
// Make newform 2142.2.a.h in Magma, downloaded from the LMFDB on 25 April 2024.
// 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_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_2142_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_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)
    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+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], [2], [1], [-4], [-2], [-1], [4], [-8], [-6], [0], [-2], [-10], [-4], [0], [-6], [4], [6], [-12], [8], [-6], [0], [12], [6], [2], [10], [8], [12], [-10], [-18], [16], [-12], [22], [-12], [-22], [8], [-2], [-20], [-8], [-6], [12], [-2], [0], [-14], [2], [-8], [12], [-16], [20], [6], [-10], [-16], [18], [4], [-2], [8], [26], [0], [-2], [-26], [20], [-6], [-28], [-24], [26], [-22], [-4], [-14], [28], [-18], [-34], [24], [-32], [-26], [20], [0], [26], [22], [30], [26], [20], [-10], [0], [-14], [8], [-12], [-18], [10], [2], [0], [-36], [16], [24], [20], [-20], [24], [2], [-10], [44], [22], [-20], [2], [12], [38], [36], [-30], [36], [14], [-8], [10], [-32], [-26], [22], [-44], [-8], [-18], [-36], [-24], [42], [12], [38], [-46], [18], [28], [-20], [-14], [-18], [32], [8], [-50], [36], [-8], [0], [-42], [6], [2], [26], [28], [-46], [38], [-28], [-46], [8], [-20], [30], [-40], [46], [6], [28], [48], [54], [-34], [4], [-40], [-28], [48], [-8], [-34], [10], [-22], [4], [-26], [8], [52], [-50], [24], [16], [-18], [34], [-62], [36], [14], [-8], [10], [16], [-10], [20], [2], [-40], [14], [-16], [-44], [30], [22], [32], [26], [54], [-44], [-38], [-32], [-46], [-20], [20], [-62], [-60], [-58], [-14], [-10], [-18], [-8], [26], [0], [-42], [34], [-60], [-30], [64], [-36], [-10], [28], [2], [42], [-40], [20], [-8], [-22], [-64], [-18], [-24], [18], [38], [56], [30], [48], [44], [-26], [-10], [-16], [40], [-20], [30], [-20], [64], [-74], [44], [32], [-46], [18], [-4], [-24], [36], [-20], [-72], [46], [-66], [8], [-64], [-12], [60], [-48], [62], [-18], [-24], [-6], [50], [-4], [-50], [-44], [18], [-6], [32], [-68], [14], [22], [62], [-28], [34], [70], [36], [26], [38], [-76], [-6], [0], [-46], [72], [-12], [14], [10], [12], [16], [-56], [40], [-42], [-44], [0], [2], [18], [24], [-34], [-54], [-12], [6], [60], [38], [-54], [0], [-38], [60], [-28], [10], [18], [48], [-76], [20], [-46], [-12], [-10], [-8], [6], [0], [2], [-50], [4], [72], [10], [28], [-32], [-14], [-18], [28], [10], [-30], [-16], [-90], [-14], [76], [-76], [16], [34], [-10], [-54], [-16], [44], [-76], [-84], [14], [78], [-70], [-32], [6], [-90], [18], [-40], [-78], [20], [14], [60], [80], [-62], [-60], [-70], [-46], [-16], [-10], [6], [48], [60], [30], [-72], [62], [-90], [-80], [28], [20], [42], [90], [-56], [42], [-52], [-92], [-96], [90], [40], [38], [52], [-64], [34], [14], [58], [-54], [-42], [-56], [30], [-84], [24], [-64], [-10], [-44], [-32], [2], [-78], [20], [68], [24], [74], [16], [-42], [4], [-6], [-90], [30], [-64], [-58], [-6], [-56], [-18], [-18], [-44], [-60], [34], [74], [-28], [12], [42], [42], [-48], [-88], [78], [24], [-78], [62], [0], [-12], [-22], [50], [4], [6], [-68], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2142_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2142_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_2142_2_a_h(: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_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_2142_2_a_h( : 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![4, 1]>,<13,R![2, 1]>],Snew);
    return Vf;
end function;