
// Make newform 4641.2.a.b in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4641_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_4641_2_a_b();".
// 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_4641_2_a_b();".
// 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_4641_a();"
function MakeCharacter_4641_a()
    N := 4641;
    order := 1;
    char_gens := [3095, 3979, 3928, 547];
    v := [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_4641_a_Hecke(Kf)
    return MakeCharacter_4641_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], [-1], [-2], [-1], [-4], [1], [1], [-4], [0], [6], [0], [-2], [2], [12], [0], [-10], [12], [-10], [4], [8], [10], [-8], [4], [10], [2], [6], [0], [4], [6], [-6], [-16], [12], [-6], [-12], [22], [8], [14], [-12], [-8], [-2], [-4], [-18], [-24], [10], [22], [-16], [-4], [8], [-12], [6], [-14], [-8], [-14], [-4], [2], [-16], [14], [8], [-26], [10], [4], [-26], [-20], [0], [-14], [-18], [-20], [18], [-12], [-2], [2], [-16], [-24], [-10], [-4], [0], [22], [30], [18], [-22], [12], [-10], [16], [18], [-32], [4], [-14], [10], [30], [-32], [20], [0], [-40], [4], [36], [16], [-2], [-6], [4], [6], [-20], [46], [20], [10], [-12], [18], [44], [-30], [0], [-30], [8], [-26], [-6], [-20], [-40], [10], [4], [8], [6], [-36], [-10], [-14], [-26], [-4], [-12], [30], [-50], [24], [32], [30], [4], [8], [40], [-10], [10], [18], [6], [20], [30], [-30], [-4], [-10], [-16], [-4], [-18], [-8], [-10], [-38], [20], [24], [-26], [-30], [4], [-48], [-12], [-24], [-8], [42], [26], [6], [-44], [-22], [8], [-52], [34], [-24], [8], [30], [-38], [6], [-60], [-2], [-32], [26], [8], [26], [28], [-42], [64], [30], [-16], [-36], [22], [58], [8], [22], [62], [-4], [50], [-48], [-6], [-36], [52], [30], [44], [26], [58], [14], [-22], [-40], [-34], [-8], [22], [2], [-36], [30], [40], [-36], [-6], [-12], [50], [54], [16], [12], [24], [-22], [-16], [18], [8], [-34], [38], [-56], [-46], [64], [36], [22], [10], [-24], [0], [12], [-50], [28], [-32], [10], [-12], [-16], [-14], [-42], [4], [-40], [-28], [-12], [-16], [46], [-22], [16], [-64], [-60], [-36], [-72], [-2], [10], [16], [-46], [-50], [-52], [-74], [60], [70], [-22], [8], [60], [-58], [6], [58], [-36], [-18], [-70], [-52], [54], [-10], [-28], [74], [-48], [-30], [-80], [68], [46], [-6], [-60], [-80], [56], [48], [-58], [12], [-72], [-30], [-42], [16], [-30], [-58], [-4], [74], [28], [-50], [-2], [56], [54], [4], [44], [-70], [-18], [-40], [12], [-12], [-70], [52], [54], [-48], [-58], [56], [-10], [-70], [4], [-64], [58], [-44], [-32], [18], [18], [-4], [-14], [-2], [16], [-6], [2], [-44], [76], [-40], [-18], [-74], [-58], [40], [-60], [-52], [28], [-82], [74], [-38], [-24], [-58], [-86], [78], [24], [-66], [12], [-18], [92], [-48], [54], [-20], [34], [78], [8], [86], [-6], [-64], [28], [-6], [96], [-82], [26], [-72], [-84], [-4], [42], [-42], [40], [-6], [-68], [20], [0], [-42], [0], [94], [-60], [-8], [-14], [-78], [26], [54], [-70], [88], [18], [-76], [-72], [80], [-26], [76], [-80], [26], [-2], [4], [28], [48], [58], [-56], [42], [4], [86], [-66], [-78], [-16], [82], [-26], [-48], [46], [2], [28], [68], [10], [-26], [-12], [-52], [-22], [30], [-48], [0], [58], [40], [-66], [22], [0], [76], [-38], [78], [-36], [-6], [84], [-80]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4641_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4641_2_a_b();".
// 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_4641_2_a_b(:prec:=1)
    chi := MakeCharacter_4641_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_4641_2_a_b();".
// 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_4641_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4641_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<5,R![2, 1]>],Snew);
    return Vf;
end function;