
// Make newform 6525.2.a.bp in Magma, downloaded from the LMFDB on 06 February 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_6525_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_6525_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_6525_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 := [3, 7, -1, -7, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [3, 5, -6, -1, 1], [6, 5, -7, -1, 1], [9, 5, -13, -1, 2]];
    Rf_basisdens := [1, 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_6525_a();"
function MakeCharacter_6525_a()
    N := 6525;
    order := 1;
    char_gens := [1451, 4177, 901];
    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_6525_a_Hecke(Kf)
    return MakeCharacter_6525_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, 1, 0, 0, 0], [0, 0, 0, 0, 0], [0, 0, 0, 0, 0], [2, 1, 0, 0, -1], [0, 1, 1, 0, 0], [2, 0, 1, 0, 2], [0, -1, 2, -1, 0], [-1, 0, -1, 1, 0], [0, 0, 1, 1, -1], [1, 0, 0, 0, 0], [2, -1, 2, 1, 1], [2, -1, 2, 2, 0], [0, 2, -1, -1, 4], [2, -2, -1, 0, -1], [0, 2, 0, -1, -2], [-3, -2, 0, -3, 1], [3, -2, 0, 0, -1], [-1, 1, 1, 0, -4], [5, 1, 2, 4, 0], [3, -2, 1, -4, -4], [2, -4, -1, 2, -2], [-1, 2, -1, -1, 5], [-6, -1, -6, 1, 4], [-3, 2, -3, 5, -5], [-4, 0, -4, -3, 4], [6, -1, 3, 0, -3], [8, 0, 0, 0, -1], [-3, 1, 0, 0, 4], [8, -1, 2, 2, -3], [-3, -4, 0, -1, -3], [-1, -3, 4, 3, -4], [0, -3, 1, 1, 2], [3, 4, -6, 3, -2], [-1, 2, -1, 7, 0], [-6, 5, 0, -3, 1], [2, 0, 7, 0, -1], [5, 0, 0, -8, -1], [8, 6, -4, -2, 0], [3, -2, 4, -4, 2], [6, -2, 3, 5, -2], [0, -5, -1, -4, -2], [-4, -1, 0, 3, -5], [0, 5, 4, 2, -1], [11, 2, 0, 3, -1], [3, 5, -3, -1, 3], [-4, -7, 4, 1, 2], [-1, -3, -2, -5, 3], [8, -9, -1, -2, 3], [-3, 9, 0, 1, -1], [2, 6, -1, 0, 0], [6, -2, 0, 9, 1], [9, -2, 2, -3, 8], [-7, -3, 2, 2, 1], [3, -2, -3, 2, 3], [3, 10, -2, 5, -1], [9, -3, -3, 0, 6], [0, 5, 6, 0, -2], [11, 7, 1, 9, 4], [-1, 2, -8, 6, 3], [-3, 0, -7, -6, 2], [20, 7, -2, 4, 2], [3, -1, 4, -10, -2], [11, -3, -1, -1, 5], [-9, 0, 1, 0, 7], [-1, 3, 7, 2, -5], [3, -7, 4, 8, 1], [5, -6, -5, -2, 5], [5, 3, -3, 0, -6], [-3, -1, -1, -1, -10], [-1, 10, -2, -7, -6], [0, 7, 3, 2, 4], [21, 5, 2, 3, -2], [11, -10, -1, 1, 3], [-16, 1, -8, -2, 3], [-13, -3, -5, -7, -3], [3, -7, 1, 1, 4], [-6, 2, 0, 5, -4], [8, -2, 1, 4, -9], [3, -1, -1, 7, 5], [-1, -13, 0, -1, -2], [-3, 7, -5, 3, -11], [-1, 4, -9, 2, -3], [-18, 0, 2, -1, -3], [-1, -4, -8, -2, 5], [-1, 5, -12, -2, 5], [6, -3, 2, 0, 1], [9, 0, -9, -4, 6], [-7, 7, -2, 6, 10], [15, -2, -1, 5, 1], [-4, -7, -1, -2, -7], [0, -4, -2, 2, 3], [-3, -9, -9, 3, 4], [-1, 5, -4, -1, 8], [6, -9, 3, 