Properties

Label 1089.35
Modulus $1089$
Conductor $363$
Order $110$
Real no
Primitive no
Minimal yes
Parity even

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([55,81]))
 
pari: [g,chi] = znchar(Mod(35,1089))
 

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 \(\chi_{363}(35,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1089.bb

\(\chi_{1089}(8,\cdot)\) \(\chi_{1089}(17,\cdot)\) \(\chi_{1089}(35,\cdot)\) \(\chi_{1089}(62,\cdot)\) \(\chi_{1089}(107,\cdot)\) \(\chi_{1089}(116,\cdot)\) \(\chi_{1089}(134,\cdot)\) \(\chi_{1089}(206,\cdot)\) \(\chi_{1089}(260,\cdot)\) \(\chi_{1089}(305,\cdot)\) \(\chi_{1089}(314,\cdot)\) \(\chi_{1089}(332,\cdot)\) \(\chi_{1089}(359,\cdot)\) \(\chi_{1089}(404,\cdot)\) \(\chi_{1089}(413,\cdot)\) \(\chi_{1089}(431,\cdot)\) \(\chi_{1089}(458,\cdot)\) \(\chi_{1089}(503,\cdot)\) \(\chi_{1089}(512,\cdot)\) \(\chi_{1089}(530,\cdot)\) \(\chi_{1089}(557,\cdot)\) \(\chi_{1089}(611,\cdot)\) \(\chi_{1089}(629,\cdot)\) \(\chi_{1089}(656,\cdot)\) \(\chi_{1089}(701,\cdot)\) \(\chi_{1089}(710,\cdot)\) \(\chi_{1089}(728,\cdot)\) \(\chi_{1089}(755,\cdot)\) \(\chi_{1089}(800,\cdot)\) \(\chi_{1089}(809,\cdot)\) ...

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

\((848,244)\) → \((-1,e\left(\frac{81}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\(1\)\(1\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{109}{110}\right)\)\(e\left(\frac{17}{110}\right)\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{41}{110}\right)\)\(e\left(\frac{43}{110}\right)\)\(e\left(\frac{52}{55}\right)\)\(e\left(\frac{32}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1089 }(35,a) \;\) at \(\;a = \) e.g. 2