Properties

Label 1045.31
Modulus $1045$
Conductor $209$
Order $30$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1045, base_ring=CyclotomicField(30))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,18,25]))
 
pari: [g,chi] = znchar(Mod(31,1045))
 

Basic properties

Modulus: \(1045\)
Conductor: \(209\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{209}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1045.bt

\(\chi_{1045}(31,\cdot)\) \(\chi_{1045}(126,\cdot)\) \(\chi_{1045}(141,\cdot)\) \(\chi_{1045}(236,\cdot)\) \(\chi_{1045}(411,\cdot)\) \(\chi_{1045}(521,\cdot)\) \(\chi_{1045}(696,\cdot)\) \(\chi_{1045}(806,\cdot)\)

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: \(\Q(\zeta_{15})\)
Fixed field: 30.0.916778600930282229451203736288894059046202165847586688659.1

Values on generators

\((837,761,496)\) → \((1,e\left(\frac{3}{5}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(12\)\(13\)\(14\)
\(-1\)\(1\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{15}\right)\)\(-1\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{19}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1045 }(31,a) \;\) at \(\;a = \) e.g. 2