Properties

Label 1856.97
Modulus $1856$
Conductor $232$
Order $28$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1856, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,23]))
 
pari: [g,chi] = znchar(Mod(97,1856))
 

Basic properties

Modulus: \(1856\)
Conductor: \(232\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{232}(213,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1856.bu

\(\chi_{1856}(97,\cdot)\) \(\chi_{1856}(417,\cdot)\) \(\chi_{1856}(801,\cdot)\) \(\chi_{1856}(1121,\cdot)\) \(\chi_{1856}(1249,\cdot)\) \(\chi_{1856}(1313,\cdot)\) \(\chi_{1856}(1377,\cdot)\) \(\chi_{1856}(1505,\cdot)\) \(\chi_{1856}(1569,\cdot)\) \(\chi_{1856}(1697,\cdot)\) \(\chi_{1856}(1761,\cdot)\) \(\chi_{1856}(1825,\cdot)\)

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: 28.0.13427827737836760536055607671312169337571202392129536.1

Values on generators

\((639,581,321)\) → \((1,-1,e\left(\frac{23}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 1856 }(97, a) \) \(-1\)\(1\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(i\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{13}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1856 }(97,a) \;\) at \(\;a = \) e.g. 2