Properties

Label 4410.937
Modulus $4410$
Conductor $245$
Order $28$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,7,18]))
 
pari: [g,chi] = znchar(Mod(937,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(202,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.cv

\(\chi_{4410}(307,\cdot)\) \(\chi_{4410}(433,\cdot)\) \(\chi_{4410}(937,\cdot)\) \(\chi_{4410}(1063,\cdot)\) \(\chi_{4410}(1693,\cdot)\) \(\chi_{4410}(2197,\cdot)\) \(\chi_{4410}(2323,\cdot)\) \(\chi_{4410}(2827,\cdot)\) \(\chi_{4410}(2953,\cdot)\) \(\chi_{4410}(3457,\cdot)\) \(\chi_{4410}(3583,\cdot)\) \(\chi_{4410}(4087,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: Number field defined by a degree 28 polynomial

Values on generators

\((3431,2647,1081)\) → \((1,i,e\left(\frac{9}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(937, a) \) \(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{9}{28}\right)\)\(1\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(-1\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{17}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(937,a) \;\) at \(\;a = \) e.g. 2