
// Make newform 6137.2.a.b in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6137_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_6137_2_a_b();".
// 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_6137_2_a_b();".
// 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_6137_a();"
function MakeCharacter_6137_a()
    N := 6137;
    order := 1;
    char_gens := [3250, 1446];
    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_6137_a_Hecke(Kf)
    return MakeCharacter_6137_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], [4], [0], [2], [1], [0], [4], [-6], [-4], [2], [6], [4], [0], [-6], [12], [-10], [-4], [4], [-6], [-12], [-4], [-10], [-2], [-10], [-8], [-8], [-6], [14], [-8], [16], [-6], [-8], [-10], [16], [-2], [24], [4], [-22], [-12], [2], [-16], [-2], [-18], [-20], [-8], [-24], [24], [6], [-6], [-16], [-18], [12], [-18], [-16], [-22], [-16], [14], [6], [-16], [-6], [12], [28], [-22], [10], [-4], [14], [32], [-18], [-30], [0], [28], [-6], [8], [24], [6], [6], [14], [-26], [8], [-22], [-12], [-2], [20], [28], [-34], [-6], [-2], [32], [12], [36], [-20], [20], [-40], [-12], [2], [-26], [36], [6], [32], [30], [4], [38], [-32], [-14], [4], [18], [24], [-10], [-20], [-26], [-6], [-48], [16], [30], [32], [8], [6], [-4], [-38], [-2], [-30], [40], [-8], [-18], [-34], [4], [40], [-50], [28], [-12], [-20], [22], [-22], [-14], [26], [32], [50], [26], [-40], [-18], [20], [48], [34], [-20], [14], [-10], [52], [-16], [-6], [-46], [-12], [-12], [-32], [4], [24], [-30], [10], [-6], [32], [22], [0], [12], [-18], [-12], [12], [46], [14], [30], [36], [-14], [-12], [58], [-40], [-22], [8], [-18], [48], [-50], [0], [0], [-62], [-42], [8], [6], [26], [-36], [38], [-12], [-2], [-16], [-28], [34], [0], [-26], [-14], [-10], [66], [-56], [10], [36], [10], [14], [-20], [-34], [16], [-60], [-6], [-44], [-46], [54], [36], [60], [-4], [38], [16], [-18], [12], [-50], [58], [36], [-66], [-28], [-20], [-26], [-42], [-60], [-32], [0], [62], [16], [-8], [6], [-68], [56], [50], [14], [-24], [32], [56], [-4], [24], [18], [-18], [28], [-28], [8], [-52], [-24], [-2], [34], [16], [-26], [-18], [4], [-2], [8], [2], [-22], [-36], [60], [-2], [10], [-62], [-36], [2], [58], [80], [6], [-6], [-4], [58], [-56], [14], [-16], [36], [-46], [-58], [76], [48], [12], [4], [10], [24], [48], [18], [66], [32], [14], [-42], [-48], [22], [-80], [-54], [54], [48], [6], [-64], [44], [-6], [-78], [-4], [48], [-40], [-2], [68], [26], [-72], [38], [-20], [-82], [-66], [4], [-40], [-10], [-20], [-4], [34], [18], [-56], [42], [34], [-64], [-6], [78], [0], [-64], [60], [30], [-58], [-54], [-12], [36], [-16], [-48], [-46], [-66], [58], [56], [-6], [90], [-30], [-8], [-78], [-24], [-2], [52], [-36], [-66], [36], [38], [78], [76], [10], [54], [-40], [-8], [18], [0], [-18], [-70], [-48], [-48], [12], [54], [-74], [72], [-74], [-52], [-64], [-36], [-6], [-72], [86], [0], [-52], [-14], [50], [70], [90], [42], [28], [-62], [8], [-84], [8], [22], [-24], [-32], [-98], [82], [-4], [36], [0], [-10], [-64], [-74], [-32], [-26], [-70], [-14], [0], [-10], [26], [28], [-82], [30], [36], [-96], [78], [-42], [-44], [-16], [-22], [-42], [68], [-28], [18], [24], [50], [50], [84], [92], [-58], [-14], [96], [10], [-20], [84]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6137_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6137_2_a_b();".
// 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_6137_2_a_b(:prec:=1)
    chi := MakeCharacter_6137_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_6137_2_a_b();".
// 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_6137_2_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6137_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>],Snew);
    return Vf;
end function;