Properties

Label 4002.77
Modulus $4002$
Conductor $2001$
Order $308$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, base_ring=CyclotomicField(308))
 
M = H._module
 
chi = DirichletCharacter(H, M([154,84,99]))
 
pari: [g,chi] = znchar(Mod(77,4002))
 

Basic properties

Modulus: \(4002\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4002.bv

\(\chi_{4002}(77,\cdot)\) \(\chi_{4002}(95,\cdot)\) \(\chi_{4002}(101,\cdot)\) \(\chi_{4002}(119,\cdot)\) \(\chi_{4002}(131,\cdot)\) \(\chi_{4002}(269,\cdot)\) \(\chi_{4002}(305,\cdot)\) \(\chi_{4002}(311,\cdot)\) \(\chi_{4002}(317,\cdot)\) \(\chi_{4002}(395,\cdot)\) \(\chi_{4002}(443,\cdot)\) \(\chi_{4002}(449,\cdot)\) \(\chi_{4002}(485,\cdot)\) \(\chi_{4002}(491,\cdot)\) \(\chi_{4002}(533,\cdot)\) \(\chi_{4002}(611,\cdot)\) \(\chi_{4002}(623,\cdot)\) \(\chi_{4002}(653,\cdot)\) \(\chi_{4002}(785,\cdot)\) \(\chi_{4002}(791,\cdot)\) \(\chi_{4002}(809,\cdot)\) \(\chi_{4002}(947,\cdot)\) \(\chi_{4002}(959,\cdot)\) \(\chi_{4002}(1001,\cdot)\) \(\chi_{4002}(1007,\cdot)\) \(\chi_{4002}(1025,\cdot)\) \(\chi_{4002}(1133,\cdot)\) \(\chi_{4002}(1139,\cdot)\) \(\chi_{4002}(1145,\cdot)\) \(\chi_{4002}(1163,\cdot)\) ...

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: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial (not computed)

Values on generators

\((2669,3133,553)\) → \((-1,e\left(\frac{3}{11}\right),e\left(\frac{9}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 4002 }(77, a) \) \(1\)\(1\)\(e\left(\frac{65}{77}\right)\)\(e\left(\frac{3}{77}\right)\)\(e\left(\frac{305}{308}\right)\)\(e\left(\frac{93}{154}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{303}{308}\right)\)\(e\left(\frac{53}{77}\right)\)\(e\left(\frac{295}{308}\right)\)\(e\left(\frac{68}{77}\right)\)\(e\left(\frac{213}{308}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4002 }(77,a) \;\) at \(\;a = \) e.g. 2