Properties

Label 1815.41
Modulus $1815$
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(1815, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,0,23]))
 
pari: [g,chi] = znchar(Mod(41,1815))
 

Basic properties

Modulus: \(1815\)
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 1815.bp

\(\chi_{1815}(41,\cdot)\) \(\chi_{1815}(101,\cdot)\) \(\chi_{1815}(116,\cdot)\) \(\chi_{1815}(206,\cdot)\) \(\chi_{1815}(266,\cdot)\) \(\chi_{1815}(281,\cdot)\) \(\chi_{1815}(326,\cdot)\) \(\chi_{1815}(371,\cdot)\) \(\chi_{1815}(431,\cdot)\) \(\chi_{1815}(446,\cdot)\) \(\chi_{1815}(491,\cdot)\) \(\chi_{1815}(536,\cdot)\) \(\chi_{1815}(611,\cdot)\) \(\chi_{1815}(656,\cdot)\) \(\chi_{1815}(701,\cdot)\) \(\chi_{1815}(761,\cdot)\) \(\chi_{1815}(776,\cdot)\) \(\chi_{1815}(821,\cdot)\) \(\chi_{1815}(866,\cdot)\) \(\chi_{1815}(926,\cdot)\) \(\chi_{1815}(986,\cdot)\) \(\chi_{1815}(1031,\cdot)\) \(\chi_{1815}(1091,\cdot)\) \(\chi_{1815}(1106,\cdot)\) \(\chi_{1815}(1151,\cdot)\) \(\chi_{1815}(1196,\cdot)\) \(\chi_{1815}(1256,\cdot)\) \(\chi_{1815}(1271,\cdot)\) \(\chi_{1815}(1316,\cdot)\) \(\chi_{1815}(1361,\cdot)\) ...

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

\((1211,727,1696)\) → \((-1,1,e\left(\frac{23}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{39}{55}\right)\)\(e\left(\frac{23}{55}\right)\)\(e\left(\frac{51}{110}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{13}{110}\right)\)\(e\left(\frac{19}{110}\right)\)\(e\left(\frac{46}{55}\right)\)\(e\left(\frac{41}{55}\right)\)\(e\left(\frac{39}{110}\right)\)\(e\left(\frac{3}{22}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{55})$
Fixed field: Number field defined by a degree 110 polynomial (not computed)