Properties

Label 1225.33
Modulus $1225$
Conductor $1225$
Order $420$
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(420))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([63,410]))
 
pari: [g,chi] = znchar(Mod(33,1225))
 

Basic properties

Modulus: \(1225\)
Conductor: \(1225\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
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.bu

\(\chi_{1225}(3,\cdot)\) \(\chi_{1225}(12,\cdot)\) \(\chi_{1225}(17,\cdot)\) \(\chi_{1225}(33,\cdot)\) \(\chi_{1225}(38,\cdot)\) \(\chi_{1225}(47,\cdot)\) \(\chi_{1225}(52,\cdot)\) \(\chi_{1225}(73,\cdot)\) \(\chi_{1225}(87,\cdot)\) \(\chi_{1225}(103,\cdot)\) \(\chi_{1225}(108,\cdot)\) \(\chi_{1225}(122,\cdot)\) \(\chi_{1225}(138,\cdot)\) \(\chi_{1225}(152,\cdot)\) \(\chi_{1225}(173,\cdot)\) \(\chi_{1225}(187,\cdot)\) \(\chi_{1225}(192,\cdot)\) \(\chi_{1225}(208,\cdot)\) \(\chi_{1225}(213,\cdot)\) \(\chi_{1225}(222,\cdot)\) \(\chi_{1225}(248,\cdot)\) \(\chi_{1225}(262,\cdot)\) \(\chi_{1225}(278,\cdot)\) \(\chi_{1225}(283,\cdot)\) \(\chi_{1225}(292,\cdot)\) \(\chi_{1225}(297,\cdot)\) \(\chi_{1225}(327,\cdot)\) \(\chi_{1225}(348,\cdot)\) \(\chi_{1225}(353,\cdot)\) \(\chi_{1225}(367,\cdot)\) ...

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((1177,101)\) → \((e\left(\frac{3}{20}\right),e\left(\frac{41}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\(1\)\(1\)\(e\left(\frac{223}{420}\right)\)\(e\left(\frac{11}{420}\right)\)\(e\left(\frac{13}{210}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{83}{140}\right)\)\(e\left(\frac{11}{210}\right)\)\(e\left(\frac{47}{105}\right)\)\(e\left(\frac{37}{420}\right)\)\(e\left(\frac{9}{140}\right)\)\(e\left(\frac{13}{105}\right)\)
value at e.g. 2