
// Make newform 1440.2.a.h in Magma, downloaded from the LMFDB on 25 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_h();".
// 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_h();".
// 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], [-2], [-2], [0], [-2], [-4], [0], [2], [-8], [-4], [-8], [-8], [8], [10], [6], [2], [-12], [-12], [-2], [-8], [4], [-12], [10], [10], [-10], [12], [10], [-6], [6], [-22], [-14], [-4], [14], [-4], [-8], [12], [16], [-6], [6], [-6], [-20], [-10], [22], [20], [4], [14], [24], [14], [-26], [20], [6], [-22], [10], [16], [-6], [28], [-28], [4], [4], [6], [32], [-28], [6], [-26], [4], [22], [24], [10], [30], [8], [6], [16], [36], [-16], [-2], [-28], [12], [-22], [18], [-2], [-16], [-34], [-4], [0], [20], [18], [-42], [-14], [12], [-12], [2], [-22], [-28], [8], [-30], [-36], [4], [2], [-8], [-14], [0], [36], [-44], [34], [-4], [18], [24], [-26], [-38], [44], [-18], [-4], [0], [48], [-12], [24], [-18], [-14], [-14], [-26], [-26], [4], [-36], [-6], [14], [-8], [-14], [-36], [-20], [-48], [-32], [-44], [20], [-18], [14], [-20], [-6], [-48], [28], [-22], [-54], [-8], [2], [-48], [28], [30], [-28], [16], [-32], [4], [4], [-40], [-40], [16], [8], [36], [10], [22], [8], [-22], [22], [18], [-18], [16], [-20], [-24], [-10], [-14], [-42], [46], [24], [14], [24], [36], [20], [38], [34], [46], [18], [58], [4], [-54], [-24], [-10], [24], [12], [-26], [36], [-50], [40], [20], [6], [-36], [-46], [2], [-32], [18], [8], [30], [16], [-8], [14], [-50], [14], [16], [-64], [-44], [60], [30], [14], [62], [-40], [20], [-62], [-10], [-52], [-48], [66], [-26], [44], [-32], [-30], [-12], [30], [66], [-40], [-6], [30], [-32], [60], [28], [-52], [28], [-72], [34], [54], [30], [36], [56], [-52], [26], [-6], [-34], [-28], [38], [30], [-44], [-24], [60], [0], [-32], [18], [18], [2], [-38], [60], [-6], [-18], [-74], [-16], [-26], [-28], [-46], [12], [54], [56], [68], [6], [46], [-80], [46], [0], [-34], [-34], [-72], [-62], [10], [50], [0], [12], [40], [-42], [-48], [20], [-54], [6], [-52], [16], [30], [-60], [78], [6], [-52], [78], [-32], [-70], [6], [16], [62], [-18], [-76], [-60], [-68], [-14], [-12], [38], [52], [64], [48], [34], [28], [-56], [-48], [38], [-82], [-4], [-26], [-40], [-28], [-42], [34], [-38], [-18], [70], [-76], [-64], [-24], [2], [46], [2], [48], [-12], [-44], [36], [-26], [-26], [-70], [-90], [56], [78], [-78], [-20], [2], [22], [-82], [-32], [60], [-70], [28], [-14], [66], [46], [-30], [78], [80], [38], [-22], [48], [40], [32], [72], [-18], [-44], [74], [18], [-74], [-42], [-42], [-44], [-96], [-30], [68], [92], [54], [-24], [-54], [36], [-46], [-42], [-6], [86], [6], [52], [96], [-80], [52], [-8], [-40], [46], [-54], [-34], [36], [-48], [58], [12], [32], [68], [-66], [-66], [90], [62], [98], [10], [-48], [4], [-68], [60], [-30], [34], [-42], [-84], [-12], [-58], [-46], [24], [94], [2], [-24], [46], [8], [72], [2], [-18], [-18], [52], [16], [20], [-48]];
    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_h();".
// 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_h(: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_h();".
// 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_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1440_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2, 1]>,<11,R![2, 1]>,<17,R![2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;