
// Make newform 2695.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2695_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_2695_2_a_l();".
// 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_2695_2_a_l();".
// 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 := [2, 1, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 2, 0, 0], [0, -4, -1, 1], [-3, -1, 1, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_2695_a();"
function MakeCharacter_2695_a()
    N := 2695;
    order := 1;
    char_gens := [2157, 1816, 981];
    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_2695_a_Hecke(Kf)
    return MakeCharacter_2695_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, 1, 0], [-1, 0, 0, -1], [-1, 0, 0, 0], [0, 0, 0, 0], [-1, 0, 0, 0], [1, 1, 0, -1], [1, 0, 0, -1], [0, -1, 2, 0], [3, -1, -1, -1], [-2, 2, 0, 0], [-4, 0, -3, 0], [-1, 1, -1, 1], [4, 0, 1, 2], [-5, 1, -1, -1], [-3, -1, 4, -3], [-3, 3, -3, -1], [-6, 2, 1, 2], [2, 1, -1, 0], [-1, 1, 5, -1], [8, 0, 0, 0], [3, 2, 0, 1], [3, 2, -1, -1], [2, 1, -2, 2], [2, 1, 0, 0], [2, -3, 0, 0], [-2, 0, -7, 0], [7, 0, 4, 3], [-6, 0, -2, -2], [-4, 0, 0, -2], [8, 0, 0, -2], [0, 0, -2, 0], [-4, 4, -4, 0], [-1, 1, -1, -3], [-2, -2, -4, 2], [2, 0, -6, 4], [-1, -2, -3, 3], [4, 1, 0, 6], [-11, 1, 3, -3], [-14, 3, -4, -2], [1, 0, 2, -5], [2, 0, -2, 6], [-6, -1, -2, 4], [10, 0, 4, 2], [-3, -3, -3, -1], [5, -1, -3, 3], [-4, 1, -1, 4], [16, 0, 2, 4], [-5, 0, 4, -9], [4, -2, -4, 0], [-6, 0, -2, -8], [7, 3, 1, 1], [0, 4, 6, 0], [10, 2, 3, 0], [2, -2, 7, -6], [10, -1, -2, 0], [4, 0, 2, -4], [4, 1, -4, 6], [-2, 0, 8, -2], [-19, -1, -1, -5], [16, -2, 2, -2], [-2, -2, 10, -6], [3, -4, 6, -3], [0, -1, -6, 0], [-4, -1, 3, 4], [-4, 4, -2, -2], [6, -6, -4, 0], [-8, 6, -6, 4], [-13, 1, -5, -3], [-1, 3, -9, 3], [12, 1, -3, 2], [-4, -2, -6, -2], [5, 0, -3, -7], [3, 7, -2, -5], [-13, -5, -5, 9], [8, -6, -4, -4], [-1, 6, -2, 3], [4, 2, 4, 6], [14, -5, 2, 0], [2, -6, 4, -8], [10, 1, 9, 4], [-2, 1, 5, 2], [-3, 0, -9, 3], [15, 0, -5, 3], [8, 2, 4, 10], [16, -4, 6, 0], [11, -1, 11, 3], [9, 4, -1, 3], [5, -1, 9, -1], [20, -2, 9, 6], [3, -3, 1, -9], [5, -6, 12, -3], [-26, 1, 0, 2], [-13, 5, -1, -1], [21, 2, 5, -3], [2, 2, 2, 6], [2, -5, -6, -2], [8, 2, 4, -2], [14, 1, -10, 4], [24, 1, 2, 8], [26, 0, 6, 4], [2, -4, -4, -2], [11, -5, 5, 1], [8, -4, 6, -4], [-12, 2, -4, 2], [0, -4, -6, 4], [-4, -3, -6, 2], [-1, -4, -6, -1], [9, -5, 4, 3], [30, -2, 14, 6], [-2, -2, 3, -4], [-18, -1, 0, 2], [-5, 1, 9, 1], [11, -1, 3, 1], [-16, -1, 1, -12], [-16, 10, -4, 0], [11, 0, 1, -3], [-3, -3, -2, 9], [5, 1, -2, 5], [12, -6, 8, -2], [-19, 4, -5, -11], [0, -7, 6, 2], [11, 1, 7, 5], [3, -1, 6, 9], [-9, -5, 9, -1], [8, 5, -5, 4], [10, -8, 6, -4], [-10, 6, 4, 8], [18, 3, 3, 2], [-11, 5, 2, -11], [-3, -8, 12, -1], [8, 0, -4, 4], [22, 0, -2, -2], [14, -6, 12, 6], [-6, -2, -2, 4], [-16, -5, 