Properties

Label 1089.26
Modulus $1089$
Conductor $363$
Order $110$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1089, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,102]))
 
pari: [g,chi] = znchar(Mod(26,1089))
 

Basic properties

Modulus: \(1089\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(26,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.z

\(\chi_{1089}(26,\cdot)\) \(\chi_{1089}(53,\cdot)\) \(\chi_{1089}(71,\cdot)\) \(\chi_{1089}(80,\cdot)\) \(\chi_{1089}(125,\cdot)\) \(\chi_{1089}(152,\cdot)\) \(\chi_{1089}(170,\cdot)\) \(\chi_{1089}(179,\cdot)\) \(\chi_{1089}(224,\cdot)\) \(\chi_{1089}(278,\cdot)\) \(\chi_{1089}(350,\cdot)\) \(\chi_{1089}(368,\cdot)\) \(\chi_{1089}(377,\cdot)\) \(\chi_{1089}(422,\cdot)\) \(\chi_{1089}(449,\cdot)\) \(\chi_{1089}(467,\cdot)\) \(\chi_{1089}(476,\cdot)\) \(\chi_{1089}(521,\cdot)\) \(\chi_{1089}(548,\cdot)\) \(\chi_{1089}(566,\cdot)\) \(\chi_{1089}(575,\cdot)\) \(\chi_{1089}(620,\cdot)\) \(\chi_{1089}(647,\cdot)\) \(\chi_{1089}(665,\cdot)\) \(\chi_{1089}(674,\cdot)\) \(\chi_{1089}(719,\cdot)\) \(\chi_{1089}(746,\cdot)\) \(\chi_{1089}(764,\cdot)\) \(\chi_{1089}(773,\cdot)\) \(\chi_{1089}(818,\cdot)\) ...

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: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)

Values on generators

\((848,244)\) → \((-1,e\left(\frac{51}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(-1\)\(1\)\(e\left(\frac{47}{110}\right)\)\(e\left(\frac{47}{55}\right)\)\(e\left(\frac{13}{110}\right)\)\(e\left(\frac{27}{55}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{36}{55}\right)\)\(e\left(\frac{101}{110}\right)\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{103}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(26,a) \;\) at \(\;a = \) e.g. 2