Properties

Label 3630.41
Modulus $3630$
Conductor $363$
Order $110$
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([55,0,23]))
 
pari: [g,chi] = znchar(Mod(41,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(363\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{363}(41,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bn

\(\chi_{3630}(41,\cdot)\) \(\chi_{3630}(101,\cdot)\) \(\chi_{3630}(281,\cdot)\) \(\chi_{3630}(371,\cdot)\) \(\chi_{3630}(431,\cdot)\) \(\chi_{3630}(491,\cdot)\) \(\chi_{3630}(611,\cdot)\) \(\chi_{3630}(701,\cdot)\) \(\chi_{3630}(761,\cdot)\) \(\chi_{3630}(821,\cdot)\) \(\chi_{3630}(1031,\cdot)\) \(\chi_{3630}(1091,\cdot)\) \(\chi_{3630}(1151,\cdot)\) \(\chi_{3630}(1271,\cdot)\) \(\chi_{3630}(1361,\cdot)\) \(\chi_{3630}(1421,\cdot)\) \(\chi_{3630}(1481,\cdot)\) \(\chi_{3630}(1601,\cdot)\) \(\chi_{3630}(1751,\cdot)\) \(\chi_{3630}(1811,\cdot)\) \(\chi_{3630}(1931,\cdot)\) \(\chi_{3630}(2021,\cdot)\) \(\chi_{3630}(2081,\cdot)\) \(\chi_{3630}(2141,\cdot)\) \(\chi_{3630}(2261,\cdot)\) \(\chi_{3630}(2351,\cdot)\) \(\chi_{3630}(2471,\cdot)\) \(\chi_{3630}(2591,\cdot)\) \(\chi_{3630}(2681,\cdot)\) \(\chi_{3630}(2741,\cdot)\) ...

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

\((1211,727,3511)\) → \((-1,1,e\left(\frac{23}{110}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{13}{110}\right)\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{3}{55}\right)\)\(e\left(\frac{54}{55}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{5}{22}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial