Properties

Label 1764.617
Modulus $1764$
Conductor $441$
Order $42$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(42)) M = H._module chi = DirichletCharacter(H, M([0,35,18]))
 
Copy content pari:[g,chi] = znchar(Mod(617,1764))
 

Basic properties

Modulus: \(1764\)
Conductor: \(441\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(42\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{441}(176,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1764.ch

\(\chi_{1764}(29,\cdot)\) \(\chi_{1764}(113,\cdot)\) \(\chi_{1764}(281,\cdot)\) \(\chi_{1764}(365,\cdot)\) \(\chi_{1764}(533,\cdot)\) \(\chi_{1764}(617,\cdot)\) \(\chi_{1764}(869,\cdot)\) \(\chi_{1764}(1037,\cdot)\) \(\chi_{1764}(1121,\cdot)\) \(\chi_{1764}(1289,\cdot)\) \(\chi_{1764}(1541,\cdot)\) \(\chi_{1764}(1625,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((883,785,1081)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1764 }(617, a) \) \(-1\)\(1\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{41}{42}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(1\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{23}{42}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{7}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1764 }(617,a) \;\) at \(\;a = \) e.g. 2