11, -3], [-4, -9, 1, -3, 1], [12, -1, -6, 6, -6], [-3, 1, -2, 0, 4], [-9, -1, 2, 7, -11], [2, -4, 10, -12, 3], [2, 9, 8, -1, -2], [2, -6, 2, 1, -9], [6, -10, 4, 7, 1], [3, -6, 4, -8, 11], [-12, -13, 5, -7, 1], [-1, -18, 2, 0, 2], [11, -1, 5, -7, -9], [-6, -5, -7, -5, 9], [-3, -9, -1, 6, -1], [-9, -10, 10, -5, -1], [-4, -12, 9, -10, 5], [-1, -1, 2, 9, -8], [-7, 4, -4, 5, 9], [27, 0, 6, -8, -2], [8, -1, 2, -15, 0], [8, -8, 5, 1, -11], [24, -9, 8, 12, 3], [29, -6, 1, 2, -1], [-12, 3, 0, -1, -4], [0, -4, -1, -1, -8], [-15, -10, 1, 5, 13], [-4, -4, 3, 10, 3], [-25, -1, -5, 4, 9], [18, 2, -11, 11, 0], [-3, 2, 2, 1, -14], [-7, 7, 0, -2, 6], [21, -10, 1, -10, 12], [-4, -6, -10, 6, 3], [15, -12, -4, -4, 6], [8, 3, 7, -7, -7], [-10, 10, 6, -12, 3], [-10, 4, -6, 15, -7], [21, -3, 11, 6, 7], [-1, -6, 6, -4, 9], [11, -4, 7, -9, 11], [-6, -8, 0, -14, 1], [2, 6, -14, 9, 2], [-3, 1, 9, -18, -1], [5, 9, -8, -3, 7], [27, -1, 7, -5, -5], [-15, -11, -11, -1, -3], [-1, 7, 7, -8, 5], [12, -14, 4, 3, 1], [-22, 6, -1, -3, -4], [-6, 10, -6, 5, 15], [-31, -1, -10, -2, 5], [-12, -1, 2, 8, 12], [-4, -9, -3, 4, -11], [-3, -1, 0, -11, 4], [-22, -11, 4, -6, -7], [-9, 2, -16, 10, -5], [5, -2, 4, -9, 10], [0, -3, 9, -9, -3], [17, 2, 3, 1, 10], [-6, 9, -3, -7, 21], [11, 4, 7, -7, -2], [6, -3, 4, -13, 15], [-13, -7, 8, -5, 2], [12, 6, -10, -10, 8], [29, -2, 4, 3, -1], [18, 7, -1, -2, -2], [-12, -1, -6, 0, -7], [-6, -4, -11, 11, 6], [-1, -1, 2, -11, 5], [-3, -4, 4, -4, -6], [-9, -4, 0, -13, -5], [-12, 6, -5, -15, -1], [-13, 7, -7, -5, 13], [-4, -6, 5, -1, -6], [-16, 18, 6, -9, 10], [-3, -1, 8, -7, -1], [-36, 9, 2, -5, 5], [14, 15, -3, 6, 6], [-21, -1, -9, -3, -12], [5, -9, -6, -1, 8], [14, -14, -4, -3, 3], [-3, 1, 16, -13, 2], [11, -5, 3, 2, -2], [-6, 10, -13, 8, -9], [17, 17, -6, 3, -6], [-4, -7, 15, -15, -7], [8, 16, 3, 3, 4], [12, -1, -6, 7, 0], [-4, -16, -1, 2, 5], [12, -1, 22, -1, -9], [9, -10, 4, -3, 15], [12, 1, 0, 11, -8], [-19, -13, -12, -5, -9], [17, -8, 6, 0, 10], [5, -11, 15, -2, 1], [-30, 5, -3, 11, 1], [11, 1, -10, -7, 6], [9, -7, -3, -24, -1], [5, 16, -2, 17, 7], [12, -8, 13, -1, 5], [3, -13, 3, -12, -7], [-18, -2, -12, -9, -8], [29, -8, 3, -8, 12], [17, 1, 10, -13, -3], [-9, -12, -1, 7, 11], [-3, -7, 11, -3, -2], [-3, -6, -5, 4, -5], [-7, -7, -4, -18, 1], [8, -5, 10, -1, 2], [2, 8, 6, -6, -13], [-33, 1, -11, -8, 6], [-6, 27, 8, 3, -11], [8, 7, 6, -10, 2], [-15, 6, -9, 1, 4], [-24, -2, -3, -1, 16], [-1, -7, 1, -18, -13], [8, 20, -9, 7, -15], [30, 10, 7, -4, 2], [-10, 10, 2, -15, -3], [15, 6, 9, -6, -10], [-3, 16, -6, -4, -9], [8, 13, 2, 2, -2], [5, 10, 6, 1, -19], [21, 