# Properties

 Label 3630.53 Modulus $3630$ Conductor $1815$ Order $220$ Real no Primitive no Minimal yes Parity even

# Learn more about

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

sage: H = DirichletGroup(3630, base_ring=CyclotomicField(220))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([110,165,212]))

pari: [g,chi] = znchar(Mod(53,3630))

## Basic properties

 Modulus: $$3630$$ Conductor: $$1815$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$220$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{1815}(53,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 3630.bt

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

$$(1211,727,3511)$$ → $$(-1,-i,e\left(\frac{53}{55}\right))$$

## Values

 $$-1$$ $$1$$ $$7$$ $$13$$ $$17$$ $$19$$ $$23$$ $$29$$ $$31$$ $$37$$ $$41$$ $$43$$ $$1$$ $$1$$ $$e\left(\frac{109}{220}\right)$$ $$e\left(\frac{127}{220}\right)$$ $$e\left(\frac{103}{220}\right)$$ $$e\left(\frac{53}{110}\right)$$ $$e\left(\frac{9}{44}\right)$$ $$e\left(\frac{21}{55}\right)$$ $$e\left(\frac{48}{55}\right)$$ $$e\left(\frac{49}{220}\right)$$ $$e\left(\frac{73}{110}\right)$$ $$e\left(\frac{15}{44}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{220})$ Fixed field: Number field defined by a degree 220 polynomial (not computed)