Properties

Label 1225.67
Modulus $1225$
Conductor $175$
Order $60$
Real no
Primitive no
Minimal no
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([39,40]))
 
pari: [g,chi] = znchar(Mod(67,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(175\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{175}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.bh

\(\chi_{1225}(67,\cdot)\) \(\chi_{1225}(128,\cdot)\) \(\chi_{1225}(177,\cdot)\) \(\chi_{1225}(263,\cdot)\) \(\chi_{1225}(312,\cdot)\) \(\chi_{1225}(373,\cdot)\) \(\chi_{1225}(422,\cdot)\) \(\chi_{1225}(508,\cdot)\) \(\chi_{1225}(667,\cdot)\) \(\chi_{1225}(753,\cdot)\) \(\chi_{1225}(802,\cdot)\) \(\chi_{1225}(863,\cdot)\) \(\chi_{1225}(912,\cdot)\) \(\chi_{1225}(998,\cdot)\) \(\chi_{1225}(1047,\cdot)\) \(\chi_{1225}(1108,\cdot)\)

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

\((1177,101)\) → \((e\left(\frac{13}{20}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{14}{15}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial