Properties

Label 4830.197
Modulus $4830$
Conductor $345$
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(4830, base_ring=CyclotomicField(44))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([22,11,0,28]))
 
pari: [g,chi] = znchar(Mod(197,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(345\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{345}(197,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.ct

\(\chi_{4830}(197,\cdot)\) \(\chi_{4830}(407,\cdot)\) \(\chi_{4830}(533,\cdot)\) \(\chi_{4830}(1037,\cdot)\) \(\chi_{4830}(1163,\cdot)\) \(\chi_{4830}(1373,\cdot)\) \(\chi_{4830}(1457,\cdot)\) \(\chi_{4830}(2003,\cdot)\) \(\chi_{4830}(2423,\cdot)\) \(\chi_{4830}(2717,\cdot)\) \(\chi_{4830}(2927,\cdot)\) \(\chi_{4830}(3137,\cdot)\) \(\chi_{4830}(3347,\cdot)\) \(\chi_{4830}(3683,\cdot)\) \(\chi_{4830}(3767,\cdot)\) \(\chi_{4830}(3893,\cdot)\) \(\chi_{4830}(4103,\cdot)\) \(\chi_{4830}(4313,\cdot)\) \(\chi_{4830}(4397,\cdot)\) \(\chi_{4830}(4733,\cdot)\)

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: 44.44.10759477646680772288556262812807675479811013525016604751482416550745256245136260986328125.1

Values on generators

\((3221,967,2761,1891)\) → \((-1,i,1,e\left(\frac{7}{11}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\(1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{41}{44}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(197,a) \;\) at \(\;a = \) e.g. 2