Properties

Label 1037.654
Modulus $1037$
Conductor $1037$
Order $120$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1037, base_ring=CyclotomicField(120))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([75,34]))
 
pari: [g,chi] = znchar(Mod(654,1037))
 

Basic properties

Modulus: \(1037\)
Conductor: \(1037\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1037.cm

\(\chi_{1037}(26,\cdot)\) \(\chi_{1037}(104,\cdot)\) \(\chi_{1037}(189,\cdot)\) \(\chi_{1037}(213,\cdot)\) \(\chi_{1037}(246,\cdot)\) \(\chi_{1037}(287,\cdot)\) \(\chi_{1037}(331,\cdot)\) \(\chi_{1037}(349,\cdot)\) \(\chi_{1037}(359,\cdot)\) \(\chi_{1037}(372,\cdot)\) \(\chi_{1037}(376,\cdot)\) \(\chi_{1037}(417,\cdot)\) \(\chi_{1037}(434,\cdot)\) \(\chi_{1037}(444,\cdot)\) \(\chi_{1037}(518,\cdot)\) \(\chi_{1037}(542,\cdot)\) \(\chi_{1037}(559,\cdot)\) \(\chi_{1037}(580,\cdot)\) \(\chi_{1037}(604,\cdot)\) \(\chi_{1037}(627,\cdot)\) \(\chi_{1037}(654,\cdot)\) \(\chi_{1037}(689,\cdot)\) \(\chi_{1037}(722,\cdot)\) \(\chi_{1037}(739,\cdot)\) \(\chi_{1037}(763,\cdot)\) \(\chi_{1037}(767,\cdot)\) \(\chi_{1037}(852,\cdot)\) \(\chi_{1037}(909,\cdot)\) \(\chi_{1037}(950,\cdot)\) \(\chi_{1037}(978,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((428,307)\) → \((e\left(\frac{5}{8}\right),e\left(\frac{17}{60}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{43}{120}\right)\)\(e\left(\frac{43}{120}\right)\)\(e\left(\frac{91}{120}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{47}{120}\right)\)\(e\left(\frac{5}{8}\right)\)
value at e.g. 2