
// Make newform 5445.2.a.bp in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_5445_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_5445_2_a_bp();".
// 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_5445_2_a_bp();".
// 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 := [1, 1, -3, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-1, -1, 1, 0], [1, -2, -1, 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_5445_a();"
function MakeCharacter_5445_a()
    N := 5445;
    order := 1;
    char_gens := [3026, 4357, 3511];
    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_5445_a_Hecke(Kf)
    return MakeCharacter_5445_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, 1, 0, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, -1, -2, 1], [0, 0, 0, 0], [0, 3, 0, -2], [-1, -1, -1, 3], [-5, -2, 1, 0], [-2, 1, 1, 0], [5, -2, -1, -2], [0, -5, 0, 0], [3, 3, 3, -7], [1, -1, 3, 1], [-6, 1, 2, 0], [-3, 1, 3, 0], [6, -1, -5, -1], [-2, 1, 4, -5], [-2, 2, -2, -1], [-5, -3, -4, 6], [-3, 3, 0, 2], [-1, -3, -2, 0], [-9, -4, -3, 6], [-1, -1, 3, -6], [-1, 0, 6, 0], [5, 0, -2, 8], [2, -3, -7, 1], [-1, 0, 4, 6], [-2, 8, 3, -7], [-5, 1, 4, -6], [-1, 1, -1, 2], [-5, -2, 3, 3], [6, -1, -6, 2], [8, -9, -4, 3], [-8, 5, 5, -7], [1, 6, 5, -7], [-12, 2, 9, 5], [6, 3, 6, -10], [4, 2, -9, -4], [-7, -5, -6, 7], [-11, -1, -2, 3], [5, -12, -2, 0], [0, 3, -8, 4], [1, -5, -6, 0], [-7, -5, 9, 2], [2, 5, -1, -7], [-4, -2, 3, -2], [-17, -1, -2, 12], [-6, -2, 2, 10], [10, 2, -7, -7], [1, 1, 13, 2], [-12, 13, 6, 3], [-4, -7, 7, 16], [-2, 10, 0, -9], [-9, -2, 1, 9], [-7, 1, 3, 14], [-1, -7, 0, -4], [4, -5, -6, -4], [-7, 7, 0, -5], [-6, -2, 0, 1], [-6, -8, 3, 16], [-17, 9, 10, -3], [-10, 1, -2, -7], [0, -5, -7, -7], [9, 13, 7, -16], [-6, 8, -5, -2], [-6, 13, 7, -4], [-2, 20, 4, -12], [-11, -6, 6, -5], [5, -3, 0, 1], [-10, -7, 0, 16], [4, 17, -2, -7], [-5, 6, 8, 0], [-3, -2, -11, 8], [-1, 6, 5, 11], [2, -2, 4, 9], [16, 5, 5, -7], [-11, -22, -6, 18], [-11, -2, 0, -3], [15, 11, 3, -13], [9, 8, 2, -3], [11, 9, -9, -5], [5, 15, 0, -17], [2, 11, 13, -15], [8, 18, -7, -17], [0, 0, -5, 8], [3, 17, -2, 3], [-16, 0, 13, 7], [-9, -8, 6, 5], [-8, 5, 3, -2], [19, -2, -9, -17], [-18, -4, 6, -9], [-8, 13, 1, -2], [-12, 12, -3, -1], [15, 5, 4, 0], [10, -12, -11, -5], [-38, 11, 5, 0], [-3, -1, -6, 9], [5, 13, 0, 3], [12, -2, -5, -12], [4, 2, 4, -1], [-2, 2, -1, -14], [-5, -20, 5, 22], [-9, -2, 9, -2], [3, 5, -13, -20], [-6, -1, 0, -16], [26, -13, -5, -8], [11, 10, -15, -16], [-36, 2, 1, 6], [5, -3, -3, 12], [24, -9, 0, -16], [2, 11, -13, 6], [11, -15, -11, 4], [0, -5, -7, 15], [21, -10, -17, -5], [-15, -16, -12, 20], [21, 12, -6, -12], [9, -8, -17, -4], [-1, -4, -2, 29], [-9, 25, -3, -6], [10, -7, -2, 9], [-18, -14, 9, 8], [-4, -20, 13, 7], [19, -4, -22, 6], [15, -11, 1, 8], [10, -2, -5, 15], [21, -23, -9, 1], [-15, 10, 7, 2], [19, -9, -15, 6], [19, 0, -18, -6], [18, -4, -14, 11], [-4, 0, -17, -8], [2, 4, -6, -30], [-5, -4, 12, 13], [-24, 0, 6, 6], [7, -7, -9, 16], [5, -18, 10, 20], [-9, -19, 12, 