Properties

Label 2205.53
Modulus $2205$
Conductor $735$
Order $84$
Real no
Primitive no
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(2205, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([42,63,20]))
 
pari: [g,chi] = znchar(Mod(53,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.ee

\(\chi_{2205}(53,\cdot)\) \(\chi_{2205}(107,\cdot)\) \(\chi_{2205}(233,\cdot)\) \(\chi_{2205}(242,\cdot)\) \(\chi_{2205}(368,\cdot)\) \(\chi_{2205}(548,\cdot)\) \(\chi_{2205}(683,\cdot)\) \(\chi_{2205}(737,\cdot)\) \(\chi_{2205}(872,\cdot)\) \(\chi_{2205}(1052,\cdot)\) \(\chi_{2205}(1178,\cdot)\) \(\chi_{2205}(1187,\cdot)\) \(\chi_{2205}(1313,\cdot)\) \(\chi_{2205}(1367,\cdot)\) \(\chi_{2205}(1493,\cdot)\) \(\chi_{2205}(1502,\cdot)\) \(\chi_{2205}(1628,\cdot)\) \(\chi_{2205}(1682,\cdot)\) \(\chi_{2205}(1808,\cdot)\) \(\chi_{2205}(1817,\cdot)\) \(\chi_{2205}(1943,\cdot)\) \(\chi_{2205}(1997,\cdot)\) \(\chi_{2205}(2123,\cdot)\) \(\chi_{2205}(2132,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1226,442,1081)\) → \((-1,-i,e\left(\frac{5}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(1\)\(1\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{17}{84}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{67}{84}\right)\)
value at e.g. 2