Properties

Label 8470.1357
Modulus $8470$
Conductor $4235$
Order $220$
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(220))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,110,204]))
 
pari: [g,chi] = znchar(Mod(1357,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(4235\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4235}(1357,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.df

\(\chi_{8470}(97,\cdot)\) \(\chi_{8470}(223,\cdot)\) \(\chi_{8470}(377,\cdot)\) \(\chi_{8470}(433,\cdot)\) \(\chi_{8470}(587,\cdot)\) \(\chi_{8470}(643,\cdot)\) \(\chi_{8470}(713,\cdot)\) \(\chi_{8470}(797,\cdot)\) \(\chi_{8470}(867,\cdot)\) \(\chi_{8470}(993,\cdot)\) \(\chi_{8470}(1147,\cdot)\) \(\chi_{8470}(1203,\cdot)\) \(\chi_{8470}(1357,\cdot)\) \(\chi_{8470}(1413,\cdot)\) \(\chi_{8470}(1483,\cdot)\) \(\chi_{8470}(1567,\cdot)\) \(\chi_{8470}(1637,\cdot)\) \(\chi_{8470}(1763,\cdot)\) \(\chi_{8470}(1917,\cdot)\) \(\chi_{8470}(1973,\cdot)\) \(\chi_{8470}(2127,\cdot)\) \(\chi_{8470}(2183,\cdot)\) \(\chi_{8470}(2253,\cdot)\) \(\chi_{8470}(2337,\cdot)\) \(\chi_{8470}(2407,\cdot)\) \(\chi_{8470}(2533,\cdot)\) \(\chi_{8470}(2687,\cdot)\) \(\chi_{8470}(2897,\cdot)\) \(\chi_{8470}(2953,\cdot)\) \(\chi_{8470}(3023,\cdot)\) ...

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: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((6777,6051,7141)\) → \((i,-1,e\left(\frac{51}{55}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{199}{220}\right)\)\(e\left(\frac{41}{220}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{27}{110}\right)\)\(e\left(\frac{43}{220}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(1357,a) \;\) at \(\;a = \) e.g. 2