Properties

Label 7623.4402
Modulus $7623$
Conductor $847$
Order $110$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, base_ring=CyclotomicField(110))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,55,71]))
 
pari: [g,chi] = znchar(Mod(4402,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(167,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.es

\(\chi_{7623}(244,\cdot)\) \(\chi_{7623}(370,\cdot)\) \(\chi_{7623}(622,\cdot)\) \(\chi_{7623}(811,\cdot)\) \(\chi_{7623}(937,\cdot)\) \(\chi_{7623}(1063,\cdot)\) \(\chi_{7623}(1315,\cdot)\) \(\chi_{7623}(1504,\cdot)\) \(\chi_{7623}(1630,\cdot)\) \(\chi_{7623}(1756,\cdot)\) \(\chi_{7623}(2008,\cdot)\) \(\chi_{7623}(2197,\cdot)\) \(\chi_{7623}(2323,\cdot)\) \(\chi_{7623}(2449,\cdot)\) \(\chi_{7623}(2701,\cdot)\) \(\chi_{7623}(2890,\cdot)\) \(\chi_{7623}(3142,\cdot)\) \(\chi_{7623}(3394,\cdot)\) \(\chi_{7623}(3583,\cdot)\) \(\chi_{7623}(3709,\cdot)\) \(\chi_{7623}(3835,\cdot)\) \(\chi_{7623}(4276,\cdot)\) \(\chi_{7623}(4402,\cdot)\) \(\chi_{7623}(4528,\cdot)\) \(\chi_{7623}(4780,\cdot)\) \(\chi_{7623}(4969,\cdot)\) \(\chi_{7623}(5095,\cdot)\) \(\chi_{7623}(5221,\cdot)\) \(\chi_{7623}(5473,\cdot)\) \(\chi_{7623}(5662,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((1,-1,e\left(\frac{71}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(4402, a) \) \(1\)\(1\)\(e\left(\frac{71}{110}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{32}{55}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{61}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(4402,a) \;\) at \(\;a = \) e.g. 2