Properties

Label 8470.41
Modulus $8470$
Conductor $847$
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(8470, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,55,23]))
 
pari: [g,chi] = znchar(Mod(41,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{847}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cx

\(\chi_{8470}(41,\cdot)\) \(\chi_{8470}(321,\cdot)\) \(\chi_{8470}(391,\cdot)\) \(\chi_{8470}(601,\cdot)\) \(\chi_{8470}(811,\cdot)\) \(\chi_{8470}(1091,\cdot)\) \(\chi_{8470}(1161,\cdot)\) \(\chi_{8470}(1581,\cdot)\) \(\chi_{8470}(1861,\cdot)\) \(\chi_{8470}(1931,\cdot)\) \(\chi_{8470}(2141,\cdot)\) \(\chi_{8470}(2351,\cdot)\) \(\chi_{8470}(2631,\cdot)\) \(\chi_{8470}(2701,\cdot)\) \(\chi_{8470}(2911,\cdot)\) \(\chi_{8470}(3121,\cdot)\) \(\chi_{8470}(3401,\cdot)\) \(\chi_{8470}(3471,\cdot)\) \(\chi_{8470}(3681,\cdot)\) \(\chi_{8470}(3891,\cdot)\) \(\chi_{8470}(4171,\cdot)\) \(\chi_{8470}(4241,\cdot)\) \(\chi_{8470}(4451,\cdot)\) \(\chi_{8470}(4661,\cdot)\) \(\chi_{8470}(4941,\cdot)\) \(\chi_{8470}(5011,\cdot)\) \(\chi_{8470}(5221,\cdot)\) \(\chi_{8470}(5431,\cdot)\) \(\chi_{8470}(5711,\cdot)\) \(\chi_{8470}(5991,\cdot)\) ...

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

\((6777,6051,7141)\) → \((1,-1,e\left(\frac{23}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{34}{55}\right)\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{47}{55}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{61}{110}\right)\)\(e\left(\frac{53}{110}\right)\)\(e\left(\frac{43}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(41,a) \;\) at \(\;a = \) e.g. 2