Properties

Label 1400.573
Modulus $1400$
Conductor $1400$
Order $20$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1400, base_ring=CyclotomicField(20))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,10,11,10]))
 
pari: [g,chi] = znchar(Mod(573,1400))
 

Basic properties

Modulus: \(1400\)
Conductor: \(1400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(20\)
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 1400.ct

\(\chi_{1400}(13,\cdot)\) \(\chi_{1400}(237,\cdot)\) \(\chi_{1400}(517,\cdot)\) \(\chi_{1400}(573,\cdot)\) \(\chi_{1400}(797,\cdot)\) \(\chi_{1400}(853,\cdot)\) \(\chi_{1400}(1077,\cdot)\) \(\chi_{1400}(1133,\cdot)\)

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: \(\Q(\zeta_{20})\)
Fixed field: 20.20.882735153125000000000000000000000000000000.1

Values on generators

\((351,701,1177,801)\) → \((1,-1,e\left(\frac{11}{20}\right),-1)\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1400 }(573,a) \;\) at \(\;a = \) e.g. 2