Properties

Label 1225.43
Modulus $1225$
Conductor $245$
Order $28$
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(1225)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([21,4]))
 
pari: [g,chi] = znchar(Mod(43,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(43,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1225.y

\(\chi_{1225}(43,\cdot)\) \(\chi_{1225}(57,\cdot)\) \(\chi_{1225}(218,\cdot)\) \(\chi_{1225}(232,\cdot)\) \(\chi_{1225}(407,\cdot)\) \(\chi_{1225}(568,\cdot)\) \(\chi_{1225}(582,\cdot)\) \(\chi_{1225}(743,\cdot)\) \(\chi_{1225}(757,\cdot)\) \(\chi_{1225}(918,\cdot)\) \(\chi_{1225}(1093,\cdot)\) \(\chi_{1225}(1107,\cdot)\)

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

\((1177,101)\) → \((-i,e\left(\frac{1}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{6}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: 28.0.17501529797217428894629579082505064100647449493408203125.1