
// Make newform 5376.2.a.bm in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5376_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_5376_2_a_bm();".
// 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_5376_2_a_bm();".
// 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, 2, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [-1, 2, 0, 0], [-2, -6, -1, 1], [3, -2, -3, 1]];
    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_5376_a();"
function MakeCharacter_5376_a()
    N := 5376;
    order := 1;
    char_gens := [2815, 5125, 1793, 4609];
    v := [1, 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_5376_a_Hecke(Kf)
    return MakeCharacter_5376_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 := [[0, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 1], [-1, 0, 0, 0], [-2, 0, -1, 0], [1, -1, 0, 0], [0, 1, 0, -1], [-1, 0, 1, 1], [-2, 1, 0, -1], [-1, 2, -1, -1], [1, 0, 1, -1], [1, 0, 1, -1], [0, 1, 0, -1], [-2, -1, 1, -1], [-2, 0, -2, -2], [-1, 0, 1, 1], [-4, 0, 0, 0], [-1, -1, 0, -2], [-2, 1, 1, 1], [4, -1, 2, -1], [2, 2, 0, 2], [-4, -2, 0, 2], [-8, 0, -2, 0], [-4, -1, 0, -3], [4, -2, -2, 0], [3, 0, -2, -1], [-3, 0, -3, -1], [-6, 0, 1, 0], [-6, 0, -2, -2], [-2, -2, 0, -2], [2, 2, 2, 0], [-6, 0, 0, -2], [0, -2, -2, 4], [-4, -2, 2, 2], [-9, 0, 1, 1], [-2, 0, 2, 2], [5, 1, 4, -2], [-2, -1, -1, 3], [10, -2, -2, 4], [-1, 2, 0, 1], [-2, -2, 3, 6], [-5, -1, -2, -2], [-4, 1, -2, 1], [11, 2, 1, 5], [13, -2, -1, 1], [10, 0, 2, 2], [-16, -1, -1, -3], [6, 4, -2, 2], [-6, -4, 0, 2], [-1, 3, 2, -2], [-4, 0, 2, 2], [4, -3, 2, 5], [8, -2, 2, 4], [-8, 4, 2, -4], [-16, 3, 0, -3], [14, 3, 0, 1], [9, -2, -2, 3], [3, -4, 3, 5], [-17, -2, 1, 3], [-14, 0, 0, -4], [-11, 2, 1, 1], [9, 2, 0, -5], [-7, 0, -1, -1], [-2, 0, -2, -2], [2, 0, 4, -4], [-7, 0, -1, -1], [-14, 3, -3, -1], [-4, 0, -2, -6], [-2, -4, -1, -4], [3, 5, -2, -4], [-12, -3, 0, 3], [-8, 1, -2, -3], [-2, 0, -2, -2], [-6, -2, 0, 0], [-12, -1, -1, -7], [-20, 2, 0, 2], [-1, 2, 3, -1], [-1, -3, -2, 0], [-12, 2, -6, -4], [6, -2, -4, -2], [-6, 0, 2, 2], [13, -2, -1, 5], [14, 1, 0, -1], [4, 0, 2, 2], [0, 0, 0, 0], [-6, 2, -3, 2], [-14, 2, 0, 2], [-2, -4, -4, 8], [5, 4, 2, -3], [-12, 2, 0, -2], [12, -4, 2, 0], [0, -4, 0, 0], [4, 0, 0, 4], [6, -4, 1, 8], [-4, -5, 3, -3], [-2, -2, 2, -4], [-21, 0, 0, -5], [0, -3, 4, 3], [12, 0, 4, 0], [1, -2, 7, 9], [4, 3, 1, -7], [-5, 0, 1, 5], [8, 4, -4, 0], [-8, -4, 2, 6], [-2, 1, 5, 1], [0, 0, -2, 2], [-2, 4, -4, -2], [0, -9, 4, 1], [-4, -1, -6, -9], [-2, 4, 0, 0], [6, 4, 6, 2], [-16, -4, -4, -4], [-16, 2, 2, -4], [-8, 4, 0, -8], [-2, 2, -6, 4], [6, -6, 4, 2], [15, 2, -1, 7], [18, 2, -2, 0], [-7, 0, 1, -5], [2, 0, -3, 8], [-3, -1, 6, 0], [5, 2, -1, -9], [3, -6, 0, -3], [10, 2, 5, 2], [0, 10, -6, -6], [1, 8, 1, 3], [26, -4, 2, 2], [-4, -6, 0, 2], [-3, 4, -3, -13], [-5, -7, 4, 12], [-6, -1, -1, -1], [-28, 1, -6, 