Properties

Label 4000.1791
Modulus $4000$
Conductor $500$
Order $50$
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(4000)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([25,0,22]))
 
pari: [g,chi] = znchar(Mod(1791,4000))
 

Basic properties

Modulus: \(4000\)
Conductor: \(500\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{500}(291,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4000.cl

\(\chi_{4000}(31,\cdot)\) \(\chi_{4000}(191,\cdot)\) \(\chi_{4000}(511,\cdot)\) \(\chi_{4000}(671,\cdot)\) \(\chi_{4000}(831,\cdot)\) \(\chi_{4000}(991,\cdot)\) \(\chi_{4000}(1311,\cdot)\) \(\chi_{4000}(1471,\cdot)\) \(\chi_{4000}(1631,\cdot)\) \(\chi_{4000}(1791,\cdot)\) \(\chi_{4000}(2111,\cdot)\) \(\chi_{4000}(2271,\cdot)\) \(\chi_{4000}(2431,\cdot)\) \(\chi_{4000}(2591,\cdot)\) \(\chi_{4000}(2911,\cdot)\) \(\chi_{4000}(3071,\cdot)\) \(\chi_{4000}(3231,\cdot)\) \(\chi_{4000}(3391,\cdot)\) \(\chi_{4000}(3711,\cdot)\) \(\chi_{4000}(3871,\cdot)\)

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

\((2751,2501,1377)\) → \((-1,1,e\left(\frac{11}{25}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\(-1\)\(1\)\(e\left(\frac{29}{50}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{21}{50}\right)\)\(e\left(\frac{12}{25}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{37}{50}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Fixed field: Number field defined by a degree 50 polynomial