Properties

Label 4002.71
Modulus $4002$
Conductor $2001$
Order $154$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, base_ring=CyclotomicField(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,14,99]))
 
pari: [g,chi] = znchar(Mod(71,4002))
 

Basic properties

Modulus: \(4002\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(71,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4002.bo

\(\chi_{4002}(35,\cdot)\) \(\chi_{4002}(71,\cdot)\) \(\chi_{4002}(167,\cdot)\) \(\chi_{4002}(179,\cdot)\) \(\chi_{4002}(209,\cdot)\) \(\chi_{4002}(353,\cdot)\) \(\chi_{4002}(473,\cdot)\) \(\chi_{4002}(515,\cdot)\) \(\chi_{4002}(593,\cdot)\) \(\chi_{4002}(647,\cdot)\) \(\chi_{4002}(671,\cdot)\) \(\chi_{4002}(731,\cdot)\) \(\chi_{4002}(767,\cdot)\) \(\chi_{4002}(821,\cdot)\) \(\chi_{4002}(863,\cdot)\) \(\chi_{4002}(905,\cdot)\) \(\chi_{4002}(995,\cdot)\) \(\chi_{4002}(1037,\cdot)\) \(\chi_{4002}(1223,\cdot)\) \(\chi_{4002}(1343,\cdot)\) \(\chi_{4002}(1559,\cdot)\) \(\chi_{4002}(1637,\cdot)\) \(\chi_{4002}(1691,\cdot)\) \(\chi_{4002}(1715,\cdot)\) \(\chi_{4002}(1733,\cdot)\) \(\chi_{4002}(1775,\cdot)\) \(\chi_{4002}(1865,\cdot)\) \(\chi_{4002}(1889,\cdot)\) \(\chi_{4002}(2063,\cdot)\) \(\chi_{4002}(2237,\cdot)\) ...

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

\((2669,3133,553)\) → \((-1,e\left(\frac{1}{11}\right),e\left(\frac{9}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 4002 }(71, a) \) \(-1\)\(1\)\(e\left(\frac{113}{154}\right)\)\(e\left(\frac{34}{77}\right)\)\(e\left(\frac{30}{77}\right)\)\(e\left(\frac{65}{77}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{23}{154}\right)\)\(e\left(\frac{36}{77}\right)\)\(e\left(\frac{29}{154}\right)\)\(e\left(\frac{27}{154}\right)\)\(e\left(\frac{129}{154}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4002 }(71,a) \;\) at \(\;a = \) e.g. 2