# Properties

 Label 3645.7 Modulus $3645$ Conductor $3645$ Order $972$ Real no Primitive yes Minimal yes Parity odd

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(972))

M = H._module

chi = DirichletCharacter(H, M([788,243]))

pari: [g,chi] = znchar(Mod(7,3645))

## Basic properties

 Modulus: $$3645$$ Conductor: $$3645$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$972$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 3645.bi

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $\Q(\zeta_{972})$ Fixed field: Number field defined by a degree 972 polynomial (not computed)

## Values on generators

$$(731,2917)$$ → $$(e\left(\frac{197}{243}\right),i)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$ $$\chi_{ 3645 }(7, a)$$ $$-1$$ $$1$$ $$e\left(\frac{59}{972}\right)$$ $$e\left(\frac{59}{486}\right)$$ $$e\left(\frac{647}{972}\right)$$ $$e\left(\frac{59}{324}\right)$$ $$e\left(\frac{104}{243}\right)$$ $$e\left(\frac{877}{972}\right)$$ $$e\left(\frac{353}{486}\right)$$ $$e\left(\frac{59}{243}\right)$$ $$e\left(\frac{1}{324}\right)$$ $$e\left(\frac{49}{162}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 3645 }(7,a) \;$$ at $$\;a =$$ e.g. 2