# Properties

 Label 2001.41 Modulus $2001$ Conductor $2001$ Order $44$ Real no Primitive yes Minimal yes Parity even

# Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(2001, base_ring=CyclotomicField(44))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([22,24,11]))

pari: [g,chi] = znchar(Mod(41,2001))

## Basic properties

 Modulus: $$2001$$ Conductor: $$2001$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$44$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 2001.bg

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Values on generators

$$(668,1132,553)$$ → $$(-1,e\left(\frac{6}{11}\right),i)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$1$$ $$1$$ $$e\left(\frac{37}{44}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{23}{44}\right)$$ $$e\left(\frac{17}{44}\right)$$ $$e\left(\frac{29}{44}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{4}{11}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{44})$$ Fixed field: Number field defined by a degree 44 polynomial