Properties

Label 2001.35
Modulus $2001$
Conductor $2001$
Order $154$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,140,33]))
 
pari: [g,chi] = znchar(Mod(35,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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 2001.bq

\(\chi_{2001}(35,\cdot)\) \(\chi_{2001}(62,\cdot)\) \(\chi_{2001}(71,\cdot)\) \(\chi_{2001}(167,\cdot)\) \(\chi_{2001}(179,\cdot)\) \(\chi_{2001}(209,\cdot)\) \(\chi_{2001}(236,\cdot)\) \(\chi_{2001}(266,\cdot)\) \(\chi_{2001}(353,\cdot)\) \(\chi_{2001}(386,\cdot)\) \(\chi_{2001}(440,\cdot)\) \(\chi_{2001}(473,\cdot)\) \(\chi_{2001}(515,\cdot)\) \(\chi_{2001}(560,\cdot)\) \(\chi_{2001}(584,\cdot)\) \(\chi_{2001}(593,\cdot)\) \(\chi_{2001}(602,\cdot)\) \(\chi_{2001}(614,\cdot)\) \(\chi_{2001}(647,\cdot)\) \(\chi_{2001}(671,\cdot)\) \(\chi_{2001}(680,\cdot)\) \(\chi_{2001}(731,\cdot)\) \(\chi_{2001}(767,\cdot)\) \(\chi_{2001}(788,\cdot)\) \(\chi_{2001}(818,\cdot)\) \(\chi_{2001}(821,\cdot)\) \(\chi_{2001}(854,\cdot)\) \(\chi_{2001}(863,\cdot)\) \(\chi_{2001}(905,\cdot)\) \(\chi_{2001}(932,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((668,1132,553)\) → \((-1,e\left(\frac{10}{11}\right),e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2001 }(35, a) \) \(-1\)\(1\)\(e\left(\frac{41}{77}\right)\)\(e\left(\frac{5}{77}\right)\)\(e\left(\frac{19}{154}\right)\)\(e\left(\frac{65}{77}\right)\)\(e\left(\frac{46}{77}\right)\)\(e\left(\frac{101}{154}\right)\)\(e\left(\frac{3}{77}\right)\)\(e\left(\frac{45}{77}\right)\)\(e\left(\frac{29}{77}\right)\)\(e\left(\frac{10}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2001 }(35,a) \;\) at \(\;a = \) e.g. 2