Properties

Label 2015.197
Modulus $2015$
Conductor $2015$
Order $60$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2015)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([15,5,46]))
 
pari: [g,chi] = znchar(Mod(197,2015))
 

Basic properties

Modulus: \(2015\)
Conductor: \(2015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(60\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2015.hn

\(\chi_{2015}(197,\cdot)\) \(\chi_{2015}(323,\cdot)\) \(\chi_{2015}(358,\cdot)\) \(\chi_{2015}(518,\cdot)\) \(\chi_{2015}(747,\cdot)\) \(\chi_{2015}(912,\cdot)\) \(\chi_{2015}(942,\cdot)\) \(\chi_{2015}(1107,\cdot)\) \(\chi_{2015}(1168,\cdot)\) \(\chi_{2015}(1202,\cdot)\) \(\chi_{2015}(1398,\cdot)\) \(\chi_{2015}(1593,\cdot)\) \(\chi_{2015}(1623,\cdot)\) \(\chi_{2015}(1722,\cdot)\) \(\chi_{2015}(1757,\cdot)\) \(\chi_{2015}(1853,\cdot)\)

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

\((807,1861,716)\) → \((i,e\left(\frac{1}{12}\right),e\left(\frac{23}{30}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(14\)
\(-1\)\(1\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{15}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{19}{60}\right)\)\(e\left(\frac{11}{30}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial