Properties

Label 1225.6
Modulus $1225$
Conductor $1225$
Order $70$
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([28,45]))
 
pari: [g,chi] = znchar(Mod(6,1225))
 

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

Galois orbit 1225.bl

\(\chi_{1225}(6,\cdot)\) \(\chi_{1225}(41,\cdot)\) \(\chi_{1225}(111,\cdot)\) \(\chi_{1225}(181,\cdot)\) \(\chi_{1225}(216,\cdot)\) \(\chi_{1225}(286,\cdot)\) \(\chi_{1225}(321,\cdot)\) \(\chi_{1225}(356,\cdot)\) \(\chi_{1225}(461,\cdot)\) \(\chi_{1225}(496,\cdot)\) \(\chi_{1225}(531,\cdot)\) \(\chi_{1225}(566,\cdot)\) \(\chi_{1225}(671,\cdot)\) \(\chi_{1225}(706,\cdot)\) \(\chi_{1225}(741,\cdot)\) \(\chi_{1225}(811,\cdot)\) \(\chi_{1225}(846,\cdot)\) \(\chi_{1225}(916,\cdot)\) \(\chi_{1225}(986,\cdot)\) \(\chi_{1225}(1021,\cdot)\) \(\chi_{1225}(1056,\cdot)\) \(\chi_{1225}(1091,\cdot)\) \(\chi_{1225}(1161,\cdot)\) \(\chi_{1225}(1196,\cdot)\)

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

\((1177,101)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{9}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{31}{70}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{47}{70}\right)\)\(e\left(\frac{57}{70}\right)\)\(e\left(\frac{16}{35}\right)\)
value at e.g. 2

Related number fields

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