Properties

Label 7220.49
Modulus $7220$
Conductor $1805$
Order $114$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7220, base_ring=CyclotomicField(114))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,57,100]))
 
pari: [g,chi] = znchar(Mod(49,7220))
 

Basic properties

Modulus: \(7220\)
Conductor: \(1805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(114\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1805}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 7220.bz

\(\chi_{7220}(49,\cdot)\) \(\chi_{7220}(349,\cdot)\) \(\chi_{7220}(729,\cdot)\) \(\chi_{7220}(809,\cdot)\) \(\chi_{7220}(1109,\cdot)\) \(\chi_{7220}(1189,\cdot)\) \(\chi_{7220}(1489,\cdot)\) \(\chi_{7220}(1569,\cdot)\) \(\chi_{7220}(1869,\cdot)\) \(\chi_{7220}(1949,\cdot)\) \(\chi_{7220}(2249,\cdot)\) \(\chi_{7220}(2329,\cdot)\) \(\chi_{7220}(2629,\cdot)\) \(\chi_{7220}(2709,\cdot)\) \(\chi_{7220}(3009,\cdot)\) \(\chi_{7220}(3089,\cdot)\) \(\chi_{7220}(3389,\cdot)\) \(\chi_{7220}(3469,\cdot)\) \(\chi_{7220}(3769,\cdot)\) \(\chi_{7220}(3849,\cdot)\) \(\chi_{7220}(4149,\cdot)\) \(\chi_{7220}(4229,\cdot)\) \(\chi_{7220}(4529,\cdot)\) \(\chi_{7220}(4609,\cdot)\) \(\chi_{7220}(4909,\cdot)\) \(\chi_{7220}(4989,\cdot)\) \(\chi_{7220}(5289,\cdot)\) \(\chi_{7220}(5369,\cdot)\) \(\chi_{7220}(5669,\cdot)\) \(\chi_{7220}(5749,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 114 polynomial (not computed)

Values on generators

\((3611,5777,6861)\) → \((1,-1,e\left(\frac{50}{57}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 7220 }(49, a) \) \(1\)\(1\)\(e\left(\frac{49}{114}\right)\)\(e\left(\frac{3}{38}\right)\)\(e\left(\frac{49}{57}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{11}{114}\right)\)\(e\left(\frac{85}{114}\right)\)\(e\left(\frac{29}{57}\right)\)\(e\left(\frac{65}{114}\right)\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{52}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 7220 }(49,a) \;\) at \(\;a = \) e.g. 2