Properties

Label 4410.3611
Modulus $4410$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4410, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([7,0,9]))
 
pari: [g,chi] = znchar(Mod(3611,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.dd

\(\chi_{4410}(41,\cdot)\) \(\chi_{4410}(461,\cdot)\) \(\chi_{4410}(671,\cdot)\) \(\chi_{4410}(1091,\cdot)\) \(\chi_{4410}(1301,\cdot)\) \(\chi_{4410}(1721,\cdot)\) \(\chi_{4410}(1931,\cdot)\) \(\chi_{4410}(2561,\cdot)\) \(\chi_{4410}(2981,\cdot)\) \(\chi_{4410}(3191,\cdot)\) \(\chi_{4410}(3611,\cdot)\) \(\chi_{4410}(4241,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 4410 }(3611, a) \) \(1\)\(1\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{17}{42}\right)\)\(e\left(\frac{6}{7}\right)\)\(-1\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{1}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{1}{21}\right)\)\(e\left(\frac{20}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4410 }(3611,a) \;\) at \(\;a = \) e.g. 2