Properties

Label 3969.2969
Modulus $3969$
Conductor $441$
Order $42$
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(3969, base_ring=CyclotomicField(42))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([35,18]))
 
pari: [g,chi] = znchar(Mod(2969,3969))
 

Basic properties

Modulus: \(3969\)
Conductor: \(441\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(176,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3969.bt

\(\chi_{3969}(134,\cdot)\) \(\chi_{3969}(512,\cdot)\) \(\chi_{3969}(701,\cdot)\) \(\chi_{3969}(1268,\cdot)\) \(\chi_{3969}(1646,\cdot)\) \(\chi_{3969}(1835,\cdot)\) \(\chi_{3969}(2213,\cdot)\) \(\chi_{3969}(2780,\cdot)\) \(\chi_{3969}(2969,\cdot)\) \(\chi_{3969}(3347,\cdot)\) \(\chi_{3969}(3536,\cdot)\) \(\chi_{3969}(3914,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: 42.0.8048184585024627239924534084535967431149761760538308558024885350095612523278220719989902827.1

Values on generators

\((2108,3727)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{3}{7}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 3969 }(2969, a) \) \(-1\)\(1\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3969 }(2969,a) \;\) at \(\;a = \) e.g. 2