# Properties

 Label 8034.235 Modulus $8034$ Conductor $103$ Order $51$ 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(8034)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,0,43]))

pari: [g,chi] = znchar(Mod(235,8034))

## Basic properties

 Modulus: $$8034$$ Conductor: $$103$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$51$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{103}(29,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 8034.ct

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

$$(5357,1237,5773)$$ → $$(1,1,e\left(\frac{43}{51}\right))$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$1$$ $$1$$ $$e\left(\frac{43}{51}\right)$$ $$e\left(\frac{19}{51}\right)$$ $$e\left(\frac{22}{51}\right)$$ $$e\left(\frac{1}{51}\right)$$ $$e\left(\frac{23}{51}\right)$$ $$e\left(\frac{4}{17}\right)$$ $$e\left(\frac{35}{51}\right)$$ $$e\left(\frac{26}{51}\right)$$ $$e\left(\frac{1}{17}\right)$$ $$e\left(\frac{11}{51}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{51})$ Fixed field: Number field defined by a degree 51 polynomial