Properties

Label 4410.3629
Modulus $4410$
Conductor $2205$
Order $42$
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([7,21,1]))
 
pari: [g,chi] = znchar(Mod(3629,4410))
 

Basic properties

Modulus: \(4410\)
Conductor: \(2205\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2205}(1424,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4410.di

\(\chi_{4410}(479,\cdot)\) \(\chi_{4410}(1139,\cdot)\) \(\chi_{4410}(1739,\cdot)\) \(\chi_{4410}(1769,\cdot)\) \(\chi_{4410}(2369,\cdot)\) \(\chi_{4410}(2399,\cdot)\) \(\chi_{4410}(2999,\cdot)\) \(\chi_{4410}(3029,\cdot)\) \(\chi_{4410}(3629,\cdot)\) \(\chi_{4410}(3659,\cdot)\) \(\chi_{4410}(4259,\cdot)\) \(\chi_{4410}(4289,\cdot)\)

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

\((3431,2647,1081)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{1}{42}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(-1\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{13}{42}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: 42.42.64499777946714835177141019992254259402911208109553981749879955329445342864388015575823925406164646148681640625.1