Properties

Label 1407.1061
Modulus $1407$
Conductor $1407$
Order $66$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(1407, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33,44,26]))
 
pari: [g,chi] = znchar(Mod(1061,1407))
 

Basic properties

Modulus: \(1407\)
Conductor: \(1407\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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 1407.cz

\(\chi_{1407}(23,\cdot)\) \(\chi_{1407}(65,\cdot)\) \(\chi_{1407}(86,\cdot)\) \(\chi_{1407}(317,\cdot)\) \(\chi_{1407}(368,\cdot)\) \(\chi_{1407}(473,\cdot)\) \(\chi_{1407}(485,\cdot)\) \(\chi_{1407}(557,\cdot)\) \(\chi_{1407}(590,\cdot)\) \(\chi_{1407}(725,\cdot)\) \(\chi_{1407}(821,\cdot)\) \(\chi_{1407}(830,\cdot)\) \(\chi_{1407}(851,\cdot)\) \(\chi_{1407}(998,\cdot)\) \(\chi_{1407}(1040,\cdot)\) \(\chi_{1407}(1061,\cdot)\) \(\chi_{1407}(1178,\cdot)\) \(\chi_{1407}(1283,\cdot)\) \(\chi_{1407}(1346,\cdot)\) \(\chi_{1407}(1376,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((470,1207,337)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{13}{33}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\(-1\)\(1\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{3}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1407 }(1061,a) \;\) at \(\;a = \) e.g. 2