
// Make newform 4400.2.a.bb in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4400_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_4400_2_a_bb();".
// 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_4400_2_a_bb();".
// 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_4400_a();"
function MakeCharacter_4400_a()
    N := 4400;
    order := 1;
    char_gens := [2751, 3301, 177, 1201];
    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_4400_a_Hecke(Kf)
    return MakeCharacter_4400_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], [2], [0], [4], [-1], [6], [-2], [-4], [6], [-2], [-8], [8], [6], [12], [10], [0], [4], [-10], [2], [8], [-2], [-4], [-4], [-14], [4], [10], [2], [-8], [-14], [-4], [8], [0], [-12], [0], [-14], [8], [20], [10], [0], [-22], [-4], [-22], [0], [14], [-6], [16], [16], [-10], [12], [-6], [-14], [-4], [6], [-12], [24], [-24], [-14], [24], [14], [22], [28], [-26], [8], [8], [0], [-28], [4], [-2], [0], [22], [-8], [-4], [-30], [26], [-20], [-30], [-6], [20], [-34], [10], [-28], [-30], [-12], [28], [16], [-14], [2], [10], [-6], [6], [2], [-4], [-6], [12], [-28], [12], [-14], [-10], [-16], [-30], [-12], [2], [24], [-30], [-28], [-4], [18], [-42], [40], [-10], [48], [-14], [4], [36], [-48], [-2], [-22], [-18], [24], [-24], [22], [14], [-42], [30], [-4], [-42], [-42], [-8], [30], [-6], [-8], [24], [8], [-28], [-22], [14], [48], [24], [12], [-10], [20], [-18], [2], [8], [-26], [48], [-14], [-10], [-20], [26], [2], [34], [-46], [48], [-38], [-48], [-40], [-18], [26], [-30], [2], [42], [32], [-36], [60], [42], [-24], [-46], [6], [0], [-40], [-62], [40], [-6], [-48], [-10], [-4], [54], [20], [-38], [-10], [-44], [-12], [-62], [14], [6], [10], [-34], [42], [20], [-20], [44], [-52], [-34], [4], [-24], [6], [56], [18], [-24], [46], [52], [-56], [38], [-36], [-68], [32], [40], [-18], [-20], [-22], [42], [50], [-46], [-20], [-30], [-28], [42], [-10], [-32], [46], [60], [30], [-14], [-48], [-30], [-36], [-48], [24], [32], [4], [-8], [-20], [22], [14], [4], [34], [-50], [-20], [24], [-54], [56], [2], [14], [18], [-12], [18], [4], [-24], [72], [38], [18], [-18], [46], [66], [0], [-54], [44], [-4], [2], [-8], [-12], [38], [-26], [24], [-20], [30], [58], [4], [-42], [-42], [-14], [20], [-40], [74], [14], [42], [-10], [-50], [48], [-8], [8], [-20], [-30], [-44], [-48], [-76], [-10], [-64], [-30], [78], [78], [42], [-40], [62], [-54], [-80], [24], [56], [-8], [26], [50], [40], [14], [12], [40], [42], [74], [-72], [-38], [-44], [18], [-30], [-42], [-24], [22], [60], [72], [52], [-14], [32], [4], [54], [-34], [2], [46], [36], [-70], [4], [34], [22], [36], [-4], [-48], [0], [-58], [34], [2], [38], [28], [-60], [72], [42], [-8], [24], [92], [38], [38], [68], [32], [36], [40], [-78], [28], [70], [30], [0], [-96], [50], [-86], [-50], [42], [-6], [76], [-74], [36], [-78], [-24], [-26], [76], [-28], [16], [90], [-36], [68], [36], [48], [-14], [-6], [26], [-10], [80], [8], [-18], [92], [10], [-80], [-84], [-16], [46], [82], [-24], [-12], [82], [48], [-6], [-32], [-90], [-68], [-42], [18], [-68], [36], [40], [-38], [0], [68], [-14], [54], [12], [-6], [-42], [50], [-52], [86], [18], [96], [-74], [48], [60], [14], [14], [-90], [52], [80], [-8], [58], [50], [20], [-84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4400_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4400_2_a_bb();".
// 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_4400_2_a_bb(:prec:=1)
    chi := MakeCharacter_4400_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_4400_2_a_bb();".
// 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_4400_2_a_bb( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4400_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>,<7,R![-4, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;