Properties

Label 2678.29
Modulus $2678$
Conductor $1339$
Order $51$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2678, base_ring=CyclotomicField(102))
 
M = H._module
 
chi = DirichletCharacter(H, M([34,86]))
 
pari: [g,chi] = znchar(Mod(29,2678))
 

Basic properties

Modulus: \(2678\)
Conductor: \(1339\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(51\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1339}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2678.bl

\(\chi_{2678}(29,\cdot)\) \(\chi_{2678}(55,\cdot)\) \(\chi_{2678}(139,\cdot)\) \(\chi_{2678}(269,\cdot)\) \(\chi_{2678}(289,\cdot)\) \(\chi_{2678}(347,\cdot)\) \(\chi_{2678}(471,\cdot)\) \(\chi_{2678}(503,\cdot)\) \(\chi_{2678}(575,\cdot)\) \(\chi_{2678}(607,\cdot)\) \(\chi_{2678}(633,\cdot)\) \(\chi_{2678}(841,\cdot)\) \(\chi_{2678}(1231,\cdot)\) \(\chi_{2678}(1277,\cdot)\) \(\chi_{2678}(1355,\cdot)\) \(\chi_{2678}(1407,\cdot)\) \(\chi_{2678}(1563,\cdot)\) \(\chi_{2678}(1595,\cdot)\) \(\chi_{2678}(1667,\cdot)\) \(\chi_{2678}(1745,\cdot)\) \(\chi_{2678}(1803,\cdot)\) \(\chi_{2678}(2109,\cdot)\) \(\chi_{2678}(2167,\cdot)\) \(\chi_{2678}(2245,\cdot)\) \(\chi_{2678}(2291,\cdot)\) \(\chi_{2678}(2395,\cdot)\) \(\chi_{2678}(2401,\cdot)\) \(\chi_{2678}(2427,\cdot)\) \(\chi_{2678}(2479,\cdot)\) \(\chi_{2678}(2505,\cdot)\) ...

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: $\Q(\zeta_{51})$
Fixed field: Number field defined by a degree 51 polynomial

Values on generators

\((1237,417)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{43}{51}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 2678 }(29, a) \) \(1\)\(1\)\(e\left(\frac{11}{51}\right)\)\(e\left(\frac{43}{51}\right)\)\(e\left(\frac{2}{51}\right)\)\(e\left(\frac{22}{51}\right)\)\(e\left(\frac{13}{17}\right)\)\(e\left(\frac{1}{17}\right)\)\(e\left(\frac{35}{51}\right)\)\(e\left(\frac{2}{17}\right)\)\(e\left(\frac{13}{51}\right)\)\(e\left(\frac{29}{51}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2678 }(29,a) \;\) at \(\;a = \) e.g. 2