Properties

Label 1400.1237
Modulus $1400$
Conductor $1400$
Order $60$
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(60))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,30,27,50]))
 
pari: [g,chi] = znchar(Mod(1237,1400))
 

Basic properties

Modulus: \(1400\)
Conductor: \(1400\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
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.dk

\(\chi_{1400}(117,\cdot)\) \(\chi_{1400}(173,\cdot)\) \(\chi_{1400}(213,\cdot)\) \(\chi_{1400}(397,\cdot)\) \(\chi_{1400}(437,\cdot)\) \(\chi_{1400}(453,\cdot)\) \(\chi_{1400}(677,\cdot)\) \(\chi_{1400}(717,\cdot)\) \(\chi_{1400}(733,\cdot)\) \(\chi_{1400}(773,\cdot)\) \(\chi_{1400}(997,\cdot)\) \(\chi_{1400}(1013,\cdot)\) \(\chi_{1400}(1053,\cdot)\) \(\chi_{1400}(1237,\cdot)\) \(\chi_{1400}(1277,\cdot)\) \(\chi_{1400}(1333,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((351,701,1177,801)\) → \((1,-1,e\left(\frac{9}{20}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{41}{60}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{13}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1400 }(1237,a) \;\) at \(\;a = \) e.g. 2