
// Make newform 7942.2.a.x in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_7942_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_7942_2_a_x();".
// 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_7942_2_a_x();".
// 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
        poly := [-4, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_7942_a();"
function MakeCharacter_7942_a()
    N := 7942;
    order := 1;
    char_gens := [5777, 6139];
    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_7942_a_Hecke(Kf)
    return MakeCharacter_7942_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, -1], [2, 0], [2, -1], [1, 0], [2, -1], [2, -1], [0, 0], [4, -1], [-2, 3], [-2, 0], [0, 2], [-2, 0], [0, 0], [8, 0], [0, -5], [0, 3], [-4, 4], [0, 3], [-6, 0], [-2, -3], [4, 0], [4, 4], [6, -2], [2, 2], [-12, 2], [-2, 6], [-8, 7], [14, -3], [-2, 2], [8, -4], [-4, 2], [10, 1], [-8, 6], [-12, 2], [-4, 0], [6, -8], [-4, 6], [16, -6], [-18, 0], [-16, 4], [16, -2], [-16, 3], [-2, -2], [8, 4], [4, 3], [8, -3], [6, -12], [0, -11], [18, -2], [2, -12], [6, 7], [-6, 10], [12, 2], [22, 0], [14, -2], [-8, 2], [18, 3], [-12, 8], [10, 6], [-4, 6], [14, -7], [12, 4], [8, -5], [2, -1], [-16, 9], [12, 1], [-6, -2], [-16, 10], [-4, 10], [-6, 11], [2, -7], [20, 4], [-6, 1], [-4, -1], [2, 0], [18, 0], [-2, 16], [18, -4], [-6, -4], [-20, 2], [12, 7], [4, -14], [-22, 6], [-16, 10], [4, 14], [2, -10], [34, 1], [-4, 12], [24, -8], [20, -4], [2, -6], [-26, -2], [-24, -6], [20, -4], [14, 3], [16, 6], [2, 2], [-8, -9], [12, -20], [12, 12], [12, 2], [12, 12], [6, 6], [20, -10], [18, 9], [-20, -8], [34, 0], [-18, 14], [-6, -6], [-12, 10], [8, 4], [14, 12], [-20, 4], [24, -8], [-6, -10], [-28, 6], [-4, -17], [10, -10], [-24, 15], [24, 1], [22, 2], [22, -11], [-8, -8], [-28, 10], [0, -16], [-22, -8], [-8, -1], [-16, 15], [-20, 18], [-4, -16], [32, 4], [10, -20], [-22, -8], [-30, -5], [-6, -3], [-20, -7], [-8, -3], [28, -5], [26, -7], [-16, 3], [20, -16], [4, -7], [-32, -7], [16, -11], [2, 2], [4, -22], [10, 0], [-20, 14], [10, -4], [42, 1], [-10, 12], [-36, 12], [32, 0], [-24, 21], [22, -16], [30, 1], [-6, 7], [-34, 13], [-22, -1], [4, -16], [6, -2], [18, 14], [8, -4], [-46, 6], [42, -4], [-18, 2], [-44, 6], [10, -6], [26, -22], [16, -5], [12, -9], [18, 14], [-34, 7], [-22, -8], [-42, 8], [8, 4], [-6, -14], [12, 10], [8, -16], [24, -7], [8, 3], [-36, 2], [14, -2], [-40, 3], [-6, 14], [-46, 5], [-24, 12], [14, 12], [22, -5], [22, 10], [4, 4], [-16, 20], [0, 1], [20, -16], [-34, -10], [-18, 13], [22, 2], [22, -1], [-2, -19], [-34, -4], [-8, 4], [-32, 19], [-34, -8], [28, -6], [-26, 24], [40, -13], [-8, -15], [-10, 17], [-12, -21], [-6, 5], [-10, -22], [-20, 20], [12, -3], [8, 18], [18, 14], [-62, 3], [-22, 10], [-26, 24], [14, -4], [-10, -16], [24, 4], [-38, -4], [36, -1], [-16, 19], [-24, 6], [-2, 2], [10, -18], [4, -2], [4, 0], [-14, 0], [32, -19], [-16, 8], [-18, -14], [4, -20], [-10, 9], [-18, 32], [0, 6], [28, -12], [-2, 16], [-24, -12], [-40, -4], [-4, -27], [-28, 3], [-10, -22], [-62, -2], [4, -20], [8, 12], [52, -8], [34, -1], [24, -8], [54, -9], [-12, -13], [22, -16], [-20, 10], [24, -18], [6, -14], [28, 8], [-24, 6], [-42, 4], [28, -18], [4, 0], [-52, 4], [62, -1], [-46, 8], [-4, -5], [-28, 9], [14, -19], [64, -1], [36, 0], [-36, 15], [-52, -7], [-26, -1], [14, -15], [-30, 2], [16, -11], [-36, -8], [58, -5], [-42, -12], [28, -12], [-48, 14], [12, 3], [-2, -13], [6, 3], [0, -8], [0, 8], [-38, 23], [-42, 4], [28, 13], [18, -30], [54, -10], [-4, 18], [10, 2], [36, 16], [18, 0], [48, -10], [-42, 6], [-26, 22], [-24, 11], [20, -28], [-18, -12], [58, -7], [46, -1], [-12, -13], [20, 4], [-22, 20], [-56, -5], [-24, 1], [-40, 16], [-32, 12], [14, 6], [-22, 2], [26, -30], [12, -24], [-26, -14], [-26, -12], [12, 20], [32, -14], [22, -7], [-22, 4], [-24, 31], [-22, -11], [-38, 19], [-6, -16], [34, 7], [66, -6], [-16, 19], [56, 5], [-20, 24], [40, 10], [0, -20], [40, 9], [-50, -13], [12, -18], [12, -22], [44, -6], [-48, 9], [-6, 30], [2, -15], [-30, 13], [-16, 5], [54, -7], [42, -5], [-18, -18], [-40, 7], [20, 0], [46, -26], [-60, -3], [0, -36], [-30, 6], [-36, 24], [-14, -8], [6, 8], [24, 6], [-14, -8], [-38, 10], [0, 35], [40, -12], [-46, -9], [-2, 0], [0, -18], [34, -8], [-42, 20], [88, -7], [52, 2], [46, 12], [-24, -6], [80, -6], [58, -4], [36, 8], [8, -31], [38, 19], [-26, 7], [-46, 14], [-58, 4], [8, 5], [38, -30], [50, 16], [-6, -27], [30, -30], [12, -30], [-22, 28], [-18, 13], [26, -28], [48, -3], [-22, -4], [-4, 29], [46, 10], [16, -26], [34, 12], [-26, -8], [-4, 25], [-52, -2], [-68, 20], [-26, -20], [46, -12], [20, 8], [-6, 2], [24, -28], [4, 6], [6, 28], [46, -12], [36, -16], [2, 24], [74, 0], [38, 11], [30, -16], [-38, -10], [28, -3], [28, 0], [14, 12], [-8, 20], [-68, 7], [-40, -12], [30, -3], [-82, 10], [48, -14], [-50, -20], [14, 33], [48, -18], [40, -18], [-34, 1], [56, -3], [8, -21], [26, -16], [20, 21], [60, -1], [22, 19], [20, 30], [-42, -21]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_7942_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_7942_2_a_x();".
// 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_7942_2_a_x(:prec:=2)
    chi := MakeCharacter_7942_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_7942_2_a_x();".
// 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_7942_2_a_x( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_7942_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-4, 1, 1]>,<5,R![-2, 1]>,<13,R![-2, -3, 1]>],Snew);
    return Vf;
end function;