Properties

Label 1470.13
Modulus $1470$
Conductor $245$
Order $28$
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(1470)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,21,22]))
 
pari: [g,chi] = znchar(Mod(13,1470))
 

Basic properties

Modulus: \(1470\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(13,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1470.bi

\(\chi_{1470}(13,\cdot)\) \(\chi_{1470}(223,\cdot)\) \(\chi_{1470}(307,\cdot)\) \(\chi_{1470}(433,\cdot)\) \(\chi_{1470}(517,\cdot)\) \(\chi_{1470}(643,\cdot)\) \(\chi_{1470}(727,\cdot)\) \(\chi_{1470}(853,\cdot)\) \(\chi_{1470}(937,\cdot)\) \(\chi_{1470}(1063,\cdot)\) \(\chi_{1470}(1147,\cdot)\) \(\chi_{1470}(1357,\cdot)\)

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

\((491,1177,1081)\) → \((1,-i,e\left(\frac{11}{14}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{11}{28}\right)\)\(1\)\(e\left(\frac{3}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(-1\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{27}{28}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{28})\)
Fixed field: 28.28.857574960063654015836849375042748140931725025177001953125.1