7, -6], [10, 2, 9, 8], [-18, 11, -4, 8], [8, 0, 2, -4], [20, 5, 0, 6], [4, -4, -6, 10], [-24, 1, -4, -16], [-4, 6, 4, 6], [-3, 1, 7, -7], [5, -7, 5, 9], [-6, 0, -4, 8], [-46, -1, -1, 2], [-15, 5, -8, -1], [1, 2, 14, -9], [-4, 0, -7, 4], [19, 5, -17, 7], [-13, 3, 1, -7], [14, 5, 10, -4], [-7, 1, 11, 1], [-22, 0, -8, 10], [19, -5, -1, 11], [-12, 0, 4, -4], [0, 0, 2, -16], [6, -3, -2, 4], [-19, 1, -8, -1], [26, -9, -1, -8], [-7, -1, -15, 1], [5, -1, 11, 7], [3, 5, -9, 19], [22, 1, 7, 2], [20, -4, 8, 10], [13, -3, 0, 13], [0, -2, 2, 8], [-17, 2, 0, 1], [4, -2, -12, 2], [-6, 5, -8, -4], [4, 4, 18, 0], [16, 1, 6, -6], [3, 8, -7, -1], [5, 3, 11, -9], [-10, -1, 9, -2], [24, 0, -12, 2], [-25, 0, -5, -1], [-7, 0, -1, -1], [-2, 3, -6, -6], [-8, 4, 3, -6], [-3, 11, -3, -7], [18, -11, 4, 2], [-30, -2, -18, -4], [-11, -3, 6, -1], [-5, 5, -3, -1], [10, -1, -2, 4], [-1, 5, 5, -19], [13, 3, -2, 9], [0, -10, 2, 6], [2, -5, -2, -2], [-10, -2, -10, 16], [-12, -6, -12, 0], [-8, 0, -2, 12], [-30, -1, -12, -4], [-19, -9, 1, 9], [26, 1, 14, 0], [18, -6, 4, 0], [27, 5, 11, -11], [-11, -3, 0, -17], [22, 2, 0, 14], [36, -6, 4, 6], [-8, -7, 24, -12], [-14, 4, -10, 8], [-28, -3, -13, 6], [12, 0, 5, 20], [-22, 6, -20, 0], [-12, 6, -1, -8], [4, -8, -2, 0], [2, 6, -6, 16], [-18, 6, 1, 6], [17, 5, 3, 19], [-6, -4, 6, -8], [5, 7, -19, 1], [1, -5, 12, -11], [-26, 11, 0, 2], [-28, 3, -2, -14], [17, 1, 13, 9], [-6, 2, -11, 0], [23, -7, -3, -13], [24, 4, -4, 2], [-18, -4, 10, -8], [2, 1, 0, 6], [19, 8, 9, -11], [-37, -7, -7, 7], [6, -9, -12, -2], [10, 8, -6, -4], [14, -4, -6, 8], [28, 8, -4, 4], [26, 0, -16, 10], [11, 10, -3, 11], [15, -1, 15, -7], [12, 4, 0, 16], [32, -2, 4, 16], [0, -10, -2, 14], [27, 3, 0, 7], [0, 3, 0, 12], [4, 6, 8, -2], [-19, -5, -11, -13], [-24, -10, 8, -4], [-32, 5, 1, 0], [-19, 5, -5, -11], [40, 4, -2, -4], [29, -1, -6, -3], [-1, -10, 3, -11], [-19, -1, 10, 7], [-4, 11, -8, 8], [-25, -7, -16, -1], [-6, 1, 9, -2], [-4, 4, 4, 0], [-8, -10, 6, 0], [-3, -13, 1, -5], [-10, 1, 5, -8], [-17, 7, -9, 11], [16, -5, -18, -2], [27, 6, -2, 13], [16, -4, 16, 12], [5, 0, 13, 3], [0, 5, -4, 0], [20, -16, 10, 2], [-27, -3, -14, -1], [-37, -1, -9, 11], [12, 10, 8, 0], [20, -4, 9, 14], [5, -12, -6, 7], [22, -5, 10, -4], [-36, 3, 13, 0], [5, 0, 5, -5], [8, 6, 28, -6], [3, 7, -25, 15], [-1, -5, -9, 13], [0, 0, 2, -10], [25, -11, 11, 1], [24, -2, 4, 18], [-4, -4, -4, -20], [-37, 4, -20, -15], [-7, 6, -6, -3], [-17, -9, -7, 15], [44, 2, 6, -10], [12, -12, 4, -22], [-8, -2, -22, 12], [22, -9, 8, 10], [0, -2, -2, -16], [-64, 1, 4, -4], [18, 5, -3, 8], [-28, 3, -12, -4], [-14, -10, 6, 10], [21, -3, -3, -13], [-29, 11, 3, 1], [-12, 1, -9, -12], [12, -7, 9, -18], [-13, 8, -9, 1], [43, -2, 2, 3], [21, -7, 7, -1], [-26, 10, -4, 10], [-21, 3, 15, 1], [28, 0, 5, -10], [-16, 10, -7, -12], [-44, -1, -6, 6], [22, -6, 4, -6], [-4, 8, 6, 16], [49, -5, 0, -5], [-35, 