Properties

Label 9900.119
Modulus $9900$
Conductor $9900$
Order $30$
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(9900)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,5,27,18]))
 
pari: [g,chi] = znchar(Mod(119,9900))
 

Basic properties

Modulus: \(9900\)
Conductor: \(9900\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
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 9900.jq

\(\chi_{9900}(119,\cdot)\) \(\chi_{9900}(2039,\cdot)\) \(\chi_{9900}(2579,\cdot)\) \(\chi_{9900}(5339,\cdot)\) \(\chi_{9900}(5459,\cdot)\) \(\chi_{9900}(5879,\cdot)\) \(\chi_{9900}(6719,\cdot)\) \(\chi_{9900}(8759,\cdot)\)

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

\((4951,5501,2377,4501)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right),e\left(\frac{3}{5}\right))\)

Values

\(-1\)\(1\)\(7\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{3}{5}\right)\)\(-1\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: Number field defined by a degree 30 polynomial