7], [-24, 6, 8, 11], [17, 6, -5, -7], [3, -2, -10, 8], [-4, 17, -3, 5], [30, 7, 0, -31], [-4, 24, -4, 1], [26, 1, -3, -35], [-7, -5, 5, 5], [-12, 20, 3, -8], [11, 10, 17, -12], [29, -20, -7, -7], [7, -3, -11, -2], [-5, -2, -8, 13], [-13, -14, -4, 14], [-14, -10, -17, 15], [0, 13, -8, 19], [11, -16, -7, -17], [2, 23, -2, -8], [-8, -8, 2, 20], [24, -8, -6, -22], [8, -14, 11, -13], [-12, 7, 13, -12], [38, 8, -5, -4], [-9, 11, 6, -19], [-17, -10, 9, 13], [-12, -11, 4, 7], [17, 7, -2, -3], [6, -15, -17, -1], [-23, -8, 14, -17], [15, 7, 1, 0], [8, 19, 2, -17], [2, 16, 25, -24], [32, 2, 3, -42], [1, -20, 14, 25], [-33, -8, 3, 24], [12, 10, -1, -4], [-8, 24, 1, 1], [18, 1, -4, -36], [-49, 3, 2, 17], [-8, 0, -17, 6], [2, 6, -1, -29], [-9, -20, 8, 16], [15, -23, -17, 11], [-22, -5, -9, 6], [9, -2, -1, 21], [-18, -34, -5, 25], [23, 30, 12, -32], [-33, -16, -8, 41], [14, 6, 2, 22], [-5, -9, -20, 11], [4, 14, 16, -8], [14, -17, -7, -1], [9, -5, -3, -13], [-11, -1, 14, -5], [-5, 6, 5, 2], [3, 16, -5, -14], [-2, -16, 22, 19], [-50, 25, 18, 20], [-32, -20, 1, 27], [19, -11, -27, 2], [7, -7, -18, -4], [2, 28, -10, -9], [-8, 15, -21, -20], [6, 10, 5, 10], [26, 20, -20, 14], [11, -7, -8, -17], [-17, -14, -9, 42], [8, 15, -22, -3], [-26, 3, 20, 28], [29, 1, 7, 7], [-21, -25, -4, 30], [31, -28, -6, -11], [28, 9, -8, 2], [-14, -1, 5, 19], [-16, -5, 13, -3], [38, -16, -3, -18], [-36, 22, 10, 0], [11, 12, 8, -13], [-20, 22, -3, 5], [36, -9, -33, 12], [-11, -1, 30, 27], [16, -15, -12, -6], [10, -13, 17, 17], [16, -37, 5, 17], [-7, 9, 4, -32], [25, -3, -12, -45], [-29, 6, 13, 2], [-18, -15, 8, -18], [31, 12, -7, -27], [14, -21, -2, 22], [1, 8, 9, -1], [44, 6, 11, -14], [15, -5, -28, 6], [4, 8, -14, -20], [-16, -9, -11, -12], [5, -14, 0, -23], [29, 6, 0, -25], [12, 30, 2, -10], [-37, 24, 4, 10], [6, -5, -7, 17], [-25, 4, 29, 10], [-7, -21, -26, 23], [44, -5, -16, 5], [-9, -16, 24, -1], [-11, -24, 14, 17], [1, -24, 13, 35], [-7, 11, 16, -15], [-15, 3, 12, -2], [31, 3, -28, 7], [-44, -4, 3, -17], [23, -22, -11, 2], [15, -16, -2, -18], [-34, -14, 2, 22], [-23, 26, -4, -20], [-15, -28, -24, 35], [-20, 18, 10, -27], [-13, 4, 3, 7], [21, -10, -5, 14], [15, -16, 10, -5], [32, -15, -18, 17], [7, -2, 21, -6], [-25, -33, 13, -1], [18, -12, 6, -30], [-18, -4, -18, -5], [-23, 33, 11, 9], [34, 1, -14, -33], [16, -13, -24, 1], [15, -18, -18, -14], [-41, 5, 7, 6], [1, -5, 4, -22], [-2, -21, -15, 30], [-60, 36, 26, 18], [24, -4, 5, -2], [39, 2, -10, -26], [20, -10, 14, -7], [-9, -13, -23, 9], [-19, 6, -13, 38], [-21, 28, 40, -9], [2, -24, -6, 10], [20, -21, 5, -22], [-9, -3, -22, -15], [3, 24, 10, -11], [4, -17, -32, 14], [-11, -2, -4, -18], [-25, -15, 3, 23], [-47, -7, 29, -9], [8, 28, -7, 7], [-22, -33, -21, 37], [-22, 27, -15, -24], [-15, -2, -2, -2], [12, 11, 21, -4], [18, 14, -10, -5], [-20, 5, -5, 25], [-2, -3, -35, 7], [36, -24, 3, 10], [30, -12, -29, 11], [-13, -10, -20, 8], [-24, -14, 35, 5], [-7, -14, 9, 56], [-4, 2, -3, -20], [-9, -4, -1, 0], [-30, 52, 21, -17], [21, 1, -26, 20], [-7, -12, 9, 40], [-40, -39, -13, 