Properties

Label 2304.13
Modulus $2304$
Conductor $2304$
Order $192$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2304)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,141,64]))
 
pari: [g,chi] = znchar(Mod(13,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(2304\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(192\)
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 2304.ca

\(\chi_{2304}(13,\cdot)\) \(\chi_{2304}(61,\cdot)\) \(\chi_{2304}(85,\cdot)\) \(\chi_{2304}(133,\cdot)\) \(\chi_{2304}(157,\cdot)\) \(\chi_{2304}(205,\cdot)\) \(\chi_{2304}(229,\cdot)\) \(\chi_{2304}(277,\cdot)\) \(\chi_{2304}(301,\cdot)\) \(\chi_{2304}(349,\cdot)\) \(\chi_{2304}(373,\cdot)\) \(\chi_{2304}(421,\cdot)\) \(\chi_{2304}(445,\cdot)\) \(\chi_{2304}(493,\cdot)\) \(\chi_{2304}(517,\cdot)\) \(\chi_{2304}(565,\cdot)\) \(\chi_{2304}(589,\cdot)\) \(\chi_{2304}(637,\cdot)\) \(\chi_{2304}(661,\cdot)\) \(\chi_{2304}(709,\cdot)\) \(\chi_{2304}(733,\cdot)\) \(\chi_{2304}(781,\cdot)\) \(\chi_{2304}(805,\cdot)\) \(\chi_{2304}(853,\cdot)\) \(\chi_{2304}(877,\cdot)\) \(\chi_{2304}(925,\cdot)\) \(\chi_{2304}(949,\cdot)\) \(\chi_{2304}(997,\cdot)\) \(\chi_{2304}(1021,\cdot)\) \(\chi_{2304}(1069,\cdot)\) ...

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

Values on generators

\((1279,2053,1793)\) → \((1,e\left(\frac{47}{64}\right),e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{77}{192}\right)\)\(e\left(\frac{65}{96}\right)\)\(e\left(\frac{145}{192}\right)\)\(e\left(\frac{35}{192}\right)\)\(e\left(\frac{9}{16}\right)\)\(e\left(\frac{57}{64}\right)\)\(e\left(\frac{91}{96}\right)\)\(e\left(\frac{77}{96}\right)\)\(e\left(\frac{127}{192}\right)\)\(e\left(\frac{13}{24}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{192})$
Fixed field: Number field defined by a degree 192 polynomial