
// Make newform 1840.2.a.i in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1840_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_1840_2_a_i();".
// 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_1840_2_a_i();".
// 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_1840_a();"
function MakeCharacter_1840_a()
    N := 1840;
    order := 1;
    char_gens := [1151, 1381, 737, 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_1840_a_Hecke(Kf)
    return MakeCharacter_1840_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], [3], [1], [2], [0], [1], [0], [0], [-1], [-3], [-3], [-8], [3], [2], [11], [-14], [8], [-4], [4], [-7], [-9], [0], [-4], [-2], [18], [18], [4], [16], [-18], [2], [-11], [9], [4], [11], [-22], [-7], [-6], [-7], [-16], [14], [-21], [12], [-2], [-1], [3], [4], [16], [16], [-2], [-2], [21], [1], [2], [-16], [5], [12], [17], [20], [-29], [6], [-10], [-24], [20], [29], [-20], [-14], [7], [26], [12], [-7], [19], [24], [-8], [-14], [-8], [-30], [-16], [25], [30], [-11], [-22], [-22], [-18], [-34], [7], [-33], [6], [-4], [13], [-4], [-42], [16], [25], [31], [25], [14], [21], [-4], [-42], [-7], [-35], [14], [16], [6], [-44], [-9], [-33], [34], [24], [37], [-16], [-16], [-48], [28], [-14], [26], [-34], [39], [-3], [-8], [30], [13], [-46], [-35], [8], [42], [-10], [-28], [6], [-8], [-17], [12], [50], [0], [29], [2], [14], [32], [-12], [26], [5], [18], [-33], [36], [6], [-6], [46], [41], [17], [17], [18], [36], [36], [15], [-32], [44], [50], [19], [-44], [12], [47], [18], [-43], [14], [48], [14], [0], [58], [-8], [-3], [36], [53], [-8], [0], [43], [-34], [-49], [-61], [44], [32], [25], [34], [-33], [30], [30], [8], [18], [12], [49], [24], [-23], [-48], [2], [-14], [6], [-14], [-32], [-56], [44], [21], [36], [0], [-17], [-44], [0], [-37], [4], [-65], [55], [19], [45], [-53], [-54], [50], [12], [54], [-17], [-27], [-46], [21], [-45], [8], [-15], [-6], [27], [-33], [36], [-45], [34], [68], [-62], [26], [50], [2], [-36], [74], [20], [24], [-15], [-69], [56], [5], [-29], [71], [-30], [-19], [-17], [-66], [-46], [10], [-12], [-6], [74], [-34], [-18], [-9], [40], [-62], [47], [31], [-4], [-34], [46], [-14], [-54], [-8], [24], [-30], [64], [23], [61], [-22], [-66], [6], [51], [-61], [52], [14], [-38], [24], [21], [26], [4], [10], [32], [0], [-38], [-40], [28], [42], [-30], [52], [-6], [28], [59], [-58], [56], [-3], [-65], [-51], [86], [48], [4], [-71], [4], [5], [12], [40], [-10], [-22], [-13], [-60], [-8], [-47], [-48], [18], [-5], [25], [4], [-9], [-70], [26], [3], [72], [-54], [16], [-14], [65], [-24], [-44], [25], [-67], [31], [61], [-80], [45], [10], [-66], [-65], [-38], [-91], [-66], [50], [-22], [6], [-36], [47], [-24], [-26], [-94], [9], [-5], [30], [-74], [32], [62], [-72], [-34], [-57], [-16], [-58], [9], [-51], [-78], [-11], [-9], [-63], [48], [-62], [-37], [-60], [39], [-56], [80], [-27], [43], [44], [34], [36], [-39], [-42], [-28], [-8], [-55], [22], [-39], [-53], [58], [74], [48], [16], [-54], [-83], [92], [-26], [-26], [98], [-30], [68], [-54], [-25], [3], [-18], [32], [15], [93], [-8], [25], [26], [84], [-91], [69], [12], [58], [-72], [33], [48], [-56], [30], [22], [22], [74], [-96], [-60], [-53], [41], [36], [62], [-11], [-91]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1840_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1840_2_a_i();".
// 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_1840_2_a_i(:prec:=1)
    chi := MakeCharacter_1840_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_1840_2_a_i();".
// 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_1840_2_a_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-3, 1]>,<7,R![-2, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;