Properties

Label 2300.17
Modulus $2300$
Conductor $575$
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(2300, base_ring=CyclotomicField(220))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,143,70]))
 
pari: [g,chi] = znchar(Mod(17,2300))
 

Basic properties

Modulus: \(2300\)
Conductor: \(575\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{575}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2300.bs

\(\chi_{2300}(17,\cdot)\) \(\chi_{2300}(33,\cdot)\) \(\chi_{2300}(37,\cdot)\) \(\chi_{2300}(53,\cdot)\) \(\chi_{2300}(97,\cdot)\) \(\chi_{2300}(113,\cdot)\) \(\chi_{2300}(153,\cdot)\) \(\chi_{2300}(217,\cdot)\) \(\chi_{2300}(237,\cdot)\) \(\chi_{2300}(273,\cdot)\) \(\chi_{2300}(297,\cdot)\) \(\chi_{2300}(313,\cdot)\) \(\chi_{2300}(333,\cdot)\) \(\chi_{2300}(337,\cdot)\) \(\chi_{2300}(373,\cdot)\) \(\chi_{2300}(433,\cdot)\) \(\chi_{2300}(477,\cdot)\) \(\chi_{2300}(497,\cdot)\) \(\chi_{2300}(513,\cdot)\) \(\chi_{2300}(517,\cdot)\) \(\chi_{2300}(573,\cdot)\) \(\chi_{2300}(613,\cdot)\) \(\chi_{2300}(617,\cdot)\) \(\chi_{2300}(677,\cdot)\) \(\chi_{2300}(697,\cdot)\) \(\chi_{2300}(733,\cdot)\) \(\chi_{2300}(753,\cdot)\) \(\chi_{2300}(773,\cdot)\) \(\chi_{2300}(797,\cdot)\) \(\chi_{2300}(833,\cdot)\) ...

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

\((1151,277,1201)\) → \((1,e\left(\frac{13}{20}\right),e\left(\frac{7}{22}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{141}{220}\right)\)\(e\left(\frac{13}{44}\right)\)\(e\left(\frac{31}{110}\right)\)\(e\left(\frac{29}{110}\right)\)\(e\left(\frac{177}{220}\right)\)\(e\left(\frac{149}{220}\right)\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{103}{110}\right)\)\(e\left(\frac{203}{220}\right)\)\(e\left(\frac{3}{110}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2300 }(17,a) \;\) at \(\;a = \) e.g. 2