Properties

Label 4100.313
Modulus $4100$
Conductor $1025$
Order $40$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4100, base_ring=CyclotomicField(40))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,38,17]))
 
pari: [g,chi] = znchar(Mod(313,4100))
 

Basic properties

Modulus: \(4100\)
Conductor: \(1025\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1025}(313,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4100.ge

\(\chi_{4100}(313,\cdot)\) \(\chi_{4100}(397,\cdot)\) \(\chi_{4100}(417,\cdot)\) \(\chi_{4100}(673,\cdot)\) \(\chi_{4100}(997,\cdot)\) \(\chi_{4100}(1037,\cdot)\) \(\chi_{4100}(1077,\cdot)\) \(\chi_{4100}(2233,\cdot)\) \(\chi_{4100}(2617,\cdot)\) \(\chi_{4100}(2753,\cdot)\) \(\chi_{4100}(2777,\cdot)\) \(\chi_{4100}(2933,\cdot)\) \(\chi_{4100}(3213,\cdot)\) \(\chi_{4100}(3637,\cdot)\) \(\chi_{4100}(3673,\cdot)\) \(\chi_{4100}(4053,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: Number field defined by a degree 40 polynomial

Values on generators

\((2051,1477,3901)\) → \((1,e\left(\frac{19}{20}\right),e\left(\frac{17}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 4100 }(313, a) \) \(1\)\(1\)\(e\left(\frac{1}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{9}{40}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{7}{20}\right)\)\(-i\)\(e\left(\frac{3}{40}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4100 }(313,a) \;\) at \(\;a = \) e.g. 2