Properties

Label 1776.241
Modulus $1776$
Conductor $37$
Order $36$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1776, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,35]))
 
pari: [g,chi] = znchar(Mod(241,1776))
 

Basic properties

Modulus: \(1776\)
Conductor: \(37\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{37}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1776.en

\(\chi_{1776}(241,\cdot)\) \(\chi_{1776}(385,\cdot)\) \(\chi_{1776}(577,\cdot)\) \(\chi_{1776}(721,\cdot)\) \(\chi_{1776}(1105,\cdot)\) \(\chi_{1776}(1201,\cdot)\) \(\chi_{1776}(1297,\cdot)\) \(\chi_{1776}(1345,\cdot)\) \(\chi_{1776}(1393,\cdot)\) \(\chi_{1776}(1441,\cdot)\) \(\chi_{1776}(1537,\cdot)\) \(\chi_{1776}(1633,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: Number field defined by a degree 36 polynomial

Values on generators

\((223,1333,593,1297)\) → \((1,1,1,e\left(\frac{35}{36}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 1776 }(241, a) \) \(-1\)\(1\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{5}{12}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1776 }(241,a) \;\) at \(\;a = \) e.g. 2