Properties

Label 3630.49
Modulus $3630$
Conductor $605$
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([0,55,14]))
 
pari: [g,chi] = znchar(Mod(49,3630))
 

Basic properties

Modulus: \(3630\)
Conductor: \(605\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(110\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{605}(49,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3630.bm

\(\chi_{3630}(49,\cdot)\) \(\chi_{3630}(169,\cdot)\) \(\chi_{3630}(229,\cdot)\) \(\chi_{3630}(289,\cdot)\) \(\chi_{3630}(379,\cdot)\) \(\chi_{3630}(499,\cdot)\) \(\chi_{3630}(559,\cdot)\) \(\chi_{3630}(619,\cdot)\) \(\chi_{3630}(709,\cdot)\) \(\chi_{3630}(829,\cdot)\) \(\chi_{3630}(889,\cdot)\) \(\chi_{3630}(949,\cdot)\) \(\chi_{3630}(1039,\cdot)\) \(\chi_{3630}(1159,\cdot)\) \(\chi_{3630}(1279,\cdot)\) \(\chi_{3630}(1369,\cdot)\) \(\chi_{3630}(1489,\cdot)\) \(\chi_{3630}(1549,\cdot)\) \(\chi_{3630}(1609,\cdot)\) \(\chi_{3630}(1699,\cdot)\) \(\chi_{3630}(1819,\cdot)\) \(\chi_{3630}(1879,\cdot)\) \(\chi_{3630}(2029,\cdot)\) \(\chi_{3630}(2149,\cdot)\) \(\chi_{3630}(2209,\cdot)\) \(\chi_{3630}(2269,\cdot)\) \(\chi_{3630}(2359,\cdot)\) \(\chi_{3630}(2479,\cdot)\) \(\chi_{3630}(2539,\cdot)\) \(\chi_{3630}(2599,\cdot)\) ...

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

\((1211,727,3511)\) → \((1,-1,e\left(\frac{7}{55}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{43}{110}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{81}{110}\right)\)\(e\left(\frac{31}{55}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{9}{55}\right)\)\(e\left(\frac{52}{55}\right)\)\(e\left(\frac{93}{110}\right)\)\(e\left(\frac{51}{55}\right)\)\(e\left(\frac{15}{22}\right)\)
value at e.g. 2

Related number fields

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