Properties

Label 40310.51
Modulus $40310$
Conductor $4031$
Order $966$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40310, base_ring=CyclotomicField(966))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,897,70]))
 
pari: [g,chi] = znchar(Mod(51,40310))
 

Basic properties

Modulus: \(40310\)
Conductor: \(4031\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(966\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4031}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 40310.fp

\(\chi_{40310}(51,\cdot)\) \(\chi_{40310}(71,\cdot)\) \(\chi_{40310}(121,\cdot)\) \(\chi_{40310}(361,\cdot)\) \(\chi_{40310}(441,\cdot)\) \(\chi_{40310}(941,\cdot)\) \(\chi_{40310}(961,\cdot)\) \(\chi_{40310}(1211,\cdot)\) \(\chi_{40310}(1281,\cdot)\) \(\chi_{40310}(1401,\cdot)\) \(\chi_{40310}(1811,\cdot)\) \(\chi_{40310}(1831,\cdot)\) \(\chi_{40310}(1861,\cdot)\) \(\chi_{40310}(1981,\cdot)\) \(\chi_{40310}(2101,\cdot)\) \(\chi_{40310}(2151,\cdot)\) \(\chi_{40310}(2271,\cdot)\) \(\chi_{40310}(2391,\cdot)\) \(\chi_{40310}(2441,\cdot)\) \(\chi_{40310}(2661,\cdot)\) \(\chi_{40310}(2761,\cdot)\) \(\chi_{40310}(2851,\cdot)\) \(\chi_{40310}(3141,\cdot)\) \(\chi_{40310}(3341,\cdot)\) \(\chi_{40310}(3721,\cdot)\) \(\chi_{40310}(3921,\cdot)\) \(\chi_{40310}(4151,\cdot)\) \(\chi_{40310}(4181,\cdot)\) \(\chi_{40310}(4211,\cdot)\) \(\chi_{40310}(4591,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 966 polynomial (not computed)

Values on generators

\((24187,19461,16821)\) → \((1,e\left(\frac{13}{14}\right),e\left(\frac{5}{69}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 40310 }(51, a) \) \(1\)\(1\)\(e\left(\frac{593}{966}\right)\)\(e\left(\frac{370}{483}\right)\)\(e\left(\frac{110}{483}\right)\)\(e\left(\frac{697}{966}\right)\)\(e\left(\frac{170}{483}\right)\)\(e\left(\frac{35}{138}\right)\)\(e\left(\frac{751}{966}\right)\)\(e\left(\frac{367}{966}\right)\)\(e\left(\frac{85}{161}\right)\)\(e\left(\frac{271}{322}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 40310 }(51,a) \;\) at \(\;a = \) e.g. 2