Properties

Label 8470.29
Modulus $8470$
Conductor $605$
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(8470, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,0,17]))
 
pari: [g,chi] = znchar(Mod(29,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.cw

\(\chi_{8470}(29,\cdot)\) \(\chi_{8470}(519,\cdot)\) \(\chi_{8470}(589,\cdot)\) \(\chi_{8470}(799,\cdot)\) \(\chi_{8470}(1009,\cdot)\) \(\chi_{8470}(1289,\cdot)\) \(\chi_{8470}(1359,\cdot)\) \(\chi_{8470}(1569,\cdot)\) \(\chi_{8470}(1779,\cdot)\) \(\chi_{8470}(2059,\cdot)\) \(\chi_{8470}(2129,\cdot)\) \(\chi_{8470}(2549,\cdot)\) \(\chi_{8470}(2829,\cdot)\) \(\chi_{8470}(2899,\cdot)\) \(\chi_{8470}(3109,\cdot)\) \(\chi_{8470}(3319,\cdot)\) \(\chi_{8470}(3599,\cdot)\) \(\chi_{8470}(3669,\cdot)\) \(\chi_{8470}(3879,\cdot)\) \(\chi_{8470}(4089,\cdot)\) \(\chi_{8470}(4369,\cdot)\) \(\chi_{8470}(4439,\cdot)\) \(\chi_{8470}(4649,\cdot)\) \(\chi_{8470}(4859,\cdot)\) \(\chi_{8470}(5139,\cdot)\) \(\chi_{8470}(5209,\cdot)\) \(\chi_{8470}(5419,\cdot)\) \(\chi_{8470}(5629,\cdot)\) \(\chi_{8470}(5909,\cdot)\) \(\chi_{8470}(5979,\cdot)\) ...

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

\((6777,6051,7141)\) → \((-1,1,e\left(\frac{17}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{91}{110}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{109}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(29,a) \;\) at \(\;a = \) e.g. 2