Properties

Label 1225.1009
Modulus $1225$
Conductor $1225$
Order $70$
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(70))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([49,30]))
 
pari: [g,chi] = znchar(Mod(1009,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
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.bj

\(\chi_{1225}(29,\cdot)\) \(\chi_{1225}(64,\cdot)\) \(\chi_{1225}(134,\cdot)\) \(\chi_{1225}(169,\cdot)\) \(\chi_{1225}(204,\cdot)\) \(\chi_{1225}(239,\cdot)\) \(\chi_{1225}(309,\cdot)\) \(\chi_{1225}(379,\cdot)\) \(\chi_{1225}(414,\cdot)\) \(\chi_{1225}(484,\cdot)\) \(\chi_{1225}(519,\cdot)\) \(\chi_{1225}(554,\cdot)\) \(\chi_{1225}(659,\cdot)\) \(\chi_{1225}(694,\cdot)\) \(\chi_{1225}(729,\cdot)\) \(\chi_{1225}(764,\cdot)\) \(\chi_{1225}(869,\cdot)\) \(\chi_{1225}(904,\cdot)\) \(\chi_{1225}(939,\cdot)\) \(\chi_{1225}(1009,\cdot)\) \(\chi_{1225}(1044,\cdot)\) \(\chi_{1225}(1114,\cdot)\) \(\chi_{1225}(1184,\cdot)\) \(\chi_{1225}(1219,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((1177,101)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{3}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{59}{70}\right)\)\(e\left(\frac{23}{70}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{37}{70}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{1}{70}\right)\)\(e\left(\frac{31}{70}\right)\)\(e\left(\frac{13}{35}\right)\)
value at e.g. 2