Properties

Label 2205.349
Modulus $2205$
Conductor $2205$
Order $42$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2205, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([28,21,27]))
 
pari: [g,chi] = znchar(Mod(349,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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 2205.dj

\(\chi_{2205}(34,\cdot)\) \(\chi_{2205}(139,\cdot)\) \(\chi_{2205}(349,\cdot)\) \(\chi_{2205}(454,\cdot)\) \(\chi_{2205}(664,\cdot)\) \(\chi_{2205}(769,\cdot)\) \(\chi_{2205}(1084,\cdot)\) \(\chi_{2205}(1294,\cdot)\) \(\chi_{2205}(1399,\cdot)\) \(\chi_{2205}(1609,\cdot)\) \(\chi_{2205}(1924,\cdot)\) \(\chi_{2205}(2029,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.601884773165316715444146020475856959985360694545262653619999023258450611352687173519068385601043701171875.1

Values on generators

\((1226,442,1081)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{9}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(-1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{11}{42}\right)\)
value at e.g. 2