3, 0, -3, 3], [18, -2, 10, 17, -4], [15, -10, -10, 19, 3], [11, -20, -10, 6, 6], [26, -4, 10, 13, -12], [6, 9, -5, 15, -3], [-10, 14, 3, -7, -12], [-18, 1, -2, -5, -14], [-1, 10, 2, -3, -15], [6, -4, 3, 20, -3], [24, 4, 7, 4, -19], [-22, -4, -6, 13, 15], [-15, -4, -8, 18, 3], [5, 0, -7, 17, -7], [-4, 11, -11, 13, -4], [11, -14, -1, 8, 11], [6, 16, 4, 15, 9], [38, 3, -6, -4, -6], [30, -4, 3, 18, 6], [-28, 6, -9, 2, 0], [3, -17, 8, -7, 17], [-21, 17, 3, 1, -7], [-18, 5, -18, -1, 2], [-6, -8, -11, 12, 7], [-4, 15, -17, -5, -3], [11, 18, 9, 5, 9], [11, 15, 15, -3, 3], [-45, 10, -7, -6, 4], [-33, -4, 6, -7, 4], [-10, 3, -14, 1, 8], [9, 19, -1, 17, -5], [14, -2, -7, 1, 0], [18, 12, 11, -10, -18], [2, 17, -1, 5, -10], [-30, 6, -1, -12, 7], [24, 4, -15, 3, 4], [-16, -6, -1, -12, 16], [3, 9, -4, -12, -5], [-9, -15, 4, -3, 13], [8, -14, 7, -6, -17], [14, -8, -2, 5, -24], [-21, 3, 8, -17, 9], [-19, 5, -16, -6, -10], [-1, -2, -11, -9, 16], [15, -3, -13, 14, -3], [-4, 1, 6, -10, 12], [-7, 8, 5, -14, 17], [-27, 7, -1, 2, 11], [20, 7, 18, 0, -18], [0, 20, 13, 2, -12], [30, 4, -2, 13, -4], [8, -3, 5, -3, 16], [-15, 24, -6, 8, -17], [11, -4, -16, -3, 9], [8, -16, 4, -2, 4], [-7, -9, -11, 0, -6], [-4, 0, 13, 17, -2], [-10, -5, -1, 13, 3], [8, -19, 9, -6, -25], [12, 10, 10, 5, 1], [11, -7, -3, -1, -4], [-10, 2, -18, 15, 5], [15, -17, 8, -5, 0], [3, 7, -7, -4, 13], [-16, 6, -13, -23, 0], [18, -6, 2, 0, 11], [5, 2, 6, 14, -9], [-10, 1, -9, -16, 6], [-6, 5, 8, 2, -7], [2, -9, 16, 4, -17], [-3, 3, 3, 14, -20], [-10, 8, -3, -6, -16], [-45, -2, -11, 6, 15], [-3, 14, 7, 3, -2], [-36, -12, -5, 0, -1], [21, 20, -8, -6, -16], [21, 3, 2, 1, -19], [-1, 4, -1, -5, 1], [9, -2, 15, 19, -1], [-19, 0, -16, 20, -3], [-6, 0, -6, -28, -3], [9, 22, 3, 14, -3], [-28, 7, -3, -13, 18], [-7, -14, -23, -2, 17], [-12, -14, -8, -11, 12], [26, 19, -2, -3, -2], [-12, 8, -2, -10, -19], [-22, 0, -5, 9, -4], [17, -27, 1, 2, 16], [-9, 3, 10, 4, -1], [-10, -12, 10, 8, -5], [21, -8, -12, 16, -1], [-1, -12, 1, -1, -8], [9, 4, -3, 5, -3], [-6, 5, -16, 7, 0], [0, 17, -9, -8, 0], [20, -9, 16, -11, 11], [0, 5, 9, -21, 5], [2, -3, -2, 2, 2], [-15, -2, -8, -24, 6], [-9, 15, -9, -6, -4], [14, 15, 1, 9, 3], [24, -14, 1, 3, 10], [-7, 4, 1, -13, -12], [23, -22, -3, -16, 14], [-15, 3, 3, 0, 5], [-22, 14, -3, 27, 1], [18, -9, 5, -13, 5], [-22, -5, 3, 9, 20], [8, 8, -5, -2, 8], [-1, -8, -22, 23, 3], [42, 19, -8, -1, 2], [9, -3, -11, -16, 28], [35, 15, -9, -7, 6], [-3, -23, -10, 7, 9], [-4, 10, -15, -3, 11], [3, -4, 13, 8, 17], [23, 0, 10, -29, -11], [6, -16, 10, -8, -16], [-16, -1, 5, -16, 7], [-24, -9, 9, 4, 4], [-13, 