Properties

Label 3600.2861
Modulus $3600$
Conductor $1200$
Order $20$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3600)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,10,16]))
 
pari: [g,chi] = znchar(Mod(2861,3600))
 

Basic properties

Modulus: \(3600\)
Conductor: \(1200\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1200}(461,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3600.dv

\(\chi_{3600}(341,\cdot)\) \(\chi_{3600}(1061,\cdot)\) \(\chi_{3600}(1421,\cdot)\) \(\chi_{3600}(1781,\cdot)\) \(\chi_{3600}(2141,\cdot)\) \(\chi_{3600}(2861,\cdot)\) \(\chi_{3600}(3221,\cdot)\) \(\chi_{3600}(3581,\cdot)\)

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

\((3151,901,2801,577)\) → \((1,-i,-1,e\left(\frac{4}{5}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\(-1\)\(1\)\(-1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{1}{5}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{20})\)
Fixed field: 20.0.49533891379200000000000000000000000000000000.1