Properties

Label 7623.1912
Modulus $7623$
Conductor $1089$
Order $165$
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(330))
 
M = H._module
 
chi = DirichletCharacter(H, M([110,0,108]))
 
pari: [g,chi] = znchar(Mod(1912,7623))
 

Basic properties

Modulus: \(7623\)
Conductor: \(1089\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(165\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1089}(823,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7623.ey

\(\chi_{7623}(169,\cdot)\) \(\chi_{7623}(295,\cdot)\) \(\chi_{7623}(400,\cdot)\) \(\chi_{7623}(421,\cdot)\) \(\chi_{7623}(526,\cdot)\) \(\chi_{7623}(610,\cdot)\) \(\chi_{7623}(652,\cdot)\) \(\chi_{7623}(841,\cdot)\) \(\chi_{7623}(862,\cdot)\) \(\chi_{7623}(988,\cdot)\) \(\chi_{7623}(1093,\cdot)\) \(\chi_{7623}(1114,\cdot)\) \(\chi_{7623}(1303,\cdot)\) \(\chi_{7623}(1345,\cdot)\) \(\chi_{7623}(1534,\cdot)\) \(\chi_{7623}(1555,\cdot)\) \(\chi_{7623}(1681,\cdot)\) \(\chi_{7623}(1786,\cdot)\) \(\chi_{7623}(1807,\cdot)\) \(\chi_{7623}(1912,\cdot)\) \(\chi_{7623}(1996,\cdot)\) \(\chi_{7623}(2038,\cdot)\) \(\chi_{7623}(2227,\cdot)\) \(\chi_{7623}(2248,\cdot)\) \(\chi_{7623}(2374,\cdot)\) \(\chi_{7623}(2479,\cdot)\) \(\chi_{7623}(2500,\cdot)\) \(\chi_{7623}(2605,\cdot)\) \(\chi_{7623}(2731,\cdot)\) \(\chi_{7623}(2920,\cdot)\) ...

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 165 polynomial (not computed)

Values on generators

\((848,4357,4600)\) → \((e\left(\frac{1}{3}\right),1,e\left(\frac{18}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 7623 }(1912, a) \) \(1\)\(1\)\(e\left(\frac{109}{165}\right)\)\(e\left(\frac{53}{165}\right)\)\(e\left(\frac{146}{165}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{119}{165}\right)\)\(e\left(\frac{106}{165}\right)\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{9}{55}\right)\)\(e\left(\frac{34}{165}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7623 }(1912,a) \;\) at \(\;a = \) e.g. 2