Properties

Label 2299.103
Modulus 22992299
Conductor 22992299
Order 330330
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(330)) M = H._module chi = DirichletCharacter(H, M([36,55]))
 
Copy content pari:[g,chi] = znchar(Mod(103,2299))
 

Basic properties

Modulus: 22992299
Conductor: 22992299
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: 330330
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2299.bq

χ2299(31,)\chi_{2299}(31,\cdot) χ2299(69,)\chi_{2299}(69,\cdot) χ2299(103,)\chi_{2299}(103,\cdot) χ2299(126,)\chi_{2299}(126,\cdot) χ2299(141,)\chi_{2299}(141,\cdot) χ2299(179,)\chi_{2299}(179,\cdot) χ2299(236,)\chi_{2299}(236,\cdot) χ2299(240,)\chi_{2299}(240,\cdot) χ2299(278,)\chi_{2299}(278,\cdot) χ2299(312,)\chi_{2299}(312,\cdot) χ2299(335,)\chi_{2299}(335,\cdot) χ2299(350,)\chi_{2299}(350,\cdot) χ2299(388,)\chi_{2299}(388,\cdot) χ2299(411,)\chi_{2299}(411,\cdot) χ2299(445,)\chi_{2299}(445,\cdot) χ2299(449,)\chi_{2299}(449,\cdot) χ2299(521,)\chi_{2299}(521,\cdot) χ2299(544,)\chi_{2299}(544,\cdot) χ2299(559,)\chi_{2299}(559,\cdot) χ2299(597,)\chi_{2299}(597,\cdot) χ2299(620,)\chi_{2299}(620,\cdot) χ2299(654,)\chi_{2299}(654,\cdot) χ2299(658,)\chi_{2299}(658,\cdot) χ2299(696,)\chi_{2299}(696,\cdot) χ2299(730,)\chi_{2299}(730,\cdot) χ2299(768,)\chi_{2299}(768,\cdot) χ2299(806,)\chi_{2299}(806,\cdot) χ2299(829,)\chi_{2299}(829,\cdot) χ2299(863,)\chi_{2299}(863,\cdot) χ2299(867,)\chi_{2299}(867,\cdot) ...

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: Q(ζ165)\Q(\zeta_{165})
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

(970,1332)(970,1332)(e(655),e(16))(e\left(\frac{6}{55}\right),e\left(\frac{1}{6}\right))

First values

aa 1-111223344556677889910101212
χ2299(103,a) \chi_{ 2299 }(103, a) 1-111e(91330)e\left(\frac{91}{330}\right)e(2330)e\left(\frac{23}{30}\right)e(91165)e\left(\frac{91}{165}\right)e(122165)e\left(\frac{122}{165}\right)e(7165)e\left(\frac{7}{165}\right)e(4255)e\left(\frac{42}{55}\right)e(91110)e\left(\frac{91}{110}\right)e(815)e\left(\frac{8}{15}\right)e(166)e\left(\frac{1}{66}\right)e(722)e\left(\frac{7}{22}\right)
Copy content sage:chi.jacobi_sum(n)
 
χ2299(103,a)   \chi_{ 2299 }(103,a) \; at   a=\;a = e.g. 2