Properties

Label 1225.1016
Modulus $1225$
Conductor $1225$
Order $35$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,20]))
 
pari: [g,chi] = znchar(Mod(1016,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(35\)
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.bd

\(\chi_{1225}(36,\cdot)\) \(\chi_{1225}(71,\cdot)\) \(\chi_{1225}(106,\cdot)\) \(\chi_{1225}(141,\cdot)\) \(\chi_{1225}(211,\cdot)\) \(\chi_{1225}(281,\cdot)\) \(\chi_{1225}(316,\cdot)\) \(\chi_{1225}(386,\cdot)\) \(\chi_{1225}(421,\cdot)\) \(\chi_{1225}(456,\cdot)\) \(\chi_{1225}(561,\cdot)\) \(\chi_{1225}(596,\cdot)\) \(\chi_{1225}(631,\cdot)\) \(\chi_{1225}(666,\cdot)\) \(\chi_{1225}(771,\cdot)\) \(\chi_{1225}(806,\cdot)\) \(\chi_{1225}(841,\cdot)\) \(\chi_{1225}(911,\cdot)\) \(\chi_{1225}(946,\cdot)\) \(\chi_{1225}(1016,\cdot)\) \(\chi_{1225}(1086,\cdot)\) \(\chi_{1225}(1121,\cdot)\) \(\chi_{1225}(1156,\cdot)\) \(\chi_{1225}(1191,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ 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 35 polynomial

Values on generators

\((1177,101)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{2}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 1225 }(1016, a) \) \(1\)\(1\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{11}{35}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{13}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{33}{35}\right)\)\(e\left(\frac{8}{35}\right)\)\(e\left(\frac{18}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1225 }(1016,a) \;\) at \(\;a = \) e.g. 2