
// Make newform 4641.2.a.c in Magma, downloaded from the LMFDB on 23 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_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_4641_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_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], [-8], [-6], [-8], [2], [6], [-4], [-12], [6], [0], [-14], [8], [-8], [2], [8], [0], [-14], [10], [6], [4], [-4], [-6], [6], [8], [-12], [-22], [-20], [6], [-20], [22], [-12], [24], [-18], [24], [10], [-12], [6], [-10], [-24], [-12], [-8], [4], [6], [14], [8], [-22], [12], [18], [4], [-10], [0], [-10], [26], [-20], [-2], [12], [8], [-10], [-2], [-24], [18], [4], [-34], [18], [-32], [8], [-10], [-12], [-20], [-18], [-2], [10], [10], [28], [6], [24], [-14], [-8], [16], [-30], [-22], [30], [-20], [12], [-24], [-16], [-8], [-4], [-16], [-26], [18], [-16], [-30], [12], [-18], [-20], [18], [-20], [34], [-32], [-6], [12], [-2], [40], [-2], [-6], [4], [36], [6], [44], [-8], [-30], [-24], [-10], [42], [30], [-36], [-44], [-34], [42], [32], [-4], [-34], [24], [-8], [-32], [22], [-30], [18], [30], [-12], [30], [6], [12], [30], [-32], [-36], [-26], [16], [14], [-54], [-8], [-16], [26], [42], [-28], [-24], [4], [0], [-8], [54], [34], [10], [60], [2], [44], [-36], [18], [-40], [16], [-22], [54], [54], [-36], [-34], [-16], [34], [-12], [10], [-36], [-10], [-4], [-10], [28], [52], [-10], [18], [-16], [22], [-42], [-44], [-50], [0], [14], [-20], [-4], [-42], [-28], [18], [-26], [-34], [-26], [-8], [62], [0], [6], [-30], [56], [46], [-16], [-40], [-54], [-16], [2], [22], [-8], [48], [-56], [-30], [0], [-46], [24], [62], [-26], [32], [18], [-32], [60], [-10], [2], [24], [-20], [-36], [46], [4], [-28], [10], [-56], [12], [-38], [-18], [12], [-16], [-12], [40], [20], [70], [46], [64], [-16], [52], [32], [-28], [22], [38], [-48], [-34], [-18], [-60], [-66], [68], [30], [42], [-24], [-80], [-2], [74], [78], [72], [-18], [42], [-68], [-66], [-70], [0], [-6], [-40], [58], [8], [36], [-34], [-46], [68], [32], [80], [-16], [6], [-84], [-52], [-38], [-50], [-16], [50], [-14], [68], [42], [-60], [-26], [-42], [-28], [86], [4], [28], [-14], [30], [0], [-12], [28], [-50], [24], [-86], [0], [70], [48], [-82], [62], [-36], [-48], [34], [56], [80], [-14], [50], [-44], [-10], [-18], [44], [-6], [26], [76], [-44], [-32], [34], [18], [-6], [-40], [24], [92], [-60], [-18], [-58], [74], [52], [22], [2], [26], [-72], [94], [-36], [18], [-48], [72], [-18], [-28], [-90], [-34], [-72], [-58], [-86], [-12], [52], [6], [-24], [58], [-62], [-60], [-36], [8], [10], [82], [28], [58], [-56], [-52], [-24], [6], [-60], [-42], [28], [8], [-14], [34], [-22], [-38], [42], [-88], [98], [-76], [-32], [52], [-50], [76], [0], [-50], [-78], [-4], [-64], [-88], [-102], [96], [-54], [-60], [54], [-10], [90], [-28], [38], [-18], [-16], [102], [34], [44], [-36], [94], [-50], [-8], [-44], [-22], [46], [80], [56], [6], [96], [94], [54], [8], [20], [90], [38], [-108], [-62], [100], [-16]];
    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_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_4641_2_a_c(: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_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_4641_2_a_c( : 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;