
// Make newform 4400.2.a.ba in Magma, downloaded from the LMFDB on 25 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_ba();".
// 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_ba();".
// 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], [0], [-1], [2], [-6], [-4], [-2], [-10], [8], [-8], [-2], [0], [2], [0], [12], [-10], [-6], [0], [-6], [-12], [16], [18], [12], [2], [-14], [12], [2], [-12], [-4], [8], [12], [-8], [-14], [-8], [-4], [-6], [20], [-2], [-4], [-6], [-16], [10], [-2], [8], [-8], [14], [24], [-6], [-10], [-12], [-2], [-20], [8], [-12], [18], [8], [-22], [-18], [32], [-14], [-4], [-32], [16], [28], [-12], [10], [-12], [-34], [-8], [-12], [26], [14], [4], [18], [-6], [28], [30], [2], [20], [2], [28], [4], [-16], [-14], [18], [-18], [2], [-34], [-6], [4], [2], [-36], [-20], [0], [-14], [22], [-4], [-22], [-16], [38], [4], [26], [-12], [4], [-22], [18], [8], [-34], [-28], [6], [-36], [4], [40], [-18], [26], [30], [-24], [32], [-42], [10], [-14], [-26], [20], [6], [-26], [24], [30], [-50], [32], [-28], [0], [12], [42], [30], [32], [-52], [-12], [-26], [-28], [-2], [34], [-4], [-26], [24], [-26], [-46], [-20], [-6], [38], [18], [-30], [4], [50], [32], [-8], [30], [-18], [-22], [-22], [46], [4], [-20], [20], [-22], [-40], [-58], [22], [0], [0], [2], [-24], [-50], [-16], [-10], [44], [-10], [40], [-22], [22], [36], [-4], [38], [22], [-10], [62], [22], [-14], [-4], [36], [-48], [52], [30], [-32], [24], [46], [24], [6], [36], [6], [-20], [-8], [54], [-4], [-44], [-56], [-60], [-50], [12], [-2], [-22], [50], [26], [-44], [2], [16], [10], [22], [16], [-66], [-60], [-34], [-46], [36], [-14], [-12], [-40], [28], [24], [28], [-16], [20], [30], [-10], [-48], [34], [-38], [-12], [-8], [-6], [32], [50], [-50], [38], [60], [58], [76], [0], [-12], [2], [-30], [62], [-18], [6], [56], [10], [72], [-28], [22], [-44], [-64], [-50], [-14], [-24], [20], [78], [-6], [40], [-30], [-58], [10], [28], [56], [14], [-42], [-78], [-18], [-74], [8], [36], [8], [40], [26], [-32], [-32], [-68], [-78], [0], [-54], [-50], [-2], [-34], [0], [-6], [-54], [-72], [-40], [64], [12], [62], [22], [-40], [54], [-68], [40], [2], [-38], [-40], [-82], [-8], [2], [-46], [-82], [-84], [38], [-20], [-8], [-20], [42], [24], [-36], [-42], [-26], [-42], [-66], [52], [26], [-48], [-42], [-42], [-20], [-44], [44], [-24], [-90], [2], [86], [-10], [40], [-68], [-72], [-38], [8], [-24], [-68], [38], [-74], [-68], [-96], [-28], [8], [50], [80], [14], [-70], [40], [-72], [-42], [10], [-70], [-70], [34], [-84], [70], [28], [-58], [-44], [-2], [-12], [-28], [76], [42], [-28], [-52], [-44], [-48], [38], [50], [62], [-90], [64], [60], [-6], [-100], [-6], [72], [52], [-4], [-34], [50], [-24], [-36], [58], [24], [-2], [80], [-26], [-28], [22], [18], [-44], [48], [-24], [-6], [0], [-20], [18], [-82], [84], [30], [66], [18], [60], [2], [-14], [-16], [-58], [48], [48], [-50], [58], [-42], [36], [-32], [72], [58], [-6], [44], [20]];
    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_ba();".
// 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_ba(: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_ba();".
// 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_ba( : 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![0, 1]>,<13,R![-2, 1]>],Snew);
    return Vf;
end function;