Properties

Label 2300.9
Modulus $2300$
Conductor $575$
Order $110$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2300, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,77,50]))
 
pari: [g,chi] = znchar(Mod(9,2300))
 

Basic properties

Modulus: \(2300\)
Conductor: \(575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{575}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2300.bo

\(\chi_{2300}(9,\cdot)\) \(\chi_{2300}(29,\cdot)\) \(\chi_{2300}(169,\cdot)\) \(\chi_{2300}(209,\cdot)\) \(\chi_{2300}(269,\cdot)\) \(\chi_{2300}(289,\cdot)\) \(\chi_{2300}(409,\cdot)\) \(\chi_{2300}(469,\cdot)\) \(\chi_{2300}(489,\cdot)\) \(\chi_{2300}(509,\cdot)\) \(\chi_{2300}(629,\cdot)\) \(\chi_{2300}(669,\cdot)\) \(\chi_{2300}(729,\cdot)\) \(\chi_{2300}(809,\cdot)\) \(\chi_{2300}(869,\cdot)\) \(\chi_{2300}(909,\cdot)\) \(\chi_{2300}(929,\cdot)\) \(\chi_{2300}(969,\cdot)\) \(\chi_{2300}(1089,\cdot)\) \(\chi_{2300}(1129,\cdot)\) \(\chi_{2300}(1189,\cdot)\) \(\chi_{2300}(1209,\cdot)\) \(\chi_{2300}(1269,\cdot)\) \(\chi_{2300}(1329,\cdot)\) \(\chi_{2300}(1369,\cdot)\) \(\chi_{2300}(1389,\cdot)\) \(\chi_{2300}(1409,\cdot)\) \(\chi_{2300}(1429,\cdot)\) \(\chi_{2300}(1589,\cdot)\) \(\chi_{2300}(1669,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ 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

\((1151,277,1201)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{5}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{19}{110}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{19}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{73}{110}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{23}{55}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{57}{110}\right)\)\(e\left(\frac{32}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2300 }(9,a) \;\) at \(\;a = \) e.g. 2