Properties

Label 3630.31
Modulus $3630$
Conductor $121$
Order $55$
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(3630)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,43]))
 
pari: [g,chi] = znchar(Mod(31,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(121\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(55\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{121}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bk

\(\chi_{3630}(31,\cdot)\) \(\chi_{3630}(91,\cdot)\) \(\chi_{3630}(181,\cdot)\) \(\chi_{3630}(301,\cdot)\) \(\chi_{3630}(361,\cdot)\) \(\chi_{3630}(421,\cdot)\) \(\chi_{3630}(631,\cdot)\) \(\chi_{3630}(691,\cdot)\) \(\chi_{3630}(751,\cdot)\) \(\chi_{3630}(841,\cdot)\) \(\chi_{3630}(961,\cdot)\) \(\chi_{3630}(1021,\cdot)\) \(\chi_{3630}(1081,\cdot)\) \(\chi_{3630}(1171,\cdot)\) \(\chi_{3630}(1351,\cdot)\) \(\chi_{3630}(1411,\cdot)\) \(\chi_{3630}(1501,\cdot)\) \(\chi_{3630}(1621,\cdot)\) \(\chi_{3630}(1681,\cdot)\) \(\chi_{3630}(1741,\cdot)\) \(\chi_{3630}(1831,\cdot)\) \(\chi_{3630}(1951,\cdot)\) \(\chi_{3630}(2011,\cdot)\) \(\chi_{3630}(2071,\cdot)\) \(\chi_{3630}(2161,\cdot)\) \(\chi_{3630}(2281,\cdot)\) \(\chi_{3630}(2341,\cdot)\) \(\chi_{3630}(2401,\cdot)\) \(\chi_{3630}(2491,\cdot)\) \(\chi_{3630}(2611,\cdot)\) ...

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

\((1211,727,3511)\) → \((1,1,e\left(\frac{43}{55}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{26}{55}\right)\)\(e\left(\frac{53}{55}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{49}{55}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{13}{55}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{6}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 55 polynomial