Properties

Label 9702.19
Modulus $9702$
Conductor $77$
Order $30$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, base_ring=CyclotomicField(30))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,25,9]))
 
pari: [g,chi] = znchar(Mod(19,9702))
 

Basic properties

Modulus: \(9702\)
Conductor: \(77\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(30\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{77}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9702.cq

\(\chi_{9702}(19,\cdot)\) \(\chi_{9702}(325,\cdot)\) \(\chi_{9702}(1207,\cdot)\) \(\chi_{9702}(4429,\cdot)\) \(\chi_{9702}(5617,\cdot)\) \(\chi_{9702}(7075,\cdot)\) \(\chi_{9702}(8263,\cdot)\) \(\chi_{9702}(8839,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{15})\)
Fixed field: \(\Q(\zeta_{77})^+\)

Values on generators

\((4313,199,5293)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9702 }(19, a) \) \(1\)\(1\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{19}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{2}{5}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9702 }(19,a) \;\) at \(\;a = \) e.g. 2