Properties

Label 1400.221
Modulus $1400$
Conductor $1400$
Order $30$
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(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,15,18,20]))
 
pari: [g,chi] = znchar(Mod(221,1400))
 

Basic properties

Modulus: \(1400\)
Conductor: \(1400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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.cx

\(\chi_{1400}(221,\cdot)\) \(\chi_{1400}(261,\cdot)\) \(\chi_{1400}(541,\cdot)\) \(\chi_{1400}(781,\cdot)\) \(\chi_{1400}(821,\cdot)\) \(\chi_{1400}(1061,\cdot)\) \(\chi_{1400}(1341,\cdot)\) \(\chi_{1400}(1381,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.30.9974033287201500125000000000000000000000000000000000000000000000.1

Values on generators

\((351,701,1177,801)\) → \((1,-1,e\left(\frac{3}{5}\right),e\left(\frac{2}{3}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1400 }(221,a) \;\) at \(\;a = \) e.g. 2