Properties

Label 8470.57
Modulus $8470$
Conductor $605$
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([55,0,122]))
 
pari: [g,chi] = znchar(Mod(57,8470))
 

Basic properties

Modulus: \(8470\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(57,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8470.dg

\(\chi_{8470}(57,\cdot)\) \(\chi_{8470}(127,\cdot)\) \(\chi_{8470}(183,\cdot)\) \(\chi_{8470}(337,\cdot)\) \(\chi_{8470}(393,\cdot)\) \(\chi_{8470}(547,\cdot)\) \(\chi_{8470}(673,\cdot)\) \(\chi_{8470}(743,\cdot)\) \(\chi_{8470}(827,\cdot)\) \(\chi_{8470}(897,\cdot)\) \(\chi_{8470}(953,\cdot)\) \(\chi_{8470}(1107,\cdot)\) \(\chi_{8470}(1163,\cdot)\) \(\chi_{8470}(1317,\cdot)\) \(\chi_{8470}(1513,\cdot)\) \(\chi_{8470}(1597,\cdot)\) \(\chi_{8470}(1723,\cdot)\) \(\chi_{8470}(1877,\cdot)\) \(\chi_{8470}(2087,\cdot)\) \(\chi_{8470}(2213,\cdot)\) \(\chi_{8470}(2283,\cdot)\) \(\chi_{8470}(2367,\cdot)\) \(\chi_{8470}(2437,\cdot)\) \(\chi_{8470}(2493,\cdot)\) \(\chi_{8470}(2647,\cdot)\) \(\chi_{8470}(2703,\cdot)\) \(\chi_{8470}(2857,\cdot)\) \(\chi_{8470}(2983,\cdot)\) \(\chi_{8470}(3053,\cdot)\) \(\chi_{8470}(3207,\cdot)\) ...

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

\((6777,6051,7141)\) → \((i,1,e\left(\frac{61}{110}\right))\)

Values

\(-1\)\(1\)\(3\)\(9\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(37\)
\(1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{167}{220}\right)\)\(e\left(\frac{93}{220}\right)\)\(e\left(\frac{29}{55}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{51}{55}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{119}{220}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8470 }(57,a) \;\) at \(\;a = \) e.g. 2