Properties

Label 3630.199
Modulus $3630$
Conductor $605$
Order $22$
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(3630)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,11,16]))
 
pari: [g,chi] = znchar(Mod(199,3630))
 

Basic properties

Modulus: \(3630\)
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 \(\chi_{605}(199,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.be

\(\chi_{3630}(199,\cdot)\) \(\chi_{3630}(529,\cdot)\) \(\chi_{3630}(859,\cdot)\) \(\chi_{3630}(1189,\cdot)\) \(\chi_{3630}(1519,\cdot)\) \(\chi_{3630}(1849,\cdot)\) \(\chi_{3630}(2509,\cdot)\) \(\chi_{3630}(2839,\cdot)\) \(\chi_{3630}(3169,\cdot)\) \(\chi_{3630}(3499,\cdot)\)

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

\((1211,727,3511)\) → \((1,-1,e\left(\frac{8}{11}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{15}{22}\right)\)
value at e.g. 2

Related number fields

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