Properties

Label 2667.29
Modulus $2667$
Conductor $381$
Order $126$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2667, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([63,0,113]))
 
pari: [g,chi] = znchar(Mod(29,2667))
 

Basic properties

Modulus: \(2667\)
Conductor: \(381\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{381}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2667.ew

\(\chi_{2667}(29,\cdot)\) \(\chi_{2667}(92,\cdot)\) \(\chi_{2667}(134,\cdot)\) \(\chi_{2667}(218,\cdot)\) \(\chi_{2667}(239,\cdot)\) \(\chi_{2667}(260,\cdot)\) \(\chi_{2667}(302,\cdot)\) \(\chi_{2667}(491,\cdot)\) \(\chi_{2667}(554,\cdot)\) \(\chi_{2667}(575,\cdot)\) \(\chi_{2667}(617,\cdot)\) \(\chi_{2667}(638,\cdot)\) \(\chi_{2667}(680,\cdot)\) \(\chi_{2667}(785,\cdot)\) \(\chi_{2667}(827,\cdot)\) \(\chi_{2667}(848,\cdot)\) \(\chi_{2667}(932,\cdot)\) \(\chi_{2667}(974,\cdot)\) \(\chi_{2667}(995,\cdot)\) \(\chi_{2667}(1226,\cdot)\) \(\chi_{2667}(1436,\cdot)\) \(\chi_{2667}(1625,\cdot)\) \(\chi_{2667}(1709,\cdot)\) \(\chi_{2667}(1835,\cdot)\) \(\chi_{2667}(1856,\cdot)\) \(\chi_{2667}(1919,\cdot)\) \(\chi_{2667}(1961,\cdot)\) \(\chi_{2667}(2087,\cdot)\) \(\chi_{2667}(2129,\cdot)\) \(\chi_{2667}(2150,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((890,1144,2416)\) → \((-1,1,e\left(\frac{113}{126}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 2667 }(29, a) \) \(1\)\(1\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{61}{126}\right)\)\(e\left(\frac{19}{63}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{73}{126}\right)\)\(e\left(\frac{1}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2667 }(29,a) \;\) at \(\;a = \) e.g. 2