Properties

Label 1225.8
Modulus $1225$
Conductor $1225$
Order $140$
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([21,120]))
 
pari: [g,chi] = znchar(Mod(8,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
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.bq

\(\chi_{1225}(8,\cdot)\) \(\chi_{1225}(22,\cdot)\) \(\chi_{1225}(78,\cdot)\) \(\chi_{1225}(92,\cdot)\) \(\chi_{1225}(113,\cdot)\) \(\chi_{1225}(127,\cdot)\) \(\chi_{1225}(162,\cdot)\) \(\chi_{1225}(183,\cdot)\) \(\chi_{1225}(253,\cdot)\) \(\chi_{1225}(267,\cdot)\) \(\chi_{1225}(288,\cdot)\) \(\chi_{1225}(302,\cdot)\) \(\chi_{1225}(323,\cdot)\) \(\chi_{1225}(337,\cdot)\) \(\chi_{1225}(358,\cdot)\) \(\chi_{1225}(372,\cdot)\) \(\chi_{1225}(428,\cdot)\) \(\chi_{1225}(463,\cdot)\) \(\chi_{1225}(477,\cdot)\) \(\chi_{1225}(498,\cdot)\) \(\chi_{1225}(512,\cdot)\) \(\chi_{1225}(533,\cdot)\) \(\chi_{1225}(547,\cdot)\) \(\chi_{1225}(603,\cdot)\) \(\chi_{1225}(617,\cdot)\) \(\chi_{1225}(652,\cdot)\) \(\chi_{1225}(673,\cdot)\) \(\chi_{1225}(708,\cdot)\) \(\chi_{1225}(722,\cdot)\) \(\chi_{1225}(778,\cdot)\) ...

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

\((1177,101)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{6}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(-1\)\(1\)\(e\left(\frac{61}{140}\right)\)\(e\left(\frac{127}{140}\right)\)\(e\left(\frac{61}{70}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{43}{140}\right)\)\(e\left(\frac{57}{70}\right)\)\(e\left(\frac{24}{35}\right)\)\(e\left(\frac{109}{140}\right)\)\(e\left(\frac{19}{140}\right)\)\(e\left(\frac{26}{35}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial