Properties

Label 7623.1271
Modulus $7623$
Conductor $7623$
Order $330$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,220,327]))
 
pari: [g,chi] = znchar(Mod(1271,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(7623\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.gc

\(\chi_{7623}(74,\cdot)\) \(\chi_{7623}(149,\cdot)\) \(\chi_{7623}(200,\cdot)\) \(\chi_{7623}(326,\cdot)\) \(\chi_{7623}(338,\cdot)\) \(\chi_{7623}(464,\cdot)\) \(\chi_{7623}(590,\cdot)\) \(\chi_{7623}(767,\cdot)\) \(\chi_{7623}(842,\cdot)\) \(\chi_{7623}(893,\cdot)\) \(\chi_{7623}(1019,\cdot)\) \(\chi_{7623}(1031,\cdot)\) \(\chi_{7623}(1157,\cdot)\) \(\chi_{7623}(1271,\cdot)\) \(\chi_{7623}(1283,\cdot)\) \(\chi_{7623}(1460,\cdot)\) \(\chi_{7623}(1535,\cdot)\) \(\chi_{7623}(1586,\cdot)\) \(\chi_{7623}(1712,\cdot)\) \(\chi_{7623}(1724,\cdot)\) \(\chi_{7623}(1850,\cdot)\) \(\chi_{7623}(1964,\cdot)\) \(\chi_{7623}(2153,\cdot)\) \(\chi_{7623}(2228,\cdot)\) \(\chi_{7623}(2279,\cdot)\) \(\chi_{7623}(2405,\cdot)\) \(\chi_{7623}(2543,\cdot)\) \(\chi_{7623}(2657,\cdot)\) \(\chi_{7623}(2669,\cdot)\) \(\chi_{7623}(2846,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 330 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{109}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1271, a) \) \(1\)\(1\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{163}{330}\right)\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{137}{330}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{119}{165}\right)\)\(e\left(\frac{191}{330}\right)\)\(e\left(\frac{157}{330}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1271,a) \;\) at \(\;a = \) e.g. 2