Properties

Label 8470.3123
Modulus $8470$
Conductor $605$
Order $44$
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(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,0,6]))
 
pari: [g,chi] = znchar(Mod(3123,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(98,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cb

\(\chi_{8470}(43,\cdot)\) \(\chi_{8470}(197,\cdot)\) \(\chi_{8470}(813,\cdot)\) \(\chi_{8470}(1583,\cdot)\) \(\chi_{8470}(1737,\cdot)\) \(\chi_{8470}(2353,\cdot)\) \(\chi_{8470}(2507,\cdot)\) \(\chi_{8470}(3123,\cdot)\) \(\chi_{8470}(3277,\cdot)\) \(\chi_{8470}(3893,\cdot)\) \(\chi_{8470}(4047,\cdot)\) \(\chi_{8470}(4663,\cdot)\) \(\chi_{8470}(4817,\cdot)\) \(\chi_{8470}(5433,\cdot)\) \(\chi_{8470}(5587,\cdot)\) \(\chi_{8470}(6203,\cdot)\) \(\chi_{8470}(6357,\cdot)\) \(\chi_{8470}(6973,\cdot)\) \(\chi_{8470}(7127,\cdot)\) \(\chi_{8470}(7897,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1

Values on generators

\((6777,6051,7141)\) → \((-i,1,e\left(\frac{3}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(i\)\(-1\)\(e\left(\frac{1}{44}\right)\)\(e\left(\frac{19}{44}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{35}{44}\right)\)\(-i\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{21}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(3123,a) \;\) at \(\;a = \) e.g. 2