Properties

Label 2310.289
Modulus $2310$
Conductor $385$
Order $30$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2310, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,10,24]))
 
pari: [g,chi] = znchar(Mod(289,2310))
 

Basic properties

Modulus: \(2310\)
Conductor: \(385\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{385}(289,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2310.de

\(\chi_{2310}(289,\cdot)\) \(\chi_{2310}(499,\cdot)\) \(\chi_{2310}(709,\cdot)\) \(\chi_{2310}(949,\cdot)\) \(\chi_{2310}(1159,\cdot)\) \(\chi_{2310}(1369,\cdot)\) \(\chi_{2310}(1549,\cdot)\) \(\chi_{2310}(2209,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial

Values on generators

\((1541,1387,661,211)\) → \((1,-1,e\left(\frac{1}{3}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 2310 }(289, a) \) \(1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{2}{5}\right)\)\(-1\)\(e\left(\frac{17}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2310 }(289,a) \;\) at \(\;a = \) e.g. 2