
// Make newform 1440.2.a.a in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1440_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_1440_2_a_a();".
// 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_1440_2_a_a();".
// 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_1440_a();"
function MakeCharacter_1440_a()
    N := 1440;
    order := 1;
    char_gens := [991, 901, 641, 577];
    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_1440_a_Hecke(Kf)
    return MakeCharacter_1440_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], [0], [-1], [-4], [0], [-2], [6], [0], [4], [2], [-8], [6], [6], [12], [12], [10], [-8], [-10], [-12], [-8], [10], [16], [-12], [6], [18], [-6], [4], [-4], [6], [6], [-4], [0], [14], [0], [10], [8], [-18], [-4], [12], [-14], [24], [-2], [8], [2], [-6], [16], [8], [-28], [28], [-2], [-18], [0], [-14], [16], [22], [4], [2], [-8], [22], [-10], [-4], [-6], [4], [8], [-22], [-30], [-8], [2], [-20], [6], [6], [0], [12], [6], [-16], [4], [26], [14], [-18], [-22], [40], [-18], [24], [18], [-16], [4], [-18], [-22], [-30], [20], [-36], [0], [-4], [-16], [32], [36], [2], [-26], [28], [38], [4], [18], [4], [38], [-40], [34], [12], [-26], [16], [-6], [12], [38], [14], [-16], [-8], [30], [-36], [-4], [50], [-24], [14], [34], [-6], [36], [40], [18], [30], [48], [-20], [-2], [-16], [4], [24], [22], [22], [-46], [-6], [-28], [-30], [-10], [-40], [-6], [4], [-20], [-10], [16], [-10], [46], [32], [-28], [-2], [-18], [-20], [28], [4], [-24], [16], [14], [-38], [-30], [-4], [-34], [-4], [0], [6], [-44], [40], [-42], [2], [42], [-24], [-10], [-40], [-22], [-32], [-58], [40], [42], [4], [-10], [28], [0], [-26], [30], [-44], [-6], [-18], [28], [-6], [24], [18], [4], [56], [-30], [60], [-34], [34], [14], [-42], [-28], [-62], [-56], [-26], [2], [-40], [-30], [0], [52], [22], [-8], [-14], [42], [-12], [-36], [64], [26], [-36], [-18], [60], [34], [-18], [32], [-18], [4], [28], [-34], [-18], [-64], [-52], [-48], [62], [-16], [-40], [38], [12], [-36], [-14], [-54], [-24], [8], [36], [-56], [52], [-42], [54], [16], [-36], [-64], [16], [-12], [-66], [-66], [-20], [-54], [-46], [-8], [62], [-28], [-6], [-22], [-12], [-68], [-34], [-34], [38], [32], [34], [22], [44], [58], [38], [-12], [58], [32], [2], [4], [-36], [-26], [-22], [32], [36], [-8], [-84], [30], [68], [-72], [34], [10], [32], [-18], [50], [60], [-2], [-64], [-50], [-30], [-8], [-54], [40], [20], [-38], [50], [0], [4], [40], [-78], [12], [38], [-48], [-74], [-44], [10], [30], [-36], [-68], [-70], [40], [56], [50], [-34], [-88], [-22], [-14], [-76], [-34], [-78], [-80], [-4], [-84], [-38], [54], [2], [16], [-44], [-40], [-36], [70], [70], [42], [-20], [-26], [46], [-54], [8], [66], [8], [-50], [-12], [56], [42], [8], [-22], [-62], [52], [30], [30], [-48], [-16], [-10], [84], [-10], [-26], [60], [24], [-28], [-70], [82], [52], [10], [48], [16], [100], [-6], [-88], [-2], [40], [-24], [2], [-2], [26], [-78], [14], [60], [-26], [-16], [-60], [40], [22], [-4], [28], [-46], [-70], [-40], [68], [-24], [26], [32], [54], [-24], [42], [70], [-26], [-20], [-2], [-86], [-104], [78], [-50], [76], [-24], [-14], [42], [-28], [24], [106], [18], [0], [12], [-74], [52], [34], [54], [-20], [-56], [-38], [-14], [100], [38], [56], [-96]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1440_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1440_2_a_a();".
// 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_1440_2_a_a(:prec:=1)
    chi := MakeCharacter_1440_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_1440_2_a_a();".
// 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_1440_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1440_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![4, 1]>,<11,R![0, 1]>,<17,R![-6, 1]>,<19,R![0, 1]>],Snew);
    return Vf;
end function;