Properties

Label 1815.16
Modulus $1815$
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(1815, base_ring=CyclotomicField(110))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,4]))
 
pari: [g,chi] = znchar(Mod(16,1815))
 

Basic properties

Modulus: \(1815\)
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 \(\chi_{121}(16,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1815.bk

\(\chi_{1815}(16,\cdot)\) \(\chi_{1815}(31,\cdot)\) \(\chi_{1815}(91,\cdot)\) \(\chi_{1815}(136,\cdot)\) \(\chi_{1815}(181,\cdot)\) \(\chi_{1815}(196,\cdot)\) \(\chi_{1815}(256,\cdot)\) \(\chi_{1815}(301,\cdot)\) \(\chi_{1815}(346,\cdot)\) \(\chi_{1815}(361,\cdot)\) \(\chi_{1815}(421,\cdot)\) \(\chi_{1815}(466,\cdot)\) \(\chi_{1815}(526,\cdot)\) \(\chi_{1815}(586,\cdot)\) \(\chi_{1815}(631,\cdot)\) \(\chi_{1815}(676,\cdot)\) \(\chi_{1815}(691,\cdot)\) \(\chi_{1815}(751,\cdot)\) \(\chi_{1815}(796,\cdot)\) \(\chi_{1815}(841,\cdot)\) \(\chi_{1815}(916,\cdot)\) \(\chi_{1815}(961,\cdot)\) \(\chi_{1815}(1006,\cdot)\) \(\chi_{1815}(1021,\cdot)\) \(\chi_{1815}(1081,\cdot)\) \(\chi_{1815}(1126,\cdot)\) \(\chi_{1815}(1171,\cdot)\) \(\chi_{1815}(1186,\cdot)\) \(\chi_{1815}(1246,\cdot)\) \(\chi_{1815}(1336,\cdot)\) ...

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

\((1211,727,1696)\) → \((1,1,e\left(\frac{2}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{2}{55}\right)\)\(e\left(\frac{4}{55}\right)\)\(e\left(\frac{14}{55}\right)\)\(e\left(\frac{6}{55}\right)\)\(e\left(\frac{37}{55}\right)\)\(e\left(\frac{16}{55}\right)\)\(e\left(\frac{8}{55}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{1}{55}\right)\)\(e\left(\frac{6}{11}\right)\)
value at e.g. 2

Related number fields

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