
// Make newform 2142.2.a.n in Magma, downloaded from the LMFDB on 14 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_2142_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_2142_2_a_n();".
// 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_2142_2_a_n();".
// 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_2142_a();"
function MakeCharacter_2142_a()
    N := 2142;
    order := 1;
    char_gens := [1667, 1837, 1261];
    v := [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_2142_a_Hecke( : Kf := Kf)
    return MakeCharacter_2142_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]];
    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], [-2], [-1], [6], [0], [1], [-2], [0], [4], [0], [8], [-2], [-4], [8], [14], [6], [-10], [0], [12], [14], [4], [6], [14], [-6], [4], [4], [-18], [4], [-2], [8], [-20], [-6], [-16], [-6], [0], [-12], [6], [-16], [-14], [-12], [-14], [16], [6], [-20], [-16], [-10], [20], [20], [4], [-18], [-16], [-18], [18], [6], [0], [-6], [-4], [16], [-30], [8], [16], [6], [0], [34], [28], [20], [-30], [10], [0], [-26], [0], [-8], [10], [2], [20], [14], [-22], [-6], [-26], [0], [10], [16], [6], [8], [0], [-22], [-26], [16], [32], [18], [-40], [16], [20], [-10], [-16], [28], [-38], [-14], [36], [-26], [-26], [10], [42], [-10], [-30], [6], [6], [-32], [-38], [-32], [-6], [-42], [4], [8], [26], [4], [32], [4], [-48], [4], [46], [22], [18], [40], [-30], [-4], [24], [16], [-36], [-24], [28], [-40], [-46], [-22], [-10], [48], [-12], [-16], [50], [0], [-36], [-12], [50], [-48], [-24], [14], [-26], [-10], [-24], [-36], [-2], [-16], [48], [30], [-20], [16], [-14], [-14], [42], [42], [14], [40], [46], [46], [-40], [44], [-26], [-42], [6], [44], [-56], [-8], [10], [20], [-38], [-50], [48], [44], [32], [-24], [32], [-16], [-42], [-48], [16], [-52], [-26], [6], [-48], [-46], [-26], [-28], [-36], [-6], [-42], [-22], [32], [30], [-48], [36], [64], [-8], [-6], [-22], [32], [20], [64], [-50], [10], [46], [0], [20], [18], [-24], [38], [-24], [42], [-24], [-6], [50], [72], [2], [-44], [22], [-26], [2], [40], [8], [-46], [22], [52], [-48], [70], [74], [44], [-62], [-52], [-42], [12], [50], [62], [68], [-10], [42], [64], [16], [12], [20], [-72], [70], [10], [-72], [-6], [8], [-24], [16], [-4], [18], [-26], [-76], [40], [30], [58], [14], [14], [-58], [-58], [58], [-66], [6], [-32], [26], [-40], [-38], [20], [-4], [58], [54], [-26], [36], [64], [-48], [-44], [-76], [40], [38], [-60], [-12], [46], [24], [-24], [-26], [-40], [-12], [42], [-32], [56], [-48], [14], [-26], [-18], [-64], [-82], [-86], [-2], [40], [-58], [0], [-66], [-16], [-44], [-22], [-16], [-40], [-86], [-18], [-8], [-90], [-14], [-64], [46], [-24], [-16], [90], [30], [-18], [48], [32], [64], [16], [-56], [-80], [94], [-14], [8], [6], [42], [14], [48], [-14], [-70], [-34], [0], [-38], [-62], [34], [4], [56], [18], [-50], [-66], [-78], [-40], [-26], [30], [-64], [56], [42], [40], [-20], [10], [-72], [-10], [70], [42], [-54], [-64], [-2], [40], [4], [-68], [6], [-52], [64], [-24], [12], [66], [-14], [-10], [-6], [46], [20], [-18], [16], [32], [-32], [16], [38], [60], [-54], [90], [-60], [-54], [96], [6], [-32], [90], [22], [18], [62], [-62], [40], [-102], [28], [48], [94], [-74], [22], [76], [2], [-42], [8], [-22], [86], [-98], [-60], [-104], [98], [36], [-6], [-6], [-36], [46], [-74], [32], [-54], [-90], [94], [48]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2142_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2142_2_a_n();".
// 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_2142_2_a_n(:prec:=1)
    chi := MakeCharacter_2142_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_2142_2_a_n();".
// 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_2142_2_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2142_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<11,R![-6, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;