
// Make newform 2205.2.a.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2205_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_2205_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_2205_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_2205_a();"
function MakeCharacter_2205_a()
    N := 2205;
    order := 1;
    char_gens := [1226, 442, 1081];
    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_2205_a_Hecke(Kf)
    return MakeCharacter_2205_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], [0], [1], [0], [2], [-6], [-6], [6], [4], [-8], [6], [-6], [-6], [0], [0], [-2], [-12], [0], [4], [14], [6], [8], [-12], [6], [-6], [-18], [0], [8], [-6], [-2], [-20], [-12], [-14], [-18], [-4], [0], [-18], [16], [-12], [-6], [-22], [12], [-14], [-6], [-22], [6], [12], [12], [-12], [-24], [-10], [14], [12], [12], [-18], [16], [-6], [-30], [26], [20], [-24], [18], [-12], [24], [-6], [22], [12], [-30], [-28], [24], [-6], [10], [12], [22], [36], [-36], [-4], [18], [16], [-24], [-12], [10], [22], [-30], [30], [20], [16], [-18], [18], [-4], [-12], [24], [12], [-14], [36], [36], [-30], [-18], [-12], [2], [-28], [-10], [24], [-40], [4], [42], [-24], [18], [-14], [-12], [12], [-38], [14], [30], [16], [-28], [36], [-12], [34], [-50], [-24], [6], [18], [4], [18], [-8], [6], [0], [36], [42], [20], [16], [-24], [-42], [6], [-12], [-30], [12], [18], [-4], [30], [8], [-4], [40], [-12], [-48], [-6], [-18], [-6], [4], [-22], [18], [-12], [-48], [12], [-22], [24], [30], [54], [-18], [20], [-34], [32], [36], [38], [0], [48], [-30], [2], [18], [10], [12], [38], [-46], [18], [6], [-28], [-28], [12], [12], [-8], [-60], [-50], [42], [-8], [-42], [18], [36], [-50], [0], [-6], [32], [-12], [18], [44], [-6], [-30], [-26], [-42], [-12], [8], [-42], [-18], [-36], [-12], [-30], [42], [-40], [20], [-6], [38], [-6], [-24], [36], [72], [-12], [-36], [-30], [8], [-62], [-42], [6], [20], [0], [-48], [-18], [6], [-38], [72], [58], [6], [6], [72], [-44], [-60], [72], [24], [-70], [26], [-24], [-32], [54], [24], [-30], [30], [48], [12], [36], [76], [4], [42], [66], [-28], [-72], [18], [-14], [2], [24], [42], [-30], [-44], [52], [72], [30], [18], [-6], [40], [30], [-68], [2], [-60], [-24], [-54], [0], [18], [36], [40], [30], [-6], [36], [48], [-40], [24], [-12], [36], [26], [-66], [22], [-42], [30], [56], [36], [62], [-60], [-10], [66], [18], [30], [-84], [-60], [-18], [58], [16], [-4], [-76], [42], [-16], [-48], [-2], [-18], [-48], [-80], [40], [12], [-76], [-60], [-36], [-22], [18], [-32], [-78], [54], [66], [-48], [-46], [-36], [-52], [-72], [-28], [-10], [30], [58], [90], [0], [20], [-24], [2], [54], [12], [-12], [-82], [22], [-30], [48], [-46], [-10], [-48], [12], [24], [18], [30], [38], [64], [48], [-86], [-30], [0], [-84], [70], [-8], [46], [-30], [-88], [-50], [12], [-90], [-18], [16], [-6], [50], [-54], [-56], [-44], [18], [74], [-60], [2], [-54], [6], [-42], [-42], [14], [36], [-86], [6], [36], [48], [-54], [20], [72], [-10], [78], [-2], [-84], [-58], [66], [-54], [-6], [4], [16], [36], [14], [-32], [30], [6], [-78], [10], [16], [24], [-12], [54], [62], [-52], [60], [50], [-6], [86], [36], [-102], [-96], [28], [42], [-8], [12], [66], [-78], [64], [56], [-6], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2205_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2205_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_2205_2_a_c(:prec:=1)
    chi := MakeCharacter_2205_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_2205_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_2205_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2205_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<11,R![-2, 1]>,<13,R![6, 1]>,<17,R![6, 1]>],Snew);
    return Vf;
end function;