Properties

Label 2205.37
Modulus $2205$
Conductor $245$
Order $84$
Real no
Primitive no
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(2205, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,64]))
 
pari: [g,chi] = znchar(Mod(37,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.ei

\(\chi_{2205}(37,\cdot)\) \(\chi_{2205}(163,\cdot)\) \(\chi_{2205}(172,\cdot)\) \(\chi_{2205}(298,\cdot)\) \(\chi_{2205}(352,\cdot)\) \(\chi_{2205}(478,\cdot)\) \(\chi_{2205}(487,\cdot)\) \(\chi_{2205}(613,\cdot)\) \(\chi_{2205}(793,\cdot)\) \(\chi_{2205}(928,\cdot)\) \(\chi_{2205}(982,\cdot)\) \(\chi_{2205}(1117,\cdot)\) \(\chi_{2205}(1297,\cdot)\) \(\chi_{2205}(1423,\cdot)\) \(\chi_{2205}(1432,\cdot)\) \(\chi_{2205}(1558,\cdot)\) \(\chi_{2205}(1612,\cdot)\) \(\chi_{2205}(1738,\cdot)\) \(\chi_{2205}(1747,\cdot)\) \(\chi_{2205}(1873,\cdot)\) \(\chi_{2205}(1927,\cdot)\) \(\chi_{2205}(2053,\cdot)\) \(\chi_{2205}(2062,\cdot)\) \(\chi_{2205}(2188,\cdot)\)

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

\((1226,442,1081)\) → \((1,i,e\left(\frac{16}{21}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{59}{84}\right)\)
value at e.g. 2