
// Make newform 2842.2.a.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2842_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_2842_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_2842_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_2842_a();"
function MakeCharacter_2842_a()
    N := 2842;
    order := 1;
    char_gens := [1277, 785];
    v := [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_2842_a_Hecke(Kf)
    return MakeCharacter_2842_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 := [[-1], [-2], [-2], [0], [4], [2], [4], [-2], [0], [-1], [2], [2], [-8], [-8], [-6], [6], [4], [-4], [-4], [8], [12], [-12], [0], [-4], [-4], [12], [16], [-12], [-14], [18], [12], [-2], [-10], [20], [18], [0], [-12], [4], [12], [-6], [-4], [-22], [20], [10], [2], [-8], [20], [4], [28], [-4], [10], [0], [-22], [2], [-6], [28], [0], [2], [-6], [22], [20], [24], [34], [10], [10], [30], [8], [-26], [-12], [-26], [22], [4], [-14], [34], [32], [-36], [26], [-14], [30], [-12], [4], [22], [-40], [-24], [-32], [32], [-38], [10], [0], [-16], [22], [-14], [-8], [20], [12], [-6], [14], [-2], [24], [22], [28], [-18], [-6], [-10], [4], [4], [0], [22], [-12], [20], [-14], [-2], [-18], [-26], [-8], [6], [28], [40], [-14], [36], [-26], [-34], [-52], [4], [16], [-34], [10], [-24], [38], [28], [16], [-8], [-8], [42], [-14], [12], [-24], [44], [28], [-6], [-36], [-42], [24], [48], [28], [30], [0], [-54], [46], [-24], [22], [48], [-12], [26], [32], [52], [-56], [42], [-34], [-10], [-24], [30], [-56], [-18], [-18], [46], [16], [-36], [50], [-40], [20], [22], [40], [54], [36], [-26], [-36], [-6], [46], [-38], [-8], [30], [-6], [2], [0], [26], [22], [2], [30], [0], [2], [44], [-4], [-32], [-32], [2], [38], [46], [26], [-48], [-26], [12], [-16], [-52], [-50], [66], [46], [52], [58], [2], [-42], [22], [-8], [26], [34], [42], [40], [0], [48], [-22], [-62], [28], [-6], [-12], [-64], [-46], [12], [36], [42], [60], [34], [16], [48], [-38], [-48], [18], [28], [6], [12], [-30], [52], [64], [16], [-74], [70], [-44], [-12], [-40], [-44], [-72], [-58], [-10], [-56], [16], [44], [-52], [26], [58], [-74], [-6], [-20], [4], [34], [-16], [-48], [10], [14], [16], [-76], [26], [54], [-76], [10], [-20], [-8], [-38], [-12], [30], [50], [0], [0], [-40], [24], [-62], [-22], [-84], [-38], [78], [-4], [-54], [-18], [-16], [-58], [2], [38], [18], [-58], [-26], [16], [70], [-12], [18], [-40], [24], [12], [58], [-32], [-6], [-16], [-46], [-4], [-30], [42], [28], [0], [54], [-22], [56], [-70], [78], [46], [54], [58], [-48], [-82], [88], [-84], [24], [-72], [-66], [-26], [18], [24], [-54], [12], [-4], [2], [-12], [-8], [-76], [14], [6], [0], [-8], [-2], [-96], [80], [52], [18], [56], [72], [46], [-50], [-64], [22], [48], [-66], [-56], [66], [-96], [-26], [-62], [-12], [-12], [-18], [-10], [-68], [-16], [-54], [72], [-64], [8], [70], [68], [34], [46], [-24], [-20], [6], [-62], [80], [54], [36], [-30], [-12], [12], [-24], [-86], [-40], [42], [-10], [54], [-68], [-62], [-12], [-46], [-64], [-8], [-68], [10], [-34], [-102], [-92], [-78], [-30], [-20], [18], [50], [-94], [60], [-62], [46], [-44], [-60], [-2], [-80], [48], [-28], [-8], [74], [6], [52], [-12], [-42], [70], [-50], [-36], [-82], [48], [-66]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2842_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2842_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_2842_2_a_a(:prec:=1)
    chi := MakeCharacter_2842_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_2842_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_2842_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2842_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<5,R![2, 1]>],Snew);
    return Vf;
end function;