Properties

Label 1815.4
Modulus $1815$
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(1815, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,55,2]))
 
pari: [g,chi] = znchar(Mod(4,1815))
 

Basic properties

Modulus: \(1815\)
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 \(\chi_{605}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1815.bm

\(\chi_{1815}(4,\cdot)\) \(\chi_{1815}(49,\cdot)\) \(\chi_{1815}(64,\cdot)\) \(\chi_{1815}(169,\cdot)\) \(\chi_{1815}(214,\cdot)\) \(\chi_{1815}(229,\cdot)\) \(\chi_{1815}(289,\cdot)\) \(\chi_{1815}(334,\cdot)\) \(\chi_{1815}(379,\cdot)\) \(\chi_{1815}(394,\cdot)\) \(\chi_{1815}(454,\cdot)\) \(\chi_{1815}(499,\cdot)\) \(\chi_{1815}(544,\cdot)\) \(\chi_{1815}(559,\cdot)\) \(\chi_{1815}(619,\cdot)\) \(\chi_{1815}(664,\cdot)\) \(\chi_{1815}(709,\cdot)\) \(\chi_{1815}(724,\cdot)\) \(\chi_{1815}(784,\cdot)\) \(\chi_{1815}(829,\cdot)\) \(\chi_{1815}(889,\cdot)\) \(\chi_{1815}(949,\cdot)\) \(\chi_{1815}(994,\cdot)\) \(\chi_{1815}(1039,\cdot)\) \(\chi_{1815}(1054,\cdot)\) \(\chi_{1815}(1114,\cdot)\) \(\chi_{1815}(1159,\cdot)\) \(\chi_{1815}(1204,\cdot)\) \(\chi_{1815}(1279,\cdot)\) \(\chi_{1815}(1324,\cdot)\) ...

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

\((1211,727,1696)\) → \((1,-1,e\left(\frac{1}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{57}{110}\right)\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{69}{110}\right)\)\(e\left(\frac{61}{110}\right)\)\(e\left(\frac{37}{110}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{43}{110}\right)\)\(e\left(\frac{28}{55}\right)\)\(e\left(\frac{17}{22}\right)\)
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)