
// Make newform 2842.2.a.c 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_c();".
// 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_c();".
// 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], [0], [0], [0], [-4], [0], [4], [-4], [0], [-1], [6], [-2], [8], [4], [-2], [-2], [10], [2], [8], [16], [0], [-4], [6], [0], [-12], [10], [-12], [4], [10], [-6], [16], [12], [-2], [-18], [-18], [-8], [14], [4], [-8], [8], [16], [16], [-12], [-14], [14], [20], [-4], [28], [-6], [22], [26], [8], [-26], [28], [-6], [4], [10], [2], [-10], [-10], [-6], [22], [-16], [18], [14], [18], [-12], [-2], [24], [4], [-34], [24], [-14], [-10], [-4], [16], [-26], [32], [10], [-4], [-10], [-10], [-32], [40], [-16], [-12], [10], [38], [18], [-16], [28], [14], [8], [-4], [-20], [10], [-36], [-30], [38], [10], [-32], [-18], [36], [6], [12], [-16], [18], [-34], [-20], [-8], [10], [-6], [30], [20], [16], [6], [-26], [-28], [-14], [36], [-40], [-50], [-18], [-16], [38], [-14], [-22], [-12], [34], [18], [28], [44], [20], [-34], [-34], [-36], [-42], [-22], [-18], [-6], [-6], [38], [-16], [20], [-38], [-6], [-30], [-10], [0], [0], [10], [-36], [24], [-26], [44], [-48], [0], [6], [22], [48], [60], [54], [20], [4], [-2], [18], [-16], [-22], [6], [-22], [-40], [-4], [-24], [46], [-16], [30], [8], [30], [6], [12], [-60], [0], [-62], [42], [24], [0], [-34], [40], [-10], [20], [50], [-36], [4], [34], [36], [-22], [62], [10], [18], [16], [18], [-32], [62], [-32], [-68], [60], [26], [-12], [30], [4], [-58], [40], [-28], [40], [66], [70], [40], [24], [-8], [-2], [-34], [-68], [18], [24], [62], [-54], [48], [8], [-62], [-48], [50], [-22], [64], [-6], [-34], [42], [-20], [-42], [68], [-54], [44], [22], [-56], [-34], [30], [-48], [-28], [-10], [40], [48], [-2], [-18], [0], [-36], [-46], [48], [42], [12], [40], [-42], [40], [20], [-76], [6], [48], [36], [30], [-52], [-20], [-54], [36], [-60], [34], [40], [52], [34], [-36], [14], [-38], [38], [24], [72], [-4], [44], [56], [-72], [26], [10], [64], [-6], [-58], [-42], [50], [56], [-82], [-56], [78], [-12], [-22], [-32], [-4], [74], [-16], [-28], [68], [-78], [12], [-4], [-80], [22], [-44], [42], [-78], [32], [48], [46], [44], [-72], [-22], [54], [32], [-42], [76], [32], [-54], [-88], [40], [22], [52], [-78], [78], [-86], [60], [-68], [20], [58], [74], [-16], [60], [8], [-26], [-38], [-46], [32], [-54], [-68], [90], [-20], [34], [14], [10], [22], [90], [-76], [-10], [20], [-6], [30], [34], [16], [86], [18], [-48], [-12], [-52], [70], [6], [-40], [26], [36], [-78], [0], [-6], [20], [-50], [-92], [-84], [16], [26], [78], [-70], [-54], [-28], [66], [62], [-36], [32], [-44], [76], [38], [-50], [-64], [-20], [36], [-8], [42], [60], [-96], [-40], [-54], [16], [90], [44], [6], [-24], [-64], [-26], [-102], [-84], [-42], [98], [18], [4], [4], [78], [42], [-80], [-104], [40], [-14], [50], [58], [4], [96], [26], [24], [76], [94], [90], [-38]];
    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_c();".
// 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_c(: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_c();".
// 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_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2842_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![0, 1]>,<5,R![0, 1]>],Snew);
    return Vf;
end function;