Properties

Label 1960.39
Modulus $1960$
Conductor $980$
Order $42$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1960, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,0,21,34]))
 
pari: [g,chi] = znchar(Mod(39,1960))
 

Basic properties

Modulus: \(1960\)
Conductor: \(980\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{980}(39,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1960.dg

\(\chi_{1960}(39,\cdot)\) \(\chi_{1960}(319,\cdot)\) \(\chi_{1960}(359,\cdot)\) \(\chi_{1960}(599,\cdot)\) \(\chi_{1960}(639,\cdot)\) \(\chi_{1960}(879,\cdot)\) \(\chi_{1960}(919,\cdot)\) \(\chi_{1960}(1159,\cdot)\) \(\chi_{1960}(1199,\cdot)\) \(\chi_{1960}(1479,\cdot)\) \(\chi_{1960}(1719,\cdot)\) \(\chi_{1960}(1759,\cdot)\)

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 42 polynomial

Values on generators

\((1471,981,1177,1081)\) → \((-1,1,-1,e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1960 }(39, a) \) \(-1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1960 }(39,a) \;\) at \(\;a = \) e.g. 2