9, -27, -5], [-1, 6, -19, -5], [15, 1, 3, -1], [-26, 4, -16, 2], [39, -1, 7, 13], [-23, 7, 1, -7], [-12, 4, 10, -6], [-14, -4, 14, -14], [-39, 3, 3, -1], [-20, 6, 8, 4], [28, 2, 8, -2], [24, 6, 22, -10], [-27, -3, -21, 5], [3, 7, -5, -5], [12, 0, -5, -14], [4, 11, -11, 16], [8, 4, -18, 16], [-42, 0, -6, -8], [6, -2, 2, 4], [-38, 2, -14, -18], [-4, 8, 8, 10], [4, 5, -3, -6], [-5, -1, -5, 15], [-3, 7, -11, 7], [18, -16, 12, -12], [-20, 0, -14, 0], [-5, -4, 26, -21], [51, -5, 11, -5], [-3, 11, -9, -13], [18, -4, -10, -12], [-1, -11, -9, -15], [-24, 3, -28, -4], [6, 0, 16, 2], [11, -2, 5, -21], [-37, -7, 4, -1], [-19, 4, -31, 3], [9, -9, 6, 3], [24, -18, 6, -14], [-26, 9, -2, 2], [-14, -8, -10, 20], [11, -1, 11, 9], [40, -13, 7, -2], [16, -16, 18, 8], [-24, -4, -10, 10], [-12, -6, 4, -32], [0, 13, 4, -6], [-25, 3, -5, 7], [-32, 3, -16, -12], [10, -4, -2, 16], [-6, -3, -16, -6], [-43, -1, 1, -5], [28, 6, 0, 6], [24, 1, -20, 12], [24, 0, -2, 18], [15, 3, -4, 1], [-6, -9, 17, -22], [14, -4, 6, -14], [27, 5, 3, -3], [19, 11, 35, -9], [1, -1, 5, -25], [20, -6, -3, -14], [-25, -7, -9, -5], [19, -2, 7, -13], [21, 5, 14, -23], [11, 5, 3, -7], [25, 8, -14, 3], [-56, 4, 6, 0], [-10, 8, -28, 4], [-10, -2, 14, 2], [-16, 16, -9, -16], [12, -2, 10, -28], [4, -14, 10, -2], [-10, -3, -10, 18], [-23, 1, 17, 3], [0, 11, -19, -4], [-41, 12, -11, -5], [9, 6, -12, -9], [16, 9, 13, -14], [-39, 4, 12, -9], [-26, -8, 12, -12], [-19, -5, 13, 3], [59, 1, -5, -5], [-45, 7, 13, -3], [-40, 19, -10, 8], [15, -4, -24, 11], [-56, 8, 2, 0], [-14, -2, -16, 0], [24, -2, 6, 20], [-7, 6, -26, 5], [-7, -20, 11, 3], [12, 7, -10, -2], [-46, 6, -4, -10], [9, -4, -8, -7], [-28, 2, -10, -4], [-11, 15, 1, -1], [0, -10, 10, 0], [-22, 2, -4, 12], [-14, -2, 24, -18], [16, 2, 12, -6], [-76, 4, -15, -2], [20, 0, 6, 18], [-12, 0, 20, -12], [30, 12, -8, -8], [10, 8, 17, -20], [36, -4, -16, 12], [-18, -2, -24, -8], [0, 2, 20, -10], [31, -10, 8, -1], [20, -7, 11, -8], [19, -7, 18, 5], [-31, -13, -3, 7], [7, -13, 25, -9], [-5, 0, 3, 11], [-23, -1, 1, 11], [-6, 15, 8, -12], [4, -4, 12, 4], [-17, -16, -14, 11], [18, 2, -20, -4], [-20, 2, -20, -4], [14, 2, -4, 8], [13, -4, -8, 15], [-5, 1, 5, -1], [2, 3, 14, -6], [20, -12, -2, 6], [8, 5, -18, 26], [31, 15, 9, -9], [-26, -2, -4, -12], [-16, -2, 19, 8], [-60, 5, 2, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2695_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2695_2_a_l();".
// 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_2695_2_a_l(:prec:=4)
    chi := MakeCharacter_2695_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_2695_2_a_l();".
// 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_2695_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2695_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![7, 8, -6, -2, 1]>,<3,R![16, -10, -8, 2, 1]>],Snew);
    return Vf;
end function;