Properties

Label 2695.1374
Modulus 26952695
Conductor 26952695
Order 4242
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2695, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,26,21]))
 
pari: [g,chi] = znchar(Mod(1374,2695))
 

Basic properties

Modulus: 26952695
Conductor: 26952695
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: 4242
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 2695.ck

χ2695(109,)\chi_{2695}(109,\cdot) χ2695(219,)\chi_{2695}(219,\cdot) χ2695(494,)\chi_{2695}(494,\cdot) χ2695(604,)\chi_{2695}(604,\cdot) χ2695(879,)\chi_{2695}(879,\cdot) χ2695(989,)\chi_{2695}(989,\cdot) χ2695(1264,)\chi_{2695}(1264,\cdot) χ2695(1374,)\chi_{2695}(1374,\cdot) χ2695(1649,)\chi_{2695}(1649,\cdot) χ2695(1759,)\chi_{2695}(1759,\cdot) χ2695(2034,)\chi_{2695}(2034,\cdot) χ2695(2144,)\chi_{2695}(2144,\cdot)

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(ζ21)\Q(\zeta_{21})
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

(2157,1816,981)(2157,1816,981)(1,e(1321),1)(-1,e\left(\frac{13}{21}\right),-1)

First values

aa 1-1112233446688991212131316161717
χ2695(1374,a) \chi_{ 2695 }(1374, a) 1-111e(221)e\left(\frac{2}{21}\right)e(542)e\left(\frac{5}{42}\right)e(421)e\left(\frac{4}{21}\right)e(314)e\left(\frac{3}{14}\right)e(27)e\left(\frac{2}{7}\right)e(521)e\left(\frac{5}{21}\right)e(1342)e\left(\frac{13}{42}\right)e(37)e\left(\frac{3}{7}\right)e(821)e\left(\frac{8}{21}\right)e(1021)e\left(\frac{10}{21}\right)
sage: chi.jacobi_sum(n)
 
χ2695(1374,a)   \chi_{ 2695 }(1374,a) \; at   a=\;a = e.g. 2