1], [-16, 8, -4, 0], [-6, 4, -2, 2], [4, -1, 4, 1], [8, 4, -2, -6], [19, 4, -2, 3], [24, -2, 6, -2], [11, -6, -4, -3], [-16, 2, -2, -4], [12, -4, 0, 4], [-17, 0, -3, -7], [18, -6, 6, 0], [22, 2, 1, 6], [1, 1, -6, -2], [20, 6, 0, 10], [3, -3, 6, 4], [-8, -1, -8, -3], [7, 4, -7, 5], [0, -9, 6, 3], [16, 4, 0, -4], [4, 3, 0, 9], [-8, 3, 3, -15], [-10, -2, 2, 8], [26, -7, 3, 1], [-2, 1, -8, -1], [4, 0, 4, 12], [-20, 9, 0, -5], [10, 0, 4, 0], [15, -2, -2, -7], [2, -6, -3, 10], [0, -4, -6, 2], [-14, 2, -6, -8], [10, 0, 2, 2], [-14, 0, -4, -8], [10, -4, 2, 10], [2, 4, 2, 2], [25, 7, -2, 4], [1, -2, -1, 11], [-23, 0, -2, -11], [2, 4, -3, 4], [3, 1, -6, 0], [24, -1, -2, -5], [-19, 10, -5, -5], [-8, -4, 0, 4], [-12, -9, 4, 5], [-10, -1, -5, -5], [-15, -4, -3, 7], [13, 0, -3, 11], [9, -1, 6, -4], [-4, -2, 4, 10], [16, 4, 0, -8], [35, 4, -1, 1], [-4, 3, -4, 9], [28, 5, 2, 5], [5, -8, 6, 9], [-20, -2, 2, 6], [-16, 2, -2, -6], [11, -2, 1, -7], [18, -2, 6, 8], [20, -6, -2, 0], [2, 10, -7, -6], [-20, 7, -9, 1], [-11, 0, -2, 1], [26, 0, -5, -4], [-16, -1, 0, 1], [7, -2, -3, -3], [-29, 4, -9, 1], [-4, -7, 4, 15], [-10, -8, -2, 6], [-37, -6, 1, -1], [-36, 4, -4, 0], [-35, 5, 2, -2], [-38, 6, 4, -2], [-16, 0, 0, -4], [17, -2, 4, 7], [20, 8, -4, -4], [34, -2, -3, -2], [-8, 0, -2, 2], [7, -4, -7, -3], [2, -8, -4, 4], [-25, 2, -10, 5], [26, -8, 6, 6], [16, 4, 4, -8], [16, -4, 8, 0], [-10, 4, 0, 4], [-6, 4, 2, -6], [16, 3, 0, -7], [26, -1, -4, 9], [-1, 2, 3, -1], [-22, 0, -6, -10], [19, -12, 3, 1], [-26, -2, -4, -6], [-4, -8, 0, 4], [0, 4, -6, -12], [-11, -4, 5, -9], [0, 5, -4, -1], [-6, 3, -8, 5], [-3, 0, 5, -5], [14, -4, 11, 4], [-10, 0, 6, -6], [17, -6, 1, -3], [16, 4, 0, -12], [12, -6, 6, 4], [-15, 14, 1, -7], [28, 0, -4, -4], [22, 2, 8, 2], [-27, 6, 3, 1], [-2, -10, 5, -6], [-14, -6, 6, 12], [2, 6, 1, -2], [-24, 0, -8, 4], [-8, -4, 0, 12], [-17, -2, 5, 3], [8, -5, -8, 5], [-40, -2, -4, 6], [-5, 0, -5, 9], [-6, 8, 0, -2], [-10, 1, -7, 1], [18, 5, 8, -9], [20, -10, 2, 6], [8, 3, -4, 13], [34, -3, 0, -1], [-26, -6, -4, 14], [15, 4, 6, -1], [-50, -2, 3, -10], [-29, 0, 3, 5], [15, 0, 5, 1], [13, -2, -4, 3], [2, 0, 4, 4], [2, 6, 2, -12], [22, 0, 1, 4], [17, -11, -2, 10], [1, 7, 6, -4], [-16, 7, 8, 5], [-12, -12, 8, -4], [21, -2, -5, 9], [-20, 13, -4, -5], [-8, -1, -3, -7], [-11, 4, 1, 11], [-49, -3, -2, 0], [26, 9, 3, 9], [-16, 10, -10, -4], [-22, 4, -10, -14], [24, -6, 2, 4], [31, 4, 7, -3], [10, 2, -1, -10], [-33, -6, 1, 7], [-19, -2, -1, -5], [22, 0, 8, 2], [-34, 8, -10, -6], [0, -2, 12, 6], [-30, 0, -6, -10], [51, -1, 0, 2], [24, 4, 8, -8], [46, -3, 0, 3], [-20, 4, 2, -6], [-9, -2, 1, -1], [15, 4, -1, 9], [8, 7, 0, 1], [-13, 6, -7, -13], [26, 4, -8, 18], [4, -10, -10, -4], [-16, -8, -4, 4], [2, 10, 4, -4], [7, 10, 0, 1], [26, -16, -6, 2], [37, 6, 6, 7], [-18, 0, 0, 2], [28, 0, 0, 0], [48, 0, -6, -2], [-23, -2, 1, 9], [-42, -2, -6, 0], [-42, 6, 