Properties

Label 2205.2116
Modulus $2205$
Conductor $49$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,2]))
 
pari: [g,chi] = znchar(Mod(2116,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.cs

\(\chi_{2205}(46,\cdot)\) \(\chi_{2205}(541,\cdot)\) \(\chi_{2205}(676,\cdot)\) \(\chi_{2205}(856,\cdot)\) \(\chi_{2205}(991,\cdot)\) \(\chi_{2205}(1171,\cdot)\) \(\chi_{2205}(1306,\cdot)\) \(\chi_{2205}(1486,\cdot)\) \(\chi_{2205}(1621,\cdot)\) \(\chi_{2205}(1801,\cdot)\) \(\chi_{2205}(1936,\cdot)\) \(\chi_{2205}(2116,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\( \chi_{ 2205 }(2116, a) \) \(1\)\(1\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{17}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2205 }(2116,a) \;\) at \(\;a = \) e.g. 2