Properties

Label 1225.16
Modulus $1225$
Conductor $1225$
Order $105$
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([42,100]))
 
pari: [g,chi] = znchar(Mod(16,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(105\)
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.bo

\(\chi_{1225}(11,\cdot)\) \(\chi_{1225}(16,\cdot)\) \(\chi_{1225}(46,\cdot)\) \(\chi_{1225}(81,\cdot)\) \(\chi_{1225}(86,\cdot)\) \(\chi_{1225}(121,\cdot)\) \(\chi_{1225}(156,\cdot)\) \(\chi_{1225}(186,\cdot)\) \(\chi_{1225}(191,\cdot)\) \(\chi_{1225}(221,\cdot)\) \(\chi_{1225}(256,\cdot)\) \(\chi_{1225}(261,\cdot)\) \(\chi_{1225}(291,\cdot)\) \(\chi_{1225}(296,\cdot)\) \(\chi_{1225}(331,\cdot)\) \(\chi_{1225}(366,\cdot)\) \(\chi_{1225}(396,\cdot)\) \(\chi_{1225}(431,\cdot)\) \(\chi_{1225}(436,\cdot)\) \(\chi_{1225}(466,\cdot)\) \(\chi_{1225}(506,\cdot)\) \(\chi_{1225}(536,\cdot)\) \(\chi_{1225}(541,\cdot)\) \(\chi_{1225}(571,\cdot)\) \(\chi_{1225}(611,\cdot)\) \(\chi_{1225}(641,\cdot)\) \(\chi_{1225}(646,\cdot)\) \(\chi_{1225}(681,\cdot)\) \(\chi_{1225}(711,\cdot)\) \(\chi_{1225}(746,\cdot)\) ...

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 105 polynomial (not computed)

Values on generators

\((1177,101)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{10}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{61}{105}\right)\)\(e\left(\frac{92}{105}\right)\)\(e\left(\frac{17}{105}\right)\)\(e\left(\frac{16}{35}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{79}{105}\right)\)\(e\left(\frac{26}{105}\right)\)\(e\left(\frac{4}{105}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{34}{105}\right)\)
value at e.g. 2