28, 1, 18, 1], [11, 7, 10, 17, -10], [8, -7, 3, -6, 21], [-3, 1, 1, 38, -1], [-15, -1, -8, 7, 24], [14, -9, -3, -29, 13], [-15, 8, -3, 10, -17], [21, -5, -4, -12, 20], [41, -9, 2, -5, -6], [-4, -5, 13, -5, -12], [-12, 26, -16, -14, -12], [-3, 5, -9, 9, -6], [2, 2, -5, -5, 12], [-7, -12, 19, 11, -10], [21, 18, -5, 2, 14], [8, -6, 14, -21, -7], [2, -16, -3, -1, -15], [18, -3, 3, 2, -4], [3, -1, -16, -23, 13], [3, 18, -9, 29, 4], [6, -9, 6, 20, -19], [30, 8, -5, 9, 16], [35, -14, 13, 7, -19], [27, 18, -15, -1, 3], [48, -6, 12, -5, 8], [0, 2, -4, -9, -24], [14, -8, 15, 0, -14], [-16, -1, 9, -6, -8], [24, 11, -10, 1, -17], [-1, 15, -3, -6, -12], [-16, 2, 2, 5, 22], [3, -19, -7, 16, 10], [-16, -3, -5, 15, -34], [21, 4, -9, -7, -10], [15, 19, 0, 2, 21], [2, -28, -8, -17, 8], [-31, -22, -8, -7, -10], [-51, 0, -29, 0, -15], [48, 6, 21, -3, -7], [44, 0, 9, -15, 16], [9, -14, 2, 9, 6], [-22, 12, 9, -11, 11], [30, 20, -10, 5, 5], [-45, -21, -6, 6, 15], [-28, 14, 13, -26, 3], [0, 10, 5, -7, 8], [23, -20, 2, -13, 29], [39, 18, -10, 0, 5], [2, 6, -16, -11, -5], [20, -25, -6, 2, 10], [-25, 3, -2, -11, 24], [21, -18, 11, 20, 3], [-31, -22, -24, 1, 7], [-6, 8, -19, 16, -26], [39, 5, -8, 4, 11], [23, 14, -8, 17, 1], [-27, 7, -15, -5, 14], [14, 15, -9, 10, -11], [-25, 8, 6, -21, -1], [18, 0, 24, -3, -22], [-1, -10, 28, -28, -6], [15, -15, 15, 9, -29], [14, -1, -7, -17, 19], [-6, -9, -6, -12, 22], [-4, 9, 11, 11, -15], [21, 0, -18, 14, 1], [3, 0, 1, 1, 13], [-31, 4, -17, -8, -17], [-43, 9, 4, 12, 2], [-24, 4, -3, 5, -3], [14, 2, 25, -11, 3], [6, -6, 7, -11, -23], [2, 17, 6, 15, -4], [3, -24, 12, 2, 10], [-12, -1, 13, -34, 12], [32, 31, -15, 11, 14], [-13, -18, 7, -4, 5], [-27, -9, 3, -4, 5], [3, -25, 14, -12, -17], [-1, -8, 13, -12, 8], [-39, -3, -10, -5, -16], [69, 9, 15, 8, 2], [-60, 8, -5, -2, 17], [-1, 8, -11, 16, 10], [18, 10, -4, 12, -25], [-9, 13, -13, -16, 4], [-34, 10, 1, -2, -4], [-12, -12, 17, -19, -3], [-9, 7, 10, 17, 3], [-3, 17, 3, -2, -11], [2, -6, 19, 22, -3], [-6, -19, 26, -7, 13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6525_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6525_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_6525_2_a_bp(:prec:=5)
    chi := MakeCharacter_6525_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_6525_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_6525_2_a_bp( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6525_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 7, -1, -7, 0, 1]>,<7,R![-1, -78, 11, 26, -10, 1]>,<11,R![27, 42, -14, -17, 2, 1]>],Snew);
    return Vf;
end function;