Properties

Label 4410.541
Modulus $4410$
Conductor $49$
Order $21$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4410)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,13]))
 
pari: [g,chi] = znchar(Mod(541,4410))
 

Basic properties

Modulus: \(4410\)
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 \(\chi_{49}(2,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.cr

\(\chi_{4410}(541,\cdot)\) \(\chi_{4410}(991,\cdot)\) \(\chi_{4410}(1171,\cdot)\) \(\chi_{4410}(1621,\cdot)\) \(\chi_{4410}(1801,\cdot)\) \(\chi_{4410}(2251,\cdot)\) \(\chi_{4410}(2881,\cdot)\) \(\chi_{4410}(3061,\cdot)\) \(\chi_{4410}(3511,\cdot)\) \(\chi_{4410}(3691,\cdot)\) \(\chi_{4410}(4141,\cdot)\) \(\chi_{4410}(4321,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Values on generators

\((3431,2647,1081)\) → \((1,1,e\left(\frac{13}{21}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{5}{7}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: \(\Q(\zeta_{49})^+\)