48], [9, 20, -29, -5], [5, 16, 6, -25], [42, -37, -29, 6], [-20, -19, -1, 38], [15, 3, -16, 0], [19, -25, 19, 20], [-45, 18, 17, 2], [10, -9, -23, -1], [-22, 31, 16, -28], [-34, 1, 19, 19], [7, 18, -24, -40], [-21, -5, 15, -22], [-47, -19, 0, 33], [20, -20, 9, -15], [25, 7, 13, -45], [2, 17, 22, -21], [10, -3, -32, 10], [-19, 0, -4, -2], [19, -23, -6, 23], [3, 20, -5, -28], [-4, 33, 12, -1], [33, -43, -20, 3], [19, -12, 15, -4], [46, -19, -27, 3], [0, 29, -1, -44], [-43, 23, 15, -7], [-37, 8, 24, 8], [-4, -3, 3, -36], [-20, 4, 11, -13], [46, -30, -20, -20], [25, -9, 4, -38], [13, -43, 3, 13], [8, -15, 31, 4], [-9, -24, 0, 33], [7, 29, 18, -24], [15, 31, -9, -47], [-49, 15, 18, 34], [2, 34, 26, -16], [-21, 0, 15, 40], [20, -15, 13, 12], [-8, -41, -10, 29], [-4, 11, 42, -28], [45, -25, -42, 9], [-23, 16, 11, 23], [-31, -3, -6, -10], [51, -1, 6, -9], [-25, 36, 16, 7], [-13, -1, 21, -24], [-36, -7, -11, 24], [-50, -27, 6, 5], [-23, 16, 7, 14], [24, 26, 17, -50], [45, 5, 4, 8], [7, -10, -30, -7], [37, 2, 7, -64], [-27, -3, 14, -39], [15, -22, -47, 7], [-2, -7, -5, 35], [26, 2, -37, 15], [39, 27, 16, -37], [-39, 9, 14, -14], [28, -25, -20, -19], [-34, 8, -3, -10], [24, -30, 4, 18], [27, 19, 38, -12], [-17, 31, -1, -24], [-54, -9, 28, -2], [1, -7, -5, 29], [58, -17, -37, 4], [-29, -1, 16, 41], [7, 16, -8, 13], [17, -19, 18, 31], [25, -23, -28, -10], [1, -16, -15, 2], [-11, 31, -17, 4], [0, -33, -4, 34], [-4, 11, -4, 2], [-35, 13, 18, 31], [15, -53, -18, 14], [-11, -28, -46, 22], [-53, -3, 12, 40], [8, 26, 15, -4], [-35, -17, -16, 12], [14, -39, 11, 12], [-10, 19, 27, -35], [39, -18, -1, 21], [-34, -8, -30, 10], [-2, 22, 46, -30], [-4, -5, -36, 20], [-3, 16, -13, 33], [-17, -1, -11, 37], [4, -9, -19, 1], [-20, -26, -11, 35], [-7, 1, 36, -20], [-2, 6, 22, 6], [-28, 38, 2, -24], [48, -37, -50, 5], [-41, -45, -23, 69], [29, -4, -27, -4], [2, 49, 11, -22], [-35, 19, -8, -16], [3, -1, -17, 20], [-25, 33, 41, -2], [-60, 52, 22, 15], [-15, -10, 33, 19], [-17, -36, 13, 3], [-33, 20, 56, -4], [-36, 23, 16, 12], [-25, -3, 16, 25], [-20, -34, 12, 25], [27, 24, 37, -33], [22, -45, -19, 12], [-67, 30, 26, 8], [-17, 9, 3, 56], [4, -13, -11, -32], [29, 17, 6, -8], [33, -2, 23, -28], [-22, 23, -13, 23], [61, 6, -9, 8], [-60, 31, 30, 4], [-48, -31, -25, 33], [12, 22, -32, 28], [8, -31, -50, 30], [-13, 31, 16, -23], [35, 13, -27, 16], [-2, 1, -3, -37], [2, -11, -15, 51], [33, 35, 3, -46]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_5445_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_5445_2_a_bp();".
// 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_5445_2_a_bp(:prec:=4)
    chi := MakeCharacter_5445_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_5445_2_a_bp();".
// 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_5445_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_5445_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1, -3, -1, 1]>,<7,R![31, -23, -11, 3, 1]>,<23,R![-11, -10, 4, 5, 1]>,<53,R![941, 311, -43, -11, 1]>],Snew);
    return Vf;
end function;