Properties

Label 2850.61
Modulus $2850$
Conductor $475$
Order $45$
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(2850, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,72,10]))
 
pari: [g,chi] = znchar(Mod(61,2850))
 

Basic properties

Modulus: \(2850\)
Conductor: \(475\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{475}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2850.ce

\(\chi_{2850}(61,\cdot)\) \(\chi_{2850}(271,\cdot)\) \(\chi_{2850}(481,\cdot)\) \(\chi_{2850}(511,\cdot)\) \(\chi_{2850}(541,\cdot)\) \(\chi_{2850}(631,\cdot)\) \(\chi_{2850}(841,\cdot)\) \(\chi_{2850}(871,\cdot)\) \(\chi_{2850}(1081,\cdot)\) \(\chi_{2850}(1111,\cdot)\) \(\chi_{2850}(1411,\cdot)\) \(\chi_{2850}(1441,\cdot)\) \(\chi_{2850}(1621,\cdot)\) \(\chi_{2850}(1681,\cdot)\) \(\chi_{2850}(1771,\cdot)\) \(\chi_{2850}(1981,\cdot)\) \(\chi_{2850}(2011,\cdot)\) \(\chi_{2850}(2191,\cdot)\) \(\chi_{2850}(2221,\cdot)\) \(\chi_{2850}(2341,\cdot)\) \(\chi_{2850}(2581,\cdot)\) \(\chi_{2850}(2761,\cdot)\) \(\chi_{2850}(2791,\cdot)\) \(\chi_{2850}(2821,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: 45.45.299215681303998835585125432825671967739342947202402933846152911778748517690473818220198154449462890625.1

Values on generators

\((1901,1027,1351)\) → \((1,e\left(\frac{4}{5}\right),e\left(\frac{1}{9}\right))\)

Values

\(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{34}{45}\right)\)\(e\left(\frac{23}{45}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{7}{9}\right)\)
value at e.g. 2