Properties

Label 2304.37
Modulus $2304$
Conductor $256$
Order $64$
Real no
Primitive no
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([0,25,0]))
 
pari: [g,chi] = znchar(Mod(37,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(256\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{256}(37,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bt

\(\chi_{2304}(37,\cdot)\) \(\chi_{2304}(109,\cdot)\) \(\chi_{2304}(181,\cdot)\) \(\chi_{2304}(253,\cdot)\) \(\chi_{2304}(325,\cdot)\) \(\chi_{2304}(397,\cdot)\) \(\chi_{2304}(469,\cdot)\) \(\chi_{2304}(541,\cdot)\) \(\chi_{2304}(613,\cdot)\) \(\chi_{2304}(685,\cdot)\) \(\chi_{2304}(757,\cdot)\) \(\chi_{2304}(829,\cdot)\) \(\chi_{2304}(901,\cdot)\) \(\chi_{2304}(973,\cdot)\) \(\chi_{2304}(1045,\cdot)\) \(\chi_{2304}(1117,\cdot)\) \(\chi_{2304}(1189,\cdot)\) \(\chi_{2304}(1261,\cdot)\) \(\chi_{2304}(1333,\cdot)\) \(\chi_{2304}(1405,\cdot)\) \(\chi_{2304}(1477,\cdot)\) \(\chi_{2304}(1549,\cdot)\) \(\chi_{2304}(1621,\cdot)\) \(\chi_{2304}(1693,\cdot)\) \(\chi_{2304}(1765,\cdot)\) \(\chi_{2304}(1837,\cdot)\) \(\chi_{2304}(1909,\cdot)\) \(\chi_{2304}(1981,\cdot)\) \(\chi_{2304}(2053,\cdot)\) \(\chi_{2304}(2125,\cdot)\) ...

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

\((1279,2053,1793)\) → \((1,e\left(\frac{25}{64}\right),1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{25}{64}\right)\)\(e\left(\frac{29}{32}\right)\)\(e\left(\frac{13}{64}\right)\)\(e\left(\frac{23}{64}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{63}{64}\right)\)\(e\left(\frac{15}{32}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{3}{64}\right)\)\(e\left(\frac{1}{8}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{64})$
Fixed field: Number field defined by a degree 64 polynomial