Properties

Label 1225.61
Modulus $1225$
Conductor $1225$
Order $210$
Real no
Primitive yes
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(1225)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([168,55]))
 
pari: [g,chi] = znchar(Mod(61,1225))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.br

\(\chi_{1225}(61,\cdot)\) \(\chi_{1225}(66,\cdot)\) \(\chi_{1225}(96,\cdot)\) \(\chi_{1225}(131,\cdot)\) \(\chi_{1225}(136,\cdot)\) \(\chi_{1225}(171,\cdot)\) \(\chi_{1225}(206,\cdot)\) \(\chi_{1225}(236,\cdot)\) \(\chi_{1225}(241,\cdot)\) \(\chi_{1225}(271,\cdot)\) \(\chi_{1225}(306,\cdot)\) \(\chi_{1225}(311,\cdot)\) \(\chi_{1225}(341,\cdot)\) \(\chi_{1225}(346,\cdot)\) \(\chi_{1225}(381,\cdot)\) \(\chi_{1225}(416,\cdot)\) \(\chi_{1225}(446,\cdot)\) \(\chi_{1225}(481,\cdot)\) \(\chi_{1225}(486,\cdot)\) \(\chi_{1225}(516,\cdot)\) \(\chi_{1225}(556,\cdot)\) \(\chi_{1225}(586,\cdot)\) \(\chi_{1225}(591,\cdot)\) \(\chi_{1225}(621,\cdot)\) \(\chi_{1225}(661,\cdot)\) \(\chi_{1225}(691,\cdot)\) \(\chi_{1225}(696,\cdot)\) \(\chi_{1225}(731,\cdot)\) \(\chi_{1225}(761,\cdot)\) \(\chi_{1225}(796,\cdot)\) ...

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

\((1177,101)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{11}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{64}{105}\right)\)\(e\left(\frac{181}{210}\right)\)\(e\left(\frac{23}{105}\right)\)\(e\left(\frac{33}{70}\right)\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{76}{105}\right)\)\(e\left(\frac{29}{105}\right)\)\(e\left(\frac{17}{210}\right)\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{46}{105}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial