Properties

Label 14157.58
Modulus $14157$
Conductor $14157$
Order $660$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(14157, base_ring=CyclotomicField(660)) M = H._module chi = DirichletCharacter(H, M([220,108,275]))
 
Copy content pari:[g,chi] = znchar(Mod(58,14157))
 

Basic properties

Modulus: \(14157\)
Conductor: \(14157\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(660\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 14157.iq

\(\chi_{14157}(58,\cdot)\) \(\chi_{14157}(115,\cdot)\) \(\chi_{14157}(223,\cdot)\) \(\chi_{14157}(466,\cdot)\) \(\chi_{14157}(526,\cdot)\) \(\chi_{14157}(553,\cdot)\) \(\chi_{14157}(643,\cdot)\) \(\chi_{14157}(691,\cdot)\) \(\chi_{14157}(808,\cdot)\) \(\chi_{14157}(817,\cdot)\) \(\chi_{14157}(994,\cdot)\) \(\chi_{14157}(1021,\cdot)\) \(\chi_{14157}(1138,\cdot)\) \(\chi_{14157}(1159,\cdot)\) \(\chi_{14157}(1285,\cdot)\) \(\chi_{14157}(1345,\cdot)\) \(\chi_{14157}(1402,\cdot)\) \(\chi_{14157}(1489,\cdot)\) \(\chi_{14157}(1510,\cdot)\) \(\chi_{14157}(1753,\cdot)\) \(\chi_{14157}(1813,\cdot)\) \(\chi_{14157}(1840,\cdot)\) \(\chi_{14157}(1930,\cdot)\) \(\chi_{14157}(1978,\cdot)\) \(\chi_{14157}(2095,\cdot)\) \(\chi_{14157}(2104,\cdot)\) \(\chi_{14157}(2281,\cdot)\) \(\chi_{14157}(2425,\cdot)\) \(\chi_{14157}(2446,\cdot)\) \(\chi_{14157}(2572,\cdot)\) ...

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{660})$
Fixed field: Number field defined by a degree 660 polynomial (not computed)

Values on generators

\((6293,3511,4357)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{55}\right),e\left(\frac{5}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 14157 }(58, a) \) \(-1\)\(1\)\(e\left(\frac{201}{220}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{347}{660}\right)\)\(e\left(\frac{41}{660}\right)\)\(e\left(\frac{163}{220}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{161}{165}\right)\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{281}{330}\right)\)\(e\left(\frac{439}{660}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 14157 }(58,a) \;\) at \(\;a = \) e.g. 2