Properties

Label 2500.11
Modulus $2500$
Conductor $2500$
Order $250$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2500, base_ring=CyclotomicField(250))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([125,238]))
 
pari: [g,chi] = znchar(Mod(11,2500))
 

Basic properties

Modulus: \(2500\)
Conductor: \(2500\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2500.t

\(\chi_{2500}(11,\cdot)\) \(\chi_{2500}(31,\cdot)\) \(\chi_{2500}(71,\cdot)\) \(\chi_{2500}(91,\cdot)\) \(\chi_{2500}(111,\cdot)\) \(\chi_{2500}(131,\cdot)\) \(\chi_{2500}(171,\cdot)\) \(\chi_{2500}(191,\cdot)\) \(\chi_{2500}(211,\cdot)\) \(\chi_{2500}(231,\cdot)\) \(\chi_{2500}(271,\cdot)\) \(\chi_{2500}(291,\cdot)\) \(\chi_{2500}(311,\cdot)\) \(\chi_{2500}(331,\cdot)\) \(\chi_{2500}(371,\cdot)\) \(\chi_{2500}(391,\cdot)\) \(\chi_{2500}(411,\cdot)\) \(\chi_{2500}(431,\cdot)\) \(\chi_{2500}(471,\cdot)\) \(\chi_{2500}(491,\cdot)\) \(\chi_{2500}(511,\cdot)\) \(\chi_{2500}(531,\cdot)\) \(\chi_{2500}(571,\cdot)\) \(\chi_{2500}(591,\cdot)\) \(\chi_{2500}(611,\cdot)\) \(\chi_{2500}(631,\cdot)\) \(\chi_{2500}(671,\cdot)\) \(\chi_{2500}(691,\cdot)\) \(\chi_{2500}(711,\cdot)\) \(\chi_{2500}(731,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 250 polynomial (not computed)

Values on generators

\((1251,1877)\) → \((-1,e\left(\frac{119}{125}\right))\)

Values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2500 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{91}{250}\right)\)\(e\left(\frac{11}{50}\right)\)\(e\left(\frac{91}{125}\right)\)\(e\left(\frac{163}{250}\right)\)\(e\left(\frac{41}{125}\right)\)\(e\left(\frac{87}{125}\right)\)\(e\left(\frac{109}{250}\right)\)\(e\left(\frac{73}{125}\right)\)\(e\left(\frac{203}{250}\right)\)\(e\left(\frac{23}{250}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2500 }(11,a) \;\) at \(\;a = \) e.g. 2