Properties

Label 4410.319
Modulus $4410$
Conductor $2205$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4410)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,21,16]))
 
pari: [g,chi] = znchar(Mod(319,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2205}(319,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.cy

\(\chi_{4410}(319,\cdot)\) \(\chi_{4410}(709,\cdot)\) \(\chi_{4410}(1339,\cdot)\) \(\chi_{4410}(1579,\cdot)\) \(\chi_{4410}(1969,\cdot)\) \(\chi_{4410}(2209,\cdot)\) \(\chi_{4410}(2599,\cdot)\) \(\chi_{4410}(2839,\cdot)\) \(\chi_{4410}(3229,\cdot)\) \(\chi_{4410}(3469,\cdot)\) \(\chi_{4410}(3859,\cdot)\) \(\chi_{4410}(4099,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{8}{21}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{5}{42}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.4213193412157217008109022143330998719897524861816838575339993162809154279468810214633478699207305908203125.2