Properties

Label 9702.29
Modulus $9702$
Conductor $4851$
Order $210$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9702, base_ring=CyclotomicField(210))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,90,147]))
 
pari: [g,chi] = znchar(Mod(29,9702))
 

Basic properties

Modulus: \(9702\)
Conductor: \(4851\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(210\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4851}(29,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 9702.fy

\(\chi_{9702}(29,\cdot)\) \(\chi_{9702}(239,\cdot)\) \(\chi_{9702}(281,\cdot)\) \(\chi_{9702}(365,\cdot)\) \(\chi_{9702}(743,\cdot)\) \(\chi_{9702}(1163,\cdot)\) \(\chi_{9702}(1289,\cdot)\) \(\chi_{9702}(1415,\cdot)\) \(\chi_{9702}(1625,\cdot)\) \(\chi_{9702}(1751,\cdot)\) \(\chi_{9702}(1877,\cdot)\) \(\chi_{9702}(2129,\cdot)\) \(\chi_{9702}(2675,\cdot)\) \(\chi_{9702}(2801,\cdot)\) \(\chi_{9702}(3011,\cdot)\) \(\chi_{9702}(3053,\cdot)\) \(\chi_{9702}(3263,\cdot)\) \(\chi_{9702}(3515,\cdot)\) \(\chi_{9702}(3935,\cdot)\) \(\chi_{9702}(4061,\cdot)\) \(\chi_{9702}(4187,\cdot)\) \(\chi_{9702}(4397,\cdot)\) \(\chi_{9702}(4439,\cdot)\) \(\chi_{9702}(4523,\cdot)\) \(\chi_{9702}(4649,\cdot)\) \(\chi_{9702}(5321,\cdot)\) \(\chi_{9702}(5447,\cdot)\) \(\chi_{9702}(5573,\cdot)\) \(\chi_{9702}(5825,\cdot)\) \(\chi_{9702}(5909,\cdot)\) ...

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: $\Q(\zeta_{105})$
Fixed field: Number field defined by a degree 210 polynomial (not computed)

Values on generators

\((4313,199,5293)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{3}{7}\right),e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 9702 }(29, a) \) \(1\)\(1\)\(e\left(\frac{13}{210}\right)\)\(e\left(\frac{37}{210}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{13}{105}\right)\)\(e\left(\frac{82}{105}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{4}{35}\right)\)\(e\left(\frac{38}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 9702 }(29,a) \;\) at \(\;a = \) e.g. 2