Properties

Label 1815.1313
Modulus $1815$
Conductor $1815$
Order $220$
Real no
Primitive yes
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(220))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([110,165,24]))
 
pari: [g,chi] = znchar(Mod(1313,1815))
 

Basic properties

Modulus: \(1815\)
Conductor: \(1815\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(220\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1815.bu

\(\chi_{1815}(38,\cdot)\) \(\chi_{1815}(47,\cdot)\) \(\chi_{1815}(53,\cdot)\) \(\chi_{1815}(92,\cdot)\) \(\chi_{1815}(113,\cdot)\) \(\chi_{1815}(137,\cdot)\) \(\chi_{1815}(152,\cdot)\) \(\chi_{1815}(158,\cdot)\) \(\chi_{1815}(203,\cdot)\) \(\chi_{1815}(212,\cdot)\) \(\chi_{1815}(218,\cdot)\) \(\chi_{1815}(257,\cdot)\) \(\chi_{1815}(278,\cdot)\) \(\chi_{1815}(302,\cdot)\) \(\chi_{1815}(317,\cdot)\) \(\chi_{1815}(368,\cdot)\) \(\chi_{1815}(377,\cdot)\) \(\chi_{1815}(383,\cdot)\) \(\chi_{1815}(422,\cdot)\) \(\chi_{1815}(443,\cdot)\) \(\chi_{1815}(467,\cdot)\) \(\chi_{1815}(482,\cdot)\) \(\chi_{1815}(488,\cdot)\) \(\chi_{1815}(533,\cdot)\) \(\chi_{1815}(542,\cdot)\) \(\chi_{1815}(548,\cdot)\) \(\chi_{1815}(587,\cdot)\) \(\chi_{1815}(647,\cdot)\) \(\chi_{1815}(653,\cdot)\) \(\chi_{1815}(698,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{220})$
Fixed field: Number field defined by a degree 220 polynomial (not computed)

Values on generators

\((1211,727,1696)\) → \((-1,-i,e\left(\frac{6}{55}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(13\)\(14\)\(16\)\(17\)\(19\)\(23\)
\(1\)\(1\)\(e\left(\frac{79}{220}\right)\)\(e\left(\frac{79}{110}\right)\)\(e\left(\frac{113}{220}\right)\)\(e\left(\frac{17}{220}\right)\)\(e\left(\frac{59}{220}\right)\)\(e\left(\frac{48}{55}\right)\)\(e\left(\frac{24}{55}\right)\)\(e\left(\frac{131}{220}\right)\)\(e\left(\frac{61}{110}\right)\)\(e\left(\frac{17}{44}\right)\)
value at e.g. 2