5, -6], [22, 1, 5, 9], [2, 8, -8, 0], [-14, 2, -1, -18], [59, 1, 4, 4], [8, 5, -2, 9], [-21, -2, 5, 15], [-2, 4, 6, 14], [-41, 4, 7, -7], [40, -4, 2, -2], [-26, 9, 3, -11], [-2, 7, 8, 5], [-32, 10, -10, -4], [-38, 8, -2, -6], [-6, -11, 8, 7], [2, 2, 4, 10], [-32, -2, -6, 0], [13, 2, 1, 17], [21, -2, 11, -1], [29, -2, 8, 11], [26, 0, -6, -2], [42, 2, 0, 2], [-2, 2, 8, 10], [0, 7, 1, -7], [-25, -10, -1, 11], [0, 11, -6, -17], [-21, 4, -13, -3], [-4, -6, -2, 10], [-31, 0, -15, -13], [-3, 12, -3, -9], [-58, 0, -2, -10], [-20, 7, -11, -11], [10, 4, 2, -2], [0, 12, 4, -4], [24, -13, 0, 5], [8, 6, -2, 8], [25, 0, -7, 7], [18, 10, 4, 12], [24, 0, 10, -6], [-23, 2, -14, -17], [-6, -4, 2, -6], [-27, -2, -7, -3], [-22, -6, -11, 2], [5, 11, -12, 8], [-4, -1, -5, -7], [-16, -4, 8, 12], [27, -8, 14, -1], [-33, -6, 3, 7], [25, -2, 15, 11], [3, -10, 9, -5], [-36, 0, 4, 4], [10, -10, -12, 16], [20, -9, 12, 5], [-48, -6, -4, -10], [22, -8, 2, 6], [44, -6, 18, 0], [-6, -13, 0, 21], [18, 2, 0, 12], [32, 1, 12, -5], [50, -7, -4, -5], [2, -4, -7, 20], [-28, 2, 2, -10], [23, -6, 9, 13], [33, -2, 18, 3], [-30, 2, 2, 0], [-37, 14, -3, -3], [-18, 4, 7, -4], [-15, 8, 11, -1], [48, -7, -6, -3], [5, 14, -13, -15], [-19, -12, 5, -9], [1, 2, 7, 13], [-14, -4, -8, -14], [-10, -1, -12, -11], [4, -12, 2, 2], [4, 3, -12, -19], [8, -6, 6, 8], [43, 2, 4, 5], [20, -6, 2, 0], [-2, -4, -2, -10], [2, 16, -8, 0], [-32, 0, 0, -12], [-46, 2, -18, -12], [-18, -12, 6, 6], [17, 1, 6, 6], [-12, -5, -3, -15], [-36, -4, 4, -4], [-15, -14, 7, 7], [-11, -8, 8, 13], [-34, 12, -3, -8], [49, -2, -3, 9], [26, -1, 0, 1], [0, 12, -6, 2], [16, 0, 8, 0], [20, 7, -8, 17], [-8, 3, 1, 5], [11, -6, -5, -13], [-1, 3, 8, 18], [-16, 14, -10, 0], [-22, -6, 6, 16], [4, 9, -12, -13], [-47, 2, 2, -9], [42, 8, 2, 10], [7, -14, 1, -1], [6, 8, -8, 0], [8, 0, -8, -4], [-62, 8, 6, -6], [-2, -16, 4, 12], [-33, -2, -5, -5], [-54, 10, -1, -2], [6, -1, -11, -9], [25, 2, -9, 7], [-19, 6, -6, -9], [-2, 9, 4, 15], [-39, 4, -7, -9], [20, -5, 4, 13], [-20, 4, 12, -20], [-23, 2, 1, -3], [-43, 11, -14, -4], [-10, 3, -8, 1], [-34, 12, 0, -10], [-26, -4, -8, -12], [-47, 14, -4, -9], [-22, -2, 13, 22], [0, 12, -2, -22], [-48, -6, -10, -6], [-6, 2, 6, -16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5376_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5376_2_a_bm();".
// 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_5376_2_a_bm(:prec:=4)
    chi := MakeCharacter_5376_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_5376_2_a_bm();".
// 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_5376_2_a_bm( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5376_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-8, 24, -14, -2, 1]>,<11,R![-200, -136, -14, 6, 1]>,<13,R![-32, 80, -20, -4, 1]>,<19,R![128, -96, -12, 8, 1]>,<29,R![2624, 32, -108, 0, 1]>],Snew);
    return Vf;
end function;