Properties

Label 4275.23
Modulus $4275$
Conductor $4275$
Order $180$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4275, base_ring=CyclotomicField(180))
 
M = H._module
 
chi = DirichletCharacter(H, M([150,99,20]))
 
pari: [g,chi] = znchar(Mod(23,4275))
 

Basic properties

Modulus: \(4275\)
Conductor: \(4275\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(180\)
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 4275.hc

\(\chi_{4275}(23,\cdot)\) \(\chi_{4275}(92,\cdot)\) \(\chi_{4275}(158,\cdot)\) \(\chi_{4275}(263,\cdot)\) \(\chi_{4275}(587,\cdot)\) \(\chi_{4275}(617,\cdot)\) \(\chi_{4275}(758,\cdot)\) \(\chi_{4275}(788,\cdot)\) \(\chi_{4275}(803,\cdot)\) \(\chi_{4275}(842,\cdot)\) \(\chi_{4275}(878,\cdot)\) \(\chi_{4275}(947,\cdot)\) \(\chi_{4275}(1013,\cdot)\) \(\chi_{4275}(1442,\cdot)\) \(\chi_{4275}(1472,\cdot)\) \(\chi_{4275}(1487,\cdot)\) \(\chi_{4275}(1562,\cdot)\) \(\chi_{4275}(1613,\cdot)\) \(\chi_{4275}(1658,\cdot)\) \(\chi_{4275}(1697,\cdot)\) \(\chi_{4275}(1733,\cdot)\) \(\chi_{4275}(1802,\cdot)\) \(\chi_{4275}(1973,\cdot)\) \(\chi_{4275}(2297,\cdot)\) \(\chi_{4275}(2327,\cdot)\) \(\chi_{4275}(2342,\cdot)\) \(\chi_{4275}(2417,\cdot)\) \(\chi_{4275}(2498,\cdot)\) \(\chi_{4275}(2513,\cdot)\) \(\chi_{4275}(2552,\cdot)\) ...

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: $\Q(\zeta_{180})$
Fixed field: Number field defined by a degree 180 polynomial (not computed)

Values on generators

\((1901,1027,1351)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{11}{20}\right),e\left(\frac{1}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(22\)
\( \chi_{ 4275 }(23, a) \) \(1\)\(1\)\(e\left(\frac{89}{180}\right)\)\(e\left(\frac{89}{90}\right)\)\(-i\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{121}{180}\right)\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{137}{180}\right)\)\(e\left(\frac{83}{180}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4275 }(23,a) \;\) at \(\;a = \) e.g. 2