Properties

Label 4410.211
Modulus $4410$
Conductor $441$
Order $21$
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([7,0,15]))
 
pari: [g,chi] = znchar(Mod(211,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(211,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.cs

\(\chi_{4410}(211,\cdot)\) \(\chi_{4410}(421,\cdot)\) \(\chi_{4410}(841,\cdot)\) \(\chi_{4410}(1051,\cdot)\) \(\chi_{4410}(1681,\cdot)\) \(\chi_{4410}(2101,\cdot)\) \(\chi_{4410}(2311,\cdot)\) \(\chi_{4410}(2731,\cdot)\) \(\chi_{4410}(3361,\cdot)\) \(\chi_{4410}(3571,\cdot)\) \(\chi_{4410}(3991,\cdot)\) \(\chi_{4410}(4201,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{5}{7}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{13}{21}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 21.21.60663096207149471029120391038078960708572561.1