
// Make newform 5376.2.a.bi in Magma, downloaded from the LMFDB on 13 June 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_bi();".
// 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_bi();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-1, 2, 0], [-4, -1, 1]];
    Rf_basisdens := [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]];
    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], [1, 0, 0], [0, 0, -1], [-1, 0, 0], [-1, 1, 1], [1, -1, 0], [-1, -1, 1], [3, 1, 0], [-1, 1, -1], [3, 1, 0], [1, -1, 0], [-3, -1, -2], [5, 1, 1], [0, 0, -2], [-2, 2, 2], [-3, -1, -2], [-2, -2, 2], [7, 1, -2], [8, 0, 0], [1, -1, -3], [10, 0, 2], [8, 0, 2], [-4, 0, -4], [11, -1, -1], [0, -2, -2], [-2, 2, 3], [-1, 1, 0], [9, -1, -1], [2, 2, 0], [2, 0, 0], [6, 2, 2], [-2, -2, -4], [6, 0, 2], [4, 0, 4], [-3, -1, 2], [6, 2, 4], [1, -1, 2], [0, 0, 4], [0, 0, 2], [-6, -2, 3], [-11, 3, 3], [3, -3, 0], [15, 1, -1], [3, 3, 2], [-5, 1, 2], [-4, -4, 0], [6, 2, -2], [8, 0, 0], [-4, 0, 2], [1, -1, 0], [10, 0, 2], [-13, -3, 3], [-6, 4, 2], [2, 2, 2], [3, -1, -1], [17, -1, 1], [-12, 4, 3], [5, -5, 0], [-1, 5, 2], [8, 2, -6], [-11, -1, 4], [2, -2, 1], [5, -1, -4], [-2, 2, 6], [4, 2, -4], [-5, 1, 2], [14, 2, 2], [16, 2, 8], [-13, -3, 1], [-1, 1, 6], [5, -3, 1], [-1, 1, 3], [-8, 0, -4], [-6, 2, 4], [-2, 2, -6], [-2, 2, 0], [19, 1, -4], [-15, -1, -2], [-2, 0, 2], [6, 0, 2], [-28, 0, 2], [-7, -5, -2], [-1, 1, -5], [-4, 2, -4], [12, -4, 0], [-5, -3, 5], [14, 0, -4], [-22, -4, 0], [-6, -2, -3], [6, 2, 6], [-2, -2, -6], [-16, 0, 6], [10, -2, -4], [-3, -5, -1], [-8, 0, -6], [-16, 0, -4], [-24, 0, 1], [-5, -5, -11], [-20, 0, 0], [17, 3, -2], [2, -2, 0], [-3, -1, 2], [2, 2, 2], [-2, 8, 6], [10, 6, -4], [6, -4, -4], [-8, -4, -6], [5, -3, -1], [29, 3, 1], [10, 0, 0], [4, 4, -4], [20, 0, 0], [-6, 0, -6], [-24, 4, 4], [-6, -2, -2], [-8, 6, 2], [15, -3, 4], [12, 4, -6], [-27, -1, 4], [-15, -1, -5], [-7, -1, -6], [9, -3, 6], [-6, -2, 3], [-9, 1, 1], [8, 4, 0], [-3, -9, 0], [-14, 2, 0], [-14, -2, 0], [-17, 1, 0], [21, 3, 4], [0, 0, 8], [15, 1, 3], [24, 0, -4], [6, -2, -8], [13, -7, -1], [-16, -2, -8], [22, -6, -1], [28, 0, 0], [18, 6, 3], [-8, -2, 0], [28, 0, -4], [9, 3, -2], [36, 4, -2], [7, 1, 5], [7, 1, 8], [-30, -2, -10], [15, 1, 6], [-27, 1, 7], [27, 1, 4], [13, 3, -7], [-16, 4, 8], [5, -3, 3], [8, 0, -2], [4, -4, 6], [-4, -4, -10], [15, 1, 3], [-2, 10, 0], [-3, 1, 5], [16, -6, -8], [36, -4, -3], [-13, -3, -11], [-8, -2, 0], [-22, -2, 2], [-10, -10, 4], [0, -2, -10], [-26, -6, 0], [6, 2, 4], [-7, -1, 6], [9, -3, -2], [-14, 6, -3], [-35, -5, 3], [9, -9, 2], [-7, -9, 1], [21, -3, -6], [-24, 0, -4], [-11, 5, 7], [2, -10, 0], [-11, -1, 4], [-27, -5, -8], [25, -1, 6], [4, 4, -10], [20, 0, 4], [-1, -3, 6], [-27, 5, 3], [11, 5, 7], [-22, -2, 1], [-16, -4, 0], [4, -8, -4], [-33, -1, -2], [-20, 4, -8], [-4, 6, -10], [-13, 5, -3], [-34, -6, 2], [22, 2, 5], [-9, 1, -3], [-23, 1, -1], [-53, -1, -6], [19, -7, -2], [-17, 3, 5], [30, 2, 2], [13, -1, -10], [-4, -4, -12], [-25, 1, 8], [6, 12, 2], [16, -4, 0], [34, -2, 5], [-4, -4, -4], [-1, 1, -11], [28, -2, -4], [15, -3, -4], [2, 12, 8], [-44, -4, 1], [-6, -2, 4], [-36, 0, 4], [32, 0, 6], [2, 4, -4], [14, -6, 4], [-17, -1, -5], [-11, -5, -15], [-5, 9, -4], [-58, -2, 0], [-9, -7, 4], [22, -8, -4], [36, 4, 12], [-4, 0, 