Properties

Label 2940.121
Modulus $2940$
Conductor $49$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,0,38]))
 
pari: [g,chi] = znchar(Mod(121,2940))
 

Basic properties

Modulus: \(2940\)
Conductor: \(49\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{49}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2940.cm

\(\chi_{2940}(121,\cdot)\) \(\chi_{2940}(541,\cdot)\) \(\chi_{2940}(781,\cdot)\) \(\chi_{2940}(1201,\cdot)\) \(\chi_{2940}(1381,\cdot)\) \(\chi_{2940}(1621,\cdot)\) \(\chi_{2940}(1801,\cdot)\) \(\chi_{2940}(2041,\cdot)\) \(\chi_{2940}(2221,\cdot)\) \(\chi_{2940}(2461,\cdot)\) \(\chi_{2940}(2641,\cdot)\) \(\chi_{2940}(2881,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((1471,1961,1177,1081)\) → \((1,1,1,e\left(\frac{19}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 2940 }(121, a) \) \(1\)\(1\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2940 }(121,a) \;\) at \(\;a = \) e.g. 2