Properties

Label 2304.53
Modulus $2304$
Conductor $768$
Order $64$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2304, base_ring=CyclotomicField(64))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,5,32]))
 
pari: [g,chi] = znchar(Mod(53,2304))
 

Basic properties

Modulus: \(2304\)
Conductor: \(768\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(64\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{768}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2304.bs

\(\chi_{2304}(53,\cdot)\) \(\chi_{2304}(125,\cdot)\) \(\chi_{2304}(197,\cdot)\) \(\chi_{2304}(269,\cdot)\) \(\chi_{2304}(341,\cdot)\) \(\chi_{2304}(413,\cdot)\) \(\chi_{2304}(485,\cdot)\) \(\chi_{2304}(557,\cdot)\) \(\chi_{2304}(629,\cdot)\) \(\chi_{2304}(701,\cdot)\) \(\chi_{2304}(773,\cdot)\) \(\chi_{2304}(845,\cdot)\) \(\chi_{2304}(917,\cdot)\) \(\chi_{2304}(989,\cdot)\) \(\chi_{2304}(1061,\cdot)\) \(\chi_{2304}(1133,\cdot)\) \(\chi_{2304}(1205,\cdot)\) \(\chi_{2304}(1277,\cdot)\) \(\chi_{2304}(1349,\cdot)\) \(\chi_{2304}(1421,\cdot)\) \(\chi_{2304}(1493,\cdot)\) \(\chi_{2304}(1565,\cdot)\) \(\chi_{2304}(1637,\cdot)\) \(\chi_{2304}(1709,\cdot)\) \(\chi_{2304}(1781,\cdot)\) \(\chi_{2304}(1853,\cdot)\) \(\chi_{2304}(1925,\cdot)\) \(\chi_{2304}(1997,\cdot)\) \(\chi_{2304}(2069,\cdot)\) \(\chi_{2304}(2141,\cdot)\) ...

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

Values on generators

\((1279,2053,1793)\) → \((1,e\left(\frac{5}{64}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{37}{64}\right)\)\(e\left(\frac{25}{32}\right)\)\(e\left(\frac{9}{64}\right)\)\(e\left(\frac{43}{64}\right)\)\(e\left(\frac{11}{16}\right)\)\(e\left(\frac{51}{64}\right)\)\(e\left(\frac{19}{32}\right)\)\(e\left(\frac{5}{32}\right)\)\(e\left(\frac{7}{64}\right)\)\(e\left(\frac{5}{8}\right)\)
value at e.g. 2