [52], [52], [-16], [-18], [2], [56], [32], [-4], [-52], [24], [62], [-78], [0], [26], [78], [4], [6], [76], [-42], [-54], [-8], [20], [38], [-34], [-46], [28], [30], [-54], [12], [-66], [46], [4], [10], [-24], [34], [8], [-20], [-2], [42], [28], [-32], [8], [8], [-10], [28], [56], [-14], [-42], [-40], [-38], [-26], [-52], [-62], [-12], [-50], [-66], [-72], [-42], [36], [68], [-22], [-18], [-24], [52], [-76], [-78], [12], [46], [-80], [-26], [-16], [-58], [-78], [-20], [-48], [-70], [76], [-24], [18], [66], [-68], [-70], [14], [-32], [-14], [34], [-76], [44], [8], [62], [-18], [22], [88], [44], [-52], [52], [-34], [-30], [-38], [-56], [54], [42], [-66], [48], [78], [36], [70], [12], [-16], [6], [-28], [-38], [-34], [-56], [22], [-30], [48], [52], [50], [16], [70], [18], [56], [-52], [-68], [42], [-42], [-56], [-38], [84], [20], [24], [62], [80], [78], [-84], [-72], [50], [-70], [74], [-10], [-62], [88], [-6], [84], [24], [-16], [22], [28], [32], [50], [-50], [-20], [12], [48], [26], [80], [50], [20], [22], [-50], [-46], [-64], [-6], [14], [-48], [46], [90], [-44], [28], [-46], [30], [-84], [-52], [-86], [-18], [-24], [-64], [2], [24], [-10], [22], [72], [-60], [26], [-42], [52], [66], [28], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2184_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2184_2_a_c();".
// 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_2184_2_a_c(:prec:=1)
    chi := MakeCharacter_2184_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_2184_2_a_c();".
// 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_2184_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2184_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![4, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;