4], [-23, 3, -2], [-9, 3, -7], [25, 7, -7], [29, 3, 4], [11, -11, -11], [-14, 2, 8], [-7, 3, -8], [-42, 2, 4], [8, 2, -8], [9, 11, 8], [16, 0, -12], [16, -2, -10], [3, 9, -2], [-41, 1, 1], [20, -4, -8], [19, -3, 1], [20, -8, 4], [-16, 8, 4], [11, -7, -6], [19, -1, 11], [-38, 6, 2], [-27, 11, 4], [16, -4, -10], [6, -6, -8], [-25, 9, 3], [20, 8, 8], [-7, -7, -1], [23, 9, 11], [42, 0, 2], [-14, -2, 7], [-43, 3, -1], [-41, 5, 6], [7, 5, 12], [-14, -2, -11], [-38, -4, -16], [4, -12, -2], [1, -1, 11], [1, 7, 4], [-25, 1, -6], [19, 11, 3], [-36, -8, -4], [-21, 1, 14], [17, -7, 5], [22, 2, -4], [9, 11, 12], [13, 3, 14], [-10, -6, -10], [30, -4, -2], [-28, -12, 8], [-40, -6, -2], [15, 1, 4], [-51, 3, -1], [-25, -3, -6], [-11, 13, 6], [56, 4, -6], [-26, -6, 8], [38, 2, 2], [-46, -2, 4], [-3, -5, 14], [4, -8, 0], [-9, -7, 3], [8, 6, -4], [1, -5, -10], [5, -5, 0], [27, 7, -9], [25, 3, -10], [-6, 2, -8], [6, 4, 14], [10, 2, 18], [2, -6, -4], [14, 10, 11], [-24, 0, 8], [-36, -4, -9], [32, 4, 10], [-8, -4, 0], [-22, -4, 12], [-27, -1, 0], [18, 6, -6], [-1, 1, 1], [-10, 10, -4], [10, 4, -12], [-23, 7, 7], [-19, -5, -12], [-21, -11, 3], [31, 5, 10], [6, 10, 16], [15, -11, 0], [2, -4, -14], [20, -4, 2], [29, -13, -7], [-14, -8, 6], [14, 6, -4], [27, 5, 23], [6, 12, -2], [14, -12, -10], [-27, -9, -12], [-31, -3, -2], [2, -2, -7], [10, -2, -8], [10, -4, -4], [-2, 0, 18], [8, 8, 0], [19, 1, 4], [17, -1, 13], [-21, -7, 8], [44, 0, 8], [13, 7, -4], [-43, -5, 4], [12, -8, -2], [-4, 12, 0], [0, -4, 6], [-8, -4, 16], [27, 7, 7], [38, 4, 2], [-23, 7, 4], [22, 6, 12], [-62, 4, 2], [-4, -4, -1], [16, -8, -4], [-39, -5, 8], [23, -7, 1], [7, -7, -8], [8, 8, -2], [16, 0, -4], [6, 10, 3], [-13, -7, 0], [17, -3, 6], [-11, 7, 2], [8, 16, 4], [-38, 2, 4], [-11, 5, 7], [30, 2, 12], [14, 14, 8], [16, -2, 12], [-47, -1, -19], [46, -2, 12], [-9, -13, -3], [-1, -15, 3], [-55, -1, -5], [-48, 12, 8], [-13, -1, -6], [20, 4, 15], [-24, 0, -14], [3, -5, 2], [59, -3, 5], [-7, 3, -12], [-9, -7, 3], [47, -3, -2], [9, 7, 20], [-3, 7, -10], [6, 6, 4], [-39, -9, -7], [30, 4, -8], [-1, 15, 7], [-24, 14, 6], [-6, -2, 15], [20, 2, 20], [12, 12, -4], [14, -12, 0], [48, 4, 8], [12, -12, 8], [38, 2, -4], [41, -1, 8], [-86, -2, -4], [12, 4, 22], [33, 9, 2], [-36, 4, 7], [21, -13, -1], [-7, 3, 0], [-23, 7, 9], [16, 6, 0], [-4, -4, 12], [-1, 7, -17], [18, 6, -20], [-13, 1, 4], [-19, 11, -2], [60, 2, 16], [-22, -2, -26], [17, 5, 9], [12, -4, 3], [-56, 0, 4], [-25, -3, 18], [18, -4, 0], [80, 4, 0], [-4, -8, -14], [68, -2, 4], [71, -3, -4], [1, -1, -1], [-8, 0, -14], [-39, -7, -22], [24, 16, -9], [-13, -3, -5], [-17, 1, -28], [43, 3, 3], [-4, 4, -6], [9, -21, -8], [-1, 9, 6], [1, 15, 13], [2, 2, -20], [66, 4, 12], [-10, -6, -19], [43, 5, -7], [2, -8, 10], [24, -4, 12], [-40, -8, 14]];
    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_bi();".
// 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_bi(:prec:=3)
    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_bi();".
// 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_bi( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5376_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-16, -14, 0, 1]>,<11,R![-96, -26, 4, 1]>,<13,R![32, -20, -4, 1]>,<19,R![48, -4, -8, 1]>,<29,R![48, -4, -8, 1]>],Snew);
    return Vf;
end function;