Properties

Label 1225.44
Modulus $1225$
Conductor $1225$
Order $210$
Real no
Primitive yes
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(1225, base_ring=CyclotomicField(210))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([189,40]))
 
pari: [g,chi] = znchar(Mod(44,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bt

\(\chi_{1225}(4,\cdot)\) \(\chi_{1225}(9,\cdot)\) \(\chi_{1225}(39,\cdot)\) \(\chi_{1225}(44,\cdot)\) \(\chi_{1225}(109,\cdot)\) \(\chi_{1225}(114,\cdot)\) \(\chi_{1225}(144,\cdot)\) \(\chi_{1225}(179,\cdot)\) \(\chi_{1225}(184,\cdot)\) \(\chi_{1225}(219,\cdot)\) \(\chi_{1225}(254,\cdot)\) \(\chi_{1225}(284,\cdot)\) \(\chi_{1225}(289,\cdot)\) \(\chi_{1225}(319,\cdot)\) \(\chi_{1225}(354,\cdot)\) \(\chi_{1225}(359,\cdot)\) \(\chi_{1225}(389,\cdot)\) \(\chi_{1225}(394,\cdot)\) \(\chi_{1225}(429,\cdot)\) \(\chi_{1225}(464,\cdot)\) \(\chi_{1225}(494,\cdot)\) \(\chi_{1225}(529,\cdot)\) \(\chi_{1225}(534,\cdot)\) \(\chi_{1225}(564,\cdot)\) \(\chi_{1225}(604,\cdot)\) \(\chi_{1225}(634,\cdot)\) \(\chi_{1225}(639,\cdot)\) \(\chi_{1225}(669,\cdot)\) \(\chi_{1225}(709,\cdot)\) \(\chi_{1225}(739,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((1177,101)\) → \((e\left(\frac{9}{10}\right),e\left(\frac{4}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{179}{210}\right)\)\(e\left(\frac{103}{210}\right)\)\(e\left(\frac{74}{105}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{103}{105}\right)\)\(e\left(\frac{2}{105}\right)\)\(e\left(\frac{41}{210}\right)\)\(e\left(\frac{27}{70}\right)\)\(e\left(\frac{43}{105}\right)\)
value at e.g. 2