Properties

Label 2300.243
Modulus $2300$
Conductor $460$
Order $44$
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(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22,33,28]))
 
pari: [g,chi] = znchar(Mod(243,2300))
 

Basic properties

Modulus: \(2300\)
Conductor: \(460\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{460}(243,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2300.bi

\(\chi_{2300}(243,\cdot)\) \(\chi_{2300}(307,\cdot)\) \(\chi_{2300}(407,\cdot)\) \(\chi_{2300}(443,\cdot)\) \(\chi_{2300}(607,\cdot)\) \(\chi_{2300}(807,\cdot)\) \(\chi_{2300}(1007,\cdot)\) \(\chi_{2300}(1043,\cdot)\) \(\chi_{2300}(1107,\cdot)\) \(\chi_{2300}(1143,\cdot)\) \(\chi_{2300}(1343,\cdot)\) \(\chi_{2300}(1407,\cdot)\) \(\chi_{2300}(1507,\cdot)\) \(\chi_{2300}(1543,\cdot)\) \(\chi_{2300}(1743,\cdot)\) \(\chi_{2300}(1807,\cdot)\) \(\chi_{2300}(1843,\cdot)\) \(\chi_{2300}(2007,\cdot)\) \(\chi_{2300}(2143,\cdot)\) \(\chi_{2300}(2243,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: 44.44.6031750835043748183809156500949648298478534300530340661248000000000000000000000000000000000.1

Values on generators

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

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{7}{44}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{21}{22}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2300 }(243,a) \;\) at \(\;a = \) e.g. 2