Properties

Label 3267.608
Modulus $3267$
Conductor $297$
Order $90$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3267, base_ring=CyclotomicField(90))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([85,72]))
 
pari: [g,chi] = znchar(Mod(608,3267))
 

Basic properties

Modulus: \(3267\)
Conductor: \(297\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(90\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{297}(14,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3267.be

\(\chi_{3267}(245,\cdot)\) \(\chi_{3267}(608,\cdot)\) \(\chi_{3267}(614,\cdot)\) \(\chi_{3267}(632,\cdot)\) \(\chi_{3267}(686,\cdot)\) \(\chi_{3267}(977,\cdot)\) \(\chi_{3267}(995,\cdot)\) \(\chi_{3267}(1049,\cdot)\) \(\chi_{3267}(1334,\cdot)\) \(\chi_{3267}(1697,\cdot)\) \(\chi_{3267}(1703,\cdot)\) \(\chi_{3267}(1721,\cdot)\) \(\chi_{3267}(1775,\cdot)\) \(\chi_{3267}(2066,\cdot)\) \(\chi_{3267}(2084,\cdot)\) \(\chi_{3267}(2138,\cdot)\) \(\chi_{3267}(2423,\cdot)\) \(\chi_{3267}(2786,\cdot)\) \(\chi_{3267}(2792,\cdot)\) \(\chi_{3267}(2810,\cdot)\) \(\chi_{3267}(2864,\cdot)\) \(\chi_{3267}(3155,\cdot)\) \(\chi_{3267}(3173,\cdot)\) \(\chi_{3267}(3227,\cdot)\)

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: Number field defined by a degree 90 polynomial

Values on generators

\((3026,244)\) → \((e\left(\frac{17}{18}\right),e\left(\frac{4}{5}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(13\)\(14\)\(16\)\(17\)
\( \chi_{ 3267 }(608, a) \) \(-1\)\(1\)\(e\left(\frac{67}{90}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{83}{90}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{16}{45}\right)\)\(e\left(\frac{41}{90}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{11}{30}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3267 }(608,a) \;\) at \(\;a = \) e.g. 2