Properties

Label 8470.309
Modulus $8470$
Conductor $605$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(8470, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,0,20]))
 
pari: [g,chi] = znchar(Mod(309,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(309,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.bo

\(\chi_{8470}(309,\cdot)\) \(\chi_{8470}(1079,\cdot)\) \(\chi_{8470}(1849,\cdot)\) \(\chi_{8470}(2619,\cdot)\) \(\chi_{8470}(4159,\cdot)\) \(\chi_{8470}(4929,\cdot)\) \(\chi_{8470}(5699,\cdot)\) \(\chi_{8470}(6469,\cdot)\) \(\chi_{8470}(7239,\cdot)\) \(\chi_{8470}(8009,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 22.22.22099245882898413967719412126414511946210986328125.1

Values on generators

\((6777,6051,7141)\) → \((-1,1,e\left(\frac{10}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(-1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{3}{22}\right)\)\(-1\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{15}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(309,a) \;\) at \(\;a = \) e.g. 2