Properties

Label 2009.48
Modulus $2009$
Conductor $287$
Order $40$
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(2009, base_ring=CyclotomicField(40))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([20,39]))
 
pari: [g,chi] = znchar(Mod(48,2009))
 

Basic properties

Modulus: \(2009\)
Conductor: \(287\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(40\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{287}(48,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2009.bj

\(\chi_{2009}(48,\cdot)\) \(\chi_{2009}(97,\cdot)\) \(\chi_{2009}(293,\cdot)\) \(\chi_{2009}(391,\cdot)\) \(\chi_{2009}(440,\cdot)\) \(\chi_{2009}(587,\cdot)\) \(\chi_{2009}(685,\cdot)\) \(\chi_{2009}(832,\cdot)\) \(\chi_{2009}(930,\cdot)\) \(\chi_{2009}(1077,\cdot)\) \(\chi_{2009}(1126,\cdot)\) \(\chi_{2009}(1224,\cdot)\) \(\chi_{2009}(1420,\cdot)\) \(\chi_{2009}(1469,\cdot)\) \(\chi_{2009}(1616,\cdot)\) \(\chi_{2009}(1910,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: 40.40.63172957949423116502957480067191906200305068755882825968063357506461803384975161.1

Values on generators

\((493,785)\) → \((-1,e\left(\frac{39}{40}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\(1\)\(1\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{19}{20}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{1}{20}\right)\)\(i\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{37}{40}\right)\)\(e\left(\frac{33}{40}\right)\